text
stringlengths 100
500k
| subset
stringclasses 4
values |
|---|---|
Quantum complexity theory
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes.
Two important quantum complexity classes are BQP and QMA.
Background
See also: Computational complexity and Complexity class
A complexity class is a collection of computational problems that can be solved by a computational model under certain resource constraints. For instance, the complexity class P is defined as the set of problems solvable by a Turing machine in polynomial time. Similarly, quantum complexity classes may be defined using quantum models of computation, such as the quantum circuit model or the equivalent quantum Turing machine. One of the main aims of quantum complexity theory is to find out how these classes relate to classical complexity classes such as P, NP, BPP, and PSPACE.
One of the reasons quantum complexity theory is studied are the implications of quantum computing for the modern Church-Turing thesis. In short the modern Church-Turing thesis states that any computational model can be simulated in polynomial time with a probabilistic Turing machine.[1][2] However, questions around the Church-Turing thesis arise in the context of quantum computing. It is unclear whether the Church-Turing thesis holds for the quantum computation model. There is much evidence that the thesis does not hold. It may not be possible for a probabilistic Turing machine to simulate quantum computation models in polynomial time.[1]
Both quantum computational complexity of functions and classical computational complexity of functions are often expressed with asymptotic notation. Some common forms of asymptotic notion of functions are $O(T(n))$, $\Omega (T(n))$, and $\Theta (T(n))$. $O(T(n))$ expresses that something is bounded above by $cT(n)$ where $c$ is a constant such that $c>0$ and $T(n)$ is a function of $n$, $\Omega (T(n))$ expresses that something is bounded below by $cT(n)$ where $c$ is a constant such that $c>0$ and $T(n)$ is a function of $n$, and $\Theta (T(n))$ expresses both $O(T(n))$ and $\Omega (T(n))$.[3] These notations also have their own names. $O(T(n))$ is called Big O notation, $\Omega (T(n))$ is called Big Omega notation, and $\Theta (T(n))$ is called Big Theta notation.
Overview of complexity classes
The important complexity classes P, BPP, BQP, PP, and PSPACE can be compared based on promise problems. A promise problem is a decision problem which has an input assumed to be selected from the set of all possible input strings. A promise problem is a pair $A=(A_{\text{yes}},A_{\text{no}})$, where $A_{\text{yes}}$ is the set of yes instances and $A_{\text{no}}$ is the set of no instances, and the intersection of these sets is empty: $A_{\text{yes}}\cap A_{\text{no}}=\varnothing $. All of the previous complexity classes contain promise problems.[4]
Complexity Class Criteria
P Promise problems for which a polynomial time deterministic Turing machine accepts all strings in $A_{\text{yes}}$ and rejects all strings in $A_{\text{no}}$[4]
BPP Promise problems for which a polynomial time Probabilistic Turing machine accepts every string in $A_{\text{yes}}$ with a probability of at least ${\frac {2}{3}}$, and accepts every string in $A_{\text{no}}$ with a probability of at most ${\frac {1}{3}}$[4]
BQP Promise problems such that for functions $a,b:\mathbb {N} \to [0,1]$, there exists a polynomial time generated family of quantum circuits $Q={\{Q_{n}:n\in \mathbb {N} \}}$, where $Q_{n}$ is a circuit which accepts $n$ qubits and gives an output of one qubit. An element $x$ of $A_{\text{yes}}$ is accepted by $Q$ with a probability greater than or equal to $a(\left\vert x\right\vert )$. An element $x$ of $A_{\text{no}}$ is accepted by $Q$ with a probability less than or equal to $b(\left\vert x\right\vert )$.[4]
PP Promise problems for which a polynomial time Probabilistic Turing machine accepts every string in $A_{\text{yes}}$ with a probability greater than ${\frac {1}{2}}$, and accepts every string in $A_{\text{no}}$ with a probability of at most ${\frac {1}{2}}$[4]
PSPACE Promise problems for which a deterministic Turing machine, that runs in polynomial space, accepts every string in $A_{\text{yes}}$ and rejects all strings in $A_{\text{no}}$[4]
BQP
Main article: BQP
BQP algorithm (1 run)
Answer
produced
Correct
answer
Yes No
Yes ≥ 2/3 ≤ 1/3
No ≤ 1/3 ≥ 2/3
The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP ("bounded error, quantum, polynomial time"). More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with error probability of at most 1/3.
As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be efficiently solved by probabilistic Turing machines with bounded error.[6] It is known that ${\mathsf {BPP\subseteq BQP}}$ and widely suspected, but not proven, that ${\mathsf {BQP\nsubseteq BPP}}$, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.[7] BQP is a subset of PP.
The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that ${\mathsf {P\subseteq BQP\subseteq PSPACE}}$; that is, the class of problems that can be efficiently solved by quantum computers includes all problems that can be efficiently solved by deterministic classical computers but does not include any problems that cannot be solved by classical computers with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems are in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that ${\mathsf {NP\nsubseteq BQP}}$; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if any NP-complete problem were in BQP, then it follows from NP-hardness that all problems in NP are in BQP).[8]
The relationship of BQP to the essential classical complexity classes can be summarized as:
${\mathsf {P\subseteq BPP\subseteq BQP\subseteq PP\subseteq PSPACE}}$
It is also known that BQP is contained in the complexity class $\color {Blue}{\mathsf {\#P}}$ (or more precisely in the associated class of decision problems ${\mathsf {P^{\#P}}}$),[8] which is a subset of PSPACE.
Simulation of quantum circuits
There is no known way to efficiently simulate a quantum computational model with a classical computer. This means that a classical computer cannot simulate a quantum computational model in polynomial time. However, a quantum circuit of $S(n)$ qubits with $T(n)$ quantum gates can be simulated by a classical circuit with $O(2^{S(n)}T(n)^{3})$ classical gates.[3] This number of classical gates is obtained by determining how many bit operations are necessary to simulate the quantum circuit. In order to do this, first the amplitudes associated with the $S(n)$ qubits must be accounted for. Each of the states of the $S(n)$ qubits can be described by a two-dimensional complex vector, or a state vector. These state vectors can also be described a linear combination of its component vectors with coefficients called amplitudes. These amplitudes are complex numbers which are normalized to one, meaning the sum of the squares of the absolute values of the amplitudes must be one.[3] The entries of the state vector are these amplitudes. Which entry each of the amplitudes are correspond to the none-zero component of the component vector which they are the coefficients of in the linear combination description. As an equation this is described as $\alpha {\begin{bmatrix}1\\0\end{bmatrix}}+\beta {\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}\alpha \\\beta \end{bmatrix}}$ or $\alpha \left\vert 1\right\rangle +\beta \left\vert 0\right\rangle ={\begin{bmatrix}\alpha \\\beta \end{bmatrix}}$ using Dirac notation. The state of the entire $S(n)$ qubit system can be described by a single state vector. This state vector describing the entire system is the tensor product of the state vectors describing the individual qubits in the system. The result of the tensor products of the $S(n)$ qubits is a single state vector which has $2^{S(n)}$ dimensions and entries that are the amplitudes associated with each basis state or component vector. Therefore, $2^{S(n)}$amplitudes must be accounted for with a $2^{S(n)}$ dimensional complex vector which is the state vector for the $S(n)$ qubit system.[9] In order to obtain an upper bound for the number of gates required to simulate a quantum circuit we need a sufficient upper bound for the amount data used to specify the information about each of the $2^{S(n)}$ amplitudes. To do this $O(T(n))$ bits of precision are sufficient for encoding each amplitude.[3] So it takes $O(2^{S(n)}T(n))$ classical bits to account for the state vector of the $S(n)$ qubit system. Next the application of the $T(n)$ quantum gates on $2^{S(n)}$ amplitudes must be accounted for. The quantum gates can be represented as $2^{S(n)}\times 2^{S(n)}$ sparse matrices.[3] So to account for the application of each of the $T(n)$ quantum gates, the state vector must be multiplied by a $2^{S(n)}\times 2^{S(n)}$ sparse matrix for each of the $T(n)$ quantum gates. Every time the state vector is multiplied by a $2^{S(n)}\times 2^{S(n)}$ sparse matrix, $O(2^{S(n)})$ arithmetic operations must be performed.[3] Therefore, there are $O(2^{S(n)}T(n)^{2})$ bit operations for every quantum gate applied to the state vector. So $O(2^{S(n)}T(n)^{2})$ classical gate are needed to simulate $S(n)$ qubit circuit with just one quantum gate. Therefore, $O(2^{S(n)}T(n)^{3})$ classical gates are needed to simulate a quantum circuit of $S(n)$ qubits with $T(n)$ quantum gates.[3] While there is no known way to efficiently simulate a quantum computer with a classical computer, it is possible to efficiently simulate a classical computer with a quantum computer. This is evident from the fact that ${\mathsf {BPP\subseteq BQP}}$.[4]
Quantum query complexity
One major advantage of using a quantum computational system instead of a classical one, is that a quantum computer may be able to give a polynomial time algorithm for some problem for which no classical polynomial time algorithm exists, but more importantly, a quantum computer may significantly decrease the calculation time for a problem that a classical computer can already solve efficiently. Essentially, a quantum computer may be able to both determine how long it will take to solve a problem, while a classical computer may be unable to do so, and can also greatly improve the computational efficiency associated with the solution to a particular problem. Quantum query complexity refers to how complex, or how many queries to the graph associated with the solution of a particular problem, are required to solve the problem. Before we delve further into query complexity, let us consider some background regarding graphing solutions to particular problems, and the queries associated with these solutions.
Query models of directed graphs
One type of problem that quantum computing can make easier to solve are graph problems. If we are to consider the amount of queries to a graph that are required to solve a given problem, let us first consider the most common types of graphs, called directed graphs, that are associated with this type of computational modelling. In brief, directed graphs are graphs where all edges between vertices are unidirectional. Directed graphs are formally defined as the graph $G=(N,E)$, where N is the set of vertices, or nodes, and E is the set of edges.[10]
Adjacency matrix model
When considering quantum computation of the solution to directed graph problems, there are two important query models to understand. First, there is the adjacency matrix model, where the graph of the solution is given by the adjacency matrix: $M\in \{0,1\}a^{n\mathrm {X} n}$, with $M_{ij}=1$, if and only if $(v_{i},v_{j})\in E$.[11]
Adjacency array model
Next, there is the slightly more complicated adjacency array model built on the idea of adjacency lists, where every vertex, $u$, is associated with an array of neighboring vertices such that $f_{i}:[d_{i}^{+}]\rightarrow [n]$, for the out-degrees of vertices $d_{i}^{+},...,d_{n}^{+}$, where $n$ is the minimum value of the upper bound of this model, and $f_{i}(j)$ returns the "$j^{th}$" vertex adjacent to $i$. Additionally, the adjacency array model satisfies the simple graph condition, $\forall i\in [n],j,j'\in [k],j\neq j':f_{i}(j)\neq f_{i}(j')$, meaning that there is only one edge between any pair of vertices, and the number of edges is minimized throughout the entire model (see Spanning tree model for more background).[11]
Quantum query complexity of certain types of graph problems
Both of the above models can be used to determine the query complexity of particular types of graphing problems, including the connectivity, strong connectivity (a directed graph version of the connectivity model), minimum spanning tree, and single source shortest path models of graphs. An important caveat to consider is that the quantum complexity of a particular type of graphing problem can change based on the query model (namely either matrix or array) used to determine the solution. The following table showing the quantum query complexities of these types of graphing problems illustrates this point well.
Quantum query complexity of certain types of graph problems
Problem Matrix model Array model
Minimum spanning tree $\Theta (n^{3/2})$ $\Theta ({\sqrt {nm}})$
Connectivity $\Theta (n^{3/2})$ $\Theta (n)$
Strong connectivity $\Theta (n^{3/2})$ $\Omega ({\sqrt {nm}})$, $O({\sqrt {nm\log(n)}})$
Single source shortest path $\Omega (n^{3/2})$, $O(n^{3/2}\log ^{2}n)$ $\Omega ({\sqrt {nm}})$, $O({\sqrt {nm}}\log ^{2}(n))$
Notice the discrepancy between the quantum query complexities associated with a particular type of problem, depending on which query model was used to determine the complexity. For example, when the matrix model is used, the quantum complexity of the connectivity model in Big O notation is $\Theta (n^{3/2})$, but when the array model is used, the complexity is $\Theta (n)$. Additionally, for brevity, we use the shorthand $m$ in certain cases, where $m=\Theta (n^{2})$.[11] The important implication here is that the efficiency of the algorithm used to solve a graphing problem is dependent on the type of query model used to model the graph.
Other types of quantum computational queries
In the query complexity model, the input can also be given as an oracle (black box). The algorithm gets information about the input only by querying the oracle. The algorithm starts in some fixed quantum state and the state evolves as it queries the oracle.
Similar to the case of graphing problems, the quantum query complexity of a black-box problem is the smallest number of queries to the oracle that are required in order to calculate the function. This makes the quantum query complexity a lower bound on the overall time complexity of a function.
Grover's algorithm
An example depicting the power of quantum computing is Grover's algorithm for searching unstructured databases. The algorithm's quantum query complexity is $ O{\left({\sqrt {N}}\right)}$, a quadratic improvement over the best possible classical query complexity $O(N)$, which is a linear search. Grover's algorithm is asymptotically optimal; in fact, it uses at most a $1+o(1)$ fraction more queries than the best possible algorithm.[12]
Deutsch-Jozsa algorithm
The Deutsch-Jozsa algorithm is a quantum algorithm designed to solve a toy problem with a smaller query complexity than is possible with a classical algorithm. The toy problem asks whether a function $f:\{0,1\}^{n}\rightarrow \{0,1\}$ is constant or balanced, those being the only two possibilities.[2] The only way to evaluate the function $f$ is to consult a black box or oracle. A classical deterministic algorithm will have to check more than half of the possible inputs to be sure of whether or not the function is constant or balanced. With $2^{n}$ possible inputs, the query complexity of the most efficient classical deterministic algorithm is $2^{n-1}+1$.[2] The Deutsch-Jozsa algorithm takes advantage of quantum parallelism to check all of the elements of the domain at once and only needs to query the oracle once, making its query complexity $1$.[2]
Other theories of quantum physics
It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an N-item database in at most $O({\sqrt[{3}]{N}})$ steps, a slight speedup over Grover's algorithm, which runs in $O({\sqrt {N}})$ steps. Note, however, that neither search method would allow quantum computers to solve NP-complete problems in polynomial time.[13] Theories of quantum gravity, such as M-theory and loop quantum gravity, may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the problem of time; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.[14][15]
See also
• Quantum computing
• Quantum Turing machine
• Polynomial hierarchy (PH)
Notes
1. Vazirani, Umesh V. (2002). "A survey of quantum complexity theory". Proceedings of Symposia in Applied Mathematics. 58: 193–217. doi:10.1090/psapm/058/1922899. ISBN 9780821820841. ISSN 2324-7088.
2. Nielsen, Michael A., 1974- (2010). Quantum computation and quantum information. Chuang, Isaac L., 1968- (10th anniversary ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 665137861.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Cleve, Richard (2000), "An Introduction to Quantum Complexity Theory", Quantum Computation and Quantum Information Theory, WORLD SCIENTIFIC, pp. 103–127, arXiv:quant-ph/9906111, Bibcode:2000qcqi.book..103C, doi:10.1142/9789810248185_0004, ISBN 978-981-02-4117-9, S2CID 958695, retrieved October 10, 2020
4. Watrous, John (2008-04-21). "Quantum Computational Complexity". arXiv:0804.3401 [quant-ph].
5. Nielsen, p. 42
6. Nielsen, Michael; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. p. 41. ISBN 978-0-521-63503-5. OCLC 174527496.
7. Nielsen, p. 201
8. Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing. 26 (5): 1411–1473. CiteSeerX 10.1.1.144.7852. doi:10.1137/S0097539796300921.
9. Häner, Thomas; Steiger, Damian S. (2017-11-12). "0.5 petabyte simulation of a 45-qubit quantum circuit". Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. New York, NY, USA: ACM. pp. 1–10. arXiv:1704.01127. doi:10.1145/3126908.3126947. ISBN 978-1-4503-5114-0. S2CID 3338733.
10. Nykamp, D.Q. "Directed Graph Definition".
11. Durr, Christoph; Heiligman, Mark; Hoyer, Peter; Mhalla, Mehdi (January 2006). "Quantum query complexity of some graph problems". SIAM Journal on Computing. 35 (6): 1310–1328. arXiv:quant-ph/0401091. doi:10.1137/050644719. ISSN 0097-5397. S2CID 27736397.
12. Zalka, Christof (1999-10-01). "Grover's quantum searching algorithm is optimal". Physical Review A. 60 (4): 2746–2751. arXiv:quant-ph/9711070. Bibcode:1999PhRvA..60.2746Z. doi:10.1103/PhysRevA.60.2746. S2CID 1542077.
13. Aaronson, Scott. "Quantum Computing and Hidden Variables" (PDF).
14. Aaronson, Scott (2005). "NP-complete Problems and Physical Reality". ACM SIGACT News. 2005. arXiv:quant-ph/0502072. Bibcode:2005quant.ph..2072A. See section 7 "Quantum Gravity": "[...] to anyone who wants a test or benchmark for a favorite quantum gravity theory,[author's footnote: That is, one without all the bother of making numerical predictions and comparing them to observation] let me humbly propose the following: can you define Quantum Gravity Polynomial-Time? [...] until we can say what it means for a 'user' to specify an 'input' and 'later' receive an 'output'—there is no such thing as computation, not even theoretically." (emphasis in original)
15. "D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". D-Wave. 25 May 2011. Archived from the original on 22 December 2020. Retrieved 30 May 2011.
References
• Nielsen, Michael; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 978-0-521-63503-5. OCLC 174527496.
• Arora, Sanjeev; Barak, Boaz (2016). Computational Complexity: A Modern Approach. Cambridge University Press. pp. 201–236. ISBN 978-0-521-42426-4.
• Watrous, John (2008). "Quantum Computational Complexity". arXiv:0804.3401v1 [quant-ph].
• Watrous J. (2009) Quantum Computational Complexity. In: Meyers R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY
External links
• MIT lectures by Scott Aaronson
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
• Quantum biology
• Quantum chemistry
• Quantum chaos
• Quantum cosmology
• Quantum differential calculus
• Quantum dynamics
• Quantum geometry
• Quantum measurement problem
• Quantum mind
• Quantum stochastic calculus
• Quantum spacetime
Technology
• Quantum algorithms
• Quantum amplifier
• Quantum bus
• Quantum cellular automata
• Quantum finite automata
• Quantum channel
• Quantum circuit
• Quantum complexity theory
• Quantum computing
• Timeline
• Quantum cryptography
• Quantum electronics
• Quantum error correction
• Quantum imaging
• Quantum image processing
• Quantum information
• Quantum key distribution
• Quantum logic
• Quantum logic gates
• Quantum machine
• Quantum machine learning
• Quantum metamaterial
• Quantum metrology
• Quantum network
• Quantum neural network
• Quantum optics
• Quantum programming
• Quantum sensing
• Quantum simulator
• Quantum teleportation
Extensions
• Casimir effect
• Quantum statistical mechanics
• Quantum field theory
• History
• Quantum gravity
• Relativistic quantum mechanics
Related
• Schrödinger's cat
• in popular culture
• EPR paradox
• Quantum mysticism
• Category
• Physics portal
• Commons
Emerging technologies
Fields
Quantum
• algorithms
• amplifier
• bus
• cellular automata
• channel
• circuit
• complexity theory
• computing
• clock
• cryptography
• post-quantum
• dynamics
• electronics
• error correction
• finite automata
• image processing
• imaging
• information
• key distribution
• logic
• logic gates
• machine
• machine learning
• metamaterial
• network
• neural network
• optics
• programming
• sensing
• simulator
• teleportation
Other
• Anti-gravity
• Acoustic levitation
• Cloak of invisibility
• Digital scent technology
• Force field
• Plasma window
• Immersive virtual reality
• Magnetic refrigeration
• Phased-array optics
• Thermoacoustic heat engine
• Category
• List
|
Wikipedia
|
Quantum cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well.
While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product.
Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry. In particular, it is ring-isomorphic to symplectic Floer homology.
Throughout this article, X is a closed symplectic manifold with symplectic form ω.
Novikov ring
Various choices of coefficient ring for the quantum cohomology of X are possible. Usually a ring is chosen that encodes information about the second homology of X. This allows the quantum cup product, defined below, to record information about pseudoholomorphic curves in X. For example, let
$H_{2}(X)=H_{2}(X,\mathbf {Z} )/\mathrm {torsion} $
be the second homology modulo its torsion. Let R be any commutative ring with unit and Λ the ring of formal power series of the form
$\lambda =\sum _{A\in H_{2}(X)}\lambda _{A}e^{A},$
where
• the coefficients $\lambda _{A}$ come from R,
• the $e^{A}$ are formal variables subject to the relation $e^{A}e^{B}=e^{A+B}$,
• for every real number C, only finitely many A with ω(A) less than or equal to C have nonzero coefficients $\lambda _{A}$.
The variable $e^{A}$ is considered to be of degree $2c_{1}(A)$, where $c_{1}$ is the first Chern class of the tangent bundle TX, regarded as a complex vector bundle by choosing any almost complex structure compatible with ω. Thus Λ is a graded ring, called the Novikov ring for ω. (Alternative definitions are common.)
Small quantum cohomology
Let
$H^{*}(X)=H^{*}(X,\mathbf {Z} )/\mathrm {torsion} $
be the cohomology of X modulo torsion. Define the small quantum cohomology with coefficients in Λ to be
$QH^{*}(X,\Lambda )=H^{*}(X)\otimes _{\mathbf {Z} }\Lambda .$
Its elements are finite sums of the form
$\sum _{i}a_{i}\otimes \lambda _{i}.$
The small quantum cohomology is a graded R-module with
$\deg(a_{i}\otimes \lambda _{i})=\deg(a_{i})+\deg(\lambda _{i}).$
The ordinary cohomology H*(X) embeds into QH*(X, Λ) via $a\mapsto a\otimes 1$, and QH*(X, Λ) is generated as a Λ-module by H*(X).
For any two cohomology classes a, b in H*(X) of pure degree, and for any A in $H_{2}(X)$, define (a∗b)A to be the unique element of H*(X) such that
$\int _{X}(a*b)_{A}\smile c=GW_{0,3}^{X,A}(a,b,c).$
(The right-hand side is a genus-0, 3-point Gromov–Witten invariant.) Then define
$a*b:=\sum _{A\in H_{2}(X)}(a*b)_{A}\otimes e^{A}.$
This extends by linearity to a well-defined Λ-bilinear map
$QH^{*}(X,\Lambda )\otimes QH^{*}(X,\Lambda )\to QH^{*}(X,\Lambda )$
called the small quantum cup product.
Geometric interpretation
The only pseudoholomorphic curves in class A = 0 are constant maps, whose images are points. It follows that
$GW_{0,3}^{X,0}(a,b,c)=\int _{X}a\smile b\smile c;$
in other words,
$(a*b)_{0}=a\smile b.$
Thus the quantum cup product contains the ordinary cup product; it extends the ordinary cup product to nonzero classes A.
In general, the Poincaré dual of (a∗b)A corresponds to the space of pseudoholomorphic curves of class A passing through the Poincaré duals of a and b. So while the ordinary cohomology considers a and b to intersect only when they meet at one or more points, the quantum cohomology records a nonzero intersection for a and b whenever they are connected by one or more pseudoholomorphic curves. The Novikov ring just provides a bookkeeping system large enough to record this intersection information for all classes A.
Example
Let X be the complex projective plane with its standard symplectic form (corresponding to the Fubini–Study metric) and complex structure. Let $\ell \in H^{2}(X)$ be the Poincaré dual of a line L. Then
$H^{*}(X)\cong \mathbf {Z} [\ell ]/\ell ^{3}.$
The only nonzero Gromov–Witten invariants are those of class A = 0 or A = L. It turns out that
$\int _{X}(\ell ^{i}*\ell ^{j})_{0}\smile \ell ^{k}=GW_{0,3}^{X,0}(\ell ^{i},\ell ^{j},\ell ^{k})=\delta (i+j+k,2)$
and
$\int _{X}(\ell ^{i}*\ell ^{j})_{L}\smile \ell ^{k}=GW_{0,3}^{X,L}(\ell ^{i},\ell ^{j},\ell ^{k})=\delta (i+j+k,5),$
where δ is the Kronecker delta. Therefore,
$\ell *\ell =\ell ^{2}e^{0}+0e^{L}=\ell ^{2},$
$\ell *\ell ^{2}=0e^{0}+1e^{L}=e^{L}.$
In this case it is convenient to rename $e^{L}$ as q and use the simpler coefficient ring Z[q]. This q is of degree $6=2c_{1}(L)$. Then
$QH^{*}(X,\mathbf {Z} [q])\cong \mathbf {Z} [\ell ,q]/(\ell ^{3}=q).$
Properties of the small quantum cup product
For a, b of pure degree,
$\deg(a*b)=\deg(a)+\deg(b)$
and
$b*a=(-1)^{\deg(a)\deg(b)}a*b.$
The small quantum cup product is distributive and Λ-bilinear. The identity element $1\in H^{0}(X)$ is also the identity element for small quantum cohomology.
The small quantum cup product is also associative. This is a consequence of the gluing law for Gromov–Witten invariants, a difficult technical result. It is tantamount to the fact that the Gromov–Witten potential (a generating function for the genus-0 Gromov–Witten invariants) satisfies a certain third-order differential equation known as the WDVV equation.
An intersection pairing
$QH^{*}(X,\Lambda )\otimes QH^{*}(X,\Lambda )\to R$
is defined by
$\left\langle \sum _{i}a_{i}\otimes \lambda _{i},\sum _{j}b_{j}\otimes \mu _{j}\right\rangle =\sum _{i,j}(\lambda _{i})_{0}(\mu _{j})_{0}\int _{X}a_{i}\smile b_{j}.$
(The subscripts 0 indicate the A = 0 coefficient.) This pairing satisfies the associativity property
$\langle a*b,c\rangle =\langle a,b*c\rangle .$
Dubrovin connection
When the base ring R is C, one can view the evenly graded part H of the vector space QH*(X, Λ) as a complex manifold. The small quantum cup product restricts to a well-defined, commutative product on H. Under mild assumptions, H with the intersection pairing $\langle ,\rangle $ is then a Frobenius algebra.
The quantum cup product can be viewed as a connection on the tangent bundle TH, called the Dubrovin connection. Commutativity and associativity of the quantum cup product then correspond to zero-torsion and zero-curvature conditions on this connection.
Big quantum cohomology
There exists a neighborhood U of 0 ∈ H such that $\langle ,\rangle $ and the Dubrovin connection give U the structure of a Frobenius manifold. Any a in U defines a quantum cup product
$*_{a}:H\otimes H\to H$
by the formula
$\langle x*_{a}y,z\rangle :=\sum _{n}\sum _{A}{\frac {1}{n!}}GW_{0,n+3}^{X,A}(x,y,z,a,\ldots ,a).$ :=\sum _{n}\sum _{A}{\frac {1}{n!}}GW_{0,n+3}^{X,A}(x,y,z,a,\ldots ,a).}
Collectively, these products on H are called the big quantum cohomology. All of the genus-0 Gromov–Witten invariants are recoverable from it; in general, the same is not true of the simpler small quantum cohomology.
Small quantum cohomology has only information of 3-point Gromov–Witten invariants, but the big quantum cohomology has of all (n ≧ 4) n-point Gromov–Witten invariants. To obtain enumerative geometrical information for some manifolds, we need to use big quantum cohomology. Small quantum cohomology would correspond to 3-point correlation functions in physics while big quantum cohomology would correspond to all of n-point correlation functions.
References
• McDuff, Dusa & Salamon, Dietmar (2004). J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications. ISBN 0-8218-3485-1.
• Fulton, W; Pandharipande, R (1996). "Notes on stable maps and quantum cohomology". arXiv:alg-geom/9608011.
• Piunikhin, Sergey; Salamon, Dietmar & Schwarz, Matthias (1996). Symplectic Floer–Donaldson theory and quantum cohomology. In C. B. Thomas (Ed.), Contact and Symplectic Geometry, pp. 171–200. Cambridge University Press. ISBN 0-521-57086-7
|
Wikipedia
|
Quantum dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula
$\phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1$
It is the same as the q-exponential function $E_{q}(x)$.
Let $u,v$ be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation $uv=qvu$. Then, the quantum dilogarithm satisfies Schützenberger's identity
$\phi (u)\phi (v)=\phi (u+v),$
Faddeev-Volkov's identity
$\phi (v)\phi (u)=\phi (u+v-vu),$
and Faddeev-Kashaev's identity
$\phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v).$
The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity.
Faddeev's quantum dilogarithm $\Phi _{b}(w)$ is defined by the following formula:
$\Phi _{b}(z)=\exp \left({\frac {1}{4}}\int _{C}{\frac {e^{-2izw}}{\sinh(wb)\sinh(w/b)}}{\frac {dw}{w}}\right),$
where the contour of integration $C$ goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. The same function can be described by the integral formula of Woronowicz:
$\Phi _{b}(x)=\exp \left({\frac {i}{2\pi }}\int _{\mathbb {R} }{\frac {\log(1+e^{tb^{2}+2\pi bx})}{1+e^{t}}}\,dt\right).$
Ludvig Faddeev discovered the quantum pentagon identity:
$\Phi _{b}({\hat {p}})\Phi _{b}({\hat {q}})=\Phi _{b}({\hat {q}})\Phi _{b}({\hat {p}}+{\hat {q}})\Phi _{b}({\hat {p}}),$
where ${\hat {p}}$ and ${\hat {q}}$ are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation
$[{\hat {p}},{\hat {q}}]={\frac {1}{2\pi i}}$
and the inversion relation
$\Phi _{b}(x)\Phi _{b}(-x)=\Phi _{b}(0)^{2}e^{\pi ix^{2}},\quad \Phi _{b}(0)=e^{{\frac {\pi i}{24}}\left(b^{2}+b^{-2}\right)}.$
The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.
The precise relationship between the q-exponential and $\Phi _{b}$ is expressed by the equality
$\Phi _{b}(z)={\frac {E_{e^{2\pi ib^{2}}}(-e^{\pi ib^{2}+2\pi zb})}{E_{e^{-2\pi i/b^{2}}}(-e^{-\pi i/b^{2}+2\pi z/b})}},$
valid for $\operatorname {Im} b^{2}>0$.
References
• Faddeev, L. D. (1994). "Current-Like Variables in Massive and Massless Integrable Models". arXiv:hep-th/9408041.
• Faddeev, L. D. (1995). "Discrete Heisenberg-Weyl group and modular group". Letters in Mathematical Physics. 34 (3): 249–254. arXiv:hep-th/9504111. Bibcode:1995LMaPh..34..249F. doi:10.1007/BF01872779. MR 1345554.
• Faddeev, L. D.; Kashaev, R. M. (1994). "Quantum dilogarithm". Modern Physics Letters A. 9 (5): 427–434. arXiv:hep-th/9310070. Bibcode:1994MPLA....9..427F. doi:10.1142/S0217732394000447. MR 1264393.
• Faddeev, L. D.; Volkov, A. Yu. (1993). "Abelian current algebra and the Virasoro algebra on the lattice". Physics Letters B. 315 (3–4): 311–318. arXiv:hep-th/9307048. Bibcode:1993PhLB..315..311F. doi:10.1016/0370-2693(93)91618-W.
• Kirillov, A. N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement. 118: 61–142. arXiv:hep-th/9408113. Bibcode:1995PThPS.118...61K. doi:10.1143/PTPS.118.61. MR 1356515.
• Schützenberger, M. P. (1953). "Une interprétation de certaines solutions de l'équation fonctionnelle: F (x + y) = F (x)F (y)". Comptes Rendus de l'Académie des Sciences de Paris. 236: 352–353.
• Woronowicz, S. L. (2000). "Quantum exponential function". Reviews in Mathematical Physics. 12 (6): 873–920. Bibcode:2000RvMaP..12..873W. doi:10.1142/S0129055X00000344. MR 1770545.
External links
• quantum dilogarithm at the nLab
|
Wikipedia
|
Quantized enveloping algebra
In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra.[1] Given a Lie algebra ${\mathfrak {g}}$, the quantum enveloping algebra is typically denoted as $U_{q}({\mathfrak {g}})$. The notation was introduced by Drinfeld and independently by Jimbo.[2]
Among the applications, studying the $q\to 0$ limit led to the discovery of crystal bases.
The case of ${\mathfrak {sl}}_{2}$
Michio Jimbo considered the algebras with three generators related by the three commutators
$[h,e]=2e,\ [h,f]=-2f,\ [e,f]=\sinh(\eta h)/\sinh \eta .$
When $\eta \to 0$, these reduce to the commutators that define the special linear Lie algebra ${\mathfrak {sl}}_{2}$. In contrast, for nonzero $\eta $, the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of ${\mathfrak {sl}}_{2}$.[3]
See also
• quantum group
References
1. Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145
2. Tjin 1992, § 5.
3. Jimbo, Michio (1985), "A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang–Baxter equation", Letters in Mathematical Physics, 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588, S2CID 123313856
• Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986, American Mathematical Society, 1: 798–820
• Tjin, T. (10 October 1992). "An introduction to quantized Lie groups and algebras". International Journal of Modern Physics A. 07 (25): 6175–6213. arXiv:hep-th/9111043. Bibcode:1992IJMPA...7.6175T. doi:10.1142/S0217751X92002805. ISSN 0217-751X. S2CID 119087306.
External links
• Quantized enveloping algebra at the nLab
• Quantized enveloping algebras at $q=1$ at MathOverflow
• Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra $U_{q}(g)$? at MathOverflow
|
Wikipedia
|
Quantum finite automaton
In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.
The automata work by receiving a finite-length string $\sigma =(\sigma _{0},\sigma _{1},\cdots ,\sigma _{k})$ of letters $\sigma _{i}$ from a finite alphabet $\Sigma $, and assigning to each such string a probability $\operatorname {Pr} (\sigma )$ indicating the probability of the automaton being in an accept state; that is, indicating whether the automaton accepted or rejected the string.
The languages accepted by QFAs are not the regular languages of deterministic finite automata, nor are they the stochastic languages of probabilistic finite automata. Study of these quantum languages remains an active area of research.
Informal description
There is a simple, intuitive way of understanding quantum finite automata. One begins with a graph-theoretic interpretation of deterministic finite automata (DFA). A DFA can be represented as a directed graph, with states as nodes in the graph, and arrows representing state transitions. Each arrow is labelled with a possible input symbol, so that, given a specific state and an input symbol, the arrow points at the next state. One way of representing such a graph is by means of a set of adjacency matrices, with one matrix for each input symbol. In this case, the list of possible DFA states is written as a column vector. For a given input symbol, the adjacency matrix indicates how any given state (row in the state vector) will transition to the next state; a state transition is given by matrix multiplication.
One needs a distinct adjacency matrix for each possible input symbol, since each input symbol can result in a different transition. The entries in the adjacency matrix must be zero's and one's. For any given column in the matrix, only one entry can be non-zero: this is the entry that indicates the next (unique) state transition. Similarly, the state of the system is a column vector, in which only one entry is non-zero: this entry corresponds to the current state of the system. Let $\Sigma $ denote the set of input symbols. For a given input symbol $\alpha \in \Sigma $, write $U_{\alpha }$ as the adjacency matrix that describes the evolution of the DFA to its next state. The set $\{U_{\alpha }|\alpha \in \Sigma \}$ then completely describes the state transition function of the DFA. Let Q represent the set of possible states of the DFA. If there are N states in Q, then each matrix $U_{\alpha }$ is N by N-dimensional. The initial state $q_{0}\in Q$ corresponds to a column vector with a one in the q0'th row. A general state q is then a column vector with a one in the q'th row. By abuse of notation, let q0 and q also denote these two vectors. Then, after reading input symbols $\alpha \beta \gamma \cdots $ from the input tape, the state of the DFA will be given by $q=\cdots U_{\gamma }U_{\beta }U_{\alpha }q_{0}.$ The state transitions are given by ordinary matrix multiplication (that is, multiply q0 by $U_{\alpha }$, etc.); the order of application is 'reversed' only because we follow the standard notation of linear algebra.
The above description of a DFA, in terms of linear operators and vectors, almost begs for generalization, by replacing the state-vector q by some general vector, and the matrices $\{U_{\alpha }\}$ by some general operators. This is essentially what a QFA does: it replaces q by a probability amplitude, and the $\{U_{\alpha }\}$ by unitary matrices. Other, similar generalizations also become obvious: the vector q can be some distribution on a manifold; the set of transition matrices become automorphisms of the manifold; this defines a topological finite automaton. Similarly, the matrices could be taken as automorphisms of a homogeneous space; this defines a geometric finite automaton.
Before moving on to the formal description of a QFA, there are two noteworthy generalizations that should be mentioned and understood. The first is the non-deterministic finite automaton (NFA). In this case, the vector q is replaced by a vector which can have more than one entry that is non-zero. Such a vector then represents an element of the power set of Q; it’s just an indicator function on Q. Likewise, the state transition matrices $\{U_{\alpha }\}$ are defined in such a way that a given column can have several non-zero entries in it. Equivalently, the multiply-add operations performed during component-wise matrix multiplication should be replaced by Boolean and-or operations, that is, so that one is working with a ring of characteristic 2.
A well-known theorem states that, for each DFA, there is an equivalent NFA, and vice versa. This implies that the set of languages that can be recognized by DFA's and NFA's are the same; these are the regular languages. In the generalization to QFAs, the set of recognized languages will be different. Describing that set is one of the outstanding research problems in QFA theory.
Another generalization that should be immediately apparent is to use a stochastic matrix for the transition matrices, and a probability vector for the state; this gives a probabilistic finite automaton. The entries in the state vector must be real numbers, positive, and sum to one, in order for the state vector to be interpreted as a probability. The transition matrices must preserve this property: this is why they must be stochastic. Each state vector should be imagined as specifying a point in a simplex; thus, this is a topological automaton, with the simplex being the manifold, and the stochastic matrices being linear automorphisms of the simplex onto itself. Since each transition is (essentially) independent of the previous (if we disregard the distinction between accepted and rejected languages), the PFA essentially becomes a kind of Markov chain.
By contrast, in a QFA, the manifold is complex projective space $\mathbb {C} P^{N}$, and the transition matrices are unitary matrices. Each point in $\mathbb {C} P^{N}$ corresponds to a quantum-mechanical probability amplitude or pure state; the unitary matrices can be thought of as governing the time evolution of the system (viz in the Schrödinger picture). The generalization from pure states to mixed states should be straightforward: A mixed state is simply a measure-theoretic probability distribution on $\mathbb {C} P^{N}$.
A worthy point to contemplate is the distributions that result on the manifold during the input of a language. In order for an automaton to be 'efficient' in recognizing a language, that distribution should be 'as uniform as possible'. This need for uniformity is the underlying principle behind maximum entropy methods: these simply guarantee crisp, compact operation of the automaton. Put in other words, the machine learning methods used to train hidden Markov models generalize to QFAs as well: the Viterbi algorithm and the forward-backward algorithm generalize readily to the QFA.
Although the study of QFA was popularized in the work of Kondacs and Watrous in 1997[1] and later by Moore and Crutchfeld,[2] they were described as early as 1971, by Ion Baianu.[3][4]
Measure-once automata
Measure-once automata were introduced by Cris Moore and James P. Crutchfield.[2] They may be defined formally as follows.
As with an ordinary finite automaton, the quantum automaton is considered to have $N$ possible internal states, represented in this case by an $N$-state qubit $|\psi \rangle $. More precisely, the $N$-state qubit $|\psi \rangle \in \mathbb {C} P^{N}$ is an element of $N$-dimensional complex projective space, carrying an inner product $\Vert \cdot \Vert $ that is the Fubini–Study metric.
The state transitions, transition matrices or de Bruijn graphs are represented by a collection of $N\times N$ unitary matrices $U_{\alpha }$, with one unitary matrix for each letter $\alpha \in \Sigma $. That is, given an input letter $\alpha $, the unitary matrix describes the transition of the automaton from its current state $|\psi \rangle $ to its next state $|\psi ^{\prime }\rangle $:
$|\psi ^{\prime }\rangle =U_{\alpha }|\psi \rangle $
Thus, the triple $(\mathbb {C} P^{N},\Sigma ,\{U_{\alpha }\;\vert \;\alpha \in \Sigma \})$ form a quantum semiautomaton.
The accept state of the automaton is given by an $N\times N$ projection matrix $P$, so that, given a $N$-dimensional quantum state $|\psi \rangle $, the probability of $|\psi \rangle $ being in the accept state is
$\langle \psi |P|\psi \rangle =\Vert P|\psi \rangle \Vert ^{2}$
The probability of the state machine accepting a given finite input string $\sigma =(\sigma _{0},\sigma _{1},\cdots ,\sigma _{k})$ is given by
$\operatorname {Pr} (\sigma )=\Vert PU_{\sigma _{k}}\cdots U_{\sigma _{1}}U_{\sigma _{0}}|\psi \rangle \Vert ^{2}$
Here, the vector $|\psi \rangle $ is understood to represent the initial state of the automaton, that is, the state the automaton was in before it started accepting the string input. The empty string $\varnothing $ is understood to be just the unit matrix, so that
$\operatorname {Pr} (\varnothing )=\Vert P|\psi \rangle \Vert ^{2}$
is just the probability of the initial state being an accepted state.
Because the left-action of $U_{\alpha }$ on $|\psi \rangle $ reverses the order of the letters in the string $\sigma $, it is not uncommon for QFAs to be defined using a right action on the Hermitian transpose states, simply in order to keep the order of the letters the same.
A regular language is accepted with probability $p$ by a quantum finite automaton, if, for all sentences $\sigma $ in the language, (and a given, fixed initial state $|\psi \rangle $), one has $p<\operatorname {Pr} (\sigma )$.
Example
Consider the classical deterministic finite automaton given by the state transition table
State Transition Table
Input
State
1 0
S1 S1 S2
S2 S2 S1
State Diagram
The quantum state is a vector, in bra–ket notation
$|\psi \rangle =a_{1}|S_{1}\rangle +a_{2}|S_{2}\rangle ={\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}$
with the complex numbers $a_{1},a_{2}$ normalized so that
${\begin{bmatrix}a_{1}^{*}\;\;a_{2}^{*}\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}=a_{1}^{*}a_{1}+a_{2}^{*}a_{2}=1$
The unitary transition matrices are
$U_{0}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}$
and
$U_{1}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$
Taking $S_{1}$ to be the accept state, the projection matrix is
$P={\begin{bmatrix}1&0\\0&0\end{bmatrix}}$
As should be readily apparent, if the initial state is the pure state $|S_{1}\rangle $ or $|S_{2}\rangle $, then the result of running the machine will be exactly identical to the classical deterministic finite state machine. In particular, there is a language accepted by this automaton with probability one, for these initial states, and it is identical to the regular language for the classical DFA, and is given by the regular expression:
$(1^{*}(01^{*}0)^{*})^{*}\,\!$
The non-classical behaviour occurs if both $a_{1}$ and $a_{2}$ are non-zero. More subtle behaviour occurs when the matrices $U_{0}$ and $U_{1}$ are not so simple; see, for example, the de Rham curve as an example of a quantum finite state machine acting on the set of all possible finite binary strings.
Measure-many automata
Measure-many automata were introduced by Kondacs and Watrous in 1997.[1] The general framework resembles that of the measure-once automaton, except that instead of there being one projection, at the end, there is a projection, or quantum measurement, performed after each letter is read. A formal definition follows.
The Hilbert space ${\mathcal {H}}_{Q}$ is decomposed into three orthogonal subspaces
${\mathcal {H}}_{Q}={\mathcal {H}}_{\text{accept}}\oplus {\mathcal {H}}_{\text{reject}}\oplus {\mathcal {H}}_{\text{non-halting}}$
In the literature, these orthogonal subspaces are usually formulated in terms of the set $Q$ of orthogonal basis vectors for the Hilbert space ${\mathcal {H}}_{Q}$. This set of basis vectors is divided up into subsets $Q_{\text{acc}}\subset Q$ and $Q_{\text{rej}}\subset Q$, such that
${\mathcal {H}}_{\text{accept}}=\operatorname {span} \{|q\rangle :|q\rangle \in Q_{\text{acc}}\}$ :|q\rangle \in Q_{\text{acc}}\}}
is the linear span of the basis vectors in the accept set. The reject space is defined analogously, and the remaining space is designated the non-halting subspace. There are three projection matrices, $P_{\text{acc}}$, $P_{\text{rej}}$ and $P_{\text{non}}$, each projecting to the respective subspace:
$P_{\text{acc}}:{\mathcal {H}}_{Q}\to {\mathcal {H}}_{\text{accept}}$
and so on. The parsing of the input string proceeds as follows. Consider the automaton to be in a state $|\psi \rangle $. After reading an input letter $\alpha $, the automaton will be in the state
$|\psi ^{\prime }\rangle =U_{\alpha }|\psi \rangle $
At this point, a measurement is performed on the state $|\psi ^{\prime }\rangle $, using the projection operators $P$, at which time its wave-function collapses into one of the three subspaces ${\mathcal {H}}_{\text{accept}}$ or ${\mathcal {H}}_{\text{reject}}$ or ${\mathcal {H}}_{\text{non-halting}}$. The probability of collapse is given by
$\operatorname {Pr} _{\text{acc}}(\sigma )=\Vert P_{\text{acc}}|\psi ^{\prime }\rangle \Vert ^{2}$
for the "accept" subspace, and analogously for the other two spaces.
If the wave function has collapsed to either the "accept" or "reject" subspaces, then further processing halts. Otherwise, processing continues, with the next letter read from the input, and applied to what must be an eigenstate of $P_{\text{non}}$. Processing continues until the whole string is read, or the machine halts. Often, additional symbols $\kappa $ and $ are adjoined to the alphabet, to act as the left and right end-markers for the string.
In the literature, the measure-many automaton is often denoted by the tuple $(Q;\Sigma ;\delta ;q_{0};Q_{\text{acc}};Q_{\text{rej}})$ ;\delta ;q_{0};Q_{\text{acc}};Q_{\text{rej}})} . Here, $Q$, $\Sigma $, $Q_{\text{acc}}$ and $Q_{\text{rej}}$ are as defined above. The initial state is denoted by $|\psi \rangle =|q_{0}\rangle $. The unitary transformations are denoted by the map $\delta $,
$\delta :Q\times \Sigma \times Q\to \mathbb {C} $
so that
$U_{\alpha }|q_{1}\rangle =\sum _{q_{2}\in Q}\delta (q_{1},\alpha ,q_{2})|q_{2}\rangle $
Relation to quantum computing
As of 2019, most quantum computers are implementations of measure-once quantum finite automata, and the software systems for programming them expose the state-preparation of $|\psi \rangle $, measurement $P$ and a choice of unitary transformations $U_{\alpha }$, such the controlled NOT gate, the Hadamard transform and other quantum logic gates, directly to the programmer.
The primary difference between real-world quantum computers and the theoretical framework presented above is that the initial state preparation cannot ever result in a point-like pure state, nor can the unitary operators be precisely applied. Thus, the initial state must be taken as a mixed state
$\rho =\int p(x)|\psi _{x}\rangle dx$
for some probability distribution $p(x)$ characterizing the ability of the machinery to prepare an initial state close to the desired initial pure state $|\psi \rangle $. This state is not stable, but suffers from some amount of quantum decoherence over time. Precise measurements are also not possible, and one instead uses positive operator-valued measures to describe the measurement process. Finally, each unitary transformation is not a single, sharply defined quantum logic gate, but rather a mixture
$U_{\alpha ,(\rho )}=\int p_{\alpha }(x)U_{\alpha ,x}dx$
for some probability distribution $p_{\alpha }(x)$ describing how well the machinery can effect the desired transformation $U_{\alpha }$.
As a result of these effects, the actual time evolution of the state cannot be taken as an infinite-precision pure point, operated on by a sequence of arbitrarily sharp transformations, but rather as an ergodic process, or more accurately, a mixing process that only concatenates transformations onto a state, but also smears the state over time.
There is no quantum analog to the push-down automaton or stack machine. This is due to the no-cloning theorem: there is no way to make a copy of the current state of the machine, push it onto a stack for later reference, and then return to it.
Geometric generalizations
The above constructions indicate how the concept of a quantum finite automaton can be generalized to arbitrary topological spaces. For example, one may take some (N-dimensional) Riemann symmetric space to take the place of $\mathbb {C} P^{N}$. In place of the unitary matrices, one uses the isometries of the Riemannian manifold, or, more generally, some set of open functions appropriate for the given topological space. The initial state may be taken to be a point in the space. The set of accept states can be taken to be some arbitrary subset of the topological space. One then says that a formal language is accepted by this topological automaton if the point, after iteration by the homeomorphisms, intersects the accept set. But, of course, this is nothing more than the standard definition of an M-automaton. The behaviour of topological automata is studied in the field of topological dynamics.
The quantum automaton differs from the topological automaton in that, instead of having a binary result (is the iterated point in, or not in, the final set?), one has a probability. The quantum probability is the (square of) the initial state projected onto some final state P; that is $\mathbf {Pr} =\vert \langle P\vert \psi \rangle \vert ^{2}$. But this probability amplitude is just a very simple function of the distance between the point $\vert P\rangle $ and the point $\vert \psi \rangle $ in $\mathbb {C} P^{N}$, under the distance metric given by the Fubini–Study metric. To recap, the quantum probability of a language being accepted can be interpreted as a metric, with the probability of accept being unity, if the metric distance between the initial and final states is zero, and otherwise the probability of accept is less than one, if the metric distance is non-zero. Thus, it follows that the quantum finite automaton is just a special case of a geometric automaton or a metric automaton, where $\mathbb {C} P^{N}$ is generalized to some metric space, and the probability measure is replaced by a simple function of the metric on that space.
See also
• Quantum Markov chain
Notes
1. Kondacs, A.; Watrous, J. (1997), "On the power of quantum finite state automata", Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pp. 66–75
2. C. Moore, J. Crutchfield, "Quantum automata and quantum grammars", Theoretical Computer Science, 237 (2000) pp 275-306.
3. I. Baianu, "Organismic Supercategories and Qualitative Dynamics of Systems" (1971), Bulletin of Mathematical Biophysics, 33 pp.339-354.
4. I. Baianu, "Categories, Functors and Quantum Automata Theory" (1971). The 4th Intl.Congress LMPS, August-Sept.1971
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
|
Wikipedia
|
Quantum graph
In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) connected at transformer stations (vertices); the differential equations would then describe the voltage along each of the lines, with boundary conditions for each edge provided at the adjacent vertices ensuring that the current added over all edges adds to zero at each vertex.
Quantum graphs were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts,[1] e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.[2]
Aside from actually solving the differential equations posed on a quantum graph for purposes of concrete applications, typical questions that arise are those of controllability (what inputs have to be provided to bring the system into a desired state, for example providing sufficient power to all houses on a power network) and identifiability (how and where one has to measure something to obtain a complete picture of the state of the system, for example measuring the pressure of a water pipe network to determine whether or not there is a leaking pipe).
Metric graphs
A metric graph is a graph consisting of a set $V$ of vertices and a set $E$ of edges where each edge $e=(v_{1},v_{2})\in E$ has been associated with an interval $[0,L_{e}]$ so that $x_{e}$ is the coordinate on the interval, the vertex $v_{1}$ corresponds to $x_{e}=0$ and $v_{2}$ to $x_{e}=L_{e}$ or vice versa. The choice of which vertex lies at zero is arbitrary with the alternative corresponding to a change of coordinate on the edge. The graph has a natural metric: for two points $x,y$ on the graph, $\rho (x,y)$ is the shortest distance between them where distance is measured along the edges of the graph.
Open graphs: in the combinatorial graph model edges always join pairs of vertices however in a quantum graph one may also consider semi-infinite edges. These are edges associated with the interval $[0,\infty )$ attached to a single vertex at $x_{e}=0$. A graph with one or more such open edges is referred to as an open graph.
Quantum graphs
Quantum graphs are metric graphs equipped with a differential (or pseudo-differential) operator acting on functions on the graph. A function $f$ on a metric graph is defined as the $|E|$-tuple of functions $f_{e}(x_{e})$ on the intervals. The Hilbert space of the graph is $\bigoplus _{e\in E}L^{2}([0,L_{e}])$ where the inner product of two functions is
$\langle f,g\rangle =\sum _{e\in E}\int _{0}^{L_{e}}f_{e}^{*}(x_{e})g_{e}(x_{e})\,dx_{e},$
$L_{e}$ may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the Laplace operator. The operator on an edge is $-{\frac {{\textrm {d}}^{2}}{{\textrm {d}}x_{e}^{2}}}$ where $x_{e}$ is the coordinate on the edge. To make the operator self-adjoint a suitable domain must be specified. This is typically achieved by taking the Sobolev space $H^{2}$ of functions on the edges of the graph and specifying matching conditions at the vertices.
The trivial example of matching conditions that make the operator self-adjoint are the Dirichlet boundary conditions, $f_{e}(0)=f_{e}(L_{e})=0$ for every edge. An eigenfunction on a finite edge may be written as
$f_{e}(x_{e})=\sin \left({\frac {n\pi x_{e}}{L_{e}}}\right)$
for integer $n$. If the graph is closed with no infinite edges and the lengths of the edges of the graph are rationally independent then an eigenfunction is supported on a single graph edge and the eigenvalues are ${\frac {n^{2}\pi ^{2}}{L_{e}^{2}}}$. The Dirichlet conditions don't allow interaction between the intervals so the spectrum is the same as that of the set of disconnected edges.
More interesting self-adjoint matching conditions that allow interaction between edges are the Neumann or natural matching conditions. A function $f$ in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero,
$\sum _{e\sim v}f'(v)=0\ ,$
where $f'(v)=f'(0)$ if the vertex $v$ is at $x=0$ and $f'(v)=-f'(L_{e})$ if $v$ is at $x=L_{e}$.
The properties of other operators on metric graphs have also been studied.
• These include the more general class of Schrödinger operators,
$\left(i{\frac {\textrm {d}}{{\textrm {d}}x_{e}}}+A_{e}(x_{e})\right)^{2}+V_{e}(x_{e})\ ,$
where $A_{e}$ is a "magnetic vector potential" on the edge and $V_{e}$ is a scalar potential.
• Another example is the Dirac operator on a graph which is a matrix valued operator acting on vector valued functions that describe the quantum mechanics of particles with an intrinsic angular momentum of one half such as the electron.
• The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the study of photonic crystals.
Theorems
All self-adjoint matching conditions of the Laplace operator on a graph can be classified according to a scheme of Kostrykin and Schrader. In practice, it is often more convenient to adopt a formalism introduced by Kuchment, see,[3] which automatically yields an operator in variational form.
Let $v$ be a vertex with $d$ edges emanating from it. For simplicity we choose the coordinates on the edges so that $v$ lies at $x_{e}=0$ for each edge meeting at $v$. For a function $f$ on the graph let
$\mathbf {f} =(f_{e_{1}}(0),f_{e_{2}}(0),\dots ,f_{e_{d}}(0))^{T},\qquad \mathbf {f} '=(f'_{e_{1}}(0),f'_{e_{2}}(0),\dots ,f'_{e_{d}}(0))^{T}.$
Matching conditions at $v$ can be specified by a pair of matrices $A$ and $B$ through the linear equation,
$A\mathbf {f} +B\mathbf {f} '=\mathbf {0} .$
The matching conditions define a self-adjoint operator if $(A,B)$ has the maximal rank $d$ and $AB^{*}=BA^{*}.$
The spectrum of the Laplace operator on a finite graph can be conveniently described using a scattering matrix approach introduced by Kottos and Smilansky .[4][5] The eigenvalue problem on an edge is,
$-{\frac {d^{2}}{dx_{e}^{2}}}f_{e}(x_{e})=k^{2}f_{e}(x_{e}).\,$
So a solution on the edge can be written as a linear combination of plane waves.
$f_{e}(x_{e})=c_{e}{\textrm {e}}^{ikx_{e}}+{\hat {c}}_{e}{\textrm {e}}^{-ikx_{e}}.\,$
where in a time-dependent Schrödinger equation $c$ is the coefficient of the outgoing plane wave at $0$ and ${\hat {c}}$ coefficient of the incoming plane wave at $0$. The matching conditions at $v$ define a scattering matrix
$S(k)=-(A+ikB)^{-1}(A-ikB).\,$
The scattering matrix relates the vectors of incoming and outgoing plane-wave coefficients at $v$, $\mathbf {c} =S(k){\hat {\mathbf {c} }}$. For self-adjoint matching conditions $S$ is unitary. An element of $\sigma _{(uv)(vw)}$ of $S$ is a complex transition amplitude from a directed edge $(uv)$ to the edge $(vw)$ which in general depends on $k$. However, for a large class of matching conditions the S-matrix is independent of $k$. With Neumann matching conditions for example
$A=\left({\begin{array}{ccccc}1&-1&0&0&\dots \\0&1&-1&0&\dots \\&&\ddots &\ddots &\\0&\dots &0&1&-1\\0&\dots &0&0&0\\\end{array}}\right),\quad B=\left({\begin{array}{cccc}0&0&\dots &0\\\vdots &\vdots &&\vdots \\0&0&\dots &0\\1&1&\dots &1\\\end{array}}\right).$
Substituting in the equation for $S$ produces $k$-independent transition amplitudes
$\sigma _{(uv)(vw)}={\frac {2}{d}}-\delta _{uw}.\,$
where $\delta _{uw}$ is the Kronecker delta function that is one if $u=w$ and zero otherwise. From the transition amplitudes we may define a $2|E|\times 2|E|$ matrix
$U_{(uv)(lm)}(k)=\delta _{vl}\sigma _{(uv)(vm)}(k){\textrm {e}}^{ikL_{(uv)}}.\,$
$U$ is called the bond scattering matrix and can be thought of as a quantum evolution operator on the graph. It is unitary and acts on the vector of $2|E|$ plane-wave coefficients for the graph where $c_{(uv)}$ is the coefficient of the plane wave traveling from $u$ to $v$. The phase ${\textrm {e}}^{ikL_{(uv)}}$ is the phase acquired by the plane wave when propagating from vertex $u$ to vertex $v$.
Quantization condition: An eigenfunction on the graph can be defined through its associated $2|E|$ plane-wave coefficients. As the eigenfunction is stationary under the quantum evolution a quantization condition for the graph can be written using the evolution operator.
$|U(k)-I|=0.\,$
Eigenvalues $k_{j}$ occur at values of $k$ where the matrix $U(k)$ has an eigenvalue one. We will order the spectrum with $0\leqslant k_{0}\leqslant k_{1}\leqslant \dots $.
The first trace formula for a graph was derived by Roth (1983). In 1997 Kottos and Smilansky used the quantization condition above to obtain the following trace formula for the Laplace operator on a graph when the transition amplitudes are independent of $k$. The trace formula links the spectrum with periodic orbits on the graph.
$d(k):=\sum _{j=0}^{\infty }\delta (k-k_{j})={\frac {L}{\pi }}+{\frac {1}{\pi }}\sum _{p}{\frac {L_{p}}{r_{p}}}A_{p}\cos(kL_{p}).$
$d(k)$ is called the density of states. The right hand side of the trace formula is made up of two terms, the Weyl term ${\frac {L}{\pi }}$ is the mean separation of eigenvalues and the oscillating part is a sum over all periodic orbits $p=(e_{1},e_{2},\dots ,e_{n})$ on the graph. $L_{p}=\sum _{e\in p}L_{e}$ is the length of the orbit and $L=\sum _{e\in E}L_{e}$ is the total length of the graph. For an orbit generated by repeating a shorter primitive orbit, $r_{p}$ counts the number of repartitions. $A_{p}=\sigma _{e_{1}e_{2}}\sigma _{e_{2}e_{3}}\dots \sigma _{e_{n}e_{1}}$ is the product of the transition amplitudes at the vertices of the graph around the orbit.
Applications
Quantum graphs were first employed in the 1930s to model the spectrum of free electrons in organic molecules like Naphthalene, see figure. As a first approximation the atoms are taken to be vertices while the σ-electrons form bonds that fix a frame in the shape of the molecule on which the free electrons are confined.
A similar problem appears when considering quantum waveguides. These are mesoscopic systems - systems built with a width on the scale of nanometers. A quantum waveguide can be thought of as a fattened graph where the edges are thin tubes. The spectrum of the Laplace operator on this domain converges to the spectrum of the Laplace operator on the graph under certain conditions. Understanding mesoscopic systems plays an important role in the field of nanotechnology.
In 1997[6] Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos, the quantum mechanics of systems that are classically chaotic. Classical motion on the graph can be defined as a probabilistic Markov chain where the probability of scattering from edge $e$ to edge $f$ is given by the absolute value of the quantum transition amplitude squared, $|\sigma _{ef}|^{2}$. For almost all finite connected quantum graphs the probabilistic dynamics is ergodic and mixing, in other words chaotic.
Quantum graphs embedded in two or three dimensions appear in the study of photonic crystals.[7] In two dimensions a simple model of a photonic crystal consists of polygonal cells of a dense dielectric with narrow interfaces between the cells filled with air. Studying dielectric modes that stay mostly in the dielectric gives rise to a pseudo-differential operator on the graph that follows the narrow interfaces.
Periodic quantum graphs like the lattice in ${\mathbb {R} }^{2}$ are common models of periodic systems and quantum graphs have been applied to the study the phenomena of Anderson localization where localized states occur at the edge of spectral bands in the presence of disorder.
See also
• Schild's Ladder, a novel dealing with a fictional quantum graph theory
• Feynman diagram
References
1. Berkolaiko, Gregory; Carlson, Robert; Kuchment, Peter; Fulling, Stephen (2006). Quantum Graphs and Their Applications (Contemporary Mathematics): Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications. Vol. 415. American Mathematical Society. ISBN 978-0821837658.
2. Freedman, Michael; Lovász, László; Schrijver, Alexander (2007). "Reflection positivity, rank connectivity, and homomorphism of graphs". Journal of the American Mathematical Society. 20 (1): 37–52. arXiv:math/0404468. doi:10.1090/S0894-0347-06-00529-7. ISSN 0894-0347. MR 2257396. S2CID 8208923.
3. Kuchment, Peter (2004). "Quantum graphs: I. Some basic structures". Waves in Random Media. 14 (1): S107–S128. doi:10.1088/0959-7174/14/1/014. ISSN 0959-7174. S2CID 16874849.
4. Kottos, Tsampikos; Smilansky, Uzy (1999). "Periodic Orbit Theory and Spectral Statistics for Quantum Graphs". Annals of Physics. 274 (1): 76–124. arXiv:chao-dyn/9812005. doi:10.1006/aphy.1999.5904. ISSN 0003-4916. S2CID 17510999.
5. Gnutzmann∥, Sven; Smilansky, Uzy (2006). "Quantum graphs: Applications to quantum chaos and universal spectral statistics". Advances in Physics. 55 (5–6): 527–625. arXiv:nlin/0605028. doi:10.1080/00018730600908042. ISSN 0001-8732. S2CID 119424306.
6. Kottos, Tsampikos; Smilansky, Uzy (1997). "Quantum Chaos on Graphs". Physical Review Letters. 79 (24): 4794–4797. doi:10.1103/PhysRevLett.79.4794. ISSN 0031-9007.
7. Kuchment, Peter; Kunyansky, Leonid (2002). "Differential Operators on Graphs and Photonic Crystals". Advances in Computational Mathematics. 16 (24): 263–290. doi:10.1023/A:1014481629504. S2CID 17506556.
|
Wikipedia
|
Quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo.
In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.
Intuitive meaning
The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the noncommutative geometry of Alain Connes. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang–Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgeny Sklyanin, Nicolai Reshetikhin and Vladimir Korepin) and related work by the Japanese School.[1] The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity.[2]
Drinfeld–Jimbo type quantum groups
One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac–Moody algebra, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra.
Let A = (aij) be the Cartan matrix of the Kac–Moody algebra, and let q ≠ 0, 1 be a complex number, then the quantum group, Uq(G), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators kλ (where λ is an element of the weight lattice, i.e. 2(λ, αi)/(αi, αi) is an integer for all i), and ei and fi (for simple roots, αi), subject to the following relations:
${\begin{aligned}k_{0}&=1\\k_{\lambda }k_{\mu }&=k_{\lambda +\mu }\\k_{\lambda }e_{i}k_{\lambda }^{-1}&=q^{(\lambda ,\alpha _{i})}e_{i}\\k_{\lambda }f_{i}k_{\lambda }^{-1}&=q^{-(\lambda ,\alpha _{i})}f_{i}\\\left[e_{i},f_{j}\right]&=\delta _{ij}{\frac {k_{i}-k_{i}^{-1}}{q_{i}-q_{i}^{-1}}}&&k_{i}=k_{\alpha _{i}},q_{i}=q^{{\frac {1}{2}}(\alpha _{i},\alpha _{i})}\\\end{aligned}}$
And for i ≠ j we have the q-Serre relations, which are deformations of the Serre relations:
${\begin{aligned}\sum _{n=0}^{1-a_{ij}}(-1)^{n}{\frac {[1-a_{ij}]_{q_{i}}!}{[1-a_{ij}-n]_{q_{i}}![n]_{q_{i}}!}}e_{i}^{n}e_{j}e_{i}^{1-a_{ij}-n}&=0\\[6pt]\sum _{n=0}^{1-a_{ij}}(-1)^{n}{\frac {[1-a_{ij}]_{q_{i}}!}{[1-a_{ij}-n]_{q_{i}}![n]_{q_{i}}!}}f_{i}^{n}f_{j}f_{i}^{1-a_{ij}-n}&=0\end{aligned}}$
where the q-factorial, the q-analog of the ordinary factorial, is defined recursively using q-number:
${\begin{aligned}{[0]}_{q_{i}}!&=1\\{[n]}_{q_{i}}!&=\prod _{m=1}^{n}[m]_{q_{i}},&&[m]_{q_{i}}={\frac {q_{i}^{m}-q_{i}^{-m}}{q_{i}-q_{i}^{-1}}}\end{aligned}}$
In the limit as q → 1, these relations approach the relations for the universal enveloping algebra U(G), where
$k_{\lambda }\to 1,\qquad {\frac {k_{\lambda }-k_{-\lambda }}{q-q^{-1}}}\to t_{\lambda }$
and tλ is the element of the Cartan subalgebra satisfying (tλ, h) = λ(h) for all h in the Cartan subalgebra.
There are various coassociative coproducts under which these algebras are Hopf algebras, for example,
${\begin{array}{lll}\Delta _{1}(k_{\lambda })=k_{\lambda }\otimes k_{\lambda }&\Delta _{1}(e_{i})=1\otimes e_{i}+e_{i}\otimes k_{i}&\Delta _{1}(f_{i})=k_{i}^{-1}\otimes f_{i}+f_{i}\otimes 1\\\Delta _{2}(k_{\lambda })=k_{\lambda }\otimes k_{\lambda }&\Delta _{2}(e_{i})=k_{i}^{-1}\otimes e_{i}+e_{i}\otimes 1&\Delta _{2}(f_{i})=1\otimes f_{i}+f_{i}\otimes k_{i}\\\Delta _{3}(k_{\lambda })=k_{\lambda }\otimes k_{\lambda }&\Delta _{3}(e_{i})=k_{i}^{-{\frac {1}{2}}}\otimes e_{i}+e_{i}\otimes k_{i}^{\frac {1}{2}}&\Delta _{3}(f_{i})=k_{i}^{-{\frac {1}{2}}}\otimes f_{i}+f_{i}\otimes k_{i}^{\frac {1}{2}}\end{array}}$
where the set of generators has been extended, if required, to include kλ for λ which is expressible as the sum of an element of the weight lattice and half an element of the root lattice.
In addition, any Hopf algebra leads to another with reversed coproduct T o Δ, where T is given by T(x ⊗ y) = y ⊗ x, giving three more possible versions.
The counit on Uq(A) is the same for all these coproducts: ε(kλ) = 1, ε(ei) = ε(fi) = 0, and the respective antipodes for the above coproducts are given by
${\begin{array}{lll}S_{1}(k_{\lambda })=k_{-\lambda }&S_{1}(e_{i})=-e_{i}k_{i}^{-1}&S_{1}(f_{i})=-k_{i}f_{i}\\S_{2}(k_{\lambda })=k_{-\lambda }&S_{2}(e_{i})=-k_{i}e_{i}&S_{2}(f_{i})=-f_{i}k_{i}^{-1}\\S_{3}(k_{\lambda })=k_{-\lambda }&S_{3}(e_{i})=-q_{i}e_{i}&S_{3}(f_{i})=-q_{i}^{-1}f_{i}\end{array}}$
Alternatively, the quantum group Uq(G) can be regarded as an algebra over the field C(q), the field of all rational functions of an indeterminate q over C.
Similarly, the quantum group Uq(G) can be regarded as an algebra over the field Q(q), the field of all rational functions of an indeterminate q over Q (see below in the section on quantum groups at q = 0). The center of quantum group can be described by quantum determinant.
Representation theory
Just as there are many different types of representations for Kac–Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups.
As is the case for all Hopf algebras, Uq(G) has an adjoint representation on itself as a module, with the action being given by
$\mathrm {Ad} _{x}\cdot y=\sum _{(x)}x_{(1)}yS(x_{(2)}),$
where
$\Delta (x)=\sum _{(x)}x_{(1)}\otimes x_{(2)}.$
Case 1: q is not a root of unity
One important type of representation is a weight representation, and the corresponding module is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector v such that kλ · v = dλv for all λ, where dλ are complex numbers for all weights λ such that
$d_{0}=1,$
$d_{\lambda }d_{\mu }=d_{\lambda +\mu },$ for all weights λ and μ.
A weight module is called integrable if the actions of ei and fi are locally nilpotent (i.e. for any vector v in the module, there exists a positive integer k, possibly dependent on v, such that $e_{i}^{k}.v=f_{i}^{k}.v=0$ for all i). In the case of integrable modules, the complex numbers dλ associated with a weight vector satisfy $d_{\lambda }=c_{\lambda }q^{(\lambda ,\nu )}$, where ν is an element of the weight lattice, and cλ are complex numbers such that
• $c_{0}=1,$
• $c_{\lambda }c_{\mu }=c_{\lambda +\mu },$ for all weights λ and μ,
• $c_{2\alpha _{i}}=1$ for all i.
Of special interest are highest-weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vector v, subject to kλ · v = dλv for all weights μ, and ei · v = 0 for all i. Similarly, a quantum group can have a lowest weight representation and lowest weight module, i.e. a module generated by a weight vector v, subject to kλ · v = dλv for all weights λ, and fi · v = 0 for all i.
Define a vector v to have weight ν if $k_{\lambda }\cdot v=q^{(\lambda ,\nu )}v$ for all λ in the weight lattice.
If G is a Kac–Moody algebra, then in any irreducible highest weight representation of Uq(G), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of U(G) with equal highest weight. If the highest weight is dominant and integral (a weight μ is dominant and integral if μ satisfies the condition that $2(\mu ,\alpha _{i})/(\alpha _{i},\alpha _{i})$ is a non-negative integer for all i), then the weight spectrum of the irreducible representation is invariant under the Weyl group for G, and the representation is integrable.
Conversely, if a highest weight module is integrable, then its highest weight vector v satisfies $k_{\lambda }\cdot v=c_{\lambda }q^{(\lambda ,\nu )}v$, where cλ · v = dλv are complex numbers such that
• $c_{0}=1,$
• $c_{\lambda }c_{\mu }=c_{\lambda +\mu },$ for all weights λ and μ,
• $c_{2\alpha _{i}}=1$ for all i,
and ν is dominant and integral.
As is the case for all Hopf algebras, the tensor product of two modules is another module. For an element x of Uq(G), and for vectors v and w in the respective modules, x ⋅ (v ⊗ w) = Δ(x) ⋅ (v ⊗ w), so that $k_{\lambda }\cdot (v\otimes w)=k_{\lambda }\cdot v\otimes k_{\lambda }.w$, and in the case of coproduct Δ1, $e_{i}\cdot (v\otimes w)=k_{i}\cdot v\otimes e_{i}\cdot w+e_{i}\cdot v\otimes w$ and $f_{i}\cdot (v\otimes w)=v\otimes f_{i}\cdot w+f_{i}\cdot v\otimes k_{i}^{-1}\cdot w.$
The integrable highest weight module described above is a tensor product of a one-dimensional module (on which kλ = cλ for all λ, and ei = fi = 0 for all i) and a highest weight module generated by a nonzero vector v0, subject to $k_{\lambda }\cdot v_{0}=q^{(\lambda ,\nu )}v_{0}$ for all weights λ, and $e_{i}\cdot v_{0}=0$ for all i.
In the specific case where G is a finite-dimensional Lie algebra (as a special case of a Kac–Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional.
In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac–Moody algebra (the highest weights are the same, as are their multiplicities).
Case 1: q is not a root of unity
Strictly, the quantum group Uq(G) is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an R-matrix. This infinite formal sum is expressible in terms of generators ei and fi, and Cartan generators tλ, where kλ is formally identified with qtλ. The infinite formal sum is the product of two factors,
$q^{\eta \sum _{j}t_{\lambda _{j}}\otimes t_{\mu _{j}}}$
and an infinite formal sum, where λj is a basis for the dual space to the Cartan subalgebra, and μj is the dual basis, and η = ±1.
The formal infinite sum which plays the part of the R-matrix has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product of two lowest weight modules. Specifically, if v has weight α and w has weight β, then
$q^{\eta \sum _{j}t_{\lambda _{j}}\otimes t_{\mu _{j}}}\cdot (v\otimes w)=q^{\eta (\alpha ,\beta )}v\otimes w,$
and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on v ⊗ W to a finite sum.
Specifically, if V is a highest weight module, then the formal infinite sum, R, has a well-defined, and invertible, action on V ⊗ V, and this value of R (as an element of End(V ⊗ V)) satisfies the Yang–Baxter equation, and therefore allows us to determine a representation of the braid group, and to define quasi-invariants for knots, links and braids.
Quantum groups at q = 0
Main article: Crystal base
Masaki Kashiwara has researched the limiting behaviour of quantum groups as q → 0, and found a particularly well behaved base called a crystal base.
Description and classification by root-systems and Dynkin diagrams
There has been considerable progress in describing finite quotients of quantum groups such as the above Uq(g) for qn = 1; one usually considers the class of pointed Hopf algebras, meaning that all subcoideals are 1-dimensional and thus there sum form a group called coradical:
• In 2002 H.-J. Schneider and N. Andruskiewitsch [3] finished their classification of pointed Hopf algebras with an abelian co-radical group (excluding primes 2, 3, 5, 7), especially as the above finite quotients of Uq(g) decompose into E′s (Borel part), dual F′s and K′s (Cartan algebra) just like ordinary Semisimple Lie algebras:
$\left({\mathfrak {B}}(V)\otimes k[\mathbf {Z} ^{n}]\otimes {\mathfrak {B}}(V^{*})\right)^{\sigma }$
Here, as in the classical theory V is a braided vector space of dimension n spanned by the E′s, and σ (a so-called cocylce twist) creates the nontrivial linking between E′s and F′s. Note that in contrast to classical theory, more than two linked components may appear. The role of the quantum Borel algebra is taken by a Nichols algebra ${\mathfrak {B}}(V)$ of the braided vectorspace.
• A crucial ingredient was I. Heckenberger's classification of finite Nichols algebras for abelian groups in terms of generalized Dynkin diagrams.[4] When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dankin diagram).
• Meanwhile, Schneider and Heckenberger[5] have generally proven the existence of an arithmetic root system also in the nonabelian case, generating a PBW basis as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used[6] on specific cases Uq(g) and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of the Weyl group of the Lie algebra g.
Compact matrix quantum groups
Main article: Compact quantum group
S. L. Woronowicz introduced compact matrix quantum groups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.
The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. By the Gelfand theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.
For a compact topological group, G, there exists a C*-algebra homomorphism Δ: C(G) → C(G) ⊗ C(G) (where C(G) ⊗ C(G) is the C*-algebra tensor product - the completion of the algebraic tensor product of C(G) and C(G)), such that Δ(f)(x, y) = f(xy) for all f ∈ C(G), and for all x, y ∈ G (where (f ⊗ g)(x, y) = f(x)g(y) for all f, g ∈ C(G) and all x, y ∈ G). There also exists a linear multiplicative mapping κ: C(G) → C(G), such that κ(f)(x) = f(x−1) for all f ∈ C(G) and all x ∈ G. Strictly, this does not make C(G) a Hopf algebra, unless G is finite. On the other hand, a finite-dimensional representation of G can be used to generate a *-subalgebra of C(G) which is also a Hopf *-algebra. Specifically, if $g\mapsto (u_{ij}(g))_{i,j}$ is an n-dimensional representation of G, then for all i, j uij ∈ C(G) and
$\Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj}.$
It follows that the *-algebra generated by uij for all i, j and κ(uij) for all i, j is a Hopf *-algebra: the counit is determined by ε(uij) = δij for all i, j (where δij is the Kronecker delta), the antipode is κ, and the unit is given by
$1=\sum _{k}u_{1k}\kappa (u_{k1})=\sum _{k}\kappa (u_{1k})u_{k1}.$
General definition
As a generalization, a compact matrix quantum group is defined as a pair (C, u), where C is a C*-algebra and $u=(u_{ij})_{i,j=1,\dots ,n}$ is a matrix with entries in C such that
• The *-subalgebra, C0, of C, which is generated by the matrix elements of u, is dense in C;
• There exists a C*-algebra homomorphism called the comultiplication Δ: C → C ⊗ C (where C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that for all i, j we have:
$\Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj}$
• There exists a linear antimultiplicative map κ: C0 → C0 (the coinverse) such that κ(κ(v*)*) = v for all v ∈ C0 and
$\sum _{k}\kappa (u_{ik})u_{kj}=\sum _{k}u_{ik}\kappa (u_{kj})=\delta _{ij}I,$
where I is the identity element of C. Since κ is antimultiplicative, then κ(vw) = κ(w) κ(v) for all v, w in C0.
As a consequence of continuity, the comultiplication on C is coassociative.
In general, C is not a bialgebra, and C0 is a Hopf *-algebra.
Informally, C can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.
Representations
A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra A is a square matrix $v=(v_{ij})_{i,j=1,\dots ,n}$ with entries in A (so v belongs to M(n, A)) such that
$\Delta (v_{ij})=\sum _{k=1}^{n}v_{ik}\otimes v_{kj}$
for all i, j and ε(vij) = δij for all i, j). Furthermore, a representation v, is called unitary if the matrix for v is unitary (or equivalently, if κ(vij) = v*ij for all i, j).
Example
An example of a compact matrix quantum group is SUμ(2), where the parameter μ is a positive real number. So SUμ(2) = (C(SUμ(2)), u), where C(SUμ(2)) is the C*-algebra generated by α and γ, subject to
$\gamma \gamma ^{*}=\gamma ^{*}\gamma ,$
$\alpha \gamma =\mu \gamma \alpha ,$
$\alpha \gamma ^{*}=\mu \gamma ^{*}\alpha ,$
$\alpha \alpha ^{*}+\mu \gamma ^{*}\gamma =\alpha ^{*}\alpha +\mu ^{-1}\gamma ^{*}\gamma =I,$
and
$u=\left({\begin{matrix}\alpha &\gamma \\-\gamma ^{*}&\alpha ^{*}\end{matrix}}\right),$
so that the comultiplication is determined by ∆(α) = α ⊗ α − γ ⊗ γ*, ∆(γ) = α ⊗ γ + γ ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(γ) = −μ−1γ, κ(γ*) = −μγ*, κ(α*) = α. Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation
$v=\left({\begin{matrix}\alpha &{\sqrt {\mu }}\gamma \\-{\frac {1}{\sqrt {\mu }}}\gamma ^{*}&\alpha ^{*}\end{matrix}}\right).$
Equivalently, SUμ(2) = (C(SUμ(2)), w), where C(SUμ(2)) is the C*-algebra generated by α and β, subject to
$\beta \beta ^{*}=\beta ^{*}\beta ,$
$\alpha \beta =\mu \beta \alpha ,$
$\alpha \beta ^{*}=\mu \beta ^{*}\alpha ,$
$\alpha \alpha ^{*}+\mu ^{2}\beta ^{*}\beta =\alpha ^{*}\alpha +\beta ^{*}\beta =I,$
and
$w=\left({\begin{matrix}\alpha &\mu \beta \\-\beta ^{*}&\alpha ^{*}\end{matrix}}\right),$
so that the comultiplication is determined by ∆(α) = α ⊗ α − μβ ⊗ β*, Δ(β) = α ⊗ β + β ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(β) = −μ−1β, κ(β*) = −μβ*, κ(α*) = α. Note that w is a unitary representation. The realizations can be identified by equating $\gamma ={\sqrt {\mu }}\beta $.
When μ = 1, then SUμ(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).
Bicrossproduct quantum groups
Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first.
The very simplest nontrivial example corresponds to two copies of R locally acting on each other and results in a quantum group (given here in an algebraic form) with generators p, K, K−1, say, and coproduct
$[p,K]=hK(K-1)$
$\Delta p=p\otimes K+1\otimes p$
$\Delta K=K\otimes K$
where h is the deformation parameter.
This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra (the Iwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated to g. For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions.
See also
• Hopf algebra
• Lie bialgebra
• Poisson–Lie group
• Quantum affine algebra
Notes
1. Schwiebert, Christian (1994), Generalized quantum inverse scattering, p. 12237, arXiv:hep-th/9412237v3, Bibcode:1994hep.th...12237S
2. Majid, Shahn (1988), "Hopf algebras for physics at the Planck scale", Classical and Quantum Gravity, 5 (12): 1587–1607, Bibcode:1988CQGra...5.1587M, CiteSeerX 10.1.1.125.6178, doi:10.1088/0264-9381/5/12/010
3. Andruskiewitsch, Schneider: Pointed Hopf algebras, New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.
4. Heckenberger: Nichols algebras of diagonal type and arithmetic root systems, Habilitation thesis 2005.
5. Heckenberger, Schneider: Root system and Weyl gruppoid for Nichols algebras, 2008.
6. Heckenberger, Schneider: Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl grupoid, 2009.
References
• Grensing, Gerhard (2013). Structural Aspects of Quantum Field Theory and Noncommutative Geometry. World Scientific. doi:10.1142/8771. ISBN 978-981-4472-69-2.
• Jagannathan, R. (2001). "Some introductory notes on quantum groups, quantum algebras, and their applications". arXiv:math-ph/0105002.
• Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0783-2, ISBN 978-0-387-94370-1, MR 1321145
• Lusztig, George (2010) [1993]. Introduction to Quantum Groups. Cambridge, MA: Birkhäuser. ISBN 978-0-817-64716-2.
• Majid, Shahn (2002), A quantum groups primer, London Mathematical Society Lecture Note Series, vol. 292, Cambridge University Press, doi:10.1017/CBO9780511549892, ISBN 978-0-521-01041-2, MR 1904789
• Majid, Shahn (January 2006), "What Is...a Quantum Group?" (PDF), Notices of the American Mathematical Society, 53 (1): 30–31, retrieved 2008-01-16
• Podles, P.; Muller, E. (1998), "Introduction to quantum groups", Reviews in Mathematical Physics, 10 (4): 511–551, arXiv:q-alg/9704002, Bibcode:1997q.alg.....4002P, doi:10.1142/S0129055X98000173, S2CID 2596718
• Shnider, Steven; Sternberg, Shlomo (1993). Quantum groups: From coalgebras to Drinfeld algebras. Graduate Texts in Mathematical Physics. Vol. 2. Cambridge, MA: International Press.
• Street, Ross (2007), Quantum groups, Australian Mathematical Society Lecture Series, vol. 19, Cambridge University Press, doi:10.1017/CBO9780511618505, ISBN 978-0-521-69524-4, MR 2294803
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
• Quantum biology
• Quantum chemistry
• Quantum chaos
• Quantum cosmology
• Quantum differential calculus
• Quantum dynamics
• Quantum geometry
• Quantum measurement problem
• Quantum mind
• Quantum stochastic calculus
• Quantum spacetime
Technology
• Quantum algorithms
• Quantum amplifier
• Quantum bus
• Quantum cellular automata
• Quantum finite automata
• Quantum channel
• Quantum circuit
• Quantum complexity theory
• Quantum computing
• Timeline
• Quantum cryptography
• Quantum electronics
• Quantum error correction
• Quantum imaging
• Quantum image processing
• Quantum information
• Quantum key distribution
• Quantum logic
• Quantum logic gates
• Quantum machine
• Quantum machine learning
• Quantum metamaterial
• Quantum metrology
• Quantum network
• Quantum neural network
• Quantum optics
• Quantum programming
• Quantum sensing
• Quantum simulator
• Quantum teleportation
Extensions
• Casimir effect
• Quantum statistical mechanics
• Quantum field theory
• History
• Quantum gravity
• Relativistic quantum mechanics
Related
• Schrödinger's cat
• in popular culture
• EPR paradox
• Quantum mysticism
• Category
• Physics portal
• Commons
Authority control: National
• Czech Republic
|
Wikipedia
|
Quantum groupoid
In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality), in its groupoid algebra or in the commutative Hopf algebroid of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous) monoidal categories, Hopf algebroids etc.
References
• Ross Street, Brian Day, "Quantum categories, star autonomy, and quantum groupoids", in "Galois Theory, Hopf Algebras, and Semiabelian Categories", Fields Institute Communications 43 (American Math. Soc. 2004) 187–226; arXiv:math/0301209
• Gabriella Böhm, "Hopf algebroids", (a chapter of) Handbook of algebra, Vol. 6, ed. by M. Hazewinkel, Elsevier 2009, 173–236 arXiv:0805.3806
• Jiang-Hua Lu, "Hopf algebroids and quantum groupoids", Int. J. Math. 7, n. 1 (1996) pp. 47–70, arXiv:q-alg/9505024, MR1369905, doi:10.1142/S0129167X96000050
|
Wikipedia
|
Quantum inverse scattering method
In quantum physics, the quantum inverse scattering method (QISM) or the algebraic Bethe ansatz is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979.[1]
It can be viewed as a quantized version of the classical inverse scattering method pioneered by Norman Zabusky and Martin Kruskal[2] used to investigate the Korteweg–de Vries equation and later other integrable partial differential equations. In both, a Lax matrix features heavily and scattering data is used to construct solutions to the original system.
While the classical inverse scattering method is used to solve integrable partial differential equations which model continuous media (for example, the KdV equation models shallow water waves), the QISM is used to solve many-body quantum systems, sometimes known as spin chains, of which the Heisenberg spin chain is the best-studied and most famous example. These are typically discrete systems, with particles fixed at different points of a lattice, but limits of results obtained by the QISM can give predictions even for field theories defined on a continuum, such as the quantum sine-Gordon model.
Discussion
The quantum inverse scattering method relates two different approaches:
1. the Bethe ansatz, a method of solving integrable quantum models in one space and one time dimension.
2. the inverse scattering transform, a method of solving classical integrable differential equations of the evolutionary type.
This method led to the formulation of quantum groups, in particular the Yangian. The center of the Yangian, given by the quantum determinant plays a prominent role in the method.
An important concept in the inverse scattering transform is the Lax representation. The quantum inverse scattering method starts by the quantization of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the algebraic Bethe ansatz.[3] This led to further progress in the understanding of quantum integrable systems, such as the quantum Heisenberg model, the quantum Nonlinear Schrödinger equation (also known as the Lieb–Liniger model or the Tonks–Girardeau gas) and the Hubbard model.
The theory of correlation functions was developed , relating determinant representations, descriptions by differential equations and the Riemann–Hilbert problem. Asymptotics of correlation functions which include space, time and temperature dependence were evaluated in 1991.
Explicit expressions for the higher conservation laws of the integrable models were obtained in 1989.
Essential progress was achieved in study of ice-type models: the bulk free energy of the six vertex model depends on boundary conditions even in the thermodynamic limit.
Main steps of the QISM
The steps can be summarized as follows (Evgeny Sklyanin 1992):
1. Take an R-matrix which solves the Yang–Baxter equation
2. Take a representation of an algebra ${\mathcal {T}}_{R}$ satisfying the RTT relations
3. Find the spectrum of the generating function $t(u)$ of the centre of ${\mathcal {T}}_{R}$
4. Find correlators.
References
1. Takhtadzhan, L A; Faddeev, Lyudvig D (31 October 1979). "THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL". Russian Mathematical Surveys. 34 (5): 11–68. doi:10.1070/RM1979v034n05ABEH003909.
2. Zabusky, N. J.; Kruskal, M. D. (9 August 1965). "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States". Physical Review Letters. 15 (6): 240–243. doi:10.1103/PhysRevLett.15.240.
3. See for example lectures by N.A. Slavnov arXiv:1804.07350
• Sklyanin, E. K. (1992). "Quantum Inverse Scattering Method. Selected Topics". arXiv:hep-th/9211111.
• Faddeev, L. (1995), "Instructive history of the quantum inverse scattering method", Acta Applicandae Mathematicae, 39 (1): 69–84, doi:10.1007/BF00994626, MR 1329554, S2CID 120648929
• Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1993), Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, ISBN 978-0-521-37320-3, MR 1245942
Integrable systems
Geometric integrability
• Frobenius integrability
• Foliation
• Liouville integrability
In classical mechanics
Examples
• Harmonic oscillator
• Central force systems
• Kepler system
• Two body problem
• Integrable tops
• Euler
• Kovalevskaya
• Lagrange
• Garnier integrable system
• Hitchin system
Theory
• Liouville–Arnold theorem
• Action-angle variables
• Superintegrable Hamiltonian system
In quantum mechanics
• Quantum harmonic oscillator
• Hydrogen atom
• Pöschl–Teller potential
Integrable PDEs/Classical integrable field theories
Examples
• KdV equation
• KdV hierarchy
• Sine-Gordon equation
• Nonlinear Schrödinger equation
• Gross–Neveu model
• Thirring model
• Kadomtsev–Petviashvili equation
Theory
• Bäcklund transformation
• Lax pairs
• Infinitely many integrals of motion
• Soliton solutions
• Topological soliton
• Inverse scattering transform
ASDYM as a master theory
• Anti-self-dual Yang–Mills equations
• Twistor correspondence
• Ward conjecture
Integrable Quantum Field theories
• Quantum Sine-Gordon
• Quantum KdV
• Quantum Liouville
• Thirring model
• Toda field theory
• Principal chiral model
Master theories
• Four-dimensional Chern–Simons theory (Lagrangian)
• Affine Gaudin models (Hamiltonian)
• Six-dimensional holomorphic Chern–Simons theory
Exactly solvable statistical lattice models
• Ising model in one- and two-dimensions
• Square ice model
• Eight-vertex model
• Hard hexagon model
• Chiral Potts model
Exactly solvable quantum spin chains
Examples
• Quantum Heisenberg model
• Gaudin model
Theory
• Algebraic Bethe ansatz
• Quantum inverse scattering method
• Yang–Baxter equation
Contributors
Classical mechanics and geometry
• Vladimir Arnold
• Leonhard Euler
• Ferdinand Georg Frobenius
• Nigel Hitchin
• Sofia Kovalevskaya
• Joseph-Louis Lagrange
• Joseph Liouville
• Siméon Denis Poisson
PDEs
• Clifford S. Gardner
• John M. Greene
• Martin David Kruskal
• Peter Lax
• Robert Miura
IQFTs
• Alexander Zamolodchikov
• Alexei Zamolodchikov
Classical and quantum statistical lattices
• Rodney Baxter
• Ludvig Faddeev
• Elliott H. Lieb
• Yang Chen-Ning
|
Wikipedia
|
Quantum invariant
In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.[1][2][3]
List of invariants
• Finite type invariant
• Kontsevich invariant
• Kashaev's invariant
• Witten–Reshetikhin–Turaev invariant (Chern–Simons)
• Invariant differential operator[4]
• Rozansky–Witten invariant
• Vassiliev knot invariant
• Dehn invariant
• LMO invariant[5]
• Turaev–Viro invariant
• Dijkgraaf–Witten invariant[6]
• Reshetikhin–Turaev invariant
• Tau-invariant
• I-Invariant
• Klein J-invariant
• Quantum isotopy invariant[7]
• Ermakov–Lewis invariant
• Hermitian invariant
• Goussarov–Habiro theory of finite-type invariant
• Linear quantum invariant (orthogonal function invariant)
• Murakami–Ohtsuki TQFT
• Generalized Casson invariant
• Casson-Walker invariant
• Khovanov–Rozansky invariant
• HOMFLY polynomial
• K-theory invariants
• Atiyah–Patodi–Singer eta invariant
• Link invariant[8]
• Casson invariant
• Seiberg–Witten invariants
• Gromov–Witten invariant
• Arf invariant
• Hopf invariant
See also
• Invariant theory
• Framed knot
• Chern–Simons theory
• Algebraic geometry
• Seifert surface
• Geometric invariant theory
References
1. Reshetikhin, N. & Turaev, V. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups". Invent. Math. 103 (1): 547. Bibcode:1991InMat.103..547R. doi:10.1007/BF01239527. S2CID 123376541.
2. Kontsevich, Maxim (1993). "Vassiliev's knot invariants". Adv. Soviet Math. 16: 137.
3. Watanabe, Tadayuki (2007). "Knotted trivalent graphs and construction of the LMO invariant from triangulations". Osaka J. Math. 44 (2): 351. Retrieved 4 December 2012.
4. Letzter, Gail (2004). "Invariant differential operators for quantum symmetric spaces, II". arXiv:math/0406194.
5. Sawon, Justin (2000). "Topological quantum field theory and hyperkähler geometry". arXiv:math/0009222.
6. "Data" (PDF). hal.archives-ouvertes.fr. 1999. Retrieved 2019-11-04.
7. "Archived copy" (PDF). knot.kaist.ac.kr. Archived from the original (PDF) on 20 July 2007. Retrieved 13 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)
8. "Invariants of 3-manifolds via link polynomials and quantum groups - Springer". doi:10.1007/BF01239527. S2CID 123376541. {{cite journal}}: Cite journal requires |journal= (help)
Further reading
• Freedman, Michael H. (1990). Topology of 4-manifolds. Princeton, N.J: Princeton University Press. ISBN 978-0691085777. OL 2220094M.
• Ohtsuki, Tomotada (December 2001). Quantum Invariants. World Scientific Publishing Company. ISBN 9789810246754. OL 9195378M.
External links
• Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev
|
Wikipedia
|
Generalizations of Pauli matrices
In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matrices. Here, a few classes of such matrices are summarized.
Multi-qubit Pauli matrices (Hermitian)
This method of generalizing the Pauli matrices refers to a generalization from a single 2-level system (qubit) to multiple such systems. In particular, the generalized Pauli matrices for a group of $N$ qubits is just the set of matrices generated by all possible products of Pauli matrices on any of the qubits.[1]
The vector space of a single qubit is $V_{1}=\mathbb {C} ^{2}$ and the vector space of $N$ qubits is $V_{N}=\left(\mathbb {C} ^{2}\right)^{\otimes N}\cong \mathbb {C} ^{2^{N}}$. We use the tensor product notation
$\sigma _{a}^{(n)}=I^{(1)}\otimes \dotsm \otimes I^{(n-1)}\otimes \sigma _{a}\otimes I^{(n+1)}\otimes \dotsm \otimes I^{(N)},\qquad a=1,2,3$
to refer to the operator on $V_{N}$ that acts as a Pauli matrix on the $n$th qubit and the identity on all other qubits. We can also use $a=0$ for the identity, i.e., for any $n$ we use $ \sigma _{0}^{(n)}=\bigotimes _{m=1}^{N}I^{(m)}$. Then the multi-qubit Pauli matrices are all matrices of the form
$\sigma _{\,{\vec {a}}}:=\bigotimes _{n=1}^{N}\sigma _{a_{n}}^{(n)}=\sigma _{a_{1}}^{(1)}\otimes \dotsm \otimes \sigma _{a_{N}}^{(N)},\qquad {\vec {a}}=(a_{1},\ldots ,a_{N})\in \{0,1,2,3\}^{\times N}$,
i.e., for ${\vec {a}}$ a vector of integers between 0 and 4. Thus there are $4^{N}$ such generalized Pauli matrices if we include the identity $ I=\bigotimes _{m=1}^{N}I^{(m)}$ and $4^{N}-1$ if we do not.
Higher spin matrices (Hermitian)
The traditional Pauli matrices are the matrix representation of the ${\mathfrak {su}}(2)$ Lie algebra generators $J_{x}$, $J_{y}$, and $J_{z}$ in the 2-dimensional irreducible representation of SU(2), corresponding to a spin-1/2 particle. These generate the Lie group SU(2).
For a general particle of spin $s=0,1/2,1,3/2,2,\ldots $, one instead utilizes the $2s+1$-dimensional irreducible representation.
Generalized Gell-Mann matrices (Hermitian)
This method of generalizing the Pauli matrices refers to a generalization from 2-level systems (Pauli matrices acting on qubits) to 3-level systems (Gell-Mann matrices acting on qutrits) and generic d-level systems (generalized Gell-Mann matrices acting on qudits).
Construction
Let Ejk be the matrix with 1 in the jk-th entry and 0 elsewhere. Consider the space of d×d complex matrices, ℂd×d, for a fixed d.
Define the following matrices,
fk,jd =
Ekj + Ejk, for k < j .
−i (Ejk − Ekj), for k > j .
hkd =
Id, the identity matrix, for k = 1,
hkd−1 ⊕ 0, for 1 < k < d .
${\sqrt {\tfrac {2}{d(d-1)}}}\left(h_{1}^{d-1}\oplus (1-d)\right)={\sqrt {\tfrac {2}{d(d-1)}}}\left(I_{d-1}\oplus (1-d)\right),$ for k = d.
The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension d.[2] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum.
The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert–Schmidt inner product on ℂd×d. By dimension count, one sees that they span the vector space of d×d complex matrices, ${\mathfrak {gl}}$(d,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of ${\mathfrak {su}}$(d ).
In dimensions d = 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.
Sylvester's generalized Pauli matrices (non-Hermitian)
A particularly notable generalization of the Pauli matrices was constructed by James Joseph Sylvester in 1882.[3] These are known as "Weyl–Heisenberg matrices" as well as "generalized Pauli matrices".[4][5]
Framing
The Pauli matrices $\sigma _{1}$ and $\sigma _{3}$ satisfy the following:
$\sigma _{1}^{2}=\sigma _{3}^{2}=I,\quad \sigma _{1}\sigma _{3}=-\sigma _{3}\sigma _{1}=e^{\pi i}\sigma _{3}\sigma _{1}.$
The so-called Walsh–Hadamard conjugation matrix is
$W={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.$
Like the Pauli matrices, W is both Hermitian and unitary. $\sigma _{1},\;\sigma _{3}$ and W satisfy the relation
$\;\sigma _{1}=W\sigma _{3}W^{*}.$
The goal now is to extend the above to higher dimensions, d.
Construction: The clock and shift matrices
See also: Generalized Clifford algebra
Fix the dimension d as before. Let ω = exp(2πi/d), a root of unity. Since ωd = 1 and ω ≠ 1, the sum of all roots annuls:
$1+\omega +\cdots +\omega ^{d-1}=0.$
Integer indices may then be cyclically identified mod d.
Now define, with Sylvester, the shift matrix
$\Sigma _{1}={\begin{bmatrix}0&0&0&\cdots &0&1\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\0&0&1&\cdots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &1&0\\\end{bmatrix}}$
and the clock matrix,
$\Sigma _{3}={\begin{bmatrix}1&0&0&\cdots &0\\0&\omega &0&\cdots &0\\0&0&\omega ^{2}&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &\omega ^{d-1}\end{bmatrix}}.$
These matrices generalize σ1 and σ3, respectively.
Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe quaternions, Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc.
These two matrices are also the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces[6][7][8] as formulated by Hermann Weyl, and they find routine applications in numerous areas of mathematical physics.[9] The clock matrix amounts to the exponential of position in a "clock" of d hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg group on a d-dimensional Hilbert space.
The following relations echo and generalize those of the Pauli matrices:
$\Sigma _{1}^{d}=\Sigma _{3}^{d}=I$
and the braiding relation,
$\Sigma _{3}\Sigma _{1}=\omega \Sigma _{1}\Sigma _{3}=e^{2\pi i/d}\Sigma _{1}\Sigma _{3},$
the Weyl formulation of the CCR, and can be rewritten as
$\Sigma _{3}\Sigma _{1}\Sigma _{3}^{d-1}\Sigma _{1}^{d-1}=\omega ~.$
On the other hand, to generalize the Walsh–Hadamard matrix W, note
$W={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&\omega ^{2-1}\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&\omega ^{d-1}\end{bmatrix}}.$
Define, again with Sylvester, the following analog matrix,[10] still denoted by W in a slight abuse of notation,
$W={\frac {1}{\sqrt {d}}}{\begin{bmatrix}1&1&1&\cdots &1\\1&\omega ^{d-1}&\omega ^{2(d-1)}&\cdots &\omega ^{(d-1)^{2}}\\1&\omega ^{d-2}&\omega ^{2(d-2)}&\cdots &\omega ^{(d-1)(d-2)}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&\omega &\omega ^{2}&\cdots &\omega ^{d-1}\end{bmatrix}}~.$
It is evident that W is no longer Hermitian, but is still unitary. Direct calculation yields
$\Sigma _{1}=W\Sigma _{3}W^{*}~,$
which is the desired analog result. Thus, W, a Vandermonde matrix, arrays the eigenvectors of Σ1, which has the same eigenvalues as Σ3.
When d = 2k, W * is precisely the discrete Fourier transform matrix, converting position coordinates to momentum coordinates and vice versa.
Definition
The complete family of d2 unitary (but non-Hermitian) independent matrices $\{\sigma _{k,j}\}_{k,j=1}^{d}$ is defined as follows:
$\sigma _{k,j}:=\left(\Sigma _{1}\right)^{k}\left(\Sigma _{3}\right)^{j}=\sum _{m=0}^{d-1}|m+k\rangle \omega ^{jm}\langle m|.$
This provides Sylvester's well-known trace-orthogonal basis for ${\mathfrak {gl}}$(d,ℂ), known as "nonions" ${\mathfrak {gl}}$(3,ℂ), "sedenions" ${\mathfrak {gl}}$(4,ℂ), etc...[11][12]
This basis can be systematically connected to the above Hermitian basis.[13] (For instance, the powers of Σ3, the Cartan subalgebra, map to linear combinations of the hkds.) It can further be used to identify ${\mathfrak {gl}}$(d,ℂ), as d → ∞, with the algebra of Poisson brackets.
Properties
With respect to the Hilbert–Schmidt inner product on operators, $\langle A,B\rangle _{\text{HS}}=\operatorname {Tr} (A^{*}B)$, Sylvester's generalized Pauli operators are orthogonal and normalized to ${\sqrt {d}}$:
$\langle \sigma _{k,j},\sigma _{k',j'}\rangle _{\text{HS}}=\delta _{kk'}\delta _{jj'}\|\sigma _{k,j}\|_{\text{HS}}^{2}=d\delta _{kk'}\delta _{jj'}$.
This can be checked directly from the above definition of $\sigma _{k,j}$.
See also
• Heisenberg group#Heisenberg group modulo an odd prime p
• Hermitian matrix
• Bloch sphere
• Discrete Fourier transform
• Generalized Clifford algebra
• Weyl–Brauer matrices
• Circulant matrix
• Shift operator
• Quantum Fourier transform
• 3D rotation group#A note on Lie algebras
Notes
1. Brown, Adam R.; Susskind, Leonard (2018-04-25). "Second law of quantum complexity". Physical Review D. 97 (8): 086015. arXiv:1701.01107. Bibcode:2018PhRvD..97h6015B. doi:10.1103/PhysRevD.97.086015. S2CID 119199949.
2. Kimura, G. (2003). "The Bloch vector for N-level systems". Physics Letters A. 314 (5–6): 339–349. arXiv:quant-ph/0301152. Bibcode:2003PhLA..314..339K. doi:10.1016/S0375-9601(03)00941-1. S2CID 119063531., Bertlmann, Reinhold A.; Philipp Krammer (2008-06-13). "Bloch vectors for qudits". Journal of Physics A: Mathematical and Theoretical. 41 (23): 235303. arXiv:0806.1174. Bibcode:2008JPhA...41w5303B. doi:10.1088/1751-8113/41/23/235303. ISSN 1751-8121. S2CID 118603188.
3. Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III . online and further.
4. Appleby, D. M. (May 2005). "Symmetric informationally complete–positive operator valued measures and the extended Clifford group". Journal of Mathematical Physics. 46 (5): 052107. arXiv:quant-ph/0412001. Bibcode:2005JMP....46e2107A. doi:10.1063/1.1896384. ISSN 0022-2488.
5. Howard, Mark; Vala, Jiri (2012-08-15). "Qudit versions of the qubit π / 8 gate". Physical Review A. 86 (2): 022316. arXiv:1206.1598. Bibcode:2012PhRvA..86b2316H. doi:10.1103/PhysRevA.86.022316. ISSN 1050-2947. S2CID 56324846.
6. Weyl, H., "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, doi:10.1007/BF02055756.
7. Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1931)
8. Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics. 6 (5): 583. Bibcode:1976FoPh....6..583S. doi:10.1007/BF00715110. S2CID 119936801.
9. For a serviceable review, see Vourdas A. (2004), "Quantum systems with finite Hilbert space", Rep. Prog. Phys. 67 267. doi:10.1088/0034-4885/67/3/R03.
10. Sylvester, J.J. (1867). "Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 34 (232): 461–475. doi:10.1080/14786446708639914.
11. Patera, J.; Zassenhaus, H. (1988). "The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type An−1". Journal of Mathematical Physics. 29 (3): 665. Bibcode:1988JMP....29..665P. doi:10.1063/1.528006.
12. Since all indices are defined cyclically mod d, $\mathrm {tr} \Sigma _{1}^{j}\Sigma _{3}^{k}\Sigma _{1}^{m}\Sigma _{3}^{n}=\omega ^{km}d~\delta _{j+m,0}\delta _{k+n,0}$.
13. Fairlie, D. B.; Fletcher, P.; Zachos, C. K. (1990). "Infinite-dimensional algebras and a trigonometric basis for the classical Lie algebras". Journal of Mathematical Physics. 31 (5): 1088. Bibcode:1990JMP....31.1088F. doi:10.1063/1.528788.
|
Wikipedia
|
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.[1]
Part of a series of articles about
Quantum mechanics
$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle $
Schrödinger equation
• Introduction
• Glossary
• History
Background
• Classical mechanics
• Old quantum theory
• Bra–ket notation
• Hamiltonian
• Interference
Fundamentals
• Complementarity
• Decoherence
• Entanglement
• Energy level
• Measurement
• Nonlocality
• Quantum number
• State
• Superposition
• Symmetry
• Tunnelling
• Uncertainty
• Wave function
• Collapse
Experiments
• Bell's inequality
• Davisson–Germer
• Double-slit
• Elitzur–Vaidman
• Franck–Hertz
• Leggett–Garg inequality
• Mach–Zehnder
• Popper
• Quantum eraser
• Delayed-choice
• Schrödinger's cat
• Stern–Gerlach
• Wheeler's delayed-choice
Formulations
• Overview
• Heisenberg
• Interaction
• Matrix
• Phase-space
• Schrödinger
• Sum-over-histories (path integral)
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
Advanced topics
• Relativistic quantum mechanics
• Quantum field theory
• Quantum information science
• Quantum computing
• Quantum chaos
• EPR paradox
• Density matrix
• Scattering theory
• Quantum statistical mechanics
• Quantum machine learning
Scientists
• Aharonov
• Bell
• Bethe
• Blackett
• Bloch
• Bohm
• Bohr
• Born
• Bose
• de Broglie
• Compton
• Dirac
• Davisson
• Debye
• Ehrenfest
• Einstein
• Everett
• Fock
• Fermi
• Feynman
• Glauber
• Gutzwiller
• Heisenberg
• Hilbert
• Jordan
• Kramers
• Pauli
• Lamb
• Landau
• Laue
• Moseley
• Millikan
• Onnes
• Planck
• Rabi
• Raman
• Rydberg
• Schrödinger
• Simmons
• Sommerfeld
• von Neumann
• Weyl
• Wien
• Wigner
• Zeeman
• Zeilinger
These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observables, which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables.
Prior to the development of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus, and increasing in complexity up to differential geometry and partial differential equations. Probability theory was used in statistical mechanics. Geometric intuition played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule, which was formulated entirely on the classical phase space.
History of the formalism
The "old quantum theory" and the need for new mathematics
In the 1890s, Planck was able to derive the blackbody spectrum, which was later used to avoid the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of electromagnetic radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Planck's constant in his honor.
In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbed photons.
All of these developments were phenomenological and challenged the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld–Wilson–Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.
In 1923, de Broglie proposed that wave–particle duality applied not only to photons but to electrons and every other physical system.
The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity.
The "new quantum theory"
Werner Heisenberg's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra. Later in the same year, Schrödinger created his wave mechanics. Schrödinger's formalism was considered easier to understand, visualize and calculate as it led to differential equations, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.
Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the Copenhagen interpretation of quantum mechanics. Schrödinger's wave function can be seen to be closely related to the classical Hamilton–Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, Paul Dirac[2] discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.
To be more precise, already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics–– the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, linear algebra was not generally popular with physicists in its present form.
Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the bra–ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in many types of generalizations of the field.
The first complete mathematical formulation of this approach, known as the Dirac–von Neumann axioms, is generally credited to John von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces) in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.
Later developments
The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases.
• Path integral formulation
• Phase-space formulation of quantum mechanics & geometric quantization
• quantum field theory in curved spacetime
• axiomatic, algebraic and constructive quantum field theory
• C*-algebra formalism
• Generalized statistical model of quantum mechanics
A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.
Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics.
Postulates of quantum mechanics
A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a phase space formulated by symplectic manifold, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a Hilbert space of states, observables are self-adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations. (It is possible, to map this Hilbert-space picture to a phase space formulation, invertibly. See below.)
The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Dirac–von Neumann axioms.[3]
Description of the state of a system
Each isolated physical system is associated with a (topologically) separable complex Hilbert space H with inner product ⟨φ|ψ⟩. Rays (that is, subspaces of complex dimension 1) in H are associated with quantum states of the system.
Postulate I
The state of an isolated physical system is represented, at a fixed time $t$, by a state vector $|\psi \rangle $ belonging to a Hilbert space ${\mathcal {H}}$ called the state space.
In other words, quantum states can be identified with equivalence classes (rays) of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state. A quantum mechanical state is a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrödinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid.[4]
Accompanying Postulate I is the composite system postulate:[5]
Composite system postulate
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
In the presence of quantum entanglement, the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or superposition, of tensor products of states of component subsystems. A subsystem in an entangled composite system generally cannot be described by a state vector (or a ray), but instead is described by a density operator; Such quantum state is known as a mixed state. The density operator of a mixed state is a trace class, nonnegative (positive semi-definite) self-adjoint operator ρ normalized to be of trace 1. In turn, any density operator of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see purification theorem).
In the absence of quantum entanglement, the quantum state of the composite system is called a separable state. The density matrix of a bipartite system in a separable state can be expressed as $\rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}$, where $\;\sum _{k}p_{k}=1$. If there is only a single non-zero $p_{k}$, then the state can be expressed just as $ \rho =\rho _{1}\otimes \rho _{2},$ and is called simply separable or product state.
Description of physical quantities
Physical observables are represented by Hermitian matrices on H. Since these operators are Hermitian, their eigenvalues are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable is discrete, then the possible results are quantized.
Postulate II.a
Every measurable physical quantity ${\mathcal {A}}$ is described by a Hermitian operator $A$ acting in the state space ${\mathcal {H}}$. This operator is an observable, meaning that its eigenvectors form a basis for ${\mathcal {H}}$. The result of measuring a physical quantity ${\mathcal {A}}$ must be one of the eigenvalues of the corresponding observable $A$.
Results of measurement
By spectral theory, we can associate a probability measure to the values of A in any state ψ. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. The expectation value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector ψ ∈ H is $\langle \psi |A|\psi \rangle $. If we represent the state ψ in the basis formed by the eigenvectors of A, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.
Postulate II.b
When the physical quantity ${\mathcal {A}}$ is measured on a system in a normalized state $|\psi \rangle $, the probability of obtaining an eigenvalue (denoted $a_{n}$ for discrete spectra and $\alpha $ for continuous spectra) of the corresponding observable $A$ is given by the amplitude squared of the appropriate wave function (projection onto corresponding eigenvector).
${\begin{aligned}\mathbb {P} (a_{n})&=|\langle a_{n}|\psi \rangle |^{2}&{\text{(Discrete, nondegenerate spectrum)}}\\\mathbb {P} (a_{n})&=\sum _{i}^{g_{n}}|\langle a_{n}^{i}|\psi \rangle |^{2}&{\text{(Discrete, degenerate spectrum)}}\\d\mathbb {P} (\alpha )&=|\langle \alpha |\psi \rangle |^{2}d\alpha &{\text{(Continuous, nondegenerate spectrum)}}\end{aligned}}$
For a mixed state ρ, the expected value of A in the state ρ is $\operatorname {tr} (A\rho )$, and the probability of obtaining an eigenvalue $a_{n}$ in a discrete, nondegenerate spectrum of the corresponding observable $A$ is given by $\mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle $.
If the eigenvalue $a_{n}$ has degenerate, orthonormal eigenvectors $\{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}$, then the projection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace:
$P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,$
and then $\mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )$.
Postulates II.a and II.b are collectively known as the Born rule of quantum mechanics.
Effect of measurement on the state
When a measurement is performed, only one result is obtained (according to some interpretations of quantum mechanics). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore the state vector must change as a result of measurement, and collapse onto the eigensubspace associated with the eigenvalue measured.
Postulate II.c
If the measurement of the physical quantity ${\mathcal {A}}$ on the system in the state $|\psi \rangle $ gives the result $a_{n}$, then the state of the system immediately after the measurement is the normalized projection of $|\psi \rangle $ onto the eigensubspace associated with $a_{n}$
$\psi \quad {\overset {a_{n}}{\Longrightarrow }}\quad {\frac {P_{n}|\psi \rangle }{\sqrt {\langle \psi |P_{n}|\psi \rangle }}}$
For a mixed state ρ, after obtaining an eigenvalue $a_{n}$ in a discrete, nondegenerate spectrum of the corresponding observable $A$, the updated state is given by $ \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}$. If the eigenvalue $a_{n}$ has degenerate, orthonormal eigenvectors $\{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}$, then the projection operator onto the eigensubspace is $P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|$.
Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of measurements, and are sometimes collectively called the measurement postulate(s).
Note that the projection-valued measures (PVM) described in the measurement postulate(s) can be generalized to positive operator-valued measures (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see Naimark's dilation theorem).
Time evolution of a system
Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the de Broglie hypothesis or path integrals.
Postulate III
The time evolution of the state vector $|\psi (t)\rangle $ is governed by the Schrödinger equation, where $H(t)$ is the observable associated with the total energy of the system (called the Hamiltonian)
$i\hbar {\frac {d}{dt}}|\psi (t)\rangle =H(t)|\psi (t)\rangle $
Equivalently, the time evolution postulate can be stated as:
Postulate III
The time evolution of a closed system is described by a unitary transformation on the initial state.
$|\psi (t)\rangle =U(t;t_{0})|\psi (t_{0})\rangle $
For a closed system in a mixed state ρ, the time evolution is $\rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})$.
The evolution of an open quantum system can be described by quantum operations (in an operator sum formalism) and quantum instruments, and generally does not have to be unitary.
Other implications of the postulates
• Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem (supersymmetry is another matter entirely).
• Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.
• One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.
• Recent research has shown[6] that the composite system postulate (tensor product postulate) can be derived from the state postulate (Postulate I) and the measurement postulates (Postulates II); Moreover, it has also been shown[7] that the measurement postulates (Postulates II) can be derived from "unitary quantum mechanics", which includes only the state postulate (Postulate I), the composite system postulate (tensor product postulate) and the unitary evolution postulate (Postulate III).
Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.
Spin
In addition to their other properties, all particles possess a quantity called spin, an intrinsic angular momentum. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position r and time t as continuous variables, ψ = ψ(r, t). For spin wavefunctions the spin is an additional discrete variable: ψ = ψ(r, t, σ), where σ takes the values;
$\sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.$
That is, the state of a single particle with spin S is represented by a (2S + 1)-component spinor of complex-valued wave functions.
Two classes of particles with very different behaviour are bosons which have integer spin (S = 0, 1, 2, ...), and fermions possessing half-integer spin (S = 1⁄2, 3⁄2, 5⁄2, ...).
Pauli's principle
The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have
Pauli principle
$\psi (\dots ,\,\mathbf {r} _{i},\sigma _{i},\,\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots )=(-1)^{2S}\cdot \psi (\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots ,\mathbf {r} _{i},\sigma _{i},\,\dots )$
i.e., on transposition of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor (−1)2S which is +1 for bosons, but (−1) for fermions. Electrons are fermions with S = 1/2; quanta of light are bosons with S = 1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension d = 2 can one construct entities where (−1)2S is replaced by an arbitrary complex number with magnitude 1, called anyons.
Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics, the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the periodic system of chemistry, are consequences of the two properties.
Mathematical structure of quantum mechanics
Pictures of dynamics
• In the so-called Schrödinger picture of quantum mechanics, the dynamics is given as follows: The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If |ψ(t)⟩ denotes the state of the system at any one time t, the following Schrödinger equation holds:
Schrödinger equation (general)
$i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle =H\left|\psi (t)\right\rangle $
where H is a densely defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and ħ is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.
Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary map U(t): H → H such that
$\left|\psi (t+s)\right\rangle =U(t)\left|\psi (s)\right\rangle $
for all times s, t. The existence of a self-adjoint Hamiltonian H such that
$U(t)=e^{-(i/\hbar )tH}$
is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that H does not depend on time and that the perturbation starts at t0 = 0; otherwise one must use the Dyson series, formally written as
$U(t)={\mathcal {T}}\left[\exp \left(-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,H(t')\right)\right],$
where ${\mathcal {T}}$ is Dyson's time-ordering symbol.
(This symbol permutes a product of noncommuting operators of the form
$B_{1}(t_{1})\cdot B_{2}(t_{2})\cdot \dots \cdot B_{n}(t_{n})$
into the uniquely determined re-ordered expression
$B_{i_{1}}(t_{i_{1}})\cdot B_{i_{2}}(t_{i_{2}})\cdot \dots \cdot B_{i_{n}}(t_{i_{n}})$
with $t_{i_{1}}\geq t_{i_{2}}\geq \dots \geq t_{i_{n}}\,.$ The result is a causal chain, the primary cause in the past on the utmost r.h.s., and finally the present effect on the utmost l.h.s. .)
• The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:
$\left|\psi \right\rangle =\left|\psi (0)\right\rangle $
$A(t)=U(-t)AU(t).$
It is then easily checked that the expected values of all observables are the same in both pictures
$\langle \psi \mid A(t)\mid \psi \rangle =\langle \psi (t)\mid A\mid \psi (t)\rangle $
and that the time-dependent Heisenberg operators satisfy
Heisenberg picture (general)
${\frac {d}{dt}}A(t)={\frac {i}{\hbar }}[H,A(t)]+{\frac {\partial A(t)}{\partial t}},$
which is true for time-dependent A = A(t). Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.
• The so-called Dirac picture or interaction picture has time-dependent states and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols:
Dirac picture
$i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle ={H}_{\rm {int}}(t)\left|\psi (t)\right\rangle $
$i\hbar {\frac {d}{dt}}A(t)=[A(t),H_{0}].$
The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. H = H0 + V, in the interaction picture it does, at least, if V does not commute with H0, since
$H_{\rm {int}}(t)\equiv e^{{(i/\hbar })tH_{0}}\,V\,e^{{(-i/\hbar })tH_{0}}.$
So the above-mentioned Dyson-series has to be used anyhow.
The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics.
Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated with the symmetry (for instance, angular or linear momentum).
Summary:
Evolution of: Picture ()
Schrödinger (S) Heisenberg (H) Interaction (I)
Ket state $|\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle $ constant $|\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle $
Observable constant $A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }$ $A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }$
Density matrix $\rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }$ constant $\rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }$
Representations
The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerations of quantization, the deformation extension from classical to quantum mechanics.
The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the Segal–Bargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann). All four are unitarily equivalent.
Time as an operator
The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" H − E, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of H − E (this requires the use of a rigged Hilbert space and a renormalization of the norm).
This is related to the quantization of constrained systems and quantization of gauge theories. It is also possible to formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).
The problem of measurement
The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement.[8] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain):
• Let A have spectral resolution
$A=\int \lambda \,d\operatorname {E} _{A}(\lambda ),$
where EA is the resolution of the identity (also called projection-valued measure) associated with A. Then the probability of the measurement outcome lying in an interval B of R is |EA(B) ψ|2. In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure
$\langle \psi \mid \operatorname {E} _{A}\psi \rangle .$
• If the measured value is contained in B, then immediately after the measurement, the system will be in the (generally non-normalized) state EA(B)ψ. If the measured value does not lie in B, replace B by its complement for the above state.
For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues λi, with corresponding eigenvectors ψi. The projection-valued measure associated with A, EA, is then
$\operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,$
where B is a Borel set containing only the single eigenvalue λi. If the system is prepared in state
$|\psi \rangle $
Then the probability of a measurement returning the value λi can be calculated by integrating the spectral measure
$\langle \psi \mid \operatorname {E} _{A}\psi \rangle $
over Bi. This gives trivially
$\langle \psi |\psi _{i}\rangle \langle \psi _{i}\mid \psi \rangle =|\langle \psi \mid \psi _{i}\rangle |^{2}.$
The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate.
A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections
$|\psi _{i}\rangle \langle \psi _{i}|$
by a finite set of positive operators
$F_{i}F_{i}^{*}$
whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {λ1 ... λn} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λi. Instead of collapsing to the (unnormalized) state
$|\psi _{i}\rangle \langle \psi _{i}|\psi \rangle $
after the measurement, the system now will be in the state
$F_{i}|\psi \rangle .$
Since the Fi Fi* operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.
The same formulation applies to general mixed states.
In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.
In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).
The relative state interpretation
An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed the "many-worlds interpretation" of quantum physics.
List of mathematical tools
Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert's Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.
The main tools include:
• linear algebra: complex numbers, eigenvectors, eigenvalues
• functional analysis: Hilbert spaces, linear operators, spectral theory
• differential equations: partial differential equations, separation of variables, ordinary differential equations, Sturm–Liouville theory, eigenfunctions
• harmonic analysis: Fourier transforms
See also: list of mathematical topics in quantum theory
Notes
1. Frederick W. Byron, Robert W. Fuller; Mathematics of classical and quantum physics; Courier Dover Publications, 1992.
2. Dirac, P. A. M. (1925). "The Fundamental Equations of Quantum Mechanics". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 109 (752): 642–653. Bibcode:1925RSPSA.109..642D. doi:10.1098/rspa.1925.0150.
3. Cohen-Tannoudji, Claude (2019). Quantum mechanics. Volume 2. Bernard Diu, Franck Laloë, Susan Reid Hemley, Nicole Ostrowsky, D. B. Ostrowsky. Weinheim. ISBN 978-3-527-82272-0. OCLC 1159410161.{{cite book}}: CS1 maint: location missing publisher (link)
4. Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623. S2CID 121930907.
5. Jauch, J. M.; Wigner, E. P.; Yanase, M. M. (1997), "Some Comments Concerning Measurements in Quantum Mechanics", Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 475–482, doi:10.1007/978-3-662-09203-3_52, ISBN 978-3-642-08179-8, retrieved 2022-03-19
6. Carcassi, Gabriele; Maccone, Lorenzo; Aidala, Christine A. (2021-03-16). "Four Postulates of Quantum Mechanics Are Three". Physical Review Letters. 126 (11): 110402. arXiv:2003.11007. doi:10.1103/PhysRevLett.126.110402. PMID 33798366. S2CID 214623241.
7. Masanes, Lluís; Galley, Thomas D.; Müller, Markus P. (2019-03-25). "The measurement postulates of quantum mechanics are operationally redundant". Nature Communications. 10 (1): 1361. doi:10.1038/s41467-019-09348-x. ISSN 2041-1723. PMC 6434053. PMID 30911009.
8. G. Greenstein and A. Zajonc
References
• J. von Neumann, Mathematical Foundations of Quantum Mechanics (1932), Princeton University Press, 1955. Reprinted in paperback form.
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
• A. Gleason, Measures on the Closed Subspaces of a Hilbert Space, Journal of Mathematics and Mechanics, 1957.
• G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004).
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin 1964 (Reprinted by Princeton University Press)
• R. Jost, The General Theory of Quantized Fields, American Mathematical Society, 1965.
• J. M. Jauch, Foundations of quantum mechanics, Addison-Wesley Publ. Cy., Reading, Massachusetts, 1968.
• G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972.
• M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972.
• T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912, Clarendon Press, Oxford and Oxford University Press, New York, 1978.
• D. Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, 42 (1979),pp. 1–70.
• R. Shankar, "Principles of Quantum Mechanics", Springer, 1980.
• E. Prugovecki, Quantum Mechanics in Hilbert Space, Dover, 1981.
• S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995.
• N. Weaver, Mathematical Quantization, Chapman & Hall/CRC 2001.
• G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, 2005.
• D. McMahon, Quantum Mechanics Demystified, 2nd Ed., McGraw-Hill Professional, 2005.
• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
• V. Moretti, Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, 2nd Edition, Springer, 2018.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• V. Moretti, Fundamental Mathematical Structures of Quantum Theory, Springer, 2019.
• K. Landsman, Foundations of Quantum Theory, Springer 2017
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
• Quantum biology
• Quantum chemistry
• Quantum chaos
• Quantum cosmology
• Quantum differential calculus
• Quantum dynamics
• Quantum geometry
• Quantum measurement problem
• Quantum mind
• Quantum stochastic calculus
• Quantum spacetime
Technology
• Quantum algorithms
• Quantum amplifier
• Quantum bus
• Quantum cellular automata
• Quantum finite automata
• Quantum channel
• Quantum circuit
• Quantum complexity theory
• Quantum computing
• Timeline
• Quantum cryptography
• Quantum electronics
• Quantum error correction
• Quantum imaging
• Quantum image processing
• Quantum information
• Quantum key distribution
• Quantum logic
• Quantum logic gates
• Quantum machine
• Quantum machine learning
• Quantum metamaterial
• Quantum metrology
• Quantum network
• Quantum neural network
• Quantum optics
• Quantum programming
• Quantum sensing
• Quantum simulator
• Quantum teleportation
Extensions
• Casimir effect
• Quantum statistical mechanics
• Quantum field theory
• History
• Quantum gravity
• Relativistic quantum mechanics
Related
• Schrödinger's cat
• in popular culture
• EPR paradox
• Quantum mysticism
• Category
• Physics portal
• Commons
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
|
Wikipedia
|
Quantum neural network
Quantum neural networks are computational neural network models which are based on the principles of quantum mechanics. The first ideas on quantum neural computation were published independently in 1995 by Subhash Kak and Ron Chrisley,[1][2] engaging with the theory of quantum mind, which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical artificial neural network models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of quantum information in order to develop more efficient algorithms.[3][4][5] One important motivation for these investigations is the difficulty to train classical neural networks, especially in big data applications. The hope is that features of quantum computing such as quantum parallelism or the effects of interference and entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments.
Most Quantum neural networks are developed as feed-forward networks. Similar to their classical counterparts, this structure intakes input from one layer of qubits, and passes that input onto another layer of qubits. This layer of qubits evaluates this information and passes on the output to the next layer. Eventually the path leads to the final layer of qubits.[6][7] The layers do not have to be of the same width, meaning they don't have to have the same number of qubits as the layer before or after it. This structure is trained on which path to take similar to classical artificial neural networks. This is discussed in a lower section. Quantum neural networks refer to three different categories: Quantum computer with classical data, classical computer with quantum data, and quantum computer with quantum data.[6]
Examples
Quantum neural network research is still in its infancy, and a conglomeration of proposals and ideas of varying scope and mathematical rigor have been put forward. Most of them are based on the idea of replacing classical binary or McCulloch-Pitts neurons with a qubit (which can be called a “quron”), resulting in neural units that can be in a superposition of the state ‘firing’ and ‘resting’.
Quantum perceptrons
A lot of proposals attempt to find a quantum equivalent for the perceptron unit from which neural nets are constructed. A problem is that nonlinear activation functions do not immediately correspond to the mathematical structure of quantum theory, since a quantum evolution is described by linear operations and leads to probabilistic observation. Ideas to imitate the perceptron activation function with a quantum mechanical formalism reach from special measurements [8][9] to postulating non-linear quantum operators (a mathematical framework that is disputed).[10][11] A direct implementation of the activation function using the circuit-based model of quantum computation has recently been proposed by Schuld, Sinayskiy and Petruccione based on the quantum phase estimation algorithm.[12]
Quantum networks
At a larger scale, researchers have attempted to generalize neural networks to the quantum setting. One way of constructing a quantum neuron is to first generalise classical neurons and then generalising them further to make unitary gates. Interactions between neurons can be controlled quantumly, with unitary gates, or classically, via measurement of the network states. This high-level theoretical technique can be applied broadly, by taking different types of networks and different implementations of quantum neurons, such as photonically implemented neurons[7][13] and quantum reservoir processor (quantum version of reservoir computing).[14] Most learning algorithms follow the classical model of training an artificial neural network to learn the input-output function of a given training set and use classical feedback loops to update parameters of the quantum system until they converge to an optimal configuration. Learning as a parameter optimisation problem has also been approached by adiabatic models of quantum computing.[15]
Quantum neural networks can be applied to algorithmic design: given qubits with tunable mutual interactions, one can attempt to learn interactions following the classical backpropagation rule from a training set of desired input-output relations, taken to be the desired output algorithm's behavior.[16][17] The quantum network thus ‘learns’ an algorithm.
Quantum associative memory
The first quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999.[18] The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a circuit-based quantum computer that simulates associative memory. The memory states (in Hopfield neural networks saved in the weights of the neural connections) are written into a superposition, and a Grover-like quantum search algorithm retrieves the memory state closest to a given input. As such, this is not a fully content-addressable memory, since only incomplete patterns can be retrieved.
The first truly content-addressable quantum memory, which can retrieve patterns also from corrupted inputs, was proposed by Carlo A. Trugenberger.[19][20][21] Both memories can store an exponential (in terms of n qubits) number of patterns but can be used only once due to the no-cloning theorem and their destruction upon measurement.
Trugenberger,[20] however, has shown that his proababilistic model of quantum associative memory can be efficiently implemented and re-used multiples times for any polynomial number of stored patterns, a large advantage with respect to classical associative memories.
Classical neural networks inspired by quantum theory
A substantial amount of interest has been given to a “quantum-inspired” model that uses ideas from quantum theory to implement a neural network based on fuzzy logic.[22]
Training
Quantum Neural Networks can be theoretically trained similarly to training classical/artificial neural networks. A key difference lies in communication between the layers of a neural networks. For classical neural networks, at the end of a given operation, the current perceptron copies its output to the next layer of perceptron(s) in the network. However, in a quantum neural network, where each perceptron is a qubit, this would violate the no-cloning theorem.[6][23] A proposed generalized solution to this is to replace the classical fan-out method with an arbitrary unitary that spreads out, but does not copy, the output of one qubit to the next layer of qubits. Using this fan-out Unitary ($U_{f}$) with a dummy state qubit in a known state (Ex. $|0\rangle $ in the computational basis), also known as an Ancilla bit, the information from the qubit can be transferred to the next layer of qubits.[7] This process adheres to the quantum operation requirement of reversibility.[7][24]
Using this quantum feed-forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden-layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed utilizes fan-out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time.[6] In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network.[24]
Cost functions
To determine the effectiveness of a neural network, a cost function is used, which essentially measures the proximity of the network’s output to the expected or desired output. In a Classical Neural Network, the weights ($w$) and biases ($b$) at each step determine the outcome of the cost function $C(w,b)$.[6] When training a Classical Neural network, the weights and biases are adjusted after each iteration, and given equation 1 below, where $y(x)$ is the desired output and $a^{\text{out}}(x)$ is the actual output, the cost function is optimized when $C(w,b)$= 0. For a quantum neural network, the cost function is determined by measuring the fidelity of the outcome state ($\rho ^{\text{out}}$) with the desired outcome state ($\phi ^{\text{out}}$), seen in Equation 2 below. In this case, the Unitary operators are adjusted after each iteration, and the cost function is optimized when C = 1.[6]
Equation 1 $C(w,b)={1 \over N}\sum _{x}{||y(x)-a^{\text{out}}(x)|| \over 2}$
Equation 2 $C={1 \over N}\sum _{x}^{N}{\langle \phi ^{\text{out}}|\rho ^{\text{out}}|\phi ^{\text{out}}\rangle }$
See also
• Differentiable programming
• Optical neural network
• Holographic associative memory
• Quantum cognition
• Quantum machine learning
References
1. Kak, S. (1995). "On quantum neural computing". Advances in Imaging and Electron Physics. 94: 259–313. doi:10.1016/S1076-5670(08)70147-2. ISBN 9780120147366.
2. Chrisley, R. (1995). "Quantum Learning". In Pylkkänen, P.; Pylkkö, P. (eds.). New directions in cognitive science: Proceedings of the international symposium, Saariselka, 4–9 August 1995, Lapland, Finland. Helsinki: Finnish Association of Artificial Intelligence. pp. 77–89. ISBN 951-22-2645-6.
3. da Silva, Adenilton J.; Ludermir, Teresa B.; de Oliveira, Wilson R. (2016). "Quantum perceptron over a field and neural network architecture selection in a quantum computer". Neural Networks. 76: 55–64. arXiv:1602.00709. Bibcode:2016arXiv160200709D. doi:10.1016/j.neunet.2016.01.002. PMID 26878722. S2CID 15381014.
4. Panella, Massimo; Martinelli, Giuseppe (2011). "Neural networks with quantum architecture and quantum learning". International Journal of Circuit Theory and Applications. 39: 61–77. doi:10.1002/cta.619. S2CID 3791858.
5. Schuld, M.; Sinayskiy, I.; Petruccione, F. (2014). "The quest for a Quantum Neural Network". Quantum Information Processing. 13 (11): 2567–2586. arXiv:1408.7005. Bibcode:2014QuIP...13.2567S. doi:10.1007/s11128-014-0809-8. S2CID 37238534.
6. Beer, Kerstin; Bondarenko, Dmytro; Farrelly, Terry; Osborne, Tobias J.; Salzmann, Robert; Scheiermann, Daniel; Wolf, Ramona (2020-02-10). "Training deep quantum neural networks". Nature Communications. 11 (1): 808. arXiv:1902.10445. Bibcode:2020NatCo..11..808B. doi:10.1038/s41467-020-14454-2. ISSN 2041-1723. PMC 7010779. PMID 32041956.
7. Wan, Kwok-Ho; Dahlsten, Oscar; Kristjansson, Hler; Gardner, Robert; Kim, Myungshik (2017). "Quantum generalisation of feedforward neural networks". npj Quantum Information. 3: 36. arXiv:1612.01045. Bibcode:2017npjQI...3...36W. doi:10.1038/s41534-017-0032-4. S2CID 51685660.
8. Perus, M. (2000). "Neural Networks as a basis for quantum associative memory". Neural Network World. 10 (6): 1001. CiteSeerX 10.1.1.106.4583.
9. Zak, M.; Williams, C. P. (1998). "Quantum Neural Nets". International Journal of Theoretical Physics. 37 (2): 651–684. doi:10.1023/A:1026656110699. S2CID 55783801.
10. Gupta, Sanjay; Zia, R.K.P. (2001). "Quantum Neural Networks". Journal of Computer and System Sciences. 63 (3): 355–383. arXiv:quant-ph/0201144. doi:10.1006/jcss.2001.1769. S2CID 206569020.
11. Faber, J.; Giraldi, G. A. (2002). "Quantum Models for Artificial Neural Network" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
12. Schuld, M.; Sinayskiy, I.; Petruccione, F. (2014). "Simulating a perceptron on a quantum computer". Physics Letters A. 379 (7): 660–663. arXiv:1412.3635. doi:10.1016/j.physleta.2014.11.061. S2CID 14288234.
13. Narayanan, A.; Menneer, T. (2000). "Quantum artificial neural network architectures and components". Information Sciences. 128 (3–4): 231–255. doi:10.1016/S0020-0255(00)00055-4. S2CID 10901562.
14. Ghosh, S.; Opala, A.; Matuszewski, M.; Paterek, P.; Liew, T. C. H. (2019). "Quantum reservoir processing". npj Quantum Information. 5: 35. arXiv:1811.10335. Bibcode:2019npjQI...5...35G. doi:10.1038/s41534-019-0149-8. S2CID 119197635.
15. Neven, H.; et al. (2008). "Training a Binary Classifier with the Quantum Adiabatic Algorithm". arXiv:0811.0416. {{cite journal}}: Cite journal requires |journal= (help)
16. Bang, J.; et al. (2014). "A strategy for quantum algorithm design assisted by machine learning". New Journal of Physics. 16 (7): 073017. arXiv:1301.1132. Bibcode:2014NJPh...16g3017B. doi:10.1088/1367-2630/16/7/073017. S2CID 55377982.
17. Behrman, E. C.; Steck, J. E.; Kumar, P.; Walsh, K. A. (2008). "Quantum Algorithm design using dynamic learning". Quantum Information and Computation. 8 (1–2): 12–29. arXiv:0808.1558. doi:10.26421/QIC8.1-2-2. S2CID 18587557.
18. Ventura, D.; Martinez, T. (1999). "A quantum associative memory based on Grover's algorithm" (PDF). Proceedings of the International Conference on Artificial Neural Networks and Genetics Algorithms: 22–27. doi:10.1007/978-3-7091-6384-9_5. ISBN 978-3-211-83364-3. S2CID 3258510. Archived from the original (PDF) on 2017-09-11.
19. Trugenberger, C. A. (2001-07-18). "Probabilistic Quantum Memories". Physical Review Letters. 87 (6). arXiv:quant-ph/0012100. doi:10.1103/physrevlett.87.067901. ISSN 0031-9007.
20. Trugenberger, Carlo A. (2002). "Quantum Pattern Recognition". Quantum Information Processing. 1 (6): 471–493.
21. Trugenberger, C. A. (2002-12-19). "Phase Transitions in Quantum Pattern Recognition". Physical Review Letters. 89 (27). arXiv:quant-ph/0204115. doi:10.1103/physrevlett.89.277903. ISSN 0031-9007.
22. Purushothaman, G.; Karayiannis, N. (1997). "Quantum Neural Networks (QNN's): Inherently Fuzzy Feedforward Neural Networks" (PDF). IEEE Transactions on Neural Networks. 8 (3): 679–93. doi:10.1109/72.572106. PMID 18255670. S2CID 1634670. Archived from the original (PDF) on 2017-09-11.
23. Nielsen, Michael A; Chuang, Isaac L (2010). Quantum computation and quantum information. Cambridge; New York: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 665137861.
24. Feynman, Richard P. (1986-06-01). "Quantum mechanical computers". Foundations of Physics. 16 (6): 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/BF01886518. ISSN 1572-9516. S2CID 122076550.
External links
• Recent review of quantum neural networks by M. Schuld, I. Sinayskiy and F. Petruccione
• Review of quantum neural networks by Wei
• Article by P. Gralewicz on the plausibility of quantum computing in biological neural networks
• Training a neural net to recognize images
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
Differentiable computing
General
• Differentiable programming
• Information geometry
• Statistical manifold
• Automatic differentiation
• Neuromorphic engineering
• Pattern recognition
• Tensor calculus
• Computational learning theory
• Inductive bias
Concepts
• Gradient descent
• SGD
• Clustering
• Regression
• Overfitting
• Hallucination
• Adversary
• Attention
• Convolution
• Loss functions
• Backpropagation
• Normalization (Batchnorm)
• Activation
• Softmax
• Sigmoid
• Rectifier
• Regularization
• Datasets
• Augmentation
• Diffusion
• Autoregression
Applications
• Machine learning
• In-context learning
• Artificial neural network
• Deep learning
• Scientific computing
• Artificial Intelligence
• Language model
• Large language model
Hardware
• IPU
• TPU
• VPU
• Memristor
• SpiNNaker
Software libraries
• TensorFlow
• PyTorch
• Keras
• Theano
• JAX
• Flux.jl
Implementations
Audio–visual
• AlexNet
• WaveNet
• Human image synthesis
• HWR
• OCR
• Speech synthesis
• Speech recognition
• Facial recognition
• AlphaFold
• DALL-E
• Midjourney
• Stable Diffusion
Verbal
• Word2vec
• Seq2seq
• BERT
• LaMDA
• Bard
• NMT
• Project Debater
• IBM Watson
• GPT-2
• GPT-3
• ChatGPT
• GPT-4
• GPT-J
• Chinchilla AI
• PaLM
• BLOOM
• LLaMA
Decisional
• AlphaGo
• AlphaZero
• Q-learning
• SARSA
• OpenAI Five
• Self-driving car
• MuZero
• Action selection
• Auto-GPT
• Robot control
People
• Yoshua Bengio
• Alex Graves
• Ian Goodfellow
• Stephen Grossberg
• Demis Hassabis
• Geoffrey Hinton
• Yann LeCun
• Fei-Fei Li
• Andrew Ng
• Jürgen Schmidhuber
• David Silver
Organizations
• Anthropic
• EleutherAI
• Google DeepMind
• Hugging Face
• OpenAI
• Meta AI
• Mila
• MIT CSAIL
Architectures
• Neural Turing machine
• Differentiable neural computer
• Transformer
• Recurrent neural network (RNN)
• Long short-term memory (LSTM)
• Gated recurrent unit (GRU)
• Echo state network
• Multilayer perceptron (MLP)
• Convolutional neural network
• Residual network
• Autoencoder
• Variational autoencoder (VAE)
• Generative adversarial network (GAN)
• Graph neural network
• Portals
• Computer programming
• Technology
• Categories
• Artificial neural networks
• Machine learning
Emerging technologies
Fields
Quantum
• algorithms
• amplifier
• bus
• cellular automata
• channel
• circuit
• complexity theory
• computing
• clock
• cryptography
• post-quantum
• dynamics
• electronics
• error correction
• finite automata
• image processing
• imaging
• information
• key distribution
• logic
• logic gates
• machine
• machine learning
• metamaterial
• network
• neural network
• optics
• programming
• sensing
• simulator
• teleportation
Other
• Anti-gravity
• Acoustic levitation
• Cloak of invisibility
• Digital scent technology
• Force field
• Plasma window
• Immersive virtual reality
• Magnetic refrigeration
• Phased-array optics
• Thermoacoustic heat engine
• Category
• List
|
Wikipedia
|
Grover's algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, refers to a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just $O({\sqrt {N}})$ evaluations of the function, where $N$ is the size of the function's domain. It was devised by Lov Grover in 1996.[1]
The analogous problem in classical computation cannot be solved in fewer than $O(N)$ evaluations (because, on average, one has to check half of the domain to get a 50% chance of finding the right input). Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function $\Omega ({\sqrt {N}})$ times, so Grover's algorithm is asymptotically optimal.[2] Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is an exponential, not polynomial, function).[3]
Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable when $N$ is large, and Grover's algorithm can be applied to speed up broad classes of algorithms.[3] Grover's algorithm could brute-force a 128-bit symmetric cryptographic key in roughly 264 iterations, or a 256-bit key in roughly 2128 iterations. It may not be the case that Grover's algorithm poses a significantly increased risk to encryption over existing classical algorithms, however.[4]
Applications and limitations
Grover's algorithm, along with variants like amplitude amplification, can be used to speed up a broad range of algorithms.[5][6][7] In particular, algorithms for NP-complete problems generally contain exhaustive search as a subroutine, which can be sped up by Grover's algorithm.[6] The current best algorithm for 3SAT is one such example. Generic constraint satisfaction problems also see quadratic speedups with Grover.[8] These algorithms do not require that the input be given in the form of an oracle, since Grover's algorithm is being applied with an explicit function, e.g. the function checking that a set of bits satisfies a 3SAT instance.
Grover's algorithm can also give provable speedups for black-box problems in quantum query complexity, including element distinctness[9] and the collision problem[10] (solved with the Brassard–Høyer–Tapp algorithm). In these types of problems, one treats the oracle function f as a database, and the goal is to use the quantum query to this function as few times as possible.
Cryptography
Grover's algorithm essentially solves the task of function inversion. Roughly speaking, if we have a function $y=f(x)$ that can be evaluated on a quantum computer, Grover's algorithm allows us to calculate $x$ when given $y$. Consequently, Grover's algorithm gives broad asymptotic speed-ups to many kinds of brute-force attacks on symmetric-key cryptography, including collision attacks and pre-image attacks.[11] However, this may not necessarily be the most efficient algorithm since, for example, the parallel rho algorithm is able to find a collision in SHA2 more efficiently than Grover's algorithm.[12]
Limitations
Grover's original paper described the algorithm as a database search algorithm, and this description is still common. The database in this analogy is a table of all of the function's outputs, indexed by the corresponding input. However, this database is not represented explicitly. Instead, an oracle is invoked to evaluate an item by its index. Reading a full database item by item and converting it into such a representation may take a lot longer than Grover's search. To account for such effects, Grover's algorithm can be viewed as solving an equation or satisfying a constraint. In such applications, the oracle is a way to check the constraint and is not related to the search algorithm. This separation usually prevents algorithmic optimizations, whereas conventional search algorithms often rely on such optimizations and avoid exhaustive search.[13] Fortunately, fast Grover's oracle implementation is possible for many constraint satisfaction and optimization problems.[14]
The major barrier to instantiating a speedup from Grover's algorithm is that the quadratic speedup achieved is too modest to overcome the large overhead of near-term quantum computers.[15] However, later generations of fault-tolerant quantum computers with better hardware performance may be able to realize these speedups for practical instances of data.
Problem description
As input for Grover's algorithm, suppose we have a function $f\colon \{0,1,\ldots ,N-1\}\to \{0,1\}$. In the "unstructured database" analogy, the domain represent indices to a database, and f(x) = 1 if and only if the data that x points to satisfies the search criterion. We additionally assume that only one index satisfies f(x) = 1, and we call this index ω. Our goal is to identify ω.
We can access f with a subroutine (sometimes called an oracle) in the form of a unitary operator Uω that acts as follows:
${\begin{cases}U_{\omega }|x\rangle =-|x\rangle &{\text{for }}x=\omega {\text{, that is, }}f(x)=1,\\U_{\omega }|x\rangle =|x\rangle &{\text{for }}x\neq \omega {\text{, that is, }}f(x)=0.\end{cases}}$
This uses the $N$-dimensional state space ${\mathcal {H}}$, which is supplied by a register with $n=\lceil \log _{2}N\rceil $ qubits. This is often written as
$U_{\omega }|x\rangle =(-1)^{f(x)}|x\rangle .$
Grover's algorithm outputs ω with probability at least 1/2 using $O({\sqrt {N}})$ applications of Uω. This probability can be made arbitrarily large by running Grover's algorithm multiple times. If one runs Grover's algorithm until ω is found, the expected number of applications is still $O({\sqrt {N}})$, since it will only be run twice on average.
Alternative oracle definition
This section compares the above oracle $U_{\omega }$ with an oracle $U_{f}$.
Uω is different from the standard quantum oracle for a function f. This standard oracle, denoted here as Uf, uses an ancillary qubit system. The operation then represents an inversion (NOT gate) on the main system conditioned by the value of f(x) from the ancillary system:
${\begin{cases}U_{f}|x\rangle |y\rangle =|x\rangle |\neg y\rangle &{\text{for }}x=\omega {\text{, that is, }}f(x)=1,\\U_{f}|x\rangle |y\rangle =|x\rangle |y\rangle &{\text{for }}x\neq \omega {\text{, that is, }}f(x)=0,\end{cases}}$
or briefly,
$U_{f}|x\rangle |y\rangle =|x\rangle |y\oplus f(x)\rangle .$
These oracles are typically realized using uncomputation.
If we are given Uf as our oracle, then we can also implement Uω, since Uω is Uf when the ancillary qubit is in the state $|-\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle -|1\rangle {\big )}=H|1\rangle $:
${\begin{aligned}U_{f}{\big (}|x\rangle \otimes |-\rangle {\big )}&={\frac {1}{\sqrt {2}}}\left(U_{f}|x\rangle |0\rangle -U_{f}|x\rangle |1\rangle \right)\\&={\frac {1}{\sqrt {2}}}\left(|x\rangle |f(x)\rangle -|x\rangle |1\oplus f(x)\rangle \right)\\&={\begin{cases}{\frac {1}{\sqrt {2}}}\left(-|x\rangle |0\rangle +|x\rangle |1\rangle \right)&{\text{if }}f(x)=1,\\{\frac {1}{\sqrt {2}}}\left(|x\rangle |0\rangle -|x\rangle |1\rangle \right)&{\text{if }}f(x)=0\end{cases}}\\&=(U_{\omega }|x\rangle )\otimes |-\rangle \end{aligned}}$
So, Grover's algorithm can be run regardless of which oracle is given.[3] If Uf is given, then we must maintain an additional qubit in the state $|-\rangle $ and apply Uf in place of Uω.
Algorithm
The steps of Grover's algorithm are given as follows:
1. Initialize the system to the uniform superposition over all states
$|s\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}|x\rangle .$
2. Perform the following "Grover iteration" $r(N)$ times:
1. Apply the operator $U_{\omega }$
2. Apply the Grover diffusion operator $U_{s}=2\left|s\right\rangle \left\langle s\right|-I$
3. Measure the resulting quantum state in the computational basis.
For the correctly chosen value of $r$, the output will be $|\omega \rangle $ with probability approaching 1 for N ≫ 1. Analysis shows that this eventual value for $r(N)$ satisfies $r(N)\leq {\Big \lceil }{\frac {\pi }{4}}{\sqrt {N}}{\Big \rceil }$.
Implementing the steps for this algorithm can be done using a number of gates linear in the number of qubits.[3] Thus, the gate complexity of this algorithm is $O(\log(N)r(N))$, or $O(\log(N))$ per iteration.
Geometric proof of correctness
There is a geometric interpretation of Grover's algorithm, following from the observation that the quantum state of Grover's algorithm stays in a two-dimensional subspace after each step. Consider the plane spanned by $|s\rangle $ and $|\omega \rangle $; equivalently, the plane spanned by $|\omega \rangle $ and the perpendicular ket $\textstyle |s'\rangle ={\frac {1}{\sqrt {N-1}}}\sum _{x\neq \omega }|x\rangle $.
Grover's algorithm begins with the initial ket $|s\rangle $, which lies in the subspace. The operator $U_{\omega }$ is a reflection at the hyperplane orthogonal to $|\omega \rangle $ for vectors in the plane spanned by $|s'\rangle $ and $|\omega \rangle $, i.e. it acts as a reflection across $|s'\rangle $. This can be seen by writing $U_{\omega }$ in the form of a Householder reflection:
$U_{\omega }=I-2|\omega \rangle \langle \omega |.$
The operator $U_{s}=2|s\rangle \langle s|-I$ is a reflection through $|s\rangle $. Both operators $U_{s}$ and $U_{\omega }$ take states in the plane spanned by $|s'\rangle $ and $|\omega \rangle $ to states in the plane. Therefore, Grover's algorithm stays in this plane for the entire algorithm.
It is straightforward to check that the operator $U_{s}U_{\omega }$ of each Grover iteration step rotates the state vector by an angle of $\theta =2\arcsin {\tfrac {1}{\sqrt {N}}}$. So, with enough iterations, one can rotate from the initial state $|s\rangle $ to the desired output state $|\omega \rangle $. The initial ket is close to the state orthogonal to $|\omega \rangle $:
$\langle s'|s\rangle ={\sqrt {\frac {N-1}{N}}}.$
In geometric terms, the angle $\theta /2$ between $|s\rangle $ and $|s'\rangle $ is given by
$\sin {\frac {\theta }{2}}={\frac {1}{\sqrt {N}}}.$
We need to stop when the state vector passes close to $|\omega \rangle $; after this, subsequent iterations rotate the state vector away from $|\omega \rangle $, reducing the probability of obtaining the correct answer. The exact probability of measuring the correct answer is
$\sin ^{2}\left({\Big (}r+{\frac {1}{2}}{\Big )}\theta \right),$
where r is the (integer) number of Grover iterations. The earliest time that we get a near-optimal measurement is therefore $r\approx \pi {\sqrt {N}}/4$.
Algebraic proof of correctness
To complete the algebraic analysis, we need to find out what happens when we repeatedly apply $U_{s}U_{\omega }$. A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm is a linear combination of $s$ and $\omega $. We can write the action of $U_{s}$ and $U_{\omega }$ in the space spanned by $\{|s\rangle ,|\omega \rangle \}$ as:
$U_{s}:a|\omega \rangle +b|s\rangle \mapsto [|\omega \rangle \,|s\rangle ]{\begin{bmatrix}-1&0\\2/{\sqrt {N}}&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}.$
$U_{\omega }:a|\omega \rangle +b|s\rangle \mapsto [|\omega \rangle \,|s\rangle ]{\begin{bmatrix}-1&-2/{\sqrt {N}}\\0&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}.$
So in the basis $\{|\omega \rangle ,|s\rangle \}$ (which is neither orthogonal nor a basis of the whole space) the action $U_{s}U_{\omega }$ of applying $U_{\omega }$ followed by $U_{s}$ is given by the matrix
$U_{s}U_{\omega }={\begin{bmatrix}-1&0\\2/{\sqrt {N}}&1\end{bmatrix}}{\begin{bmatrix}-1&-2/{\sqrt {N}}\\0&1\end{bmatrix}}={\begin{bmatrix}1&2/{\sqrt {N}}\\-2/{\sqrt {N}}&1-4/N\end{bmatrix}}.$
This matrix happens to have a very convenient Jordan form. If we define $t=\arcsin(1/{\sqrt {N}})$, it is
$U_{s}U_{\omega }=M{\begin{bmatrix}e^{2it}&0\\0&e^{-2it}\end{bmatrix}}M^{-1}$ where $M={\begin{bmatrix}-i&i\\e^{it}&e^{-it}\end{bmatrix}}.$
It follows that r-th power of the matrix (corresponding to r iterations) is
$(U_{s}U_{\omega })^{r}=M{\begin{bmatrix}e^{2rit}&0\\0&e^{-2rit}\end{bmatrix}}M^{-1}.$
Using this form, we can use trigonometric identities to compute the probability of observing ω after r iterations mentioned in the previous section,
$\left|{\begin{bmatrix}\langle \omega |\omega \rangle &\langle \omega |s\rangle \end{bmatrix}}(U_{s}U_{\omega })^{r}{\begin{bmatrix}0\\1\end{bmatrix}}\right|^{2}=\sin ^{2}\left((2r+1)t\right).$
Alternatively, one might reasonably imagine that a near-optimal time to distinguish would be when the angles 2rt and −2rt are as far apart as possible, which corresponds to $2rt\approx \pi /2$, or $r=\pi /4t=\pi /4\arcsin(1/{\sqrt {N}})\approx \pi {\sqrt {N}}/4$. Then the system is in state
$[|\omega \rangle \,|s\rangle ](U_{s}U_{\omega })^{r}{\begin{bmatrix}0\\1\end{bmatrix}}\approx [|\omega \rangle \,|s\rangle ]M{\begin{bmatrix}i&0\\0&-i\end{bmatrix}}M^{-1}{\begin{bmatrix}0\\1\end{bmatrix}}=|\omega \rangle {\frac {1}{\cos(t)}}-|s\rangle {\frac {\sin(t)}{\cos(t)}}.$
A short calculation now shows that the observation yields the correct answer ω with error $O\left({\frac {1}{N}}\right)$.
Extensions and variants
Multiple matching entries
If, instead of 1 matching entry, there are k matching entries, the same algorithm works, but the number of iterations must be $ {\frac {\pi }{4}}{\left({\frac {N}{k}}\right)^{1/2}}$instead of $ {\frac {\pi }{4}}{N^{1/2}}.$
There are several ways to handle the case if k is unknown.[16] A simple solution performs optimally up to a constant factor: run Grover's algorithm repeatedly for increasingly small values of k, e.g., taking k = N, N/2, N/4, ..., and so on, taking $k=N/2^{t}$ for iteration t until a matching entry is found.
With sufficiently high probability, a marked entry will be found by iteration $t=\log _{2}(N/k)+c$ for some constant c. Thus, the total number of iterations taken is at most
${\frac {\pi }{4}}{\Big (}1+{\sqrt {2}}+{\sqrt {4}}+\cdots +{\sqrt {\frac {N}{k2^{c}}}}{\Big )}=O{\big (}{\sqrt {N/k}}{\big )}.$
A version of this algorithm is used in order to solve the collision problem.[17][18]
Quantum partial search
A modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004.[19] In partial search, one is not interested in finding the exact address of the target item, only the first few digits of the address. Equivalently, we can think of "chunking" the search space into blocks, and then asking "in which block is the target item?". In many applications, such a search yields enough information if the target address contains the information wanted. For instance, to use the example given by L. K. Grover, if one has a list of students organized by class rank, we may only be interested in whether a student is in the lower 25%, 25–50%, 50–75% or 75–100% percentile.
To describe partial search, we consider a database separated into $K$ blocks, each of size $b=N/K$. The partial search problem is easier. Consider the approach we would take classically – we pick one block at random, and then perform a normal search through the rest of the blocks (in set theory language, the complement). If we don't find the target, then we know it's in the block we didn't search. The average number of iterations drops from $N/2$ to $(N-b)/2$.
Grover's algorithm requires $ {\frac {\pi }{4}}{\sqrt {N}}$ iterations. Partial search will be faster by a numerical factor that depends on the number of blocks $K$. Partial search uses $n_{1}$ global iterations and $n_{2}$ local iterations. The global Grover operator is designated $G_{1}$ and the local Grover operator is designated $G_{2}$.
The global Grover operator acts on the blocks. Essentially, it is given as follows:
1. Perform $j_{1}$ standard Grover iterations on the entire database.
2. Perform $j_{2}$ local Grover iterations. A local Grover iteration is a direct sum of Grover iterations over each block.
3. Perform one standard Grover iteration.
The optimal values of $j_{1}$ and $j_{2}$ are discussed in the paper by Grover and Radhakrishnan. One might also wonder what happens if one applies successive partial searches at different levels of "resolution". This idea was studied in detail by Vladimir Korepin and Xu, who called it binary quantum search. They proved that it is not in fact any faster than performing a single partial search.
Optimality
Grover's algorithm is optimal up to sub-constant factors. That is, any algorithm that accesses the database only by using the operator Uω must apply Uω at least a $1-o(1)$ fraction as many times as Grover's algorithm.[20] The extension of Grover's algorithm to k matching entries, π(N/k)1/2/4, is also optimal.[17] This result is important in understanding the limits of quantum computation.
If the Grover's search problem was solvable with logc N applications of Uω, that would imply that NP is contained in BQP, by transforming problems in NP into Grover-type search problems. The optimality of Grover's algorithm suggests that quantum computers cannot solve NP-Complete problems in polynomial time, and thus NP is not contained in BQP.
It has been shown that a class of non-local hidden variable quantum computers could implement a search of an $N$-item database in at most $O({\sqrt[{3}]{N}})$ steps. This is faster than the $O({\sqrt {N}})$ steps taken by Grover's algorithm.[21]
See also
• Amplitude amplification
• Brassard–Høyer–Tapp algorithm (for solving the collision problem)
• Shor's algorithm (for factorization)
• Quantum walk search
Notes
1. Grover, Lov K. (1996-07-01). "A fast quantum mechanical algorithm for database search". Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96. STOC '96. Philadelphia, Pennsylvania, USA: Association for Computing Machinery. pp. 212–219. arXiv:quant-ph/9605043. Bibcode:1996quant.ph..5043G. doi:10.1145/237814.237866. ISBN 978-0-89791-785-8. S2CID 207198067.
2. Bennett C.H.; Bernstein E.; Brassard G.; Vazirani U. (1997). "The strengths and weaknesses of quantum computation". SIAM Journal on Computing. 26 (5): 1510–1523. arXiv:quant-ph/9701001. doi:10.1137/s0097539796300933. S2CID 13403194.
3. Nielsen, Michael A. (2010). Quantum computation and quantum information. Isaac L. Chuang. Cambridge: Cambridge University Press. pp. 276–305. ISBN 978-1-107-00217-3. OCLC 665137861.
4. Daniel J. Bernstein (2010-03-03). "Grover vs. McEliece" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
5. Grover, Lov K. (1998). "A framework for fast quantum mechanical algorithms". In Vitter, Jeffrey Scott (ed.). Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23–26, 1998. Association for Computing Machinery. pp. 53–62. arXiv:quant-ph/9711043. doi:10.1145/276698.276712.
6. Ambainis, A. (2004-06-01). "Quantum search algorithms". ACM SIGACT News. 35 (2): 22–35. doi:10.1145/992287.992296. ISSN 0163-5700. S2CID 11326499.
7. Jordan, Stephen. "Quantum Algorithm Zoo". quantumalgorithmzoo.org. Retrieved 2021-04-21.
8. Cerf, Nicolas J.; Grover, Lov K.; Williams, Colin P. (2000-05-01). "Nested Quantum Search and NP-Hard Problems". Applicable Algebra in Engineering, Communication and Computing. 10 (4): 311–338. doi:10.1007/s002000050134. ISSN 1432-0622. S2CID 311132.
9. Ambainis, Andris (2007-01-01). "Quantum Walk Algorithm for Element Distinctness". SIAM Journal on Computing. 37 (1): 210–239. doi:10.1137/S0097539705447311. ISSN 0097-5397. S2CID 6581885.
10. Brassard, Gilles; Høyer, Peter; Tapp, Alain (1998). "Quantum Cryptanalysis of Hash and Claw-Free Functions". In Lucchesi, Claudio L.; Moura, Arnaldo V. (eds.). LATIN '98: Theoretical Informatics, Third Latin American Symposium, Campinas, Brazil, April, 20-24, 1998, Proceedings. Lecture Notes in Computer Science. Vol. 1380. Springer. pp. 163–169. arXiv:quant-ph/9705002. doi:10.1007/BFb0054319.
11. Post-quantum cryptography. Daniel J. Bernstein, Johannes Buchmann, Erik, Dipl.-Math Dahmén. Berlin: Springer. 2009. ISBN 978-3-540-88702-7. OCLC 318545517.{{cite book}}: CS1 maint: others (link)
12. Bernstein, Daniel J. (2021-04-21). "Cost analysis of hash collisions: Will quantum computers make SHARCS obsolete?" (PDF). Conference Proceedings for Special-purpose Hardware for Attacking Cryptographic Systems (SHARCS '09). 09: 105–117.
13. Viamontes G.F.; Markov I.L.; Hayes J.P. (2005), "Is Quantum Search Practical?" (PDF), Computing in Science and Engineering, 7 (3): 62–70, arXiv:quant-ph/0405001, Bibcode:2005CSE.....7c..62V, doi:10.1109/mcse.2005.53, S2CID 8929938
14. Sinitsyn N. A.; Yan B. (2023). "Topologically protected Grover's oracle for the Partition Problem". arXiv:2304.10488 [quant-ph].
15. Babbush, Ryan; McClean, Jarrod R.; Newman, Michael; Gidney, Craig; Boixo, Sergio; Neven, Hartmut (2021-03-29). "Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage". PRX Quantum. 2 (1): 010103. arXiv:2011.04149. doi:10.1103/PRXQuantum.2.010103.
16. Aaronson, Scott (April 19, 2021). "Introduction to Quantum Information Science Lecture Notes" (PDF).{{cite web}}: CS1 maint: url-status (link)
17. Michel Boyer; Gilles Brassard; Peter Høyer; Alain Tapp (1998), "Tight Bounds on Quantum Searching", Fortschr. Phys., 46 (4–5): 493–506, arXiv:quant-ph/9605034, Bibcode:1998ForPh..46..493B, doi:10.1002/3527603093.ch10, ISBN 9783527603091
18. Andris Ambainis (2004), "Quantum search algorithms", SIGACT News, 35 (2): 22–35, arXiv:quant-ph/0504012, Bibcode:2005quant.ph..4012A, doi:10.1145/992287.992296, S2CID 11326499
19. L.K. Grover; J. Radhakrishnan (2005-02-07). "Is partial quantum search of a database any easier?". arXiv:quant-ph/0407122v4.
20. Zalka, Christof (1999-10-01). "Grover's quantum searching algorithm is optimal". Physical Review A. 60 (4): 2746–2751. arXiv:quant-ph/9711070. Bibcode:1999PhRvA..60.2746Z. doi:10.1103/PhysRevA.60.2746. S2CID 1542077.
21. Aaronson, Scott. "Quantum Computing and Hidden Variables" (PDF).
References
• Grover L.K.: A fast quantum mechanical algorithm for database search, Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 212
• Grover L.K.: From Schrödinger's equation to quantum search algorithm, American Journal of Physics, 69(7): 769–777, 2001. Pedagogical review of the algorithm and its history.
• Grover L.K.: QUANTUM COMPUTING: How the weird logic of the subatomic world could make it possible for machines to calculate millions of times faster than they do today The Sciences, July/August 1999, pp. 24–30.
• Nielsen, M.A. and Chuang, I.L. Quantum computation and quantum information. Cambridge University Press, 2000. Chapter 6.
• What's a Quantum Phone Book?, Lov Grover, Lucent Technologies
External links
Wikiquote has quotations related to Grover's algorithm.
• Davy Wybiral. "Quantum Circuit Simulator". Archived from the original on 2017-01-16. Retrieved 2017-01-13.
• Craig Gidney (2013-03-05). "Grover's Quantum Search Algorithm".
• François Schwarzentruber (2013-05-18). "Grover's algorithm".
• Alexander Prokopenya. "Quantum Circuit Implementing Grover's Search Algorithm". Wolfram Alpha.
• "Quantum computation, theory of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Roberto Maestre (2018-05-11). "Grover's Algorithm implemented in R and C". GitHub.
• Bernhard Ömer. "QCL - A Programming Language for Quantum Computers". Retrieved 2022-04-30. Implemented in /qcl-0.6.4/lib/grover.qcl
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
|
Wikipedia
|
Quantum computing
A quantum computer is a computer that exploits quantum mechanical phenomena. At small scales, physical matter exhibits properties of both particles and waves, and quantum computing leverages this behavior, specifically quantum superposition and entanglement, using specialized hardware that supports the preparation and manipulation of quantum states. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations exponentially faster than any modern "classical" computer. In particular, a large-scale quantum computer could break widely used encryption schemes and aid physicists in performing physical simulations; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications.
The basic unit of information in quantum computing is the qubit, similar to the bit in traditional digital electronics. Unlike a classical bit, a qubit can exist in a superposition of its two "basis" states, which loosely means that it is in both states simultaneously. When measuring a qubit, the result is a probabilistic output of a classical bit. If a quantum computer manipulates the qubit in a particular way, wave interference effects can amplify the desired measurement results. The design of quantum algorithms involves creating procedures that allow a quantum computer to perform calculations efficiently and quickly.
Physically engineering high-quality qubits has proven challenging. If a physical qubit is not sufficiently isolated from its environment, it suffers from quantum decoherence, introducing noise into calculations. National governments have invested heavily in experimental research that aims to develop scalable qubits with longer coherence times and lower error rates. Two of the most promising technologies are superconductors (which isolate an electrical current by eliminating electrical resistance) and ion traps (which confine a single ion using electromagnetic fields).
In principle, a non-quantum (classical) computer can solve the same computational problems as a quantum computer, given enough time. Quantum advantage comes in the form of time complexity rather than computability, and quantum complexity theory shows that some quantum algorithms for carefully selected tasks require exponentially fewer computational steps than the best known non-quantum algorithms. Such tasks can in theory be solved on a large-scale quantum computer whereas classical computers would not finish computations in any reasonable amount of time. However, quantum speedup is not universal or even typical across computational tasks, since basic tasks such as sorting are proven to not allow any asymptotic quantum speedup. Claims of quantum supremacy have drawn significant attention to the discipline, but are demonstrated on contrived tasks, while near-term practical use cases remain limited.
Optimism about quantum computing is fueled by a broad range of new theoretical hardware possibilities facilitated by quantum physics, but the improving understanding of quantum computing limitations counterbalances this optimism. In particular, quantum speedups have been traditionally estimated for noiseless quantum computers, whereas the impact of noise and the use of quantum error-correction can undermine low-polynomial speedups.
History
For many years, the fields of quantum mechanics and computer science formed distinct academic communities.[2] Modern quantum theory developed in the 1920s to explain the wave–particle duality observed at atomic scales,[3] and digital computers emerged in the following decades to replace human computers for tedious calculations.[4] Both disciplines had practical applications during World War II; computers played a major role in wartime cryptography,[5] and quantum physics was essential for the nuclear physics used in the Manhattan Project.[6]
As physicists applied quantum mechanical models to computational problems and swapped digital bits for qubits, the fields of quantum mechanics and computer science began to converge. In 1980, Paul Benioff introduced the quantum Turing machine, which uses quantum theory to describe a simplified computer.[7] When digital computers became faster, physicists faced an exponential increase in overhead when simulating quantum dynamics,[8] prompting Yuri Manin and Richard Feynman to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation.[9][10][11] In a 1984 paper, Charles Bennett and Gilles Brassard applied quantum theory to cryptography protocols and demonstrated that quantum key distribution could enhance information security.[12][13]
Quantum algorithms then emerged for solving oracle problems, such as Deutsch's algorithm in 1985,[14] the Bernstein–Vazirani algorithm in 1993,[15] and Simon's algorithm in 1994.[16] These algorithms did not solve practical problems, but demonstrated mathematically that one could gain more information by querying a black box with a quantum state in superposition, sometimes referred to as quantum parallelism.[17] Peter Shor built on these results with his 1994 algorithms for breaking the widely used RSA and Diffie–Hellman encryption protocols,[18] which drew significant attention to the field of quantum computing.[19] In 1996, Grover's algorithm established a quantum speedup for the widely applicable unstructured search problem.[20][21] The same year, Seth Lloyd proved that quantum computers could simulate quantum systems without the exponential overhead present in classical simulations,[22] validating Feynman's 1982 conjecture.[23]
Over the years, experimentalists have constructed small-scale quantum computers using trapped ions and superconductors.[24] In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology,[25][26] and subsequent experiments have increased the number of qubits and reduced error rates.[24] In 2019, Google AI and NASA announced that they had achieved quantum supremacy with a 54-qubit machine, performing a computation that is impossible for any classical computer.[27][28][29] However, the validity of this claim is still being actively researched.[30][31]
The threshold theorem shows how increasing the number of qubits can mitigate errors,[32] yet fully fault-tolerant quantum computing remains "a rather distant dream".[33] According to some researchers, noisy intermediate-scale quantum (NISQ) machines may have specialized uses in the near future, but noise in quantum gates limits their reliability.[33]
Investment in quantum computing research has increased in the public and private sectors.[34][35] As one consulting firm summarized,[36]
... investment dollars are pouring in, and quantum-computing start-ups are proliferating. ... While quantum computing promises to help businesses solve problems that are beyond the reach and speed of conventional high-performance computers, use cases are largely experimental and hypothetical at this early stage.
With focus on business management’s point of view, the potential applications of quantum computing into four major categories are cybersecurity, data analytics and artificial intelligence, optimization and simulation, and data management and searching.[37]
Quantum information processing
Computer engineers typically describe a modern computer's operation in terms of classical electrodynamics. Within these "classical" computers, some components (such as semiconductors and random number generators) may rely on quantum behavior, but these components are not isolated from their environment, so any quantum information quickly decoheres. While programmers may depend on probability theory when designing a randomized algorithm, quantum mechanical notions like superposition and interference are largely irrelevant for program analysis.
Quantum programs, in contrast, rely on precise control of coherent quantum systems. Physicists describe these systems mathematically using linear algebra. Complex numbers model probability amplitudes, vectors model quantum states, and matrices model the operations that can be performed on these states. Programming a quantum computer is then a matter of composing operations in such a way that the resulting program computes a useful result in theory and is implementable in practice.
As physicist Charlie Bennett describes the relationship between quantum and classical computers,[38]
A classical computer is a quantum computer ... so we shouldn't be asking about "where do quantum speedups come from?" We should say, "well, all computers are quantum. ... Where do classical slowdowns come from?"
Quantum information
The qubit serves as the basic unit of quantum information. It represents a two-state system, just like a classical bit, except that it can exist in a superposition of its two states.[39] In one sense, a superposition is like a probability distribution over the two values.[40] However, a quantum computation can be influenced by both values at once, inexplicable by either state individually. In this sense, a "superposed" qubit stores both values simultaneously.[17]
A two-dimensional vector mathematically represents a qubit state. Physicists typically use Dirac notation for quantum mechanical linear algebra, writing |ψ⟩ 'ket psi' for a vector labeled ψ. Because a qubit is a two-state system, any qubit state takes the form α|0⟩ + β|1⟩, where |0⟩ and |1⟩ are the standard basis states,[lower-alpha 1] and α and β are the probability amplitudes. If either α or β is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a quantum state vector acts similarly to a (classical) probability vector, with one key difference: unlike probabilities, probability amplitudes are not necessarily positive numbers.[40] Negative amplitudes allow for destructive wave interference.[lower-alpha 2]
When a qubit is measured in the standard basis, the result is a classical bit. The Born rule describes the norm-squared correspondence between amplitudes and probabilities—when measuring a qubit α|0⟩ + β|1⟩, the state collapses to |0⟩ with probability |α|2, or to |1⟩ with probability |β|2. Any valid qubit state has coefficients α and β such that |α|2 + |β|2 = 1. As an example, measuring the qubit 1/√2|0⟩ + 1/√2|1⟩ would produce either |0⟩ or |1⟩ with equal probability.
Each additional qubit doubles the dimension of the state space. As an example, the vector 1/√2|00⟩ + 1/√2|01⟩ represents a two-qubit state, a tensor product of the qubit |0⟩ with the qubit 1/√2|0⟩ + 1/√2|1⟩. This vector inhabits a four-dimensional vector space spanned by the basis vectors |00⟩, |01⟩, |10⟩, and |11⟩. The Bell state 1/√2|00⟩ + 1/√2|11⟩ is impossible to decompose into the tensor product of two individual qubits—the two qubits are entangled because their probability amplitudes are correlated. In general, the vector space for an n-qubit system is 2n-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2100 classical values.
Unitary operators
The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix
$X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.$
Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus
$X|0\rangle =|1\rangle $ and $X|1\rangle =|0\rangle $.
The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are
$|00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.$ :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}
The CNOT gate can then be represented using the following matrix:
$\operatorname {CNOT} :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.$ :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}
As a mathematical consequence of this definition, $ \operatorname {CNOT} |00\rangle =|00\rangle $, $ \operatorname {CNOT} |01\rangle =|01\rangle $, $ \operatorname {CNOT} |10\rangle =|11\rangle $, and $ \operatorname {CNOT} |11\rangle =|10\rangle $. In other words, the CNOT applies a NOT gate ($ X$ from before) to the second qubit if and only if the first qubit is in the state $ |1\rangle $. If the first qubit is $ |0\rangle $, nothing is done to either qubit.
In summary, quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.
Quantum parallelism
Quantum parallelism refers to the ability of quantum computers to evaluate a function for multiple input values simultaneously. This can be achieved by preparing a quantum system in a superposition of input states, and applying a unitary transformation that encodes the function to be evaluated. The resulting state encodes the function's output values for all input values in the superposition, allowing for the computation of multiple outputs simultaneously. This property is key to the speedup of many quantum algorithms.[17]
Quantum programming
There are a number of models of computation for quantum computing, distinguished by the basic elements in which the computation is decomposed.
Gate array
A quantum gate array decomposes computation into a sequence of few-qubit quantum gates. A quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.
Any quantum computation (which is, in the above formalism, any unitary matrix of size $2^{n}\times 2^{n}$ over $n$ qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.
Measurement-based quantum computing
A measurement-based quantum computer decomposes computation into a sequence of Bell state measurements and single-qubit quantum gates applied to a highly entangled initial state (a cluster state), using a technique called quantum gate teleportation.
Adiabatic quantum computing
An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.[42]
Topological quantum computing
A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice.[43]
Quantum Turing machine
A quantum Turing machine is the quantum analog of a Turing machine.[7] All of these models of computation—quantum circuits,[44] one-way quantum computation,[45] adiabatic quantum computation,[46] and topological quantum computation[47]—have been shown to be equivalent to the quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical.
Communication
Quantum cryptography enables new ways to transmit data securely; for example, quantum key distribution uses entangled quantum states to establish secure cryptographic keys.[48] When a sender and receiver exchange quantum states, they can guarantee that an adversary does not intercept the message, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change.[49] With appropriate cryptographic protocols, the sender and receiver can thus establish shared private information resistant to eavesdropping.[12][50]
Modern fiber-optic cables can transmit quantum information over relatively short distances. Ongoing experimental research aims to develop more reliable hardware (such as quantum repeaters), hoping to scale this technology to long-distance quantum networks with end-to-end entanglement. Theoretically, this could enable novel technological applications, such as distributed quantum computing and enhanced quantum sensing.[51][52]
Algorithms
Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.[53]
Quantum algorithms that offer more than a polynomial speedup over the best-known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing discrete logarithms, solving Pell's equation, and more generally solving the hidden subgroup problem for abelian finite groups.[53] These algorithms depend on the primitive of the quantum Fourier transform. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, but evidence suggests that this is unlikely.[54] Certain oracle problems like Simon's problem and the Bernstein–Vazirani problem do give provable speedups, though this is in the quantum query model, which is a restricted model where lower bounds are much easier to prove and doesn't necessarily translate to speedups for practical problems.
Other problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain Jones polynomials, and the quantum algorithm for linear systems of equations have quantum algorithms appearing to give super-polynomial speedups and are BQP-complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that no quantum algorithm gives a super-polynomial speedup, which is believed to be unlikely.[55]
Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.[53] Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.[21]
Simulation of quantum systems
Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, quantum simulation may be an important application of quantum computing.[56] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[57] In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.[58][59]
About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry (even though naturally occurring organisms also produce ammonia). Quantum simulations might be used to understand this process and increase the energy efficiency of production.[60] It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process[61] by the mid 2020s[62] although some have predicted it will take longer.[63]
Post-quantum cryptography
A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes).[64] By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors.[65] This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.
Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of post-quantum cryptography.[66][67] Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory.[66][68] Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem.[69] It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case,[70] meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size).
Search problems
Main article: Grover's algorithm
The most well-known example of a problem that allows for a polynomial quantum speedup is unstructured search, which involves finding a marked item out of a list of $n$ items in a database. This can be solved by Grover's algorithm using $O({\sqrt {n}})$ queries to the database, quadratically fewer than the $\Omega (n)$ queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Many examples of provable quantum speedups for query problems are based on Grover's algorithm, including Brassard, Høyer, and Tapp's algorithm for finding collisions in two-to-one functions,[71] and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.[72]
Problems that can be efficiently addressed with Grover's algorithm have the following properties:[73][74]
1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. There exists a boolean function that evaluates each input and determines whether it is the correct answer
For problems with all these properties, the running time of Grover's algorithm on a quantum computer scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied[75] is Boolean satisfiability problem, where the database through which the algorithm iterates is that of all possible answers. An example and possible application of this is a password cracker that attempts to guess a password. Breaking symmetric ciphers with this algorithm is of interest to government agencies.[76]
Quantum annealing
Quantum annealing relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. Adiabatic optimization may be helpful for solving computational biology problems.[77]
Machine learning
Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.[78][33]
For example, the quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.[79][33] Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.[80][81][82]
Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems[22] and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models[83] including quantum GANs[84] may eventually be developed into ultimate generative chemistry algorithms.
Engineering
As of 2023, classical computers outperform quantum computers for all real-world applications. While current quantum computers may speed up solutions to particular mathematical problems, they give no computational advantage for practical tasks. For many tasks there is no promise of useful quantum speedup, and some tasks probably prohibit any quantum speedup. Scientists and engineers are exploring multiple technologies for quantum computing hardware and hope to develop scalable quantum architectures, but serious obstacles remain.[85][86]
Challenges
There are a number of technical challenges in building a large-scale quantum computer.[87] Physicist David DiVincenzo has listed these requirements for a practical quantum computer:[88]
• Physically scalable to increase the number of qubits
• Qubits that can be initialized to arbitrary values
• Quantum gates that are faster than decoherence time
• Universal gate set
• Qubits that can be read easily.
Sourcing parts for quantum computers is also very difficult. Superconducting quantum computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.[89]
The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers that enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.[90]
Decoherence
One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[91] Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator[92]) in order to prevent significant decoherence.[93] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.[94]
As a result, time-consuming tasks may render some quantum algorithms inoperable, as attempting to maintain the state of qubits for a long enough duration will eventually corrupt the superpositions.[95]
These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.
As described by the threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often-cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing.
Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction.[96] With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds. However, the encoding and error-correction overheads increase the size of a real fault-tolerant quantum computer by several orders of magnitude. Careful estimates[97][98] show that at least 3 million physical qubits would factor 2,048-bit integer in 5 months on a fully error-corrected trapped-ion quantum computer. In terms of the number of physical qubits, to date, this remains the lowest estimate[99] for practically useful integer factorization problem sizing 1,024-bit or larger.
Another approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads, and relying on braid theory to form stable logic gates.[100][101]
Quantum supremacy
Physicist John Preskill coined the term quantum supremacy to describe the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers.[102][103][104] The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.[105]
In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the Sycamore quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world's fastest computer.[28][106][107] This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,[108][109] and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Sycamore and classical supercomputers[110][111][112] and even beating it.[113][114][115]
In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer, Jiuzhang, to demonstrate quantum supremacy.[116][117][118] The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.[119]
Claims of quantum supremacy have generated hype around quantum computing,[120] but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.[85][121]
Skepticism
Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 Nature spotlight article summarised current quantum computers as being "For now, [good for] absolutely nothing".[85] The article elaborated that quantum computers are yet to be more useful or efficient than conventional computers in any case, though it also argued that in the long term such computers are likely to be useful. A 2023 Communications of the ACM article[86] found that current quantum computing algorithms are "insufficient for practical quantum advantage without significant improvements across the software/hardware stack". It argues that the most promising candidates for achieving speedup with quantum computers are "small-data problems", for example in chemistry and materials science. However, the article also concludes that a large range of the potential applications it considered, such as machine learning, "will not achieve quantum advantage with current quantum algorithms in the foreseeable future", and it identified I/O constraints that make speedup unlikely for "big data problems, unstructured linear systems, and database search based on Grover's algorithm".
This state of affairs can be traced to several current and long-term considerations.
• Conventional computer hardware and algorithms are not only optimized for practical tasks, but are still improving rapidly, particularly GPU accelerators.
• Current quantum computing hardware generates only a limited amount of entanglement before getting overwhelmed by noise and does not rule out practical simulation on conventional computers, possibly except for contrived cases.
• Quantum algorithms provide speedup over conventional algorithms only for some tasks, and matching these tasks with practical applications proved challenging. Some promising tasks and applications require resources far beyond those available today.[122][123] In particular, processing large amounts of non-quantum data is a challenge for quantum computers.[86]
• Some promising algorithms have been "dequantized", i.e., their non-quantum analogues with similar complexity have been found.
• If quantum error correction is used to scale quantum computers to practical applications, its overhead may undermine speedup offered by many quantum algorithms.[86]
• Complexity analysis of algorithms sometimes makes abstract assumptions that do not hold in applications. For example, input data may not already be available encoded in quantum states, and "oracle functions" used in Grover's algorithm often have internal structure that can be exploited for faster algorithms.
In particular, building computers with large numbers of qubits may be futile if those qubits are not connected well enough and cannot maintain sufficiently high degree of entanglement for long time. When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations. Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms (which may be unknown) and show that some quantum algorithms asymptomatically improve upon those bounds.
Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons.
Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.[124] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[125] Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.[126][127][128] Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:
"So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10300... Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never."[129][130]
Candidates for physical realizations
A practical quantum computer must use a physical system as a programmable quantum register.[131] Researchers are exploring several technologies as candidates for reliable qubit implementations.[132] Superconductors and trapped ions are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.[133]
Theory
Computability
Further information: Computability theory
Any computational problem solvable by a classical computer is also solvable by a quantum computer.[134] Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.
Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. More formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem, and the existence of quantum computers does not disprove the Church–Turing thesis.[135]
Complexity
Main article: Quantum complexity theory
While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers.
The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.[136] It is known that ${\mathsf {BPP\subseteq BQP}}$ and is widely suspected that ${\mathsf {BQP\subsetneq BPP}}$, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.[137]
The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that ${\mathsf {P\subseteq BQP\subseteq PSPACE}}$; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that ${\mathsf {NP\nsubseteq BQP}}$; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).[138]
See also
• D-Wave Systems – Canadian quantum computing company
• Electronic quantum holography
• Glossary of quantum computing
• IARPA
• List of emerging technologies
• List of quantum processors
• Magic state distillation – Quantum computing algorithm
• Natural computing
• Optical computing
• Quantum bus
• Quantum cognition
• Quantum volume
• Quantum weirdness – Unintuitive aspects of quantum mechanics
• Rigetti Computing
• Supercomputer
• Theoretical computer science
• Unconventional computing
• Valleytronics
Notes
1. The standard basis is also the computational basis.[41]
2. In general, probability amplitudes are complex numbers.
References
1. Russell, John (10 January 2019). "IBM Quantum Update: Q System One Launch, New Collaborators, and QC Center Plans". HPCwire. Retrieved 9 January 2023.
2. Aaronson 2013, p. 132.
3. Bhatta, Varun S. (10 May 2020). "Plurality of Wave–Particle Duality" (PDF). Current Science. 118 (9): 1365. doi:10.18520/cs/v118/i9/1365-1374. ISSN 0011-3891. S2CID 216143449.
4. Ceruzzi, Paul E. (2012). Computing: A Concise History. Cambridge, Massachusetts. pp. 3, 46. ISBN 978-0-262-31038-3. OCLC 796812982.{{cite book}}: CS1 maint: location missing publisher (link)
5. Hodges, Andrew (2014). Alan Turing: The Enigma. Princeton University Press. p. xviii. ISBN 9780691164724.
6. Mårtensson-Pendrill, Ann-Marie (1 November 2006). "The Manhattan project—a part of physics history". Physics Education. 41 (6): 493–501. Bibcode:2006PhyEd..41..493M. doi:10.1088/0031-9120/41/6/001. ISSN 0031-9120. S2CID 120294023.
7. Benioff, Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics. 22 (5): 563–591. Bibcode:1980JSP....22..563B. doi:10.1007/bf01011339. S2CID 122949592.
8. Buluta, Iulia; Nori, Franco (2 October 2009). "Quantum Simulators". Science. 326 (5949): 108–111. Bibcode:2009Sci...326..108B. doi:10.1126/science.1177838. ISSN 0036-8075. PMID 19797653. S2CID 17187000.
9. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 10 May 2013. Retrieved 4 March 2013.
10. Feynman, Richard (June 1982). "Simulating Physics with Computers" (PDF). International Journal of Theoretical Physics. 21 (6/7): 467–488. Bibcode:1982IJTP...21..467F. doi:10.1007/BF02650179. S2CID 124545445. Archived from the original (PDF) on 8 January 2019. Retrieved 28 February 2019.
11. Nielsen & Chuang 2010, p. 214.
12. Bennett, Charles H.; Brassard, Gilles (December 1984). Quantum cryptography: Public key distribution and coin tossing. IEEE International Conference on Computers, Systems & Signal Processing. Bangalore. pp. 175–179. arXiv:2003.06557. doi:10.1016/j.tcs.2014.05.025.
13. Brassard, G. (2005). "Brief history of quantum cryptography: A personal perspective". IEEE Information Theory Workshop on Theory and Practice in Information-Theoretic Security, 2005. Awaji Island, Japan: IEEE. pp. 19–23. arXiv:quant-ph/0604072. doi:10.1109/ITWTPI.2005.1543949. ISBN 978-0-7803-9491-9. S2CID 16118245.
14. Deutsch, D. (8 July 1985). "Quantum theory, the Church–Turing principle and the universal quantum computer". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 400 (1818): 97–117. Bibcode:1985RSPSA.400...97D. doi:10.1098/rspa.1985.0070. ISSN 0080-4630. S2CID 1438116.
15. Bernstein, Ethan; Vazirani, Umesh (1993). "Quantum complexity theory". Proceedings of the twenty-fifth annual ACM symposium on Theory of computing - STOC '93. San Diego, California, United States: ACM Press. pp. 11–20. doi:10.1145/167088.167097. ISBN 978-0-89791-591-5. S2CID 676378.
16. Simon, D. R. (1994). "On the power of quantum computation". Proceedings 35th Annual Symposium on Foundations of Computer Science. Santa Fe, New Mexico, USA: IEEE Comput. Soc. Press. pp. 116–123. doi:10.1109/SFCS.1994.365701. ISBN 978-0-8186-6580-6. S2CID 7457814.
17. Nielsen & Chuang 2010, p. 30-32.
18. Shor 1994.
19. National Academies of Sciences, Engineering, and Medicine 2019, p. 15.
20. Grover, Lov K. (1996). A fast quantum mechanical algorithm for database search. ACM symposium on Theory of computing. Philadelphia: ACM Press. pp. 212–219. arXiv:quant-ph/9605043. doi:10.1145/237814.237866. ISBN 978-0-89791-785-8.
21. Nielsen & Chuang 2010, p. 7.
22. Lloyd, Seth (23 August 1996). "Universal Quantum Simulators". Science. 273 (5278): 1073–1078. Bibcode:1996Sci...273.1073L. doi:10.1126/science.273.5278.1073. ISSN 0036-8075. PMID 8688088. S2CID 43496899.
23. Cao, Yudong; Romero, Jonathan; Olson, Jonathan P.; Degroote, Matthias; Johnson, Peter D.; et al. (9 October 2019). "Quantum Chemistry in the Age of Quantum Computing". Chemical Reviews. 119 (19): 10856–10915. arXiv:1812.09976. doi:10.1021/acs.chemrev.8b00803. ISSN 0009-2665. PMID 31469277. S2CID 119417908.
24. National Academies of Sciences, Engineering, and Medicine 2019, pp. 164–169.
25. Chuang, Isaac L.; Gershenfeld, Neil; Kubinec, Markdoi (April 1998). "Experimental Implementation of Fast Quantum Searching". Physical Review Letters. American Physical Society. 80 (15): 3408–3411. Bibcode:1998PhRvL..80.3408C. doi:10.1103/PhysRevLett.80.3408.
26. Holton, William Coffeen. "quantum computer". Encyclopedia Britannica. Encyclopædia Britannica. Retrieved 4 December 2021.
27. Gibney, Elizabeth (23 October 2019). "Hello quantum world! Google publishes landmark quantum supremacy claim". Nature. 574 (7779): 461–462. Bibcode:2019Natur.574..461G. doi:10.1038/d41586-019-03213-z. PMID 31645740.
28. Lay summary: Martinis, John; Boixo, Sergio (23 October 2019). "Quantum Supremacy Using a Programmable Superconducting Processor". Nature. Google AI. 574 (7779): 505–510. arXiv:1910.11333. Bibcode:2019Natur.574..505A. doi:10.1038/s41586-019-1666-5. PMID 31645734. S2CID 204836822. Retrieved 27 April 2022.
• Journal article: Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; et al. (23 October 2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779): 505–510. arXiv:1910.11333. Bibcode:2019Natur.574..505A. doi:10.1038/s41586-019-1666-5. PMID 31645734. S2CID 204836822.
29. Aaronson, Scott (30 October 2019). "Opinion | Why Google's Quantum Supremacy Milestone Matters". The New York Times. ISSN 0362-4331. Retrieved 25 September 2021.
30. Pednault, Edwin (22 October 2019). "On 'Quantum Supremacy'". IBM Research Blog. Retrieved 9 February 2021.
31. Pan, Feng; Zhang, Pan (4 March 2021). "Simulating the Sycamore quantum supremacy circuits". arXiv:2103.03074 [quant-ph].
32. Nielsen & Chuang 2010, p. 481.
33. Preskill, John (6 August 2018). "Quantum Computing in the NISQ era and beyond". Quantum. 2: 79. arXiv:1801.00862. Bibcode:2018Quant...2...79P. doi:10.22331/q-2018-08-06-79. S2CID 44098998.
34. Gibney, Elizabeth (2 October 2019). "Quantum gold rush: the private funding pouring into quantum start-ups". Nature. 574 (7776): 22–24. Bibcode:2019Natur.574...22G. doi:10.1038/d41586-019-02935-4. PMID 31578480.
35. Rodrigo, Chris Mills (12 February 2020). "Trump budget proposal boosts funding for artificial intelligence, quantum computing". The Hill. Retrieved 11 July 2021.
36. Biondi, Matteo; Heid, Anna; Henke, Nicolaus; Mohr, Niko; Pautasso, Lorenzo; et al. (14 December 2021). "Quantum computing use cases are getting real—what you need to know". McKinsey & Company. Retrieved 1 April 2022.
37. Leong, Kelvin; Sung, Anna (November 2022). "What Business Managers Should Know About Quantum Computing?" (PDF). Journal of Interdisciplinary Sciences. Retrieved 13 August 2023.
38. Bennett, Charlie (31 July 2020). Information Is Quantum: How Physics Helped Explain the Nature of Information and What Can Be Done With It (Videotape). Event occurs at 1:08:22 – via YouTube.
39. Nielsen & Chuang 2010, p. 13.
40. Aaronson 2013, p. 110.
41. Mermin 2007, p. 18.
42. Das, A.; Chakrabarti, B. K. (2008). "Quantum Annealing and Analog Quantum Computation". Rev. Mod. Phys. 80 (3): 1061–1081. arXiv:0801.2193. Bibcode:2008RvMP...80.1061D. CiteSeerX 10.1.1.563.9990. doi:10.1103/RevModPhys.80.1061. S2CID 14255125.
43. Nayak, Chetan; Simon, Steven; Stern, Ady; Das Sarma, Sankar (2008). "Nonabelian Anyons and Quantum Computation". Reviews of Modern Physics. 80 (3): 1083–1159. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/RevModPhys.80.1083. S2CID 119628297.
44. Chi-Chih Yao, A. (1993). "Quantum circuit complexity". Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science. pp. 352–361. doi:10.1109/SFCS.1993.366852. ISBN 0-8186-4370-6. S2CID 195866146.
45. Raussendorf, Robert; Browne, Daniel E.; Briegel, Hans J. (25 August 2003). "Measurement-based quantum computation on cluster states". Physical Review A. 68 (2): 022312. arXiv:quant-ph/0301052. Bibcode:2003PhRvA..68b2312R. doi:10.1103/PhysRevA.68.022312. S2CID 6197709.
46. Aharonov, Dorit; van Dam, Wim; Kempe, Julia; Landau, Zeph; Lloyd, Seth; Regev, Oded (1 January 2008). "Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation". SIAM Review. 50 (4): 755–787. arXiv:quant-ph/0405098. Bibcode:2008SIAMR..50..755A. doi:10.1137/080734479. ISSN 0036-1445. S2CID 1503123.
47. Freedman, Michael H.; Larsen, Michael; Wang, Zhenghan (1 June 2002). "A Modular Functor Which is Universal for Quantum Computation". Communications in Mathematical Physics. 227 (3): 605–622. arXiv:quant-ph/0001108. Bibcode:2002CMaPh.227..605F. doi:10.1007/s002200200645. ISSN 0010-3616. S2CID 8990600.
48. Pirandola, S.; Andersen, U. L.; Banchi, L.; Berta, M.; Bunandar, D.; Colbeck, R.; Englund, D.; Gehring, T.; Lupo, C.; Ottaviani, C.; Pereira, J. L.; Razavi, M.; Shamsul Shaari, J.; Tomamichel, M.; Usenko, V. C. (14 December 2020). "Advances in quantum cryptography". Advances in Optics and Photonics. 12 (4): 1017. arXiv:1906.01645. Bibcode:2020AdOP...12.1012P. doi:10.1364/AOP.361502. ISSN 1943-8206. S2CID 174799187.
49. Xu, Feihu; Ma, Xiongfeng; Zhang, Qiang; Lo, Hoi-Kwong; Pan, Jian-Wei (26 May 2020). "Secure quantum key distribution with realistic devices". Reviews of Modern Physics. 92 (2): 025002-3. arXiv:1903.09051. Bibcode:2020RvMP...92b5002X. doi:10.1103/RevModPhys.92.025002. S2CID 210942877.
50. Xu, Guobin; Mao, Jianzhou; Sakk, Eric; Wang, Shuangbao Paul (22 March 2023). "An Overview of Quantum-Safe Approaches: Quantum Key Distribution and Post-Quantum Cryptography". 2023 57th Annual Conference on Information Sciences and Systems (CISS). IEEE. p. 3. doi:10.1109/CISS56502.2023.10089619. ISBN 978-1-6654-5181-9.
51. Kozlowski, Wojciech; Wehner, Stephanie (25 September 2019). "Towards Large-Scale Quantum Networks". Proceedings of the Sixth Annual ACM International Conference on Nanoscale Computing and Communication. ACM. pp. 1–7. arXiv:1909.08396. doi:10.1145/3345312.3345497. ISBN 978-1-4503-6897-1.
52. Guo, Xueshi; Breum, Casper R.; Borregaard, Johannes; Izumi, Shuro; Larsen, Mikkel V.; Gehring, Tobias; Christandl, Matthias; Neergaard-Nielsen, Jonas S.; Andersen, Ulrik L. (23 December 2019). "Distributed quantum sensing in a continuous-variable entangled network". Nature Physics. 16 (3): 281–284. arXiv:1905.09408. doi:10.1038/s41567-019-0743-x. ISSN 1745-2473. S2CID 256703226.
53. Jordan, Stephen (14 October 2022) [22 April 2011]. "Quantum Algorithm Zoo". Archived from the original on 29 April 2018.
54. Aaronson, Scott; Arkhipov, Alex (6 June 2011). "The computational complexity of linear optics". Proceedings of the forty-third annual ACM symposium on Theory of computing. San Jose, California: Association for Computing Machinery. pp. 333–342. arXiv:1011.3245. doi:10.1145/1993636.1993682. ISBN 978-1-4503-0691-1.
55. Nielsen & Chuang 2010, p. 42.
56. Norton, Quinn (15 February 2007). "The Father of Quantum Computing". Wired.
57. Ambainis, Andris (Spring 2014). "What Can We Do with a Quantum Computer?". Institute for Advanced Study.
58. Chang, Kenneth (14 June 2023). "Quantum Computing Advance Begins New Era, IBM Says – A quantum computer came up with better answers to a physics problem than a conventional supercomputer". The New York Times. Archived from the original on 14 June 2023. Retrieved 15 June 2023.
59. Kim, Youngseok; et al. (14 June 2023). "Evidence for the utility of quantum computing before fault tolerance". Nature. 618 (7965): 500–505. Bibcode:2023Natur.618..500K. doi:10.1038/s41586-023-06096-3. PMC 10266970. PMID 37316724.
60. Morello, Andrea (21 November 2018). Lunch & Learn: Quantum Computing. Sibos TV. Archived from the original on 11 December 2021. Retrieved 4 February 2021 – via YouTube.
61. Ruane, Jonathan; McAfee, Andrew; Oliver, William D. (1 January 2022). "Quantum Computing for Business Leaders". Harvard Business Review. ISSN 0017-8012. Retrieved 12 April 2023.
62. Budde, Florian; Volz, Daniel (12 July 2019). "Quantum computing and the chemical industry | McKinsey". www.mckinsey.com. McKinsey and Company. Retrieved 12 April 2023.
63. Bourzac, Katherine (30 October 2017). "Chemistry is quantum computing's killer app". cen.acs.org. American Chemical Society. Retrieved 12 April 2023.
64. Lenstra, Arjen K. (2000). "Integer Factoring" (PDF). Designs, Codes and Cryptography. 19 (2/3): 101–128. doi:10.1023/A:1008397921377. S2CID 9816153. Archived from the original (PDF) on 10 April 2015.
65. Nielsen & Chuang 2010, p. 216.
66. Bernstein, Daniel J. (2009). "Introduction to post-quantum cryptography". Post-Quantum Cryptography. Berlin, Heidelberg: Springer. pp. 1–14. doi:10.1007/978-3-540-88702-7_1. ISBN 978-3-540-88701-0. S2CID 61401925.
67. See also pqcrypto.org, a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing.
68. McEliece, R. J. (January 1978). "A Public-Key Cryptosystem Based On Algebraic Coding Theory" (PDF). DSNPR. 44: 114–116. Bibcode:1978DSNPR..44..114M.
69. Kobayashi, H.; Gall, F. L. (2006). "Dihedral Hidden Subgroup Problem: A Survey". Information and Media Technologies. 1 (1): 178–185. doi:10.2197/ipsjdc.1.470.
70. Bennett, Charles H.; Bernstein, Ethan; Brassard, Gilles; Vazirani, Umesh (October 1997). "Strengths and Weaknesses of Quantum Computing". SIAM Journal on Computing. 26 (5): 1510–1523. arXiv:quant-ph/9701001. Bibcode:1997quant.ph..1001B. doi:10.1137/s0097539796300933. S2CID 13403194.
71. Brassard, Gilles; Høyer, Peter; Tapp, Alain (2016). "Quantum Algorithm for the Collision Problem". In Kao, Ming-Yang (ed.). Encyclopedia of Algorithms. New York, NY: Springer. pp. 1662–1664. arXiv:quant-ph/9705002. doi:10.1007/978-1-4939-2864-4_304. ISBN 978-1-4939-2864-4. S2CID 3116149.
72. Farhi, Edward; Goldstone, Jeffrey; Gutmann, Sam (23 December 2008). "A Quantum Algorithm for the Hamiltonian NAND Tree". Theory of Computing. 4 (1): 169–190. doi:10.4086/toc.2008.v004a008. ISSN 1557-2862. S2CID 8258191.
73. Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. pp. 242–244. ISBN 978-1-84628-887-6.
74. Grover, Lov (29 May 1996). "A fast quantum mechanical algorithm for database search". arXiv:quant-ph/9605043.
75. Ambainis, Ambainis (June 2004). "Quantum search algorithms". ACM SIGACT News. 35 (2): 22–35. arXiv:quant-ph/0504012. Bibcode:2005quant.ph..4012A. doi:10.1145/992287.992296. S2CID 11326499.
76. Rich, Steven; Gellman, Barton (1 February 2014). "NSA seeks to build quantum computer that could crack most types of encryption". The Washington Post.
77. Outeiral, Carlos; Strahm, Martin; Morris, Garrett; Benjamin, Simon; Deane, Charlotte; Shi, Jiye (2021). "The prospects of quantum computing in computational molecular biology". WIREs Computational Molecular Science. 11. arXiv:2005.12792. doi:10.1002/wcms.1481. S2CID 218889377.
78. Biamonte, Jacob; Wittek, Peter; Pancotti, Nicola; Rebentrost, Patrick; Wiebe, Nathan; Lloyd, Seth (September 2017). "Quantum machine learning". Nature. 549 (7671): 195–202. arXiv:1611.09347. Bibcode:2017Natur.549..195B. doi:10.1038/nature23474. ISSN 0028-0836. PMID 28905917. S2CID 64536201.
79. Harrow, Aram; Hassidim, Avinatan; Lloyd, Seth (2009). "Quantum algorithm for solving linear systems of equations". Physical Review Letters. 103 (15): 150502. arXiv:0811.3171. Bibcode:2009PhRvL.103o0502H. doi:10.1103/PhysRevLett.103.150502. PMID 19905613. S2CID 5187993.
80. Benedetti, Marcello; Realpe-Gómez, John; Biswas, Rupak; Perdomo-Ortiz, Alejandro (9 August 2016). "Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning". Physical Review A. 94 (2): 022308. arXiv:1510.07611. Bibcode:2016PhRvA..94b2308B. doi:10.1103/PhysRevA.94.022308.
81. Ajagekar, Akshay; You, Fengqi (5 December 2020). "Quantum computing assisted deep learning for fault detection and diagnosis in industrial process systems". Computers & Chemical Engineering. 143: 107119. arXiv:2003.00264. doi:10.1016/j.compchemeng.2020.107119. ISSN 0098-1354. S2CID 211678230.
82. Ajagekar, Akshay; You, Fengqi (1 December 2021). "Quantum computing based hybrid deep learning for fault diagnosis in electrical power systems". Applied Energy. 303: 117628. doi:10.1016/j.apenergy.2021.117628. ISSN 0306-2619.
83. Gao, Xun; Anschuetz, Eric R.; Wang, Sheng-Tao; Cirac, J. Ignacio; Lukin, Mikhail D. (2022). "Enhancing Generative Models via Quantum Correlations". Physical Review X. 12 (2): 021037. arXiv:2101.08354. Bibcode:2022PhRvX..12b1037G. doi:10.1103/PhysRevX.12.021037. S2CID 231662294.
84. Li, Junde; Topaloglu, Rasit; Ghosh, Swaroop (9 January 2021). "Quantum Generative Models for Small Molecule Drug Discovery". arXiv:2101.03438 [cs.ET].
85. Brooks, Michael (24 May 2023). "Quantum computers: what are they good for?". Nature. 617 (7962): S1–S3. Bibcode:2023Natur.617S...1B. doi:10.1038/d41586-023-01692-9. PMID 37225885. S2CID 258847001.
86. Torsten Hoefler; Thomas Häner; Matthias Troyer (May 2023). "Disentangling Hype from Practicality: On Realistically Achieving Quantum Advantage". Communications of the ACM.
87. Dyakonov, Mikhail (15 November 2018). "The Case Against Quantum Computing". IEEE Spectrum.
88. DiVincenzo, David P. (13 April 2000). "The Physical Implementation of Quantum Computation". Fortschritte der Physik. 48 (9–11): 771–783. arXiv:quant-ph/0002077. Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. S2CID 15439711.
89. Giles, Martin (17 January 2019). "We'd have more quantum computers if it weren't so hard to find the damn cables". MIT Technology Review. Retrieved 17 May 2021.
90. Pauka SJ, Das K, Kalra B, Moini A, Yang Y, Trainer M, Bousquet A, Cantaloube C, Dick N, Gardner GC, Manfra MJ, Reilly DJ (2021). "A cryogenic CMOS chip for generating control signals for multiple qubits". Nature Electronics. 4 (4): 64–70. arXiv:1912.01299. doi:10.1038/s41928-020-00528-y. S2CID 231715555.
91. DiVincenzo, David P. (1995). "Quantum Computation". Science. 270 (5234): 255–261. Bibcode:1995Sci...270..255D. CiteSeerX 10.1.1.242.2165. doi:10.1126/science.270.5234.255. S2CID 220110562.
92. Zu, H.; Dai, W.; de Waele, A.T.A.M. (2022). "Development of Dilution refrigerators – A review". Cryogenics. 121. Bibcode:2022Cryo..121....1Z. doi:10.1016/j.cryogenics.2021.103390. ISSN 0011-2275. S2CID 244005391.
93. Jones, Nicola (19 June 2013). "Computing: The quantum company". Nature. 498 (7454): 286–288. Bibcode:2013Natur.498..286J. doi:10.1038/498286a. PMID 23783610.
94. Vepsäläinen, Antti P.; Karamlou, Amir H.; Orrell, John L.; Dogra, Akshunna S.; Loer, Ben; et al. (August 2020). "Impact of ionizing radiation on superconducting qubit coherence". Nature. 584 (7822): 551–556. arXiv:2001.09190. Bibcode:2020Natur.584..551V. doi:10.1038/s41586-020-2619-8. ISSN 1476-4687. PMID 32848227. S2CID 210920566.
95. Amy, Matthew; Matteo, Olivia; Gheorghiu, Vlad; Mosca, Michele; Parent, Alex; Schanck, John (30 November 2016). "Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3". arXiv:1603.09383 [quant-ph].
96. Dyakonov, M. I. (14 October 2006). S. Luryi; Xu, J.; Zaslavsky, A. (eds.). "Is Fault-Tolerant Quantum Computation Really Possible?". Future Trends in Microelectronics. Up the Nano Creek: 4–18. arXiv:quant-ph/0610117. Bibcode:2006quant.ph.10117D.
97. Ahsan, Muhammad (2015). Architecture Framework for Trapped-ion Quantum Computer based on Performance Simulation Tool. OCLC 923881411.
98. Ahsan, Muhammad; Meter, Rodney Van; Kim, Jungsang (28 December 2016). "Designing a Million-Qubit Quantum Computer Using a Resource Performance Simulator". ACM Journal on Emerging Technologies in Computing Systems. 12 (4): 39:1–39:25. doi:10.1145/2830570. ISSN 1550-4832. S2CID 1258374.
99. Gidney, Craig; Ekerå, Martin (15 April 2021). "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits". Quantum. 5: 433. arXiv:1905.09749. Bibcode:2021Quant...5..433G. doi:10.22331/q-2021-04-15-433. ISSN 2521-327X. S2CID 162183806.
100. Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2003). "Topological quantum computation". Bulletin of the American Mathematical Society. 40 (1): 31–38. arXiv:quant-ph/0101025. doi:10.1090/S0273-0979-02-00964-3. MR 1943131.
101. Monroe, Don (1 October 2008). "Anyons: The breakthrough quantum computing needs?". New Scientist.
102. Preskill, John (26 March 2012). "Quantum computing and the entanglement frontier". arXiv:1203.5813 [quant-ph].
103. Preskill, John (6 August 2018). "Quantum Computing in the NISQ era and beyond". Quantum. 2: 79. arXiv:1801.00862. Bibcode:2018Quant...2...79P. doi:10.22331/q-2018-08-06-79.
104. Boixo, Sergio; Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan; et al. (2018). "Characterizing Quantum Supremacy in Near-Term Devices". Nature Physics. 14 (6): 595–600. arXiv:1608.00263. Bibcode:2018NatPh..14..595B. doi:10.1038/s41567-018-0124-x. S2CID 4167494.
105. Savage, Neil (5 July 2017). "Quantum Computers Compete for "Supremacy"". Scientific American.
106. Giles, Martin (20 September 2019). "Google researchers have reportedly achieved 'quantum supremacy'". MIT Technology Review. Retrieved 15 May 2020.
107. Tavares, Frank (23 October 2019). "Google and NASA Achieve Quantum Supremacy". NASA. Retrieved 16 November 2021.
108. Pednault, Edwin; Gunnels, John A.; Nannicini, Giacomo; Horesh, Lior; Wisnieff, Robert (22 October 2019). "Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits". arXiv:1910.09534 [quant-ph].
109. Cho, Adrian (23 October 2019). "IBM casts doubt on Google's claims of quantum supremacy". Science. doi:10.1126/science.aaz6080. ISSN 0036-8075. S2CID 211982610.
110. Liu, Yong (Alexander); Liu, Xin (Lucy); Li, Fang (Nancy); Fu, Haohuan; Yang, Yuling; et al. (14 November 2021). "Closing the "quantum supremacy" gap". Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. SC '21. New York, New York: Association for Computing Machinery. pp. 1–12. arXiv:2110.14502. doi:10.1145/3458817.3487399. ISBN 978-1-4503-8442-1. S2CID 239036985.
111. Bulmer, Jacob F. F.; Bell, Bryn A.; Chadwick, Rachel S.; Jones, Alex E.; Moise, Diana; et al. (28 January 2022). "The boundary for quantum advantage in Gaussian boson sampling". Science Advances. 8 (4): eabl9236. arXiv:2108.01622. Bibcode:2022SciA....8.9236B. doi:10.1126/sciadv.abl9236. ISSN 2375-2548. PMC 8791606. PMID 35080972.
112. McCormick, Katie (10 February 2022). "Race Not Over Between Classical and Quantum Computers". Physics. 15: 19. Bibcode:2022PhyOJ..15...19M. doi:10.1103/Physics.15.19. S2CID 246910085.
113. Pan, Feng; Chen, Keyang; Zhang, Pan (2022). "Solving the Sampling Problem of the Sycamore Quantum Circuits". Physical Review Letters. 129 (9): 090502. arXiv:2111.03011. Bibcode:2022PhRvL.129i0502P. doi:10.1103/PhysRevLett.129.090502. PMID 36083655. S2CID 251755796.
114. Cho, Adrian (2 August 2022). "Ordinary computers can beat Google's quantum computer after all". doi:10.1126/science.ade2364. {{cite journal}}: Cite journal requires |journal= (help)
115. "Google's 'quantum supremacy' usurped by researchers using ordinary supercomputer". TechCrunch. 5 August 2022. Retrieved 7 August 2022.
116. Ball, Philip (3 December 2020). "Physicists in China challenge Google's 'quantum advantage'". Nature. 588 (7838): 380. Bibcode:2020Natur.588..380B. doi:10.1038/d41586-020-03434-7. PMID 33273711.
117. Garisto, Daniel. "Light-based Quantum Computer Exceeds Fastest Classical Supercomputers". Scientific American. Retrieved 7 December 2020.
118. Conover, Emily (3 December 2020). "The new light-based quantum computer Jiuzhang has achieved quantum supremacy". Science News. Retrieved 7 December 2020.
119. Zhong, Han-Sen; Wang, Hui; Deng, Yu-Hao; Chen, Ming-Cheng; Peng, Li-Chao; et al. (3 December 2020). "Quantum computational advantage using photons". Science. 370 (6523): 1460–1463. arXiv:2012.01625. Bibcode:2020Sci...370.1460Z. doi:10.1126/science.abe8770. ISSN 0036-8075. PMID 33273064. S2CID 227254333.
120. Roberson, Tara M. (21 May 2020). "{{subst:title case|Can hype be a force for good?}}". Public Understanding of Science. 29 (5): 544–552. doi:10.1177/0963662520923109. ISSN 0963-6625. PMID 32438851. S2CID 218831653.
121. Cavaliere, Fabio; Mattsson, John; Smeets, Ben (September 2020). "The security implications of quantum cryptography and quantum computing". Network Security. 2020 (9): 9–15. doi:10.1016/S1353-4858(20)30105-7. ISSN 1353-4858. S2CID 222349414.
122. Monroe, Don (December 2022). "Quantum Computers and the Universe". Communications of the ACM.
123. Swayne, Matt (20 June 2023). "PsiQuantum Sees 700x Reduction in Computational Resource Requirements to Break Elliptic Curve Cryptography With a Fault Tolerant Quantum Computer". The Quanrum Insider.
124. Unruh, Bill (1995). "Maintaining coherence in Quantum Computers". Physical Review A. 51 (2): 992–997. arXiv:hep-th/9406058. Bibcode:1995PhRvA..51..992U. doi:10.1103/PhysRevA.51.992. PMID 9911677. S2CID 13980886.
125. Davies, Paul (6 March 2007). "The implications of a holographic universe for quantum information science and the nature of physical law". arXiv:quant-ph/0703041.
126. Regan, K. W. (23 April 2016). "Quantum Supremacy and Complexity". Gödel's Lost Letter and P=NP.
127. Kalai, Gil (May 2016). "The Quantum Computer Puzzle" (PDF). Notices of the AMS. 63 (5): 508–516.
128. Rinott, Yosef; Shoham, Tomer; Kalai, Gil (13 July 2021). "Statistical Aspects of the Quantum Supremacy Demonstration". arXiv:2008.05177 [quant-ph].
129. Dyakonov, Mikhail (15 November 2018). "The Case Against Quantum Computing". IEEE Spectrum. Retrieved 3 December 2019.
130. Dyakonov, Mikhail (24 March 2020). Will We Ever Have a Quantum Computer?. Springer. ISBN 9783030420185. Retrieved 22 May 2020.
131. Tacchino, Francesco; Chiesa, Alessandro; Carretta, Stefano; Gerace, Dario (19 December 2019). "Quantum Computers as Universal Quantum Simulators: State‐of‐the‐Art and Perspectives". Advanced Quantum Technologies. 3 (3): 1900052. arXiv:1907.03505. doi:10.1002/qute.201900052. ISSN 2511-9044. S2CID 195833616.
132. National Academies of Sciences, Engineering, and Medicine 2019, p. 127.
133. National Academies of Sciences, Engineering, and Medicine 2019, p. 114.
134. Nielsen & Chuang 2010, p. 29.
135. Nielsen & Chuang 2010, p. 126.
136. Nielsen & Chuang 2010, p. 41.
137. Nielsen & Chuang 2010, p. 201.
138. Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing. 26 (5): 1411–1473. CiteSeerX 10.1.1.144.7852. doi:10.1137/S0097539796300921.
Further reading
Textbooks
• Aaronson, Scott (2013). Quantum Computing Since Democritus. Cambridge University Press. doi:10.1017/CBO9780511979309. ISBN 978-0-521-19956-8. OCLC 829706638.
• Akama, Seiki (2014). Elements of Quantum Computing: History, Theories and Engineering Applications. Springer. doi:10.1007/978-3-319-08284-4. ISBN 978-3-319-08284-4. OCLC 884786739.
• Benenti, Giuliano; Casati, Giulio; Rossini, Davide; Strini, Giuliano (2019). Principles of Quantum Computation and Information: A Comprehensive Textbook (2nd ed.). doi:10.1142/10909. ISBN 978-981-3237-23-0. OCLC 1084428655. S2CID 62280636.
• Bernhardt, Chris (2019). Quantum Computing for Everyone. MIT Press. ISBN 978-0-262-35091-4. OCLC 1082867954.
• Hidary, Jack D. (2021). Quantum Computing: An Applied Approach (2nd ed.). doi:10.1007/978-3-030-83274-2. ISBN 978-3-03-083274-2. OCLC 1272953643. S2CID 238223274.
• Hiroshi, Imai; Masahito, Hayashi, eds. (2006). Quantum Computation and Information: From Theory to Experiment. Topics in Applied Physics. Vol. 102. doi:10.1007/3-540-33133-6. ISBN 978-3-540-33133-9.
• Hughes, Ciaran; Isaacson, Joshua; Perry, Anastasia; Sun, Ranbel F.; Turner, Jessica (2021). Quantum Computing for the Quantum Curious (PDF). doi:10.1007/978-3-030-61601-4. ISBN 978-3-03-061601-4. OCLC 1244536372. S2CID 242566636.
• Jaeger, Gregg (2007). Quantum Information: An Overview. doi:10.1007/978-0-387-36944-0. ISBN 978-0-387-36944-0. OCLC 186509710.
• Johnston, Eric R.; Harrigan, Nic; Gimeno-Segovia, Mercedes (2019). Programming Quantum Computers: Essential Algorithms and Code Samples. O'Reilly Media, Incorporated. ISBN 978-1-4920-3968-6. OCLC 1111634190.
• Kaye, Phillip; Laflamme, Raymond; Mosca, Michele (2007). An Introduction to Quantum Computing. OUP Oxford. ISBN 978-0-19-857000-4. OCLC 85896383.
• Kitaev, Alexei Yu.; Shen, Alexander H.; Vyalyi, Mikhail N. (2002). Classical and Quantum Computation. American Mathematical Soc. ISBN 978-0-8218-3229-5. OCLC 907358694.
• Mermin, N. David (2007). Quantum Computer Science: An Introduction. doi:10.1017/CBO9780511813870. ISBN 978-0-511-34258-5. OCLC 422727925.
• National Academies of Sciences, Engineering, and Medicine (2019). Grumbling, Emily; Horowitz, Mark (eds.). Quantum Computing: Progress and Prospects. Washington, DC. doi:10.17226/25196. ISBN 978-0-309-47970-7. OCLC 1091904777. S2CID 125635007.{{cite book}}: CS1 maint: location missing publisher (link)
• Nielsen, Michael; Chuang, Isaac (2010). Quantum Computation and Quantum Information (10th anniversary ed.). doi:10.1017/CBO9780511976667. ISBN 978-0-511-99277-3. OCLC 700706156. S2CID 59717455.
• Stolze, Joachim; Suter, Dieter (2004). Quantum Computing: A Short Course from Theory to Experiment. doi:10.1002/9783527617760. ISBN 978-3-527-61776-0. OCLC 212140089.
• Susskind, Leonard; Friedman, Art (2014). Quantum Mechanics: The Theoretical Minimum. Theoretical Minimum. New York: Basic Books. ISBN 978-0-465-08061-8.
• Wichert, Andreas (2020). Principles of Quantum Artificial Intelligence: Quantum Problem Solving and Machine Learning (2nd ed.). doi:10.1142/11938. ISBN 978-981-12-2431-7. OCLC 1178715016. S2CID 225498497.
• Wong, Thomas (2022). Introduction to Classical and Quantum Computing (PDF). Rooted Grove. ISBN 979-8-9855931-0-5. OCLC 1308951401.
• Zeng, Bei; Chen, Xie; Zhou, Duan-Lu; Wen, Xiao-Gang (2019). Quantum Information Meets Quantum Matter. arXiv:1508.02595. doi:10.1007/978-1-4939-9084-9. ISBN 978-1-4939-9084-9. OCLC 1091358969. S2CID 118528258.
Academic papers
• Abbot, Derek; Doering, Charles R.; Caves, Carlton M.; Lidar, Daniel M.; Brandt, Howard E.; et al. (2003). "Dreams versus Reality: Plenary Debate Session on Quantum Computing". Quantum Information Processing. 2 (6): 449–472. arXiv:quant-ph/0310130. doi:10.1023/B:QINP.0000042203.24782.9a. hdl:2027.42/45526. S2CID 34885835.
• Berthiaume, Andre (1 December 1998). "Quantum Computation". Solution Manual for Quantum Mechanics. pp. 233–234. doi:10.1142/9789814541893_0016. ISBN 978-981-4541-88-6. S2CID 128255429. {{cite book}}: |website= ignored (help)
• DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". Fortschritte der Physik. 48 (9–11): 771–783. arXiv:quant-ph/0002077. Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. S2CID 15439711.
• DiVincenzo, David P. (1995). "Quantum Computation". Science. 270 (5234): 255–261. Bibcode:1995Sci...270..255D. CiteSeerX 10.1.1.242.2165. doi:10.1126/science.270.5234.255. S2CID 220110562. Table 1 lists switching and dephasing times for various systems.
• Feynman, Richard (1982). "Simulating physics with computers". International Journal of Theoretical Physics. 21 (6–7): 467–488. Bibcode:1982IJTP...21..467F. CiteSeerX 10.1.1.45.9310. doi:10.1007/BF02650179. S2CID 124545445.
• Jeutner, Valentin (2021). "The Quantum Imperative: Addressing the Legal Dimension of Quantum Computers". Morals & Machines. 1 (1): 52–59. doi:10.5771/2747-5174-2021-1-52. S2CID 236664155.
• Krantz, P.; Kjaergaard, M.; Yan, F.; Orlando, T. P.; Gustavsson, S.; Oliver, W. D. (17 June 2019). "A Quantum Engineer's Guide to Superconducting Qubits". Applied Physics Reviews. 6 (2): 021318. arXiv:1904.06560. Bibcode:2019ApPRv...6b1318K. doi:10.1063/1.5089550. ISSN 1931-9401. S2CID 119104251.
• Mitchell, Ian (1998). "Computing Power into the 21st Century: Moore's Law and Beyond".
• Shor, Peter W. (1994). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Symposium on Foundations of Computer Science. Santa Fe, New Mexico: IEEE. pp. 124–134. doi:10.1109/SFCS.1994.365700. ISBN 978-0-8186-6580-6.
• Simon, Daniel R. (1994). "On the Power of Quantum Computation". Institute of Electrical and Electronics Engineers Computer Society Press.
External links
• Media related to Quantum computer at Wikimedia Commons
• Learning materials related to Quantum computing at Wikiversity
• Stanford Encyclopedia of Philosophy: "Quantum Computing" by Amit Hagar and Michael E. Cuffaro.
• "Quantum computation, theory of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Quantum computing for the very curious by Andy Matuschak and Michael Nielsen
Lectures
• Quantum computing for the determined – 22 video lectures by Michael Nielsen
• Video Lectures by David Deutsch
• Lectures at the Institut Henri Poincaré (slides and videos)
• Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008)
• Lomonaco, Sam. Four Lectures on Quantum Computing given at Oxford University in July 2006
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
Emerging technologies
Fields
Quantum
• algorithms
• amplifier
• bus
• cellular automata
• channel
• circuit
• complexity theory
• computing
• clock
• cryptography
• post-quantum
• dynamics
• electronics
• error correction
• finite automata
• image processing
• imaging
• information
• key distribution
• logic
• logic gates
• machine
• machine learning
• metamaterial
• network
• neural network
• optics
• programming
• sensing
• simulator
• teleportation
Other
• Anti-gravity
• Acoustic levitation
• Cloak of invisibility
• Digital scent technology
• Force field
• Plasma window
• Immersive virtual reality
• Magnetic refrigeration
• Phased-array optics
• Thermoacoustic heat engine
• Category
• List
Computer science
Note: This template roughly follows the 2012 ACM Computing Classification System.
Hardware
• Printed circuit board
• Peripheral
• Integrated circuit
• Very Large Scale Integration
• Systems on Chip (SoCs)
• Energy consumption (Green computing)
• Electronic design automation
• Hardware acceleration
Computer systems organization
• Computer architecture
• Embedded system
• Real-time computing
• Dependability
Networks
• Network architecture
• Network protocol
• Network components
• Network scheduler
• Network performance evaluation
• Network service
Software organization
• Interpreter
• Middleware
• Virtual machine
• Operating system
• Software quality
Software notations and tools
• Programming paradigm
• Programming language
• Compiler
• Domain-specific language
• Modeling language
• Software framework
• Integrated development environment
• Software configuration management
• Software library
• Software repository
Software development
• Control variable
• Software development process
• Requirements analysis
• Software design
• Software construction
• Software deployment
• Software engineering
• Software maintenance
• Programming team
• Open-source model
Theory of computation
• Model of computation
• Formal language
• Automata theory
• Computability theory
• Computational complexity theory
• Logic
• Semantics
Algorithms
• Algorithm design
• Analysis of algorithms
• Algorithmic efficiency
• Randomized algorithm
• Computational geometry
Mathematics of computing
• Discrete mathematics
• Probability
• Statistics
• Mathematical software
• Information theory
• Mathematical analysis
• Numerical analysis
• Theoretical computer science
Information systems
• Database management system
• Information storage systems
• Enterprise information system
• Social information systems
• Geographic information system
• Decision support system
• Process control system
• Multimedia information system
• Data mining
• Digital library
• Computing platform
• Digital marketing
• World Wide Web
• Information retrieval
Security
• Cryptography
• Formal methods
• Security services
• Intrusion detection system
• Hardware security
• Network security
• Information security
• Application security
Human–computer interaction
• Interaction design
• Social computing
• Ubiquitous computing
• Visualization
• Accessibility
Concurrency
• Concurrent computing
• Parallel computing
• Distributed computing
• Multithreading
• Multiprocessing
Artificial intelligence
• Natural language processing
• Knowledge representation and reasoning
• Computer vision
• Automated planning and scheduling
• Search methodology
• Control method
• Philosophy of artificial intelligence
• Distributed artificial intelligence
Machine learning
• Supervised learning
• Unsupervised learning
• Reinforcement learning
• Multi-task learning
• Cross-validation
Graphics
• Animation
• Rendering
• Photograph manipulation
• Graphics processing unit
• Mixed reality
• Virtual reality
• Image compression
• Solid modeling
Applied computing
• E-commerce
• Enterprise software
• Computational mathematics
• Computational physics
• Computational chemistry
• Computational biology
• Computational social science
• Computational engineering
• Computational healthcare
• Digital art
• Electronic publishing
• Cyberwarfare
• Electronic voting
• Video games
• Word processing
• Operations research
• Educational technology
• Document management
• Category
• Outline
• WikiProject
• Commons
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
• Quantum biology
• Quantum chemistry
• Quantum chaos
• Quantum cosmology
• Quantum differential calculus
• Quantum dynamics
• Quantum geometry
• Quantum measurement problem
• Quantum mind
• Quantum stochastic calculus
• Quantum spacetime
Technology
• Quantum algorithms
• Quantum amplifier
• Quantum bus
• Quantum cellular automata
• Quantum finite automata
• Quantum channel
• Quantum circuit
• Quantum complexity theory
• Quantum computing
• Timeline
• Quantum cryptography
• Quantum electronics
• Quantum error correction
• Quantum imaging
• Quantum image processing
• Quantum information
• Quantum key distribution
• Quantum logic
• Quantum logic gates
• Quantum machine
• Quantum machine learning
• Quantum metamaterial
• Quantum metrology
• Quantum network
• Quantum neural network
• Quantum optics
• Quantum programming
• Quantum sensing
• Quantum simulator
• Quantum teleportation
Extensions
• Casimir effect
• Quantum statistical mechanics
• Quantum field theory
• History
• Quantum gravity
• Relativistic quantum mechanics
Related
• Schrödinger's cat
• in popular culture
• EPR paradox
• Quantum mysticism
• Category
• Physics portal
• Commons
Authority control: National
• Israel
• United States
• Czech Republic
|
Wikipedia
|
Quantum phase estimation algorithm
In quantum computing, the quantum phase estimation algorithm is a quantum algorithm to estimate the phase corresponding to an eigenvalue of a given unitary operator. Because the eigenvalues of a unitary operator always have unit modulus, they are characterized by their phase, and therefore the algorithm can be equivalently described as retrieving either the phase or the eigenvalue itself. The algorithm was initially introduced by Alexei Kitaev in 1995.[1][2]: 246
Phase estimation is frequently used as a subroutine in other quantum algorithms, such as Shor's algorithm,[2]: 131 the quantum algorithm for linear systems of equations, and the quantum counting algorithm.
Formal description of the problem
Let $U$ be a unitary operator acting on an $m$-qubit register. Unitarity implies that all the eigenvalues of $U$ have unit modulus, and can therefore be characterized by their phase. Thus if $|\psi \rangle $ is an eigenvector of $U$, then $U|\psi \rangle =e^{2\pi i\theta }\left|\psi \right\rangle $ for some $\theta \in \mathbb {R} $. Due to the periodicity of the complex exponential, we can always assume $0\leq \theta <1$.
Our goal is to find a good approximation to $\theta $ with a small number of gates and with high probability. The quantum phase estimation algorithm achieves this under the assumptions of having oracular access to $U$, and having $|\psi \rangle $ available as a quantum state.
More precisely, the algorithm returns an approximation for $\theta $, with high probability within additive error $\varepsilon $, using $O(\log(1/\varepsilon ))$ qubits (without counting the ones used to encode the eigenvector state) and $O(1/\varepsilon )$ controlled-U operations.
The algorithm
Setup
The input consists of two registers (namely, two parts): the upper $n$ qubits comprise the first register, and the lower $m$ qubits are the second register.
The initial state of the system is:
$|0\rangle ^{\otimes n}|\psi \rangle .$
After applying n-bit Hadamard gate operation $H^{\otimes n}$ on the first register, the state becomes:
${\frac {1}{2^{\frac {n}{2}}}}(|0\rangle +|1\rangle )^{\otimes n}|\psi \rangle ={\frac {1}{2^{n/2}}}\sum _{j=0}^{2^{n}-1}|j\rangle |\psi \rangle $.
Let $U$ be a unitary operator with eigenvector $|\psi \rangle $ such that $U|\psi \rangle =e^{2\pi i\theta }|\psi \rangle $. Thus,
$U^{2^{j}}|\psi \rangle =e^{2\pi i2^{j}\theta }|\psi \rangle $.
Overall, the transformation implemented on the two registers by the controlled gates applying $U,U^{2},U^{2^{2}},\ldots ,U^{2^{n-1}}$ is
$|k\rangle |\psi \rangle \mapsto |k\rangle U^{k}|\psi \rangle $
This can be seen by the decomposition of $k$ into its bitstring $k_{n-1}k_{n-2}\ldots k_{1}k_{0}$ and binary representation $2^{n-1}k_{n-1}+2^{n-2}k_{n-2}+\ldots +2k_{1}+k_{0}$, where $k_{n-1},\ldots ,k_{1},k_{0}\in \{0,1\}$. Clearly, $U^{k}$ becomes
$U^{2^{n-1}k_{n-1}+\ldots +2^{2}k_{2}+2k_{1}+k_{0}}=U^{2^{n-1}k_{n-1}}\ldots U^{2^{2}k_{2}}U^{2k_{1}}U^{k_{0}}$
Each $U^{2^{j}k_{j}}$ will only apply if the qubit $k_{j}$ is $1$, implying that it is controlled by that bit. Therefore the overall transformation to $|k\rangle U^{k}|\psi \rangle $ is equivalent to the controlled $U^{2^{j}}$ gates from each $j$-th qubit. Therefore, the state will be transformed by the controlled $U^{2^{j}}$ gates like so:
${\frac {1}{\sqrt {2^{n}}}}\sum _{j=0}^{2^{n}-1}|j\rangle |\psi \rangle \mapsto {\frac {1}{\sqrt {2^{n}}}}\sum _{j=0}^{2^{n}-1}e^{2\pi ij\theta }|j\rangle |\psi \rangle $
At this point, the second register with the eigenvector is not needed. It can be reused again in another run of phase estimation. The state without $|\psi \rangle $ is
${\frac {1}{\sqrt {2^{n}}}}\sum _{j=0}^{2^{n}-1}e^{2\pi ij\theta }|j\rangle $
Apply inverse quantum Fourier transform
Applying the inverse quantum Fourier transform on
${\frac {1}{2^{\frac {n}{2}}}}\sum _{k=0}^{2^{n}-1}e^{2\pi i\theta k}|k\rangle $
yields
${\frac {1}{2^{\frac {n}{2}}}}\sum _{k=0}^{2^{n}-1}e^{2\pi i\theta k}\left({\frac {1}{2^{\frac {n}{2}}}}\sum _{x=0}^{2^{n}-1}e^{\frac {-2\pi ikx}{2^{n}}}|x\rangle \right)={\frac {1}{2^{n}}}\sum _{x=0}^{2^{n}-1}\sum _{k=0}^{2^{n}-1}e^{-{\frac {2\pi ik}{2^{n}}}\left(x-2^{n}\theta \right)}|x\rangle .$
We can approximate the value of $\theta \in [0,1]$ by rounding $2^{n}\theta $ to the nearest integer. This means that $2^{n}\theta =a+2^{n}\delta ,$ where $a$ is the nearest integer to $2^{n}\theta ,$ and the difference $2^{n}\delta $ satisfies $0\leqslant |2^{n}\delta |\leqslant {\tfrac {1}{2}}$.
Using this decomposition we can rewrite the state as $ \sum _{x=0}^{2^{n}-1}c_{x}|x\rangle ,$ where
$c_{x}\equiv {\frac {1}{2^{n}}}\sum _{k=0}^{2^{n}-1}e^{-{\frac {2\pi ik}{2^{n}}}(x-2^{n}\theta )}={\frac {1}{2^{n}}}\sum _{k=0}^{2^{n}-1}e^{-{\frac {2\pi ik}{2^{n}}}\left(x-a\right)}e^{2\pi i\delta k}.$
Measurement
Performing a measurement in the computational basis on the first register yields the outcome $|y\rangle $ with probability
$\Pr(y)=|c_{y}|^{2}=\left|{\frac {1}{2^{n}}}\sum _{k=0}^{2^{n}-1}e^{{\frac {-2\pi ik}{2^{n}}}(y-a)}e^{2\pi i\delta k}\right|^{2}.$
It follows that $\operatorname {Pr} (a)=1$ if $\delta =0$, that is, when $\theta $ can be written as $\theta =a/2^{n}$, one always finds the outcome $y=a$. On the other hand, if $\delta \neq 0$, the probability reads
$\operatorname {Pr} (a)={\frac {1}{2^{2n}}}\left|\sum _{k=0}^{2^{n}-1}e^{2\pi i\delta k}\right|^{2}={\frac {1}{2^{2n}}}\left|{\frac {1-{e^{2\pi i2^{n}\delta }}}{1-{e^{2\pi i\delta }}}}\right|^{2}.$
From this expression we can see that $\Pr(a)\geqslant {\frac {4}{\pi ^{2}}}\approx 0.405$ when $\delta \neq 0$. To see this, we observe that from the definition of $\delta $ we have the inequality $|\delta |\leqslant {\tfrac {1}{2^{n+1}}}$, and thus:[3]: 157 [4]: 348
${\begin{aligned}\Pr(a)&={\frac {1}{2^{2n}}}\left|{\frac {1-{e^{2\pi i2^{n}\delta }}}{1-{e^{2\pi i\delta }}}}\right|^{2}&&{\text{for }}\delta \neq 0\\[6pt]&={\frac {1}{2^{2n}}}\left|{\frac {2\sin \left(\pi 2^{n}\delta \right)}{2\sin(\pi \delta )}}\right|^{2}&&\left|1-e^{2ix}\right|^{2}=4\left|\sin(x)\right|^{2}\\[6pt]&={\frac {1}{2^{2n}}}{\frac {\left|\sin \left(\pi 2^{n}\delta \right)\right|^{2}}{|\sin(\pi \delta )|^{2}}}\\[6pt]&\geqslant {\frac {1}{2^{2n}}}{\frac {\left|\sin \left(\pi 2^{n}\delta \right)\right|^{2}}{|\pi \delta |^{2}}}&&|\sin(\pi \delta )|\leqslant |\pi \delta |\\[6pt]&\geqslant {\frac {1}{2^{2n}}}{\frac {|2\cdot 2^{n}\delta |^{2}}{|\pi \delta |^{2}}}&&|2\cdot 2^{n}\delta |\leqslant |\sin(\pi 2^{n}\delta )|{\text{ for }}|\delta |\leqslant {\frac {1}{2^{n+1}}}\\[6pt]&\geqslant {\frac {4}{\pi ^{2}}}.\end{aligned}}$
We conclude that the algorithm always provides the best $n$-bit estimate of $\theta $ with high probability. By adding a number of extra qubits on the order of $O(\log(1/\epsilon ))$ and truncating the extra qubits the probability can increase to $1-\epsilon $.[4]
Toy examples
Consider the simplest possible instance of the algorithm, where only $n=1$ qubit, on top of the qubits required to encode $|\psi \rangle $, is involved. Suppose the eigenvalue of $|\psi \rangle $ reads $\lambda =e^{2\pi i\theta }$. The first part of the algorithm generates the one-qubit state $ |\phi \rangle \equiv {\frac {1}{\sqrt {2}}}(|0\rangle +\lambda |1\rangle )$. Applying the inverse QFT amounts in this case to applying a Hadamard gate. The final outcome probabilities are thus $p_{\pm }=|\langle \pm |\phi \rangle |^{2}$ where $ |\pm \rangle \equiv {\frac {1}{\sqrt {2}}}(|0\rangle \pm |1\rangle )$, or more explicitly,
$p_{\pm }={\frac {|1\pm \lambda |^{2}}{4}}={\frac {1\pm \cos(2\pi \theta )}{2}}.$
Suppose $\lambda =1$, meaning $|\phi \rangle =|+\rangle $. Then $p_{+}=1$, $p_{-}=0$, and we recover deterministically the precise value of $\lambda $ from the measurement outcomes. The same applies if $\lambda =-1$.
If on the other hand $\lambda =e^{2\pi i/3}$, then $p_{\pm }=[1\pm \cos(2\pi /3)]/2$, that is, $p_{+}=1/4$ and $p_{-}=3/4$. In this case the result is not deterministic, but we still find the outcome $|-\rangle $ as more likely, compatibly with the fact that $2/3$ is close to 1 than to 0.
More generally, if $\lambda =e^{2\pi i\theta }$, then $p_{+}\geq 1/2$ if and only if $|\theta |\leq 1/4$. This is consistent with the results above because in the cases $\lambda =\pm 1$, corresponding to $\theta =0,1/2$, the phase is retrieved deterministically, adn the ther phases are retrieved with higher accuracy the more they are close to these two.
See also
• Shor's algorithm
• Quantum counting algorithm
• Parity measurement
References
1. Kitaev, A. Yu (1995-11-20). "Quantum measurements and the Abelian Stabilizer Problem". arXiv:quant-ph/9511026.
2. Nielsen, Michael A. & Isaac L. Chuang (2001). Quantum computation and quantum information (Repr. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 978-0521635035.
3. Benenti, Guiliano; Casati, Giulio; Strini, Giuliano (2004). Principles of quantum computation and information (Reprinted. ed.). New Jersey [u.a.]: World Scientific. ISBN 978-9812388582.
4. Cleve, R.; Ekert, A.; Macchiavello, C.; Mosca, M. (8 January 1998). "Quantum algorithms revisited". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 454 (1969): 339–354. arXiv:quant-ph/9708016. Bibcode:1998RSPSA.454..339C. doi:10.1098/rspa.1998.0164. S2CID 16128238.
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
|
Wikipedia
|
Noncommutative projective geometry
In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.
Examples
• The quantum plane, the most basic example, is the quotient ring of the free ring:
$k\langle x,y\rangle /(yx-qxy)$
• More generally, the quantum polynomial ring is the quotient ring:
$k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})$
Proj construction
By definition, the Proj of a graded ring R is the quotient category of the category of finitely generated graded modules over R by the subcategory of torsion modules. If R is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of R in this sense is equivalent to the category of coherent sheaves on the usual Proj of R. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring.
See also
• Elliptic algebra
• Calabi–Yau algebra
• Sklyanin algebra
References
• Ajitabh, Kaushal (1994), Modules over regular algebras and quantum planes (PDF) (Ph.D. thesis)
• Artin, Michael (1992), "Geometry of quantum planes", Contemporary Mathematics, 124: 1–15, MR 1144023
• Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].
|
Wikipedia
|
Quantum walk
Quantum walks are quantum analogues of classical random walks. In contrast to the classical random walk, where the walker occupies definite states and the randomness arises due to stochastic transitions between states, in quantum walks randomness arises through: (1) quantum superposition of states, (2) non-random, reversible unitary evolution and (3) collapse of the wave function due to state measurements.
As with classical random walks, quantum walks admit formulations in both discrete time and continuous time.
Motivation
Quantum walks are motivated by the widespread use of classical random walks in the design of randomized algorithms, and are part of several quantum algorithms. For some oracular problems, quantum walks provide an exponential speedup over any classical algorithm.[1][2] Quantum walks also give polynomial speedups over classical algorithms for many practical problems, such as the element distinctness problem,[3] the triangle finding problem,[4] and evaluating NAND trees.[5] The well-known Grover search algorithm can also be viewed as a quantum walk algorithm.
Relation to classical random walks
Quantum walks exhibit very different features from classical random walks. In particular, they do not converge to limiting distributions and due to the power of quantum interference they may spread significantly faster or slower than their classical equivalents.
Continuous time
Continuous-time quantum walks arise when one replaces the continuum spatial domain in the Schrödinger equation with a discrete set. That is, instead of having a quantum particle propagate in a continuum, one restricts the set of possible position states to the vertex set $V$ of some graph $G=(V,E)$ which can be either finite or countably infinite. Under particular conditions, continuous-time quantum walks can provide a model for universal quantum computation.[6]
Relation to non-relativistic Schrödinger dynamics
Consider the dynamics of a non-relativistic, spin-less free quantum particle with mass $m$ propagating on an infinite one-dimensional spatial domain. The particle's motion is completely described by its wave function $\psi :\mathbb {R} \times \mathbb {R} _{\geq 0}\to \mathbb {C} $ :\mathbb {R} \times \mathbb {R} _{\geq 0}\to \mathbb {C} } which satisfies the one-dimensional, free particle Schrödinger equation
${\textbf {i}}\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi }{\partial x^{2}}}$
where ${\textbf {i}}={\sqrt {-1}}$ and $\hbar $ is the reduced Planck's constant. Now suppose that only the spatial part of the domain is discretized, $\mathbb {R} $ being replaced with $\mathbb {Z} _{\Delta x}\equiv \{\ldots ,-2\,\Delta x,-\Delta x,0,\Delta x,2\,\Delta x,\ldots \}$ where $\Delta x$ is the separation between the spatial sites the particle can occupy. The wave function becomes the map $\psi :\mathbb {Z} _{\Delta x}\times \mathbb {R} _{\geq 0}\to \mathbb {C} $ :\mathbb {Z} _{\Delta x}\times \mathbb {R} _{\geq 0}\to \mathbb {C} } and the second spatial partial derivative becomes the discrete laplacian
${\frac {\partial ^{2}\psi }{\partial x^{2}}}\to {\frac {L_{\mathbb {Z} }\psi (j\,\Delta x,t)}{\Delta x^{2}}}\equiv {\frac {\psi \left((j+1)\,\Delta x,t\right)-2\psi \left(j\,\Delta x,t\right)+\psi \left((j-1)\,\Delta x,t\right)}{\Delta x^{2}}}$
The evolution equation for a continuous time quantum walk on $\mathbb {Z} _{\Delta x}$ is thus
${\textbf {i}}{\frac {\partial \psi }{\partial t}}=-\omega _{\Delta x}L_{\mathbb {Z} }\psi $
where $\omega _{\Delta x}\equiv \hbar /2m\,\Delta x^{2}$is a characteristic frequency. This construction naturally generalizes to the case that the discretized spatial domain is an arbitrary graph $G=(V,E)$ and the discrete laplacian $L_{\mathbb {Z} }$is replaced by the graph Laplacian $L_{G}\equiv D_{G}-A_{G}$where $D_{G}$ and $A_{G}$ are the degree matrix and the adjacency matrix, respectively. Common choices of graphs that show up in the study of continuous time quantum walks are the d-dimensional lattices $\mathbb {Z} ^{d}$, cycle graphs $\mathbb {Z} /N\mathbb {Z} $, d-dimensional discrete tori $(\mathbb {Z} /N\mathbb {Z} )^{d}$, the d-dimensional hypercube $\mathbb {Q} ^{d}$and random graphs.
Discrete time
Discrete-time quantum walks on $\mathbb {Z} $
The evolution of a quantum walk in discrete time is specified by the product of two unitary operators: (1) a "coin flip" operator and (2) a conditional shift operator, which are applied repeatedly. The following example is instructive here.[7] Imagine a particle with a spin-1/2-degree of freedom propagating on a linear array of discrete sites. If the number of such sites is countably infinite, we identify the state space with $\mathbb {Z} $. The particle's state can then be described by a product state
$|\Psi \rangle =|s\rangle \otimes |\psi \rangle $
consisting of an internal spin state
$|s\rangle \in {\mathcal {H}}_{C}=\left\{a_{\uparrow }|{\uparrow }\rangle +a_{\downarrow }|{\downarrow }\rangle :a_{\uparrow /\downarrow }\in \mathbb {C} \right\}$
and a position state
$|\psi \rangle \in {\mathcal {H}}_{P}=\left\{\sum _{x\in \mathbb {Z} }\alpha _{x}|x\rangle :\sum _{x\in \mathbb {Z} }|\alpha _{x}|^{2}<\infty \right\}$ :\sum _{x\in \mathbb {Z} }|\alpha _{x}|^{2}<\infty \right\}}
where ${\mathcal {H}}_{C}=\mathbb {C} ^{2}$ is the "coin space" and ${\mathcal {H}}_{P}=\ell ^{2}(\mathbb {Z} )$ is the space of physical quantum position states. The product $\otimes $ in this setting is the Kronecker (tensor) product. The conditional shift operator for the quantum walk on the line is given by
$S=|{\uparrow }\rangle \langle {\uparrow }|\otimes \sum \limits _{i}|i+1\rangle \langle i|+|{\downarrow }\rangle \langle {\downarrow }|\otimes \sum \limits _{i}|i-1\rangle \langle i|,$
i.e. the particle jumps right if it has spin up and left if it has spin down. Explicitly, the conditional shift operator acts on product states according to
$S(|{\uparrow }\rangle \otimes |i\rangle )=|{\uparrow }\rangle \otimes |i+1\rangle $
$S(|{\downarrow }\rangle \otimes |i\rangle )=|{\downarrow }\rangle \otimes |i-1\rangle $
If we first rotate the spin with some unitary transformation $C:{\mathcal {H}}_{C}\to {\mathcal {H}}_{C}$ and then apply $S$, we get a non-trivial quantum motion on $\mathbb {Z} $. A popular choice for such a transformation is the Hadamard gate $C=H$, which, with respect to the standard z-component spin basis, has matrix representation
$H={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&\;\;1\\1&-1\end{pmatrix}}$
When this choice is made for the coin flip operator, the operator itself is called the "Hadamard coin" and the resulting quantum walk is called the "Hadamard walk". If the walker is initialized at the origin and in the spin-up state, a single time step of the Hadamard walk on $\mathbb {Z} $ is
$|{\uparrow }\rangle \otimes |0\rangle \;\,{\overset {H}{\longrightarrow }}\;\,{\frac {1}{\sqrt {2}}}(|{\uparrow }\rangle +|{\downarrow }\rangle )\otimes |0\rangle \;\,{\overset {S}{\longrightarrow }}\;\,{\frac {1}{\sqrt {2}}}(|{\uparrow }\rangle \otimes |1\rangle +|{\downarrow }\rangle \otimes |{-1}\rangle ).$
Measurement of the system's state at this point would reveal an up spin at position 1 or a down spin at position −1, both with probability 1/2. Repeating the procedure would correspond to a classical simple random walk on $\mathbb {Z} $. In order to observe non-classical motion, no measurement is performed on the state at this point (and therefore do not force a collapse of the wave function). Instead, repeat the procedure of rotating the spin with the coin flip operator and conditionally jumping with $S$. This way, quantum correlations are preserved and different position states can interfere with one another. This gives a drastically different probability distribution than the classical random walk (Gaussian distribution) as seen in the figure to the right. Spatially one sees that the distribution is not symmetric: even though the Hadamard coin gives both up and down spin with equal probability, the distribution tends to drift to the right when the initial spin is $|{\uparrow }\rangle $. This asymmetry is entirely due to the fact that the Hadamard coin treats the $|{\uparrow }\rangle $ and $|{\downarrow }\rangle $ state asymmetrically. A symmetric probability distribution arises if the initial state is chosen to be
$|\Psi _{0}^{\text{symm}}\rangle ={\frac {1}{\sqrt {2}}}(|{\uparrow }\rangle -{\textbf {i}}|{\downarrow }\rangle )\otimes |0\rangle $
Dirac equation
Consider what happens when we discretize a massive Dirac operator over one spatial dimension. In the absence of a mass term, we have left-movers and right-movers. They can be characterized by an internal degree of freedom, "spin" or a "coin". When we turn on a mass term, this corresponds to a rotation in this internal "coin" space. A quantum walk corresponds to iterating the shift and coin operators repeatedly.
This is very much like Richard Feynman's model of an electron in 1 (one) spatial and 1 (one) time dimension. He summed up the zigzagging paths, with left-moving segments corresponding to one spin (or coin), and right-moving segments to the other. See Feynman checkerboard for more details.
The transition probability for a 1-dimensional quantum walk behaves like the Hermite functions which (1) asymptotically oscillate in the classically allowed region, (2) is approximated by the Airy function around the wall of the potential, and (3) exponentially decay in the classically hidden region.[8]
Realization
Atomic lattice is the leading quantum platform in terms of scalability. Coined and coinless discrete-time quantum-walk could be realized in the atomic lattice via a distance-selective spin-exchange interaction.[9] Remarkably the platform preserves the coherence over couple of hundred sites and steps in 1, 2 or 3 dimensions in the spatial space. The long-range dipolar interaction allows designing periodic boundary conditions, facilitating the QW over topological surfaces.[9]
See also
• Path integral formulation
• Quantum walk search
References
1. A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman, Exponential algorithmic speedup by quantum walk, Proc. 35th ACM Symposium on Theory of Computing, pp. 59–68, 2003, arXiv:quant-ph/0209131.
2. A. M. Childs, L. J. Schulman, and U. V. Vazirani, Quantum algorithms for hidden nonlinear structures, Proc. 48th IEEE Symposium on Foundations of Computer Science, pp. 395–404, 2007, arXiv:0705.2784.
3. Andris Ambainis, Quantum walk algorithm for element distinctness, SIAM J. Comput. 37 (2007), no. 1, 210–239, arXiv:quant-ph/0311001, preliminary version in FOCS 2004.
4. F. Magniez, M. Santha, and M. Szegedy, Quantum algorithms for the triangle problem, Proc. 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 1109–1117, 2005, quant-ph/0310134.
5. E. Farhi, J. Goldstone, and S. Gutmann, A quantum algorithm for the Hamiltonian NAND tree, Theory of Computing 4 (2008), no. 1, 169–190, quant-ph/0702144
6. Andrew M. Childs, "Universal Computation by Quantum Walk".
7. Kempe, Julia (1 July 2003). "Quantum random walks – an introductory overview". Contemporary Physics. 44 (4): 307–327. arXiv:quant-ph/0303081. Bibcode:2003ConPh..44..307K. doi:10.1080/00107151031000110776. ISSN 0010-7514. S2CID 17300331.
8. T. Sunada and T. Tate, Asymptotic behavior of quantum walks on the line, Journal of Functional Analysis 262 (2012) 2608–2645
9. Khazali, Mohammadsadegh (3 March 2022). "Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction". Quantum. 6: 664. arXiv:2101.11412. Bibcode:2022Quant...6..664K. doi:10.22331/q-2022-03-03-664. ISSN 2521-327X. S2CID 246635019.
Further reading
• Julia Kempe (2003). "Quantum random walks – an introductory overview". Contemporary Physics. 44 (4): 307–327. arXiv:quant-ph/0303081. Bibcode:2003ConPh..44..307K. doi:10.1080/00107151031000110776. S2CID 17300331.
• Andris Ambainis (2003). "Quantum walks and their algorithmic applications". International Journal of Quantum Information. 1 (4): 507–518. arXiv:quant-ph/0403120. doi:10.1142/S0219749903000383. S2CID 10324299.
• Miklos Santha (2008). "Quantum walk based search algorithms". 5th Theory and Applications of Models of Computation (TAMC), Xian, April, LNCS 4978. 5 (8): 31–46. arXiv:0808.0059. Bibcode:2008arXiv0808.0059S.
• Salvador E. Venegas-Andraca (2012). "Quantum walks: a comprehensive review". Quantum Information Processing. 11 (5): 1015–1106. arXiv:1201.4780. doi:10.1007/s11128-012-0432-5. S2CID 27676690.
• Salvador E. Venegas-Andraca (2008). Quantum Walks for Computer Scientists. ISBN 978-1598296563.
• Kia Manouchehri, Jingbo Wang (2014). Physical Implementation of Quantum Walks. ISBN 978-3-642-36014-5.
External links
• International Workshop on Mathematical and Physical Foundations of Discrete Time Quantum Walk
• Quantum walk
|
Wikipedia
|
Quantum revival
In quantum mechanics, the quantum revival [1] is a periodic recurrence of the quantum wave function from its original form during the time evolution either many times in space as the multiple scaled fractions in the form of the initial wave function (fractional revival) or approximately or exactly to its original form from the beginning (full revival). The quantum wave function periodic in time exhibits therefore the full revival every period. The phenomenon of revivals is most readily observable for the wave functions being well localized wave packets at the beginning of the time evolution for example in the hydrogen atom. For Hydrogen, the fractional revivals show up as multiple angular Gaussian bumps around the circle drawn by the radial maximum of leading circular state component (that with the highest amplitude in the eigenstate expansion) of the original localized state and the full revival as the original Gaussian .[2] The full revivals are exact for the infinite quantum well, harmonic oscillator or the hydrogen atom, while for shorter times are approximate for the hydrogen atom and a lot of quantum systems.[3]
The plot of collapses and revivals of quantum oscillations of the JCM atomic inversion.[4]
Example - arbitrary truncated wave function of the quantum system with rational energies
Consider a quantum system with the energies $E_{i}$ and the eigenstates $\psi _{i}$
$H\psi _{i}=E_{i}\psi _{i}$
and let the energies be the rational fractions of some constant $C$
$E_{i}=C{M_{i} \over N_{i}}$
(for example for hydrogen atom $M_{i}=1$, $N_{i}=i^{2}$, $C=-13.6eV$.
Then the truncated (till $\mathbb {N} _{max}$ of states) solution of the time dependent Schrödinger equation is
$\Psi (t)=\sum _{i=0}^{\mathbb {N} _{max}}a_{i}e^{-i{{E_{i}} \over \hbar }t}\psi _{i}$
.
Let $L_{cm}$ be to lowest common multiple of all $N_{i}$ and $L_{cd}$ greatest common divisor of all $M_{i}$ then for each $N_{i}$ the ${L_{cm}}/N_{i}$ is an integer, for each $M_{i}$ the ${M_{i}}/L_{cd}$ is an integer, $2\pi M_{i}{L_{cm}}/(N_{i}L_{cd})$ is the full multiple of $2\pi $ angle and
$\Psi (t)=\Psi (t+T)$
after the full revival time time
$T={2\pi \hbar \over {L_{cd}C}}L_{cm}$.
For the quantum system as small as Hydrogen and $\mathbb {N} _{max}$ as small as 100 it may take quadrillions of years till it will fully revive. Especially once created by fields the Trojan wave packet in a hydrogen atom exists without any external fields stroboscopically and eternally repeating itself after sweeping almost the whole hypercube of quantum phases exactly every full revival time.
The striking consequence is that no finite-bit computer can propagate the numerical wave function accurately for the arbitrarily long time. If the processor number is n-bit long floating point number then the number can be stored by the computer only with the finite accuracy after the comma and the energy is (up to 8 digits after the comma) for example 2.34576893 = 234576893/100000000 and as the finite fraction it is exactly rational and the full revival occurs for any wave function of any quantum system after the time $t/2\pi =100000000$ which is its maximum exponent and so on that may not be true for all quantum systems or all stationary quantum systems undergo the full and exact revival numerically.
In the system with the rational energies i.e. where the quantum exact full revival exists its existence immediately proves the quantum Poincaré recurrence theorem and the time of the full quantum revival equals to the Poincaré recurrence time. While the rational numbers are dense in real numbers and the arbitrary function of the quantum number can be approximated arbitrarily exactly with Padé approximants with the coefficients of arbitrary decimal precision for the arbitrarily long time each quantum system therefore revives almost exactly. It also means that the Poincaré recurrence and the full revival is mathematically the same thing [5] and it is commonly accepted that the recurrence is called the full revival if it occurs after the reasonable and physically measurable time that is possible to be detected by the realistic apparatus and this happens due to a very special energy spectrum having a large basic energy spacing gap of which the energies are arbitrary (not necessarily harmonic) multiples.
References
1. J.H. Eberly; N.B. Narozhny & J.J. Sanchez-Mondragon (1980). "Periodic spontaneous collapse and revival in a simple quantum model". Phys. Rev. Lett. 44 (20): 1323–1326. Bibcode:1980PhRvL..44.1323E. doi:10.1103/PhysRevLett.44.1323.
2. Z. Dacic Gaeta & C. R. Stroud, Jr. (1990). "Classical and quantum mechanical dynamics of quasiclassical state of a hydrogen atom". Phys. Rev. A. 42 (11): 6308–6313. Bibcode:1990PhRvA..42.6308G. doi:10.1103/PhysRevA.42.6308. PMID 9903927.
3. Zhang, Jiang-Min; Haque, Masudul (2014). "Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight binding model". Scienceopen Research. arXiv:1404.4280. doi:10.14293/S2199-1006.1.SOR-PHYS.A2CEM4.v1. S2CID 57487218.
4. A. A. Karatsuba; E. A. Karatsuba (2009). "A resummation formula for collapse and revival in the Jaynes–Cummings model". J. Phys. A: Math. Theor. 42 (19): 195304, 16. Bibcode:2009JPhA...42s5304K. doi:10.1088/1751-8113/42/19/195304. S2CID 120269208.
5. Bocchieri, P.; Loinger, A. (1957). "Quantum Recurrence Theorem". Phys. Rev. 107 (2): 337–338. Bibcode:1957PhRv..107..337B. doi:10.1103/PhysRev.107.337.
|
Wikipedia
|
Quantum signal processing
Quantum Signal Processing is a Hamiltonian simulation algorithm with optimal lower bounds in query complexity. It linearizes the operator of a quantum walk using eigenvalue transformation. The quantum walk takes a constant number of queries. So quantum signal processing's cost depends on the constant number of calls to the quantum walk operator, number of single qubit quantum gates that aid in the eigenvalue transformation and an ancilla qubit.[1]
Eigenvalue transformation
Given a unitary $W|u_{i}\rangle =e^{i\theta _{i}}|u_{i}\rangle $, calculate $A=f(W)=\sum _{i}e^{if(\theta _{i})}\left|u_{i}\right\rangle \langle u_{i}|$. For example, if $W={\begin{bmatrix}e^{i\theta _{1}}&0&0\\0&e^{i\theta _{2}}&0\\0&0&e^{i\theta _{3}}\end{bmatrix}}$, $A={\begin{bmatrix}e^{if(\theta _{1})}&0&0\\0&e^{if(\theta _{2})}&0\\0&0&e^{if(\theta _{3})}\end{bmatrix}}$. [1]
Algorithm
Input: Given a Hamiltonian $H$, define a quantum walk operator $W$ using 2 d-sparse oracles $O_{H}$ and $O_{F}$. $O_{H}$ accepts inputs $j$ and $k$ ($j$ is the row of the Hamiltonian and $k$ is the column) and outputs $\langle j|H\left|k\right\rangle $, so querying $O_{H}=|j\rangle |k\rangle |l\rangle =|j\rangle |k\rangle |l\oplus H_{j,k}\rangle $. $O_{F}$ accepts inputs $j$ and $l$ and computes the $l^{\text{th}}$ non-zero element in the $j^{\text{th}}$ row of $H$. [2]
Output: $e^{iHt}$
1. Create an input state $\theta $
2. Define a controlled-gate, $c-W$
3. Repeatedly apply single qubit gates to the ancilla followed applications of $c-W$ to the register that contains $\theta $ $O(td||H||_{\text{max}}+{\frac {\log {\frac {1}{\epsilon }}}{\log \log {\frac {1}{\epsilon }}}})$ times.
References
1. Low, Guang Hao; Chuang, Isaac (2017). "Optimal Hamiltonian Simulation by Quantum Signal Processing". Physical Review Letters. 118 (1): 010501. arXiv:1606.02685. Bibcode:2017PhRvL.118a0501L. doi:10.1103/PhysRevLett.118.010501. PMID 28106413. S2CID 1118993.
2. Guan Hao Low (January 17, 2017). Optimal Hamiltonian simulation by quantum signal processing (YouTube). Retrieved September 9, 2019.
|
Wikipedia
|
Quantum stochastic calculus
Quantum stochastic calculus is a generalization of stochastic calculus to noncommuting variables.[1] The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing measurement, as in quantum trajectories.[2]: 148 Just as the Lindblad master equation provides a quantum generalization to the Fokker–Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical Langevin equations.
For the remainder of this article stochastic calculus will be referred to as classical stochastic calculus, in order to clearly distinguish it from quantum stochastic calculus.
Heat baths
An important physical scenario in which a quantum stochastic calculus is needed is the case of a system interacting with a heat bath. It is appropriate in many circumstances to model the heat bath as an assembly of harmonic oscillators. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following Hamiltonian:[3]: 42, 45
$H=H_{\mathrm {sys} }(\mathbf {Z} )+{\frac {1}{2}}\sum _{n}\left((p_{n}-\kappa _{n}X)^{2}+\omega _{n}^{2}q_{n}^{2}\right)\,,$
where $H_{\mathrm {sys} }$ is the system Hamiltonian, $\mathbf {Z} $ is a vector containing the system variables corresponding to a finite number of degrees of freedom, $n$ is an index for the different bath modes, $\omega _{n}$ is the frequency of a particular mode, $p_{n}$ and $q_{n}$ are bath operators for a particular mode, $X$ is a system operator, and $\kappa _{n}$ quantifies the coupling between the system and a particular bath mode.
In this scenario the equation of motion for an arbitrary system operator $Y$ is called the quantum Langevin equation and may be written as:[3]: 46–47
${\dot {Y}}(t)={\frac {i}{\hbar }}[H_{\mathrm {sys} },Y(t)]-{\frac {i}{2\hbar }}\left[X,\left\{Y(t),\xi (t)-\int _{t_{0}}^{t}f(t-t_{0}){\dot {X}}(t^{\prime })\mathrm {d} t^{\prime }-f(t-t_{0})X(t_{0})\right\}\right]\,,$
where $[\cdot ,\cdot ]$ and $\{\cdot ,\cdot \}$ denote the commutator and anticommutator (respectively), the memory function $f$ is defined as:
$f(t)\equiv \sum _{n}\kappa _{n}^{2}\cos(\omega _{n}t)\,,$
and the time dependent noise operator $\xi $ is defined as:
$\xi (t)\equiv i\sum _{n}\kappa _{n}{\sqrt {\frac {\hbar \omega _{n}}{2}}}\left(-a_{n}(t_{0})e^{-i\omega _{n}(t-t_{0})}+a_{n}^{\dagger }(t_{0})e^{i\omega _{n}(t-t_{0})}\right)\,,$
where the bath annihilation operator $a_{n}$ is defined as:
$a_{n}\equiv {\frac {\omega _{n}q_{n}+ip_{n}}{\sqrt {2\hbar \omega _{n}}}}\,.$
Oftentimes this equation is more general than is needed, and further approximations are made to simplify the equation.
White noise formalism
For many purposes it is convenient to make approximations about the nature of the heat bath in order to achieve a white noise formalism. In such a case the interaction may be modeled by the Hamiltonian $H=H_{\mathrm {sys} }+H_{B}+H_{\mathrm {int} }$ where:[4]: 3762
$H_{B}=\hbar \int _{-\infty }^{\infty }\mathrm {d} \omega \,\omega b^{\dagger }(\omega )b(\omega )\,,$
and
$H_{\mathrm {int} }=i\hbar \int _{-\infty }^{\infty }\mathrm {d} \omega \,\kappa (\omega )\left(b^{\dagger }(\omega )c-c^{\dagger }b(\omega )\right)\,,$
where $b(\omega )$ are annihilation operators for the bath with the commutation relation $[b(\omega ),b^{\dagger }(\omega ^{\prime })]=\delta (\omega -\omega ^{\prime })$, $c$ is an operator on the system, $\kappa (\omega )$ quantifies the strength of the coupling of the bath modes to the system, and $H_{\mathrm {sys} }$ describes the free system evolution.[3]: 148 This model uses the rotating wave approximation and extends the lower limit of $\omega $ to $-\infty $ in order to admit a mathematically simple white noise formalism. The coupling strengths are also usually simplified to a constant in what is sometimes called the first Markov approximation:[4]: 3763
$\kappa (\omega )={\sqrt {\frac {\gamma }{2\pi }}}\,.$
Systems coupled to a bath of harmonic oscillators can be thought of as being driven by a noise input and radiating a noise output.[3]: 43 The input noise operator at time $t$ is defined by:[3]: 150 [4]: 3763
$b_{\mathrm {in} }(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\mathrm {d} \omega \,e^{-i\omega (t-t_{0})}b_{0}(\omega )\,,$
where $b_{0}(\omega )=\left.b(\omega )\right\vert _{t=t_{0}}$, since this operator is expressed in the Heisenberg picture. Satisfaction of the commutation relation $[b_{\mathrm {in} }(t),b_{\mathrm {in} }^{\dagger }(t^{\prime })]=\delta (t-t^{\prime })$ allows the model to have a strict correspondence with a Markovian master equation.[2]: 142
In the white noise setting described so far, the quantum Langevin equation for an arbitrary system operator $a$ takes a simpler form:[4]: 3763
${\dot {a}}=-{\frac {i}{\hbar }}[a,H_{\mathrm {sys} }]-[a,c^{\dagger }]\left({\frac {\gamma }{2}}c+{\sqrt {\gamma }}b_{\mathrm {in} }(t)\right)+\left({\frac {\gamma }{2}}c^{\dagger }+{\sqrt {\gamma }}b_{\mathrm {in} }^{\dagger }(t)\right)[a,c]\,.$ (WN1)
For the case most closely corresponding to classical white noise, the input to the system is described by a density operator giving the following expectation value:[3]: 154
$\langle b_{\mathrm {in} }^{\dagger }(t)b_{\mathrm {in} }(t^{\prime })\rangle _{\rho _{\mathrm {in} }}=N\delta (t-t^{\prime })\,.$
(WN2)
Quantum Wiener process
In order to define quantum stochastic integration, it is important to define a quantum Wiener process:[3]: 155 [4]: 3765
$B(t,t_{0})=\int _{t_{0}}^{t}b_{\mathrm {in} }(t^{\prime })\mathrm {d} t^{\prime }\,.$
This definition gives the quantum Wiener process the commutation relation $[B(t,t_{0}),B^{\dagger }(t,t_{0})]=t-t_{0}$. The property of the bath annihilation operators in (WN2) implies that the quantum Wiener process has an expectation value of:
$\langle B^{\dagger }(t,t_{0})B(t,t_{0})\rangle _{\rho (t,t_{0})}=N(t-t_{0})\,.$
The quantum Wiener processes are also specified such that their quasiprobability distributions are Gaussian by defining the density operator:
$\rho (t,t_{0})=(1-e^{-\kappa })\exp \left[-{\frac {\kappa B^{\dagger }(t,t_{0})B(t,t_{0})}{t-t_{0}}}\right]\,,$
where $N=1/(e^{\kappa }-1)$.[4]: 3765
Quantum stochastic integration
The stochastic evolution of system operators can also be defined in terms of the stochastic integration of given equations.
Quantum Itô integral
The quantum Itô integral of a system operator $g(t)$ is given by:[3]: 155
$(\mathbf {I} )\int _{t_{0}}^{t}g(t^{\prime })\mathrm {d} B(t^{\prime })=\lim _{n\to \infty }\sum _{i=1}^{n}g(t_{i})\left(B(t_{i+1},t_{0})-B(t_{i},t_{0})\right)\,,$
where the bold (I) preceding the integral stands for Itô. One of the characteristics of defining the integral in this way is that the increments $\mathrm {d} B$ and $\mathrm {d} B^{\dagger }$ commute with the system operator.
Itô quantum stochastic differential equation
In order to define the Itô QSDE, it is necessary to know something about the bath statistics.[3]: 159 In the context of the white noise formalism described earlier, the Itô QSDE can be defined as:[3]: 156
$(\mathbf {I} )\,\mathrm {d} a=-{\frac {i}{\hbar }}[a,H_{\mathrm {sys} }]\mathrm {d} t+\gamma \left((N+1){\mathcal {D}}[c^{\dagger }]a+N{\mathcal {D}}[c]a\right)\mathrm {d} t-{\sqrt {\gamma }}\left([a,c^{\dagger }]\mathrm {d} B(t)-\mathrm {d} B^{\dagger }(t)[a,c]\right)\,,$
where the equation has been simplified using the Lindblad superoperator:[2]: 105
${\mathcal {D}}[A]a\equiv AaA^{\dagger }-{\frac {1}{2}}\left(A^{\dagger }Aa+aA^{\dagger }A\right)\,.$
This differential equation is interpreted as defining the system operator $a$ as the quantum Itô integral of the right hand side, and is equivalent to the Langevin equation (WN1).[4]: 3765
Quantum Stratonovich integral
The quantum Stratonovich integral of a system operator $g(t)$ is given by:[3]: 157
$(\mathbf {S} )\int _{t_{0}}^{t}g(t^{\prime })\mathrm {d} B(t^{\prime })=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {g(t_{i})+g(t_{i+1})}{2}}\left(B(t_{i+1},t_{0})-B(t_{i},t_{0})\right)\,,$
where the bold (S) preceding the integral stands for Stratonovich. Unlike the Itô formulation, the increments in the Stratonovich integral do not commute with the system operator, and it can be shown that:[3]
$(\mathbf {S} )\int _{t_{0}}^{t}g(t^{\prime })\mathrm {d} B(t^{\prime })-(\mathbf {S} )\int _{t_{0}}^{t}\mathrm {d} B(t^{\prime })g(t^{\prime })={\frac {\sqrt {\gamma }}{2}}\int _{t_{0}}^{t}\mathrm {d} t^{\prime }\,[g(t^{\prime }),c(t^{\prime })]\,.$
Stratonovich quantum stochastic differential equation
The Stratonovich QSDE can be defined as:[3]: 158
$(\mathbf {S} )\,\mathrm {d} a=-{\frac {i}{\hbar }}[a,H_{\mathrm {sys} }]\mathrm {d} t-{\frac {\gamma }{2}}\left([a,c^{\dagger }]c-c^{\dagger }[a,c]\right)\mathrm {d} t-{\sqrt {\gamma }}\left([a,c^{\dagger }]\mathrm {d} B(t)-\mathrm {d} B^{\dagger }(t)[a,c]\right)\,.$
This differential equation is interpreted as defining the system operator $a$ as the quantum Stratonovich integral of the right hand side, and is in the same form as the Langevin equation (WN1).[4]: 3766–3767
Relation between Itô and Stratonovich integrals
The two definitions of quantum stochastic integrals relate to one another in the following way, assuming a bath with $N$ defined as before:[3]
$(\mathbf {S} )\int _{t_{0}}^{t}g(t^{\prime })\mathrm {d} B(t^{\prime })=(\mathbf {I} )\int _{t_{0}}^{t}g(t^{\prime })\mathrm {d} B(t^{\prime })+{\frac {1}{2}}{\sqrt {\gamma }}N\int _{t_{0}}^{t}\mathrm {d} t^{\prime }\,[g(t^{\prime }),c(t^{\prime })]\,.$
Calculus rules
Just as with classical stochastic calculus, the appropriate product rule can be derived for Itô and Stratonovich integration, respectively:[3]: 156, 159
$(\mathbf {I} )\,\mathrm {d} (ab)=a\,\mathrm {d} b+b\,\mathrm {d} a+\mathrm {d} a\,\mathrm {d} b\,,$
$(\mathbf {S} )\,\mathrm {d} (ab)=a\,\mathrm {d} b+\mathrm {d} a\,b\,.$
As is the case in classical stochastic calculus, the Stratonovich form is the one which preserves the ordinary calculus (which in this case is noncommuting). A peculiarity in the quantum generalization is the necessity to define both Itô and Stratonovitch integration in order to prove that the Stratonovitch form preserves the rules of noncommuting calculus.[3]: 155
Quantum trajectories
Quantum trajectories can generally be thought of as the path through Hilbert space that the state of a quantum system traverses over time. In a stochastic setting, these trajectories are often conditioned upon measurement results. The unconditioned Markovian evolution of a quantum system (averaged over all possible measurement outcomes) is given by a Lindblad equation. In order to describe the conditioned evolution in these cases, it is necessary to unravel the Lindblad equation by choosing a consistent QSDE. In the case where the conditioned system state is always pure, the unraveling could be in the form of a stochastic Schrödinger equation (SSE). If the state may become mixed, then it is necessary to use a stochastic master equation (SME).[2]: 148
Example unravelings
Consider the following Lindblad master equation for a system interacting with a vacuum bath:[2]: 145
${\dot {\rho }}={\mathcal {D}}[c]\rho -i[H_{\mathrm {sys} },\rho ]\,.$
This describes the evolution of the system state averaged over the outcomes of any particular measurement that might be made on the bath. The following SME describes the evolution of the system conditioned on the results of a continuous photon-counting measurement performed on the bath:
$\mathrm {d} \rho _{I}(t)=\left(\mathrm {d} N(t){\mathcal {G}}[c]-\mathrm {d} t{\mathcal {H}}[iH_{\mathrm {sys} }+{\frac {1}{2}}c^{\dagger }c]\right)\rho _{I}(t)\,,$
where
${\begin{array}{rcl}{\mathcal {G}}[r]\rho &\equiv &{\frac {r\rho r^{\dagger }}{\operatorname {Tr} [r\rho r^{\dagger }]}}-\rho \\{\mathcal {H}}[r]\rho &\equiv &r\rho +\rho r^{\dagger }-\operatorname {Tr} [r\rho +\rho r^{\dagger }]\rho \end{array}}$
are nonlinear superoperators and $N(t)$ is the photocount, indicating how many photons have been detected at time $t$ and giving the following jump probability:[2]: 152, 155
$\operatorname {E} [\mathrm {d} N(t)]=\mathrm {d} t\operatorname {Tr} [c^{\dagger }c\rho _{I}(t)]\,,$
where $\operatorname {E} [\cdot ]$ denotes the expected value. Another type of measurement that could be made on the bath is homodyne detection, which results in quantum trajectories given by the following SME:
$\mathrm {d} \rho _{J}(t)=-i[H_{\mathrm {sys} },\rho _{J}(t)]\mathrm {d} t+\mathrm {d} t{\mathcal {D}}[c]\rho _{J}(t)+\mathrm {d} W(t){\mathcal {H}}[c]\rho _{J}(t)\,,$
where $\mathrm {d} W(t)$ is a Wiener increment satisfying:[2]: 161
${\begin{array}{rcl}\mathrm {d} W(t)^{2}&=&\mathrm {d} t\\\operatorname {E} [\mathrm {d} W(t)]&=&0\,.\end{array}}$
Although these two SMEs look wildly different, calculating their expected evolution shows that they are both indeed unravelings of the same Lindlad master equation:
$\operatorname {E} [\mathrm {d} \rho _{I}(t)]=\operatorname {E} [\mathrm {d} \rho _{J}(t)]={\dot {\rho }}\mathrm {d} t\,.$
Computational considerations
One important application of quantum trajectories is reducing the computational resources required to simulate a master equation. For a Hilbert space of dimension d, the amount of real numbers required to store the density matrix is of order d2, and the time required to compute the master equation evolution is of order d4. Storing the state vector for a SSE, on the other hand, only requires an amount of real numbers of order d, and the time to compute trajectory evolution is only of order d2. The master equation evolution can then be approximated by averaging over many individual trajectories simulated using the SSE, a technique sometimes referred to as the Monte Carlo wave-function approach.[5] Although the number of calculated trajectories n must be very large in order to accurately approximate the master equation, good results can be obtained for trajectory counts much less than d2. Not only does this technique yield faster computation time, but it also allows for the simulation of master equations on machines that do not have enough memory to store the entire density matrix.[2]: 153
References
1. Hudson, R. L.; Parthasarathy, K. R. (1984-09-01). "Quantum Ito's Formula and Stochastic Evolutions". Communications in Mathematical Physics. 93 (3): 301–323. Bibcode:1984CMaPh..93..301H. doi:10.1007/BF01258530. S2CID 122848524.
2. Wiseman, Howard M.; Milburn, Gerard J. (2010). Quantum Measurement and Control. New York: Cambridge University Press. ISBN 978-0-521-80442-4.
3. Gardiner, C. W.; Zoller, P. (2010). Quantum Noise. Springer Series in Synergetics (3rd ed.). Berlin Heidelberg: Springer-Verlag. ISBN 978-3-642-06094-6.
4. Gardiner, C. W.; Collett, M. J. (June 1985). "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation". Physical Review A. 31 (6): 3761–3774. Bibcode:1985PhRvA..31.3761G. doi:10.1103/PhysRevA.31.3761. PMID 9895956.
5. Dalibard, Jean; Castin, Yvan; Mølmer, Klaus (Feb 1992). "Wave-function approach to dissipative processes in quantum optics". Phys. Rev. Lett. American Physical Society. 68 (5): 580–583. arXiv:0805.4002. Bibcode:1992PhRvL..68..580D. doi:10.1103/PhysRevLett.68.580. PMID 10045937.
|
Wikipedia
|
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics.[1]: xi QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.
Quantum field theory
Feynman diagram
History
Background
• Field theory
• Electromagnetism
• Weak force
• Strong force
• Quantum mechanics
• Special relativity
• General relativity
• Gauge theory
• Yang–Mills theory
Symmetries
• Symmetry in quantum mechanics
• C-symmetry
• P-symmetry
• T-symmetry
• Lorentz symmetry
• Poincaré symmetry
• Gauge symmetry
• Explicit symmetry breaking
• Spontaneous symmetry breaking
• Noether charge
• Topological charge
Tools
• Anomaly
• Background field method
• BRST quantization
• Correlation function
• Crossing
• Effective action
• Effective field theory
• Expectation value
• Feynman diagram
• Lattice field theory
• LSZ reduction formula
• Partition function
• Propagator
• Quantization
• Regularization
• Renormalization
• Vacuum state
• Wick's theorem
Equations
• Dirac equation
• Klein–Gordon equation
• Proca equations
• Wheeler–DeWitt equation
• Bargmann–Wigner equations
Standard Model
• Quantum electrodynamics
• Electroweak interaction
• Quantum chromodynamics
• Higgs mechanism
Incomplete theories
• String theory
• Supersymmetry
• Technicolor
• Theory of everything
• Quantum gravity
Scientists
• Anderson
• Anselm
• Bargmann
• Becchi
• Belavin
• Berezin
• Bethe
• Bjorken
• Bleuer
• Bogoliubov
• Brodsky
• Brout
• Buchholz
• Cachazo
• Callan
• Coleman
• Dashen
• DeWitt
• Dirac
• Doplicher
• Dyson
• Englert
• Faddeev
• Fadin
• Fermi
• Feynman
• Fierz
• Fock
• Frampton
• Fritzsch
• Fröhlich
• Fredenhagen
• Furry
• Glashow
• Gelfand
• Gell-Mann
• Goldstone
• Gribov
• Gross
• Gupta
• Guralnik
• Haag
• Heisenberg
• Hepp
• Higgs
• Hagen
• 't Hooft
• Ivanenko
• Jackiw
• Jona-Lasinio
• Jordan
• Jost
• Källén
• Kendall
• Kinoshita
• Klebanov
• Kontsevich
• Kuraev
• Landau
• Lee
• Lehmann
• Leutwyler
• Lipatov
• Łopuszański
• Low
• Lüders
• Maiani
• Majorana
• Maldacena
• Migdal
• Mills
• Møller
• Naimark
• Nambu
• Neveu
• Nishijima
• Oehme
• Oppenheimer
• Osterwalder
• Parisi
• Pauli
• Peskin
• Polyakov
• Pomeranchuk
• Popov
• Proca
• Rubakov
• Ruelle
• Salam
• Schrader
• Schwarz
• Schwinger
• Segal
• Seiberg
• Semenoff
• Shifman
• Shirkov
• Skyrme
• Stora
• Stueckelberg
• Sudarshan
• Symanzik
• Thirring
• Tomonaga
• Tyutin
• Vainshtein
• Veltman
• Virasoro
• Ward
• Weinberg
• Weisskopf
• Wentzel
• Wess
• Wetterich
• Weyl
• Wick
• Wightman
• Wigner
• Wilczek
• Wilson
• Witten
• Yang
• Yukawa
• Zamolodchikov
• Zamolodchikov
• Zee
• Zimmermann
• Zinn-Justin
• Zuber
• Zumino
QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory in quantum mechanics.
History
Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory.
Theoretical background
Quantum field theory results from the combination of classical field theory, quantum mechanics, and special relativity.[1]: xi A brief overview of these theoretical precursors follows.
The earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica. The force of gravity as described by Newton is an "action at a distance"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley, however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact."[2]: 4 It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a vector in the case of gravitational field) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick.[3]: 18
Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.[2][4]: 301 [5]: 2
The theory of classical electromagnetism was completed in 1864 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, and electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light. Action-at-a-distance was thus conclusively refuted.[2]: 19
Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different wavelengths.[6] Max Planck's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators. This process of restricting energies to discrete values is called quantization.[7]: Ch.2 Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called photons (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.[6]
In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave–particle duality, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances.[6] Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contributions from Max Planck, Louis de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, and Wolfgang Pauli.[3]: 22–23
In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity, built on Maxwell's electromagnetism. New rules, called Lorentz transformations, were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred.[3]: 19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying quantum mechanics could explain the stimulated emission of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain spontaneous emission, where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to linear operators.[6]
Quantum electrodynamics
Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.[8]: 1
Through the works of Born, Heisenberg, and Pascual Jordan in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via canonical quantization by treating the electromagnetic field as a set of quantum harmonic oscillators.[8]: 1 With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world.[3]: 22
In his seminal 1927 paper The quantum theory of the emission and absorption of radiation, Dirac coined the term quantum electrodynamics (QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric current density and the electromagnetic vector potential. Using first-order perturbation theory, he successfully explained the phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the ground state). Therefore, even in a perfect vacuum, there remains an oscillating electromagnetic field having zero-point energy. It is this quantum fluctuation of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the scattering of photons, resonance fluorescence and non-relativistic Compton scattering. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations.[6]: 71
In 1928, Dirac wrote down a wave equation that described relativistic electrons—the Dirac equation. It had the following important consequences: the spin of an electron is 1/2; the electron g-factor is 2; it led to the correct Sommerfeld formula for the fine structure of the hydrogen atom; and it could be used to derive the Klein–Nishina formula for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.[6]: 71–72
The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and quantum fields (such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the excited states of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, Eugene Wigner, Heisenberg, Pauli, and Enrico Fermi discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for beta decay known as Fermi's interaction. Atomic nuclei do not contain electrons per se, but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom.[3]: 22–23
It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of antimatter. Indeed, the evidence for positrons was discovered in 1932 by Carl David Anderson in cosmic rays. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called pair production; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the Dirac hole theory.[6]: 72 [3]: 23 QFT naturally incorporated antiparticles in its formalism.[3]: 24
Infinities and renormalization
Robert Oppenheimer showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron self-energy and the vacuum zero-point energy of the electron and photon fields,[6] suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.[3]: 25 It was not until 20 years later that a systematic approach to remove such infinities was developed.
A series of papers was published between 1934 and 1938 by Ernst Stueckelberg that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.[6]
Faced with these infinities, John Archibald Wheeler and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called S-matrix theory. Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of observables (e.g. the energy of an atom) in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945, Richard Feynman and Wheeler daringly suggested abandoning QFT altogether and proposed action-at-a-distance as the mechanism of particle interactions.[3]: 26
In 1947, Willis Lamb and Robert Retherford measured the minute difference in the 2S1/2 and 2P1/2 energy levels of the hydrogen atom, also called the Lamb shift. By ignoring the contribution of photons whose energy exceeds the electron mass, Hans Bethe successfully estimated the numerical value of the Lamb shift.[6][3]: 28 Subsequently, Norman Myles Kroll, Lamb, James Bruce French, and Victor Weisskopf again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations.[6]
The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga. The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as renormalization and can be applied to arbitrary order in perturbation theory.[6] As Tomonaga said in his Nobel lecture:
Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with [the] Americans'.[9]
By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment (the deviation of the electron g-factor from 2) and vacuum polarization. These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities".[6]
At the same time, Feynman introduced the path integral formulation of quantum mechanics and Feynman diagrams.[8]: 2 The latter can be used to visually and intuitively organize and to help compute terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the scattering amplitude of the interaction represented by the diagram.[1]: 5
It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.[8]: 2
Non-renormalizability
Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades.[3]: 30
The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the Fermi theory of the weak interaction, are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities.[3]: 30
The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the coupling constant, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the fine-structure constant α ≈ 1/137, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the strong interaction is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods.[3]: 31
With these difficulties looming, many theorists began to turn away from QFT. Some focused on symmetry principles and conservation laws, while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations.[3]: 31
Source theory
Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory,[10] but in 1951[11][12] he found a way around the problem of the infinities with a new method using external sources as currents coupled to gauge fields.[13] Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the classical process of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his source theory in 1966[14] then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields.[15][16][17] Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.[15]
In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general.[18] Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities.[19]
Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury.[20] The neglect of source theory by the physics community was a major disappointment for Schwinger:
The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger[15]
See "the shoes incident" between J. Schwinger and S. Weinberg.[21]
Standard-Model
In 1954, Yang Chen-Ning and Robert Mills generalized the local symmetry of QED, leading to non-Abelian gauge theories (also known as Yang–Mills theories), which are based on more complicated local symmetry groups.[22]: 5 In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of "charge" interact via the exchange of massless gauge bosons. Unlike photons, these gauge bosons themselves carry charge.[3]: 32 [23]
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, Abdus Salam and John Clive Ward arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.[24]
Peter Higgs, Robert Brout, François Englert, Gerald Guralnik, Carl Hagen, and Tom Kibble proposed in their famous Physical Review Letters papers that the gauge symmetry in Yang–Mills theories could be broken by a mechanism called spontaneous symmetry breaking, through which originally massless gauge bosons could acquire mass.[22]: 5–6
By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, Steven Weinberg wrote down in 1967 a theory describing electroweak interactions between all leptons and the effects of the Higgs boson. His theory was at first mostly ignored,[24][22]: 6 until it was brought back to light in 1971 by Gerard 't Hooft's proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to quarks in 1970 by Glashow, John Iliopoulos, and Luciano Maiani, marking its completion.[24]
Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler discovered in 1971 that certain phenomena involving the strong interaction could also be explained by non-Abelian gauge theory. Quantum chromodynamics (QCD) was born. In 1973, David Gross, Frank Wilczek, and Hugh David Politzer showed that non-Abelian gauge theories are "asymptotically free", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) [22]: 11 Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible.[3]: 32
These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the Standard Model of elementary particles.[25] The Standard Model successfully describes all fundamental interactions except gravity, and its many predictions have been met with remarkable experimental confirmation in subsequent decades.[8]: 3 The Higgs boson, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at CERN, marking the complete verification of the existence of all constituents of the Standard Model.[26]
Other developments
The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The 't Hooft–Polyakov monopole was discovered theoretically by 't Hooft and Alexander Polyakov, flux tubes by Holger Bech Nielsen and Poul Olesen, and instantons by Polyakov and coauthors. These objects are inaccessible through perturbation theory.[8]: 4
Supersymmetry also appeared in the same period. The first supersymmetric QFT in four dimensions was built by Yuri Golfand and Evgeny Likhtman in 1970, but their result failed to garner widespread interest due to the Iron Curtain. Supersymmetry only took off in the theoretical community after the work of Julius Wess and Bruno Zumino in 1973.[8]: 7
Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of quantum gravity led to the development of string theory,[8]: 6 itself a type of two-dimensional QFT with conformal symmetry.[27] Joël Scherk and John Schwarz first proposed in 1974 that string theory could be the quantum theory of gravity.[28]
Condensed-matter-physics
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to many-body systems in condensed matter physics.
Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of Yoichiro Nambu's application of superconductor theory to elementary particles, while the concept of renormalization came out of the study of second-order phase transitions in matter.[29]
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle—phonons. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems.[30]
Gauge theory is used to describe the quantization of magnetic flux in superconductors, the resistivity in the quantum Hall effect, as well as the relation between frequency and voltage in the AC Josephson effect.[30]
Principles
For simplicity, natural units are used in the following sections, in which the reduced Planck constant ħ and the speed of light c are both set to one.
Classical fields
A classical field is a function of spatial and time coordinates.[31] Examples include the gravitational field in Newtonian gravity g(x, t) and the electric field E(x, t) and magnetic field B(x, t) in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom.[31][32]
Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.
Canonical quantization and path integrals are two common formulations of QFT.[33]: 61 To motivate the fundamentals of QFT, an overview of classical field theory follows.
The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as ϕ(x, t), where x is the position vector, and t is the time. Suppose the Lagrangian of the field, $L$, is
$L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],$
where ${\mathcal {L}}$ is the Lagrangian density, ${\dot {\phi }}$ is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian:[1]: 16
${\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,$
we obtain the equations of motion for the field, which describe the way it varies in time and space:
$\left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.$
This is known as the Klein–Gordon equation.[1]: 17
The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows:
$\phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),$
where a is a complex number (normalized by convention), * denotes complex conjugation, and ωp is the frequency of the normal mode:
$\omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.$
Thus each normal mode corresponding to a single p can be seen as a classical harmonic oscillator with frequency ωp.[1]: 21,26
Canonical quantization
The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.
The displacement of a classical harmonic oscillator is described by
$x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},$
where a is a complex number (normalized by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label x of a quantum field.
For a quantum harmonic oscillator, x(t) is promoted to a linear operator ${\hat {x}}(t)$:
${\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.$
Complex numbers a and a* are replaced by the annihilation operator ${\hat {a}}$ and the creation operator ${\hat {a}}^{\dagger }$, respectively, where † denotes Hermitian conjugation. The commutation relation between the two is
$\left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.$
The Hamiltonian of the simple harmonic oscillator can be written as
${\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .$
The vacuum state $|0\rangle $, which is the lowest energy state, is defined by
${\hat {a}}|0\rangle =0$
and has energy ${\frac {1}{2}}\hbar \omega $ One can easily check that $[{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega ,$ which implies that ${\hat {a}}^{\dagger }$ increases the energy of the simple harmonic oscillator by $\hbar \omega $. For example, the state ${\hat {a}}^{\dagger }|0\rangle $ is an eigenstate of energy $3\hbar \omega /2$. Any energy eigenstate state of a single harmonic oscillator can be obtained from $|0\rangle $ by successively applying the creation operator ${\hat {a}}^{\dagger }$:[1]: 20 and any state of the system can be expressed as a linear combination of the states
$|n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .$
A similar procedure can be applied to the real scalar field ϕ, by promoting it to a quantum field operator ${\hat {\phi }}$, while the annihilation operator ${\hat {a}}_{\mathbf {p} }$, the creation operator ${\hat {a}}_{\mathbf {p} }^{\dagger }$ and the angular frequency $w_{\mathbf {p} }$are now for a particular p:
${\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).$
Their commutation relations are:[1]: 21
$\left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }\right]=\left[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,$
where δ is the Dirac delta function. The vacuum state $|0\rangle $ is defined by
${\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .$
Any quantum state of the field can be obtained from $|0\rangle $ by successively applying creation operators ${\hat {a}}_{\mathbf {p} }^{\dagger }$ (or by a linear combination of such states), e.g. [1]: 22
$\left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .$
While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems.[34] The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization.[1]: 19
The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields,[1]: 52 vector fields (e.g. the electromagnetic field), and even strings.[35] However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary.
The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field:[1]: 77
${\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},$
where μ is a spacetime index, $\partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}$, etc. The summation over the index μ has been omitted following the Einstein notation. If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.
Path integrals
The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state $|\phi _{I}\rangle $ at time t = 0 to some final state $|\phi _{F}\rangle $ at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian (i.e. generator of time evolution), then[33]: 10
$\langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .$
Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral:[1]: 282 [33]: 12
$\langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},$
where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H via Legendre transformation. The initial and final conditions of the path integral are respectively
$\phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.$
In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.
Two-point correlation function
In calculations, one often encounters expression like
$\langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle $
in the free or interacting theory, respectively. Here, $x$ and $y$ are position four-vectors, $T$ is the time ordering operator that shuffles its operands so the time-components $x^{0}$ and $y^{0}$ increase from right to left, and $|\Omega \rangle $ is the ground state (vacuum state) of the interacting theory, different from the free ground state $|0\rangle $. This expression represents the probability amplitude for the field to propagate from y to x, and goes by multiple names, like the two-point propagator, two-point correlation function, two-point Green's function or two-point function for short.[1]: 82
The free two-point function, also known as the Feynman propagator, can be found for the real scalar field by either canonical quantization or path integrals to be[1]: 31,288 [33]: 23
$\langle 0|T\{\phi (x)\phi (y)\}|0\rangle \equiv D_{F}(x-y)=\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.$
In an interacting theory, where the Lagrangian or Hamiltonian contains terms $L_{I}(t)$ or $H_{I}(t)$ that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function.
In canonical quantization, the two-point correlation function can be written as:[1]: 87
$\langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\left\langle 0\left|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }{\left\langle 0\left|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }},$
where ε is an infinitesimal number and ϕI is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in $\phi ^{4}$-theory, the interacting term of the Hamiltonian is $ H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}$,[1]: 84 and the expansion of the two-point correlator in terms of $\lambda $ becomes
$\langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }\sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.$
This perturbation expansion expresses the interacting two-point function in terms of quantities $\langle 0|\cdots |0\rangle $ that are evaluated in the free theory.
In the path integral formulation, the two-point correlation function can be written[1]: 284
$\langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}{\int {\mathcal {D}}\phi \,\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},$
where ${\mathcal {L}}$ is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in λ, reducing the interacting two-point function to quantities in the free theory.
Wick's theorem further reduce any n-point correlation function in the free theory to a sum of products of two-point correlation functions. For example,
${\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle &=\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}$
Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.[1]: 90 This makes the Feynman propagator one of the most important quantities in quantum field theory.
Feynman diagram
Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the λ1 term in the two-point correlation function in the ϕ4 theory is
${\frac {-i\lambda }{4!}}\int d^{4}z\,\langle 0|T\{\phi (x)\phi (y)\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .$
After applying Wick's theorem, one of the terms is
$12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z).$
This term can instead be obtained from the Feynman diagram
.
The diagram consists of
• external vertices connected with one edge and represented by dots (here labeled $x$ and $y$).
• internal vertices connected with four edges and represented by dots (here labeled $z$).
• edges connecting the vertices and represented by lines.
Every vertex corresponds to a single $\phi $ field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:
1. For every internal vertex $z_{i}$, write down a factor $ -i\lambda \int d^{4}z_{i}$.
2. For every edge that connects two vertices $z_{i}$ and $z_{j}$, write down a factor $D_{F}(z_{i}-z_{j})$.
3. Divide by the symmetry factor of the diagram.
With the symmetry factor $2$, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.[1]: 91–94
In order to compute the n-point correlation function to the k-th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,
$\langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle $
is equal to the sum of (expressions corresponding to) all connected diagrams with n external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the ϕ4 interaction theory discussed above, every vertex must have four legs.[1]: 98
In realistic applications, the scattering amplitude of a certain interaction or the decay rate of a particle can be computed from the S-matrix, which itself can be found using the Feynman diagram method.[1]: 102–115
Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.[33]: 44 Lines whose end points are vertices can be thought of as the propagation of virtual particles.[1]: 31
Renormalization
Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.
Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ, have no physical meaning — m, λ, and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ, obtain expressions for the physical quantities, and then take the limit Λ → ∞. This is an example of regularization, a class of methods to treat divergences in QFT, with Λ being the regulator.
The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ4 theory, the field strength is first redefined:
$\phi =Z^{1/2}\phi _{r},$
where ϕ is the bare field, ϕr is the renormalized field, and Z is a constant to be determined. The Lagrangian density becomes:
${\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},$
where mr and λr are the experimentally measurable, renormalized, mass and coupling constant, respectively, and
$\delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}$
are constants to be determined. The first three terms are the ϕ4 Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or dimensional regularization); call the regulator Λ. Compute Feynman diagrams, in which divergent terms will depend on Λ. Then, define δZ, δm, and δλ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained.[1]: 323–326
It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalizable QFT,[1]: 719–727 while quantum gravity is non-renormalizable.[1]: 798 [33]: 421
Renormalization group
The renormalization group, developed by Kenneth Wilson, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.[1]: 393 The way in which each parameter changes with scale is described by its β function.[1]: 417 Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation.[1]: 410–411
As an example, the coupling constant in QED, namely the elementary charge e, has the following β function:
$\beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O{\mathord {\left(e^{5}\right)}},$
where Λ is the energy scale under which the measurement of e is performed. This differential equation implies that the observed elementary charge increases as the scale increases.[36] The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.[1]: 420
The coupling constant g in quantum chromodynamics, a non-Abelian gauge theory based on the symmetry group SU(3), has the following β function:
$\beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O{\mathord {\left(g^{5}\right)}},$
where Nf is the number of quark flavours. In the case where Nf ≤ 16 (the Standard Model has Nf = 6), the coupling constant g decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom.[1]: 531
Conformal field theories (CFTs) are special QFTs that admit conformal symmetry. They are insensitive to changes in the scale, as all their coupling constants have vanishing β function. (The converse is not true, however — the vanishing of all β functions does not imply conformal symmetry of the theory.)[37] Examples include string theory[27] and N = 4 supersymmetric Yang–Mills theory.[38]
According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off Λ, i.e. that the theory is no longer valid at energies higher than Λ, and all degrees of freedom above the scale Λ are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable effective field theory.[1]: 402–403 The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.[8]: 2 According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off Λ from calculations in such a theory merely indicates that new physical phenomena appear at scales above Λ, where a new theory is necessary.[33]: 156
Other theories
The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and ϕ4 theory of the real scalar field. A similar process can be done for other types of fields, including the complex scalar field, the vector field, and the Dirac field, as well as other types of interaction terms, including the electromagnetic interaction and the Yukawa interaction.
As an example, quantum electrodynamics contains a Dirac field ψ representing the electron field and a vector field Aμ representing the electromagnetic field (photon field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:
${\mathcal {L}}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },$
where γμ are Dirac matrices, ${\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}$, and $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ is the electromagnetic field strength. The parameters in this theory are the (bare) electron mass m and the (bare) elementary charge e. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.[1]: 78
Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of positrons, while those pointing backward in time represent the propagation of electrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.
Gauge symmetry
If the following transformation to the fields is performed at every spacetime point x (a local transformation), then the QED Lagrangian remains unchanged, or invariant:
$\psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},$
where α(x) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory.[1]: 482–483 Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations $e^{i\alpha (x)}$ and $e^{i\alpha '(x)}$ is yet another symmetry transformation $e^{i[\alpha (x)+\alpha '(x)]}$. For any α(x), $e^{i\alpha (x)}$ is an element of the U(1) group, thus QED is said to have U(1) gauge symmetry.[1]: 496 The photon field Aμ may be referred to as the U(1) gauge boson.
U(1) is an Abelian group, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups, giving rise to non-Abelian gauge theories (also known as Yang–Mills theories).[1]: 489 Quantum chromodynamics, which describes the strong interaction, is a non-Abelian gauge theory with an SU(3) gauge symmetry. It contains three Dirac fields ψi, i = 1,2,3 representing quark fields as well as eight vector fields Aa,μ, a = 1,...,8 representing gluon fields, which are the SU(3) gauge bosons.[1]: 547 The QCD Lagrangian density is:[1]: 490–491
${\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},$
where Dμ is the gauge covariant derivative:
$D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},$
where g is the coupling constant, ta are the eight generators of SU(3) in the fundamental representation (3×3 matrices),
$F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},$
and fabc are the structure constants of SU(3). Repeated indices i,j,a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:
$\psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu }^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger }(x),$
where U(x) is an element of SU(3) at every spacetime point x:
$U(x)=e^{i\alpha (x)^{a}t^{a}}.$
The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density ${\mathcal {L}}[\phi ,\partial _{\mu }\phi ]$ under a certain local transformation of the fields, the measure $ \int {\mathcal {D}}\phi $ of the path integral may change.[33]: 243 For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1), in which all anomalies exactly cancel.[1]: 705–707
The theoretical foundation of general relativity, the equivalence principle, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group.[39]
Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding conservation law.[1]: 17–18 [33]: 73 For example, the U(1) symmetry of QED implies charge conservation.[40]
Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field Aμ, being a four-vector, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarization. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing Aμ can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.[33]: 168
To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.[1]: 512–515 A more rigorous generalization of the Faddeev–Popov procedure is given by BRST quantization.[1]: 517
Spontaneous symmetry-breaking
Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.[1]: 347
To illustrate the mechanism, consider a linear sigma model containing N real scalar fields, described by the Lagrangian density:
${\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\phi ^{i}\right)\left(\partial ^{\mu }\phi ^{i}\right)+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}\left(\phi ^{i}\phi ^{i}\right)^{2},$
where μ and λ are real parameters. The theory admits an O(N) global symmetry:
$\phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).$
The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field ϕ0 satisfying
$\phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.$
Without loss of generality, let the ground state be in the N-th direction:
$\phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).$
The original N fields can be rewritten as:
$\phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),$
and the original Lagrangian density as:
${\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\pi ^{k}\right)\left(\partial ^{\mu }\pi ^{k}\right)+{\frac {1}{2}}\left(\partial _{\mu }\sigma \right)\left(\partial ^{\mu }\sigma \right)-{\frac {1}{2}}\left(2\mu ^{2}\right)\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}\left(\pi ^{k}\pi ^{k}\right)^{2},$
where k = 1, ..., N − 1. The original O(N) global symmetry is no longer manifest, leaving only the subgroup O(N − 1). The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.[1]: 349–350
Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, O(N) has N(N − 1)/2 continuous symmetries (the dimension of its Lie algebra), while O(N − 1) has (N − 1)(N − 2)/2. The number of broken symmetries is their difference, N − 1, which corresponds to the N − 1 massless fields πk.[1]: 351
On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.[1]: 743–744
In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures.[33]: 199 In the Standard Model of elementary particles, the W and Z bosons, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson, a process called the Higgs mechanism.[1]: 690
Supersymmetry
All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesized the existence of a type of symmetry, called supersymmetry, that relates bosons and fermions.[1]: 795 [33]: 443
The Standard Model obeys Poincaré symmetry, whose generators are the spacetime translations Pμ and the Lorentz transformations Jμν.[41]: 58–60 In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators Qα, called supercharges, which themselves transform as Weyl fermions.[1]: 795 [33]: 444 The symmetry group generated by all these generators is known as the super-Poincaré group. In general there can be more than one set of supersymmetry generators, QαI, I = 1, ..., N, which generate the corresponding N = 1 supersymmetry, N = 2 supersymmetry, and so on.[1]: 795 [33]: 450 Supersymmetry can also be constructed in other dimensions,[42] most notably in (1+1) dimensions for its application in superstring theory.[43]
The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.[33]: 448 Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), N = 4 supersymmetric Yang–Mills theory,[33]: 450 and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa.[33]: 444
If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity.[44]
Supersymmetry is a potential solution to many current problems in physics. For example, the hierarchy problem of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the grand unified scale or the Planck scale—can be resolved by relating the Higgs field and its super-partner, the Higgsino. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of dark matter.[1]: 796–797 [45]
Nevertheless, as of 2018, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.[1]: 797 [33]: 443
Other spacetimes
The ϕ4 theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime.
In condensed matter physics, QFT is used to describe (2+1)-dimensional electron gases.[46] In high-energy physics, string theory is a type of (1+1)-dimensional QFT,[33]: 452 [27] while Kaluza–Klein theory uses gravity in extra dimensions to produce gauge theories in lower dimensions.[33]: 428–429
In Minkowski space, the flat metric ημν is used to raise and lower spacetime indices in the Lagrangian, e.g.
$A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,$
where ημν is the inverse of ημν satisfying ημρηρν = δμν. For QFTs in curved spacetime on the other hand, a general metric (such as the Schwarzschild metric describing a black hole) is used:
$A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,$
where gμν is the inverse of gμν. For a real scalar field, the Lagrangian density in a general spacetime background is
${\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),$
where g = det(gμν), and ∇μ denotes the covariant derivative.[47] The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.
Topological quantum field theory
The correlation functions and physical predictions of a QFT depend on the spacetime metric gμν. For a special class of QFTs called topological quantum field theories (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric.[48]: 36 QFTs in curved spacetime generally change according to the geometry (local structure) of the spacetime background, while TQFTs are invariant under spacetime diffeomorphisms but are sensitive to the topology (global structure) of spacetime. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime. Chern–Simons theory is an example of TQFT and has been used to construct models of quantum gravity.[49] Applications of TQFT include the fractional quantum Hall effect and topological quantum computers.[50]: 1–5 The world line trajectory of fractionalized particles (known as anyons) can form a link configuration in the spacetime,[51] which relates the braiding statistics of anyons in physics to the link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.[52]
Perturbative and non-perturbative methods
Using perturbation theory, the total effect of a small interaction term can be approximated order by order by a series expansion in the number of virtual particles participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a Feynman diagram. The electromagnetic force between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the W and Z bosons carry the weak interaction, while gluons carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as 't Hooft–Polyakov monopole, domain wall, flux tube, and instanton.[8] Examples of QFTs that are completely solvable non-perturbatively include minimal models of conformal field theory[53] and the Thirring model.[54]
Mathematical rigor
In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem, there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined.[55]
However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, Kevin Costello's monograph Renormalization and Effective Field Theory[56] provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of Kadanoff, Wilson, and Polchinski, together with the Batalin-Vilkovisky approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory,[57] can be given a sound mathematical interpretation from their finite-dimensional analogues.[58]
Since the 1950s,[59] theoretical physicists and mathematicians have attempted to organize all QFTs into a set of axioms, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory, a subfield of mathematical physics,[60]: 2 which has led to such results as CPT theorem, spin–statistics theorem, and Goldstone's theorem,[59] and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions,[61] the three-dimensional scalar field theories with a quartic interaction, etc.[62]
Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms.[63]
Algebraic quantum field theory is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag–Kastler axioms.[60]: 2–3 One way to construct theories satisfying Wightman axioms is to use Osterwalder–Schrader axioms, which give the necessary and sufficient conditions for a real time theory to be obtained from an imaginary time theory by analytic continuation (Wick rotation).[60]: 10
Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills theories as set out by the above axioms. The full problem statement is as follows.[64]
Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on $\mathbb {R} ^{4}$ and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).
See also
• Abraham–Lorentz force
• AdS/CFT correspondence
• Axiomatic quantum field theory
• Introduction to quantum mechanics
• Common integrals in quantum field theory
• Conformal field theory
• Constructive quantum field theory
• Dirac's equation
• Form factor (quantum field theory)
• Feynman diagram
• Green–Kubo relations
• Green's function (many-body theory)
• Group field theory
• Lattice field theory
• List of quantum field theories
• Local quantum field theory
• Noncommutative quantum field theory
• Quantization of a field
• Quantum electrodynamics
• Quantum field theory in curved spacetime
• Quantum chromodynamics
• Quantum flavordynamics
• Quantum hadrodynamics
• Quantum hydrodynamics
• Quantum triviality
• Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
• Relationship between string theory and quantum field theory
• Schwinger–Dyson equation
• Static forces and virtual-particle exchange
• Symmetry in quantum mechanics
• Theoretical and experimental justification for the Schrödinger equation
• Topological quantum field theory
• Ward–Takahashi identity
• Wheeler–Feynman absorber theory
• Wigner's classification
• Wigner's theorem
References
1. Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. ISBN 978-0-201-50397-5.
2. Hobson, Art (2013). "There are no particles, there are only fields". American Journal of Physics. 81 (211): 211–223. arXiv:1204.4616. Bibcode:2013AmJPh..81..211H. doi:10.1119/1.4789885. S2CID 18254182.
3. Weinberg, Steven (1977). "The Search for Unity: Notes for a History of Quantum Field Theory". Daedalus. 106 (4): 17–35. JSTOR 20024506.
4. John L. Heilbron (14 February 2003). The Oxford Companion to the History of Modern Science. Oxford University Press. ISBN 978-0-19-974376-6.
5. Joseph John Thomson (1893). Notes on Recent Researches in Electricity and Magnetism: Intended as a Sequel to Professor Clerk-Maxwell's 'Treatise on Electricity and Magnetism'. Dawsons.
6. Weisskopf, Victor (November 1981). "The development of field theory in the last 50 years". Physics Today. 34 (11): 69–85. Bibcode:1981PhT....34k..69W. doi:10.1063/1.2914365.
7. Werner Heisenberg (1999). Physics and Philosophy: The Revolution in Modern Science. Prometheus Books. ISBN 978-1-57392-694-2.
8. Shifman, M. (2012). Advanced Topics in Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-19084-8.
9. Tomonaga, Shinichiro (1966). "Development of Quantum Electrodynamics". Science. 154 (3751): 864–868. Bibcode:1966Sci...154..864T. doi:10.1126/science.154.3751.864. PMID 17744604.
10. Mehra and Milton (2000). Climbing the Mountain: The scientific biography of Julian Schwinger. Oxford University Press. p. 454.
11. Schwinger, Julian (July 1951). "On the Green's functions of quantized fields. I". Proceedings of the National Academy of Sciences. 37 (7): 452–455. doi:10.1073/pnas.37.7.452. ISSN 0027-8424. PMC 1063400. PMID 16578383.
12. Schwinger, Julian (July 1951). "On the Green's functions of quantized fields. II". Proceedings of the National Academy of Sciences. 37 (7): 455–459. doi:10.1073/pnas.37.7.455. ISSN 0027-8424. PMC 1063401. PMID 16578384.
13. Schweber, Silvan S. (2005-05-31). "The sources of Schwinger's Green's functions". Proceedings of the National Academy of Sciences. 102 (22): 7783–7788. doi:10.1073/pnas.0405167101. ISSN 0027-8424. PMC 1142349. PMID 15930139.
14. Schwinger, Julian (1966). "Particles and Sources". Phys Rev. 152 (4): 1219. Bibcode:1966PhRv..152.1219S. doi:10.1103/PhysRev.152.1219.
15. Schwinger, Julian (1998). Particles, Sources and Fields vol. 1. Reading, MA: Perseus Books. p. xi. ISBN 0-7382-0053-0.
16. Schwinger, Julian (1998). Particles, sources, and fields. 2 (1. print ed.). Reading, Mass: Advanced Book Program, Perseus Books. ISBN 978-0-7382-0054-5.
17. Schwinger, Julian (1998). Particles, sources, and fields. 3 (1. print ed.). Reading, Mass: Advanced Book Program, Perseus Books. ISBN 978-0-7382-0055-2.
18. C.R. Hagen; et al., eds. (1967). Proc of the 1967 Int. Conference on Particles and Fields. NY: Interscience. p. 128.
19. Mehra and Milton (2000). Climbing the Mountain: The scientific biography of Julian Schwinger. Oxford University Press. p. 467.
20. Schwinger, Julian (1998). Particles, Sources and Fields vol. 1. Reading, MA: Perseus Bookks. pp. 82–85.
21. Mehra, Jagdish; Milton, Kimball A. (2005). Climbing the mountain: the scientific biography of Julian Schwinger (Reprinted ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-852745-9.
22. 't Hooft, Gerard (2015-03-17). "The Evolution of Quantum Field Theory". The Standard Theory of Particle Physics. Advanced Series on Directions in High Energy Physics. Vol. 26. pp. 1–27. arXiv:1503.05007. Bibcode:2016stpp.conf....1T. doi:10.1142/9789814733519_0001. ISBN 978-981-4733-50-2. S2CID 119198452.
23. Yang, C. N.; Mills, R. L. (1954-10-01). "Conservation of Isotopic Spin and Isotopic Gauge Invariance". Physical Review. 96 (1): 191–195. Bibcode:1954PhRv...96..191Y. doi:10.1103/PhysRev.96.191.
24. Coleman, Sidney (1979-12-14). "The 1979 Nobel Prize in Physics". Science. 206 (4424): 1290–1292. Bibcode:1979Sci...206.1290C. doi:10.1126/science.206.4424.1290. JSTOR 1749117. PMID 17799637.
25. Sutton, Christine. "Standard model". britannica.com. Encyclopædia Britannica. Retrieved 2018-08-14.
26. Kibble, Tom W. B. (2014-12-12). "The Standard Model of Particle Physics". arXiv:1412.4094 [physics.hist-ph].
27. Polchinski, Joseph (2005). String Theory. Vol. 1. Cambridge University Press. ISBN 978-0-521-67227-6.
28. Schwarz, John H. (2012-01-04). "The Early History of String Theory and Supersymmetry". arXiv:1201.0981 [physics.hist-ph].
29. "Common Problems in Condensed Matter and High Energy Physics" (PDF). science.energy.gov. Office of Science, U.S. Department of Energy. 2015-02-02. Retrieved 2018-07-18.
30. Wilczek, Frank (2016-04-19). "Particle Physics and Condensed Matter: The Saga Continues". Physica Scripta. 2016 (T168): 014003. arXiv:1604.05669. Bibcode:2016PhST..168a4003W. doi:10.1088/0031-8949/T168/1/014003. S2CID 118439678.
31. Tong 2015, Chapter 1
32. In fact, its number of degrees of freedom is uncountable, because the vector space dimension of the space of continuous (differentiable, real analytic) functions on even a finite dimensional Euclidean space is uncountable. On the other hand, subspaces (of these function spaces) that one typically considers, such as Hilbert spaces (e.g. the space of square integrable real valued functions) or separable Banach spaces (e.g. the space of continuous real-valued functions on a compact interval, with the uniform convergence norm), have denumerable (i. e. countably infinite) dimension in the category of Banach spaces (though still their Euclidean vector space dimension is uncountable), so in these restricted contexts, the number of degrees of freedom (interpreted now as the vector space dimension of a dense subspace rather than the vector space dimension of the function space of interest itself) is denumerable.
33. Zee, A. (2010). Quantum Field Theory in a Nutshell. Princeton University Press. ISBN 978-0-691-01019-9.
34. Fock, V. (1932-03-10). "Konfigurationsraum und zweite Quantelung". Zeitschrift für Physik (in German). 75 (9–10): 622–647. Bibcode:1932ZPhy...75..622F. doi:10.1007/BF01344458. S2CID 186238995.
35. Becker, Katrin; Becker, Melanie; Schwarz, John H. (2007). String Theory and M-Theory. Cambridge University Press. p. 36. ISBN 978-0-521-86069-7.
36. Fujita, Takehisa (2008-02-01). "Physics of Renormalization Group Equation in QED". arXiv:hep-th/0606101.
37. Aharony, Ofer; Gur-Ari, Guy; Klinghoffer, Nizan (2015-05-19). "The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants". Journal of High Energy Physics. 2015 (5): 31. arXiv:1501.06664. Bibcode:2015JHEP...05..031A. doi:10.1007/JHEP05(2015)031. S2CID 115167208.
38. Kovacs, Stefano (1999-08-26). "N = 4 supersymmetric Yang–Mills theory and the AdS/SCFT correspondence". arXiv:hep-th/9908171.
39. Veltman, M. J. G. (1976). Methods in Field Theory, Proceedings of the Les Houches Summer School, Les Houches, France, 1975.
40. Brading, Katherine A. (March 2002). "Which symmetry? Noether, Weyl, and conservation of electric charge". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 33 (1): 3–22. Bibcode:2002SHPMP..33....3B. CiteSeerX 10.1.1.569.106. doi:10.1016/S1355-2198(01)00033-8.
41. Weinberg, Steven (1995). The Quantum Theory of Fields. Cambridge University Press. ISBN 978-0-521-55001-7.
42. de Wit, Bernard; Louis, Jan (1998-02-18). "Supersymmetry and Dualities in various dimensions". arXiv:hep-th/9801132.
43. Polchinski, Joseph (2005). String Theory. Vol. 2. Cambridge University Press. ISBN 978-0-521-67228-3.
44. Nath, P.; Arnowitt, R. (1975). "Generalized Super-Gauge Symmetry as a New Framework for Unified Gauge Theories". Physics Letters B. 56 (2): 177. Bibcode:1975PhLB...56..177N. doi:10.1016/0370-2693(75)90297-x.
45. Munoz, Carlos (2017-01-18). "Models of Supersymmetry for Dark Matter". EPJ Web of Conferences. 136: 01002. arXiv:1701.05259. Bibcode:2017EPJWC.13601002M. doi:10.1051/epjconf/201713601002. S2CID 55199323.
46. Morandi, G.; Sodano, P.; Tagliacozzo, A.; Tognetti, V. (2000). Field Theories for Low-Dimensional Condensed Matter Systems. Springer. ISBN 978-3-662-04273-1.
47. Parker, Leonard E.; Toms, David J. (2009). Quantum Field Theory in Curved Spacetime. Cambridge University Press. p. 43. ISBN 978-0-521-87787-9.
48. Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008-12-11). "Undergraduate Lecture Notes in Topological Quantum Field Theory". arXiv:0810.0344v5 [math-th].
49. Carlip, Steven (1998). Quantum Gravity in 2+1 Dimensions. Cambridge University Press. pp. 27–29. doi:10.1017/CBO9780511564192. ISBN 9780511564192.
50. Carqueville, Nils; Runkel, Ingo (2018). "Introductory lectures on topological quantum field theory". Banach Center Publications. 114: 9–47. arXiv:1705.05734. doi:10.4064/bc114-1. S2CID 119166976.
51. Witten, Edward (1989). "Quantum Field Theory and the Jones Polynomial". Communications in Mathematical Physics. 121 (3): 351–399. Bibcode:1989CMaPh.121..351W. doi:10.1007/BF01217730. MR 0990772. S2CID 14951363.
52. Putrov, Pavel; Wang, Juven; Yau, Shing-Tung (2017). "Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions". Annals of Physics. 384 (C): 254–287. arXiv:1612.09298. Bibcode:2017AnPhy.384..254P. doi:10.1016/j.aop.2017.06.019. S2CID 119578849.
53. Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997). Conformal Field Theory. Springer. ISBN 978-1-4612-7475-9.
54. Thirring, W. (1958). "A Soluble Relativistic Field Theory?". Annals of Physics. 3 (1): 91–112. Bibcode:1958AnPhy...3...91T. doi:10.1016/0003-4916(58)90015-0.
55. Haag, Rudolf (1955). "On Quantum Field Theories" (PDF). Dan Mat Fys Medd. 29 (12).
56. Kevin Costello, Renormalization and Effective Field Theory, Mathematical Surveys and Monographs Volume 170, American Mathematical Society, 2011, ISBN 978-0-8218-5288-0
57. Gerald B. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians, Mathematical Surveys and Monographs Volume 149, American Mathematical Society, 2008, ISBN 0821847058 | chapter=8
58. Nguyen, Timothy (2016). "The perturbative approach to path integrals: A succinct mathematical treatment". J. Math. Phys. 57 (9): 092301. arXiv:1505.04809. Bibcode:2016JMP....57i2301N. doi:10.1063/1.4962800. S2CID 54813572.
59. Buchholz, Detlev (2000). "Current Trends in Axiomatic Quantum Field Theory". Quantum Field Theory. Lecture Notes in Physics. 558: 43–64. arXiv:hep-th/9811233. Bibcode:2000LNP...558...43B. doi:10.1007/3-540-44482-3_4. ISBN 978-3-540-67972-1. S2CID 5052535.
60. Summers, Stephen J. (2016-03-31). "A Perspective on Constructive Quantum Field Theory". arXiv:1203.3991v2 [math-ph].
61. Simon, Barry (1974). The P(phi)_2 Euclidean (quantum) field theory. Princeton, New Jersey: Princeton University Press. ISBN 0-691-08144-1. OCLC 905864308.
62. Glimm, James; Jaffe, Arthur (1987). Quantum Physics : a Functional Integral Point of View. New York, NY: Springer New York. ISBN 978-1-4612-4728-9. OCLC 852790676.
63. Sati, Hisham; Schreiber, Urs (2012-01-06). "Survey of mathematical foundations of QFT and perturbative string theory". arXiv:1109.0955v2 [math-ph].
64. Jaffe, Arthur; Witten, Edward. "Quantum Yang–Mills Theory" (PDF). Clay Mathematics Institute. Retrieved 2018-07-18.
Bibliography
• Streater, R.; Wightman, A. (1964). PCT, Spin and Statistics and all That. W. A. Benjamin.
• Osterwalder, K.; Schrader, R. (1973). "Axioms for Euclidean Green's functions". Communications in Mathematical Physics. 31 (2): 83–112. Bibcode:1973CMaPh..31...83O. doi:10.1007/BF01645738. S2CID 189829853.
• Osterwalder, K.; Schrader, R. (1975). "Axioms for Euclidean Green's functions II". Communications in Mathematical Physics. 42 (3): 281–305. Bibcode:1975CMaPh..42..281O. doi:10.1007/BF01608978. S2CID 119389461.
Further reading
General readers
• Pais, A. (1994) [1986]. Inward Bound: Of Matter and Forces in the Physical World (reprint ed.). Oxford, New York, Toronto: Oxford University Press. ISBN 978-0198519973.
• Schweber, S. S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press. ISBN 9780691033273.
• Feynman, R.P. (2001) [1964]. The Character of Physical Law. MIT Press. ISBN 978-0-262-56003-0.
• Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6.
• Gribbin, J. (1998). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicolson. ISBN 978-0-297-81752-9.
Introductory texts
• McMahon, D. (2008). Quantum Field Theory. McGraw-Hill. ISBN 978-0-07-154382-8.
• Bogolyubov, N.; Shirkov, D. (1982). Quantum Fields. Benjamin Cummings. ISBN 978-0-8053-0983-6.
• Frampton, P.H. (2000). Gauge Field Theories. Frontiers in Physics (2nd ed.). Wiley.; 2008, 3rd edition. ISBN 3527408355.
• Greiner, W.; Müller, B. (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0.
• Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory. McGraw-Hill. ISBN 978-0-07-032071-0.
• Kane, G.L. (1987). Modern Elementary Particle Physics. Perseus Group. ISBN 978-0-201-11749-3.
• Kleinert, H.; Schulte-Frohlinde, Verena (2001). Critical Properties of φ4-Theories. World Scientific. ISBN 978-981-02-4658-7.
• Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation (PDF). World Scientific. ISBN 978-981-279-170-2.
• Lancaster, T., & Blundell, S. J. (2014). Quantum field theory for the gifted amateur. OUP Oxford. ISBN 9780199699339
• Loudon, R. (1983). The Quantum Theory of Light. Oxford University Press. ISBN 978-0-19-851155-7.
• Mandl, F.; Shaw, G. (1993). Quantum Field Theory. John Wiley & Sons. ISBN 978-0-471-94186-6.
• Ryder, L.H. (1985). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-33859-2.
• Schwartz, M.D. (2014). Quantum Field Theory and the Standard Model. Cambridge University Press. ISBN 978-1107034730. Archived from the original on 2018-03-22. Retrieved 2020-05-13.
• Ynduráin, F.J. (1996). Relativistic Quantum Mechanics and Introduction to Field Theory (1st ed.). Springer. Bibcode:1996rqmi.book.....Y. doi:10.1007/978-3-642-61057-8. ISBN 978-3-540-60453-2.
• Greiner, W.; Reinhardt, J. (1996). Field Quantization. Springer. ISBN 978-3-540-59179-5.
• Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. ISBN 978-0-201-50397-5.
• Scharf, Günter (2014) [1989]. Finite Quantum Electrodynamics: The Causal Approach (third ed.). Dover Publications. ISBN 978-0486492735.
• Srednicki, M. (2007). Quantum Field Theory. Cambridge University Press. ISBN 978-0521-8644-97.
• Tong, David (2015). "Lectures on Quantum Field Theory". Retrieved 2016-02-09.
• Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. ISBN 978-0691140346.
Advanced texts
• Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3.
• Bogoliubov, N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. (1990). General Principles of Quantum Field Theory. Kluwer Academic Publishers. ISBN 978-0-7923-0540-8.
• Weinberg, S. (1995). The Quantum Theory of Fields. Vol. 1. Cambridge University Press. ISBN 978-0521550017.
External links
• Media related to Quantum field theory at Wikimedia Commons
One-dimensional quantum field theory on Wikiversity
• "Quantum field theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Stanford Encyclopedia of Philosophy: "Quantum Field Theory", by Meinard Kuhlmann.
• Siegel, Warren, 2005. Fields. arXiv:hep-th/9912205.
• Quantum Field Theory by P. J. Mulders
Quantum field theories
Theories
• Algebraic QFT
• Axiomatic QFT
• Conformal field theory
• Lattice field theory
• Noncommutative QFT
• Gauge theory
• QFT in curved spacetime
• String theory
• Supergravity
• Thermal QFT
• Topological QFT
• Two-dimensional conformal field theory
Models
Regular
• Born–Infeld
• Euler–Heisenberg
• Ginzburg–Landau
• Non-linear sigma
• Proca
• Quantum electrodynamics
• Quantum chromodynamics
• Quartic interaction
• Scalar electrodynamics
• Scalar chromodynamics
• Soler
• Yang–Mills
• Yang–Mills–Higgs
• Yukawa
Low dimensional
• 2D Yang–Mills
• Bullough–Dodd
• Gross–Neveu
• Schwinger
• Sine-Gordon
• Thirring
• Thirring–Wess
• Toda
Conformal
• 2D free massless scalar
• Liouville
• Minimal
• Polyakov
• Wess–Zumino–Witten
Supersymmetric
• Wess–Zumino
• N = 1 super Yang–Mills
• Seiberg–Witten
• Super QCD
Superconformal
• 6D (2,0)
• ABJM
• N = 4 super Yang–Mills
Supergravity
• Higher dimensional
• N = 8
• Pure 4D N = 1
Topological
• BF
• Chern–Simons
Particle theory
• Chiral
• Fermi
• MSSM
• Nambu–Jona-Lasinio
• NMSSM
• Standard Model
• Stueckelberg
Related
• Casimir effect
• Cosmic string
• History
• Loop quantum gravity
• Loop quantum cosmology
• On shell and off shell
• Quantum chaos
• Quantum dynamics
• Quantum foam
• Quantum fluctuations
• links
• Quantum gravity
• links
• Quantum hadrodynamics
• Quantum hydrodynamics
• Quantum information
• Quantum information science
• links
• Quantum logic
• Quantum thermodynamics
See also: Template:Quantum mechanics topics
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
• Quantum biology
• Quantum chemistry
• Quantum chaos
• Quantum cosmology
• Quantum differential calculus
• Quantum dynamics
• Quantum geometry
• Quantum measurement problem
• Quantum mind
• Quantum stochastic calculus
• Quantum spacetime
Technology
• Quantum algorithms
• Quantum amplifier
• Quantum bus
• Quantum cellular automata
• Quantum finite automata
• Quantum channel
• Quantum circuit
• Quantum complexity theory
• Quantum computing
• Timeline
• Quantum cryptography
• Quantum electronics
• Quantum error correction
• Quantum imaging
• Quantum image processing
• Quantum information
• Quantum key distribution
• Quantum logic
• Quantum logic gates
• Quantum machine
• Quantum machine learning
• Quantum metamaterial
• Quantum metrology
• Quantum network
• Quantum neural network
• Quantum optics
• Quantum programming
• Quantum sensing
• Quantum simulator
• Quantum teleportation
Extensions
• Casimir effect
• Quantum statistical mechanics
• Quantum field theory
• History
• Quantum gravity
• Relativistic quantum mechanics
Related
• Schrödinger's cat
• in popular culture
• EPR paradox
• Quantum mysticism
• Category
• Physics portal
• Commons
Quantum gravity
Central concepts
• AdS/CFT correspondence
• Ryu–Takayanagi conjecture
• Causal patch
• Gravitational anomaly
• Graviton
• Holographic principle
• IR/UV mixing
• Planck units
• Quantum foam
• Trans-Planckian problem
• Weinberg–Witten theorem
• Faddeev–Popov ghost
• Batalin-Vilkovisky formalism
• CA-duality
Toy models
• 2+1D topological gravity
• CGHS model
• Jackiw–Teitelboim gravity
• Liouville gravity
• RST model
• Topological quantum field theory
Quantum field theory
in curved spacetime
• Bunch–Davies vacuum
• Hawking radiation
• Semiclassical gravity
• Unruh effect
Black holes
• Black hole complementarity
• Black hole information paradox
• Black-hole thermodynamics
• Bekenstein bound
• Bousso's holographic bound
• Cosmic censorship hypothesis
• ER = EPR
• Firewall (physics)
• Gravitational singularity
Approaches
String theory
• Bosonic string theory
• M-theory
• Supergravity
• Superstring theory
Canonical quantum gravity
• Loop quantum gravity
• Wheeler–DeWitt equation
Euclidean quantum gravity
• Hartle–Hawking state
Others
• Causal dynamical triangulation
• Causal sets
• Noncommutative geometry
• Spin foam
• Group field theory
• Superfluid vacuum theory
• Twistor theory
• Dual graviton
Applications
• Quantum cosmology
• Eternal inflation
• Multiverse
• FRW/CFT duality
• See also: Template:Quantum mechanics topics
Major branches of physics
Divisions
• Pure
• Applied
• Engineering
Approaches
• Experimental
• Theoretical
• Computational
Classical
• Classical mechanics
• Newtonian
• Analytical
• Celestial
• Continuum
• Acoustics
• Classical electromagnetism
• Classical optics
• Ray
• Wave
• Thermodynamics
• Statistical
• Non-equilibrium
Modern
• Relativistic mechanics
• Special
• General
• Nuclear physics
• Quantum mechanics
• Particle physics
• Atomic, molecular, and optical physics
• Atomic
• Molecular
• Modern optics
• Condensed matter physics
Interdisciplinary
• Astrophysics
• Atmospheric physics
• Biophysics
• Chemical physics
• Geophysics
• Materials science
• Mathematical physics
• Medical physics
• Ocean physics
• Quantum information science
Related
• History of physics
• Nobel Prize in Physics
• Philosophy of physics
• Physics education
• Timeline of physics discoveries
Standard Model
Background
• Particle physics
• Fermions
• Gauge boson
• Higgs boson
• Quantum field theory
• Gauge theory
• Strong interaction
• Color charge
• Quantum chromodynamics
• Quark model
• Electroweak interaction
• Weak interaction
• Quantum electrodynamics
• Fermi's interaction
• Weak hypercharge
• Weak isospin
Constituents
• CKM matrix
• Spontaneous symmetry breaking
• Higgs mechanism
• Mathematical formulation of the Standard Model
Beyond the
Standard Model
Evidence
• Hierarchy problem
• Dark matter
• Cosmological constant
• problem
• Strong CP problem
• Neutrino oscillation
Theories
• Technicolor
• Kaluza–Klein theory
• Grand Unified Theory
• Theory of everything
Supersymmetry
• MSSM
• NMSSM
• Split supersymmetry
• Supergravity
Quantum gravity
• String theory
• Superstring theory
• Loop quantum gravity
• Causal dynamical triangulation
• Canonical quantum gravity
• Superfluid vacuum theory
• Twistor theory
Experiments
• Gran Sasso
• INO
• LHC
• SNO
• Super-K
• Tevatron
• Category
• Commons
Authority control: National
• France
• BnF data
• Germany
• Israel
• United States
• Japan
• Czech Republic
|
Wikipedia
|
Critical three-state Potts model
The three-state Potts CFT, also known as the $\mathbb {Z} _{3}$ parafermion CFT, is a conformal field theory in two dimensions. It is a minimal model with central charge $c=4/5$. It is considered to be the simplest minimal model with a non-diagonal partition function in Virasoro characters, as well as the simplest non-trivial CFT with the W-algebra as a symmetry.[1][2][3]
Properties
The critical three-state Potts model has a central charge of $c=4/5$, and thus belongs to the discrete family of unitary minimal models with central charge less than one. These conformal field theories are fully classified and for the most part well-understood.
The modular partition function of the critical three-state Potts model is given by
$Z=|\chi _{1,1}+\chi _{4,1}|^{2}+|\chi _{2,1}+\chi _{3,1}|^{2}+2|\chi _{4,3}|^{2}+2|\chi _{3,3}|^{2}$
Here $\chi _{r,s}(q)\equiv {\textrm {Tr}}_{(r,s)}(q^{L_{0}-c/24})$ refers to the Virasoro character, found by taking the trace over the Verma module generated from the Virasoro primary operator labeled by integers $r,s$. The labeling $(r,s)$ is a standard convention for primary operators of the $c<1$ minimal models.
Furthermore, the critical three-state Potts model is symmetric not only under the Virasoro algebra, but also under an enlarged algebra called the W-algebra that includes the Virasoro algebra as well as some spin-3 currents. The local holomorphic W primaries are given by $1,\epsilon ,\sigma _{1},\sigma _{2},\psi _{1},\psi _{2}$. The local antiholomorphic W primaries similarly are given by $1,{\bar {\epsilon }},{\bar {\sigma }}_{1},{\bar {\sigma }}_{2},{\bar {\psi }}_{1},{\bar {\psi }}_{2}$ with the same scaling dimensions. Each field in the theory is either a combination of a holomorphic and antiholomorphic W-algebra primary field, or a descendant of such a field generated by acting with W-algebra generators. Some primaries of the Virasoro algebra, such as the $(3,1)$ primary, are not primaries of the W algebra.
Chiral W primaries in the critical 3-state Potts model
PrimaryDimension$\mathbb {Z} _{3}$ charge Kac Label
$1$00 (1,1)+(4,1)
$\epsilon $2/50 (2,1)+(3,1)
$\psi _{1}$2/31 (1,3)
$\psi _{2}$2/3-1 (1,3)
$\sigma _{1}$1/151 (3,3)
$\sigma _{2}$1/15-1 (3,3)
The partition function is diagonal when expressed in terms of W-algebra characters (where traces are taken over irreducible representations of the W algebra, instead of over irreducible representations of the Virasoro algebra). Since $\chi _{1}=\chi _{1,1}+\chi _{4,1}$ and $\chi _{\epsilon }=\chi _{2,1}+\chi _{3,1}$, we can write
$Z=|\chi _{1}|^{2}+|\chi _{\epsilon }|^{2}+|\chi _{\psi _{1}}|^{2}+|\chi _{\psi _{2}}|^{2}+|\chi _{\sigma _{1}}|^{2}+|\chi _{\sigma _{2}}|^{2}$
The operators $\sigma _{1},\sigma _{2},\psi _{1},\psi _{2}$ are charged under the action of a global $\mathbb {Z} _{3}$ symmetry. That is, under a global global $\mathbb {Z} _{3}$ transformation, they pick up phases $\sigma _{a}\to e^{2\pi ia/3}\sigma _{a}$ and $\psi _{a}\to e^{2\pi ia/3}\psi _{a}$ for $a=1,2$. The fusion rules governing the operator product expansions involving these fields respect the action of this $\mathbb {Z} _{3}$ transformation. There is also a charge conjugation symmetry that interchanges $\sigma _{1}\leftrightarrow \sigma _{2},\psi _{1}\leftrightarrow \psi _{2}$. Sometimes the notation $\sigma ,\sigma ^{\dagger },\psi ,\psi ^{\dagger }$ is used in the literature instead of $\sigma _{1},\sigma _{2},\psi _{1},\psi _{2}$.
The critical three-state Potts model is one of the two modularly invariant conformal field theories that exist with central charge $c=4/5$. The other such theory is the tetracritical Ising model, which has a diagonal partition function in terms of Virasoro characters. It is possible to obtain the critical three-state Potts model from the tetracritical Ising model by applying a $\mathbb {Z} _{2}$ orbifold transformation to the latter.
Lattice Hamiltonians
The critical three-state Potts conformal field theory can be realised as the low energy effective theory at the phase transition of the one-dimensional quantum three-state Potts model.
The Hamiltonian of the quantum three-state Potts model is given by
$H=-J(\sum _{\langle i,j\rangle }(Z_{i}^{\dagger }Z_{j}+Z_{i}Z_{j}^{\dagger })+g\sum _{j}(X_{j}+X_{j}^{\dagger }))$
Here $J$ and $g$ are positive parameters. The first term couples degrees of freedom on nearest neighbour sites in the lattice. $X$ and $Z$ are $3\times 3$ clock matrices satisfying $X^{3}=Z^{3}=1$ and same-site commutation relation $ZX=\omega XZ$ where $\omega =-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}$.
This Hamiltonian is symmetric under any permutation of the three $Z$ eigenstates on each site, as long as the same permutation is done on every site. Thus it is said to have a global $S_{3}$ symmetry. A $\mathbb {Z} _{3}$ subgroup of this symmetry is generated by the unitary operator $\prod _{j}X_{j}$.
In one dimension, the model has two gapped phases, the ordered phase and the disordered phase. The ordered phase occurs at $0<g<1$ and is characterised by a nonzero ground state expectation value of the order parameter $Z_{j}$ at any site $j$. The ground state in this phase explicitly breaks the global $S_{3}$ symmetry and is thus three-fold degenerate. The disordered phase occurs at $g>1$ and is characterised by a single ground state. In between these two phases is a phase transition at $g=1$. At this particular value of $g$, the Hamiltonian is gapless with a ground state energy of $E_{0}=-({\frac {4}{3}}+{\frac {2{\sqrt {3}}}{\pi }})JL$, where $L$ is the length of the chain. In other words, in the limit of an infinitely long chain, the lowest energy eigenvalues of the Hamiltonian are spaced infinitesimally close to each other. As is the case for most one dimensional gapless theories, it is possible to describe the low energy physics of the 3-state Potts model using a 1+1 dimensional conformal field theory; in this particular lattice model that conformal field theory is none other than the critical three-state Potts model.
Lattice operator correspondence
Under the flow of renormalisation group, lattice operators in the quantum three-state Potts model flow to fields in the conformal field theory. In general, understanding which operators flow to what fields is difficult and not obvious. Analytical and numerical arguments suggest a correspondence between a few lattice operators and CFT fields as follows.[3] Lattice indices $j$ map to the corresponding field positions $z,{\bar {z}}$ in space-time, and non-universal real number prefactors are ignored.
• $Z_{j}\sim \Phi _{\sigma _{1},{\bar {\sigma }}_{1}}(z,{\bar {z}})$, the ${\frac {2}{15}}$-dimensional field composed of holomorphic and anti-holomorphic parts $\sigma _{1}(z)$ and ${\bar {\sigma }}_{1}({\bar {z}})$
• $Z_{j}^{\dagger }\sim \Phi _{\sigma _{2},{\bar {\sigma }}_{2}}(z,{\bar {z}})$
• $Z_{j}Z_{j+1}^{\dagger }-{\frac {1}{2}}(X_{j}+X_{j+1})+{\textrm {h.c.}}\sim \Phi _{\epsilon ,{\bar {\epsilon }}}(z,{\bar {z}})$ . As can be seen in the lattice language, adding this operator to every site of the Hamiltonian has the effect of tuning $g$ away from 1. This operator is called the thermal operator, because in the classical statistical mechanics analog of the quantum lattice model, tuning $g$ would be equivalent to changing temperature away from the critical temperature.
• $-Z_{j}Z_{j+1}^{\dagger }-{\frac {1}{2}}(X_{j}+X_{j+1})+{\textrm {h.c.}}+{\frac {4}{3}}+{\frac {2{\sqrt {3}}}{\pi }}\sim T(z)+{\bar {T}}({\bar {z}})$, the dimension-2 stress-energy tensor field.
• $Z_{j}(2-3\omega ^{2}X_{j}-3\omega X_{j}^{2})-2Z_{j}^{\dagger }(Z_{j-1}^{\dagger }+Z_{j+1}^{\dagger })\sim \psi _{1}(z){\bar {\psi }}_{1}({\bar {z}})$
References
1. "Boundary critical phenomena in the three-state Potts model" (PDF). www.kitp.ucsb.edu.
2. Di Francesco, Philippe; Mathieu, Pierre; Senechal, David (1997). Conformal Field Theory. Springer. p. 365. ISBN 0-387-94785-X.
3. Mong, Roger S K; Clarke, David J; Alicea, Jason; Lindner, Netanel H; Fendley, Paul (October 27, 2014). "Parafermionic conformal field theory on the lattice". Journal of Physics A: Mathematical and Theoretical. 47 (45): 452001. doi:10.1088/1751-8113/47/45/452001. S2CID 437648.
|
Wikipedia
|
Quantum topology
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.
Part of a series of articles about
Quantum mechanics
$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle $
Schrödinger equation
• Introduction
• Glossary
• History
Background
• Classical mechanics
• Old quantum theory
• Bra–ket notation
• Hamiltonian
• Interference
Fundamentals
• Complementarity
• Decoherence
• Entanglement
• Energy level
• Measurement
• Nonlocality
• Quantum number
• State
• Superposition
• Symmetry
• Tunnelling
• Uncertainty
• Wave function
• Collapse
Experiments
• Bell's inequality
• Davisson–Germer
• Double-slit
• Elitzur–Vaidman
• Franck–Hertz
• Leggett–Garg inequality
• Mach–Zehnder
• Popper
• Quantum eraser
• Delayed-choice
• Schrödinger's cat
• Stern–Gerlach
• Wheeler's delayed-choice
Formulations
• Overview
• Heisenberg
• Interaction
• Matrix
• Phase-space
• Schrödinger
• Sum-over-histories (path integral)
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
Advanced topics
• Relativistic quantum mechanics
• Quantum field theory
• Quantum information science
• Quantum computing
• Quantum chaos
• EPR paradox
• Density matrix
• Scattering theory
• Quantum statistical mechanics
• Quantum machine learning
Scientists
• Aharonov
• Bell
• Bethe
• Blackett
• Bloch
• Bohm
• Bohr
• Born
• Bose
• de Broglie
• Compton
• Dirac
• Davisson
• Debye
• Ehrenfest
• Einstein
• Everett
• Fock
• Fermi
• Feynman
• Glauber
• Gutzwiller
• Heisenberg
• Hilbert
• Jordan
• Kramers
• Pauli
• Lamb
• Landau
• Laue
• Moseley
• Millikan
• Onnes
• Planck
• Rabi
• Raman
• Rydberg
• Schrödinger
• Simmons
• Sommerfeld
• von Neumann
• Weyl
• Wien
• Wigner
• Zeeman
• Zeilinger
Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of one space within another such as knots and links in three-dimensional space. This bra–ket notation of kets and bras can be generalised, becoming maps of vector spaces associated with topological spaces that allow tensor products.[1]
Topological entanglement involving linking and braiding can be intuitively related to quantum entanglement.[1]
See also
• Topological quantum field theory
• Reshetikhin–Turaev invariant
References
1. Kauffman, Louis H.; Baadhio, Randy A. (1993). Quantum Topology. River Edge, NJ: World Scientific. ISBN 981-02-1544-4.
External links
• Quantum Topology, a journal published by EMS Publishing House
|
Wikipedia
|
Quantum Turing machine
A quantum Turing machine (QTM) or universal quantum computer is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algorithm can be expressed formally as a particular quantum Turing machine. However, the computationally equivalent quantum circuit is a more common model.[1][2]: 2
Turing machines
Machine
• Turing machine equivalents
• Turing machine examples
• Turing machine gallery
Variants
• Alternating Turing machine
• Neural Turing machine
• Nondeterministic Turing machine
• Quantum Turing machine
• Post–Turing machine
• Probabilistic Turing machine
• Multitape Turing machine
• Multi-track Turing machine
• Symmetric Turing machine
• Total Turing machine
• Unambiguous Turing machine
• Universal Turing machine
• Zeno machine
Science
• Alan Turing
• Category:Turing machine
Quantum Turing machines can be related to classical and probabilistic Turing machines in a framework based on transition matrices. That is, a matrix can be specified whose product with the matrix representing a classical or probabilistic machine provides the quantum probability matrix representing the quantum machine. This was shown by Lance Fortnow.[3]
Informal sketch
Unsolved problem in physics:
Is a universal quantum computer sufficient to efficiently simulate an arbitrary physical system?
(more unsolved problems in physics)
A way of understanding the quantum Turing machine (QTM) is that it generalizes the classical Turing machine (TM) in the same way that the quantum finite automaton (QFA) generalizes the deterministic finite automaton (DFA). In essence, the internal states of a classical TM are replaced by pure or mixed states in a Hilbert space; the transition function is replaced by a collection of unitary matrices that map the Hilbert space to itself.[4]
That is, a classical Turing machine is described by a 7-tuple $M=\langle Q,\Gamma ,b,\Sigma ,\delta ,q_{0},F\rangle $.
For a three-tape quantum Turing machine (one tape holding the input, a second tape holding intermediate calculation results, and a third tape holding output):
• The set of states $Q$ is replaced by a Hilbert space.
• The tape alphabet symbols $\Gamma $ are likewise replaced by a Hilbert space (usually a different Hilbert space than the set of states).
• The blank symbol $b\in \Gamma $ is an element of the Hilbert space.
• The input and output symbols $\Sigma $ are usually taken as a discrete set, as in the classical system; thus, neither the input nor output to a quantum machine need be a quantum system itself.
• The transition function $\delta :\Sigma \times Q\otimes \Gamma \to \Sigma \times Q\otimes \Gamma \times \{L,R\}$ :\Sigma \times Q\otimes \Gamma \to \Sigma \times Q\otimes \Gamma \times \{L,R\}} is a generalization of a transition monoid and is understood to be a collection of unitary matrices that are automorphisms of the Hilbert space $Q$.
• The initial state $q_{0}\in Q$ may be either a mixed state or a pure state.
• The set $F$ of final or accepting states is a subspace of the Hilbert space $Q$.
The above is merely a sketch of a quantum Turing machine, rather than its formal definition, as it leaves vague several important details: for example, how often a measurement is performed; see for example, the difference between a measure-once and a measure-many QFA. This question of measurement affects the way in which writes to the output tape are defined.
History
In 1980 and 1982, physicist Paul Benioff published articles[5][6] that first described a quantum mechanical model of Turing machines. A 1985 article written by Oxford University physicist David Deutsch further developed the idea of quantum computers by suggesting that quantum gates could function in a similar fashion to traditional digital computing binary logic gates.[4]
Iriyama, Ohya, and Volovich have developed a model of a linear quantum Turing machine (LQTM). This is a generalization of a classical QTM that has mixed states and that allows irreversible transition functions. These allow the representation of quantum measurements without classical outcomes.[7]
A quantum Turing machine with postselection was defined by Scott Aaronson, who showed that the class of polynomial time on such a machine (PostBQP) is equal to the classical complexity class PP.[8]
See also
• Quantum simulator § Solving physics problems
References
1. Andrew Yao (1993). Quantum circuit complexity. 34th Annual Symposium on Foundations of Computer Science. pp. 352–361.
2. Abel Molina; John Watrous (2018). "Revisiting the simulation of quantum Turing machines by quantum circuits". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 475 (2226). arXiv:1808.01701. doi:10.1098/rspa.2018.0767. PMC 6598068. PMID 31293355.
3. Fortnow, Lance (2003). "One Complexity Theorist's View of Quantum Computing". Theoretical Computer Science. 292 (3): 597–610. arXiv:quant-ph/0003035. doi:10.1016/S0304-3975(01)00377-2. S2CID 18657540.
4. Deutsch, David (July 1985). "Quantum theory, the Church-Turing principle and the universal quantum computer" (PDF). Proceedings of the Royal Society A. 400 (1818): 97–117. Bibcode:1985RSPSA.400...97D. CiteSeerX 10.1.1.41.2382. doi:10.1098/rspa.1985.0070. S2CID 1438116. Archived from the original (PDF) on 2008-11-23.
5. Benioff, Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics. 22 (5): 563–591. Bibcode:1980JSP....22..563B. doi:10.1007/bf01011339. S2CID 122949592.
6. Benioff, P. (1982). "Quantum mechanical hamiltonian models of turing machines". Journal of Statistical Physics. 29 (3): 515–546. Bibcode:1982JSP....29..515B. doi:10.1007/BF01342185. S2CID 14956017.
7. Simon Perdrix; Philippe Jorrand (2007-04-04). "Classically Controlled Quantum Computation". Math. Struct. In Comp. Science. 16 (4): 601–620. arXiv:quant-ph/0407008. doi:10.1017/S096012950600538X. S2CID 16142327. Also: Simon Perdrix and Philippe Jorrand (2006). "Classically-Controlled Quantum Computation" (PDF). Math. Struct. In Comp. Science. 16 (4): 601–620. arXiv:quant-ph/0407008. CiteSeerX 10.1.1.252.1823. doi:10.1017/S096012950600538X. S2CID 16142327.
8. Aaronson, Scott (2005). "Quantum computing, postselection, and probabilistic polynomial-time". Proceedings of the Royal Society A. 461 (2063): 3473–3482. arXiv:quant-ph/0412187. Bibcode:2005RSPSA.461.3473A. doi:10.1098/rspa.2005.1546. S2CID 1770389. Preprint available at .
Further reading
• Molina, Abel; Watrous, John (2018). "Revisiting the simulation of quantum Turing machines by quantum circuits". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 475 (2226). arXiv:1808.01701. doi:10.1098/rspa.2018.0767. PMC 6598068. PMID 31293355.
• Iriyama, Satoshi; Ohya, Masanori; Volovich, Igor (2004). "Generalized Quantum Turing Machine and its Application to the SAT Chaos Algorithm". arXiv:quant-ph/0405191.
• Deutsch, D. (1985). "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 400 (1818): 97–117. Bibcode:1985RSPSA.400...97D. CiteSeerX 10.1.1.41.2382. doi:10.1098/rspa.1985.0070. JSTOR 2397601. S2CID 1438116.
External links
• The quantum computer – history
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
|
Wikipedia
|
Quantum ergodicity
In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to initial conditions. Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to energy eigenstates of a quantized ergodic Hamiltonian tend to a uniform distribution in the classical phase space. This is consistent with the intuition that the flows of ergodic systems are equidistributed in phase space. By contrast, classical completely integrable systems generally have periodic orbits in phase space, and this is exhibited in a variety of ways in the high-energy limit of the eigenstates: typically, some form of concentration occurs in the semiclassical limit $\hbar \rightarrow 0$.
The model case of a Hamiltonian is the geodesic Hamiltonian on the cotangent bundle of a compact Riemannian manifold. The quantization of the geodesic flow is given by the fundamental solution of the Schrödinger equation
$U_{t}=\exp(it{\sqrt {\Delta }})$
where ${\sqrt {\Delta }}$ is the square root of the Laplace–Beltrami operator. The quantum ergodicity theorem of Shnirelman 1974, Zelditch, and Yves Colin de Verdière states that a compact Riemannian manifold whose unit tangent bundle is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the nth eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as n → ∞ in a subset of the natural numbers of natural density equal to one. Quantum ergodicity can be formulated as a non-commutative analogue of the classical ergodicity (T. Sunada).
Since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space. Thus, when translating the concept of ergodicity to the quantum realm, it is natural to assume that the eigenstates of the quantum chaotic system would fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit $\hbar \rightarrow 0$. The quantum ergodicity theorems of Shnirelman, Zelditch, and Yves Colin de Verdière proves that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. However, the quantum ergodicity theorem leaves open the possibility of eigenfunctions become sparse with serious holes as $\hbar \rightarrow 0$, leaving large but not macroscopic gaps on the energy manifolds in the phase space. In particular, the theorem allows the existence of a subset of macroscopically nonerdodic states which on the other hand must approach zero measure, i.e., the contribution of this set goes towards zero percent of all eigenstates when $\hbar \rightarrow 0$.[5]
For example, the theorem do not exclude quantum scarring, as the phase space volume of the scars also gradually vanishes in this limit.[1][5][6][2] A quantum eigenstate is scarred by periodic orbit if its probability density is on the classical invariant manifolds near and all along that periodic orbit is systematically enhanced above the classical, statistically expected density along that orbit.[5] In a simplified manner, a quantum scar refers to an eigenstate of whose probability density is enhanced in the neighborhood of a classical periodic orbit when the corresponding classical system is chaotic. In conventional scarring, the responsive periodic orbit is unstable.[1][5][6][2] The instability is a decisive point that separates quantum scars from a more trivial finding that the probability density is enhanced near stable periodic orbits due to the Bohr's correspondence principle. The latter can be viewed as a purely classical phenomenon, whereas in the former quantum interference is important. On the other hand, in the perturbation-induced quantum scarring,[3][7][8][9][4] some of the high-energy eigenstates of a locally perturbed quantum dot contain scars of short periodic orbits of the corresponding unperturbed system. Even though similar in appearance to ordinary quantum scars, these scars have a fundamentally different origin.,[3][7][4] In this type of scarring, there are no periodic orbits in the perturbed classical counterpart or they are too unstable to cause a scar in a conventional sense. Conventional and perturbation-induced scars are both a striking visual example of classical-quantum correspondence and of a quantum suppression of chaos (see the figure). In particular, scars are a significant correction to the assumption that the corresponding eigenstates of a classically chaotic Hamiltonian are only featureless and random. In some sense, scars can be considered as an eigenstate counterpart to the quantum ergodicity theorem of how short periodic orbits provide corrections to the universal random matrix theory eigenvalue statistics.
See also
• Eigenstate thermalization hypothesis
• Ergodic hypothesis
• Quantum chaos
• Scar (physics)
External links
• Shnirelman theorem, Scholarpedia article
References
1. Heller, Eric J. (1984-10-15). "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits". Physical Review Letters. 53 (16): 1515–1518. Bibcode:1984PhRvL..53.1515H. doi:10.1103/PhysRevLett.53.1515.
2. Kaplan, L.; Heller, E. J. (1998-04-10). "Linear and Nonlinear Theory of Eigenfunction Scars". Annals of Physics. 264 (2): 171–206. arXiv:chao-dyn/9809011. Bibcode:1998AnPhy.264..171K. doi:10.1006/aphy.1997.5773. ISSN 0003-4916. S2CID 120635994.
3. Keski-Rahkonen, J.; Ruhanen, A.; Heller, E. J.; Räsänen, E. (2019-11-21). "Quantum Lissajous Scars". Physical Review Letters. 123 (21): 214101. arXiv:1911.09729. Bibcode:2019PhRvL.123u4101K. doi:10.1103/PhysRevLett.123.214101. PMID 31809168. S2CID 208248295.
4. Keski-Rahkonen, Joonas (2020). Quantum Chaos in Disordered Two-Dimensional Nanostructures. Tampere University. ISBN 978-952-03-1699-0.
5. Heller, Eric Johnson (2018). The semiclassical way to dynamics and spectroscopy. Princeton: Princeton University Press. ISBN 978-1-4008-9029-3. OCLC 1034625177.
6. Kaplan, L (1999-01-01). "Scars in quantum chaotic wavefunctions". Nonlinearity. 12 (2): R1–R40. doi:10.1088/0951-7715/12/2/009. ISSN 0951-7715. S2CID 250793219.
7. Luukko, Perttu J. J.; Drury, Byron; Klales, Anna; Kaplan, Lev; Heller, Eric J.; Räsänen, Esa (2016-11-28). "Strong quantum scarring by local impurities". Scientific Reports. 6 (1): 37656. arXiv:1511.04198. Bibcode:2016NatSR...637656L. doi:10.1038/srep37656. ISSN 2045-2322. PMC 5124902. PMID 27892510.
8. Keski-Rahkonen, J.; Luukko, P. J. J.; Kaplan, L.; Heller, E. J.; Räsänen, E. (2017-09-20). "Controllable quantum scars in semiconductor quantum dots". Physical Review B. 96 (9): 094204. arXiv:1710.00585. Bibcode:2017PhRvB..96i4204K. doi:10.1103/PhysRevB.96.094204. S2CID 119083672.
9. Keski-Rahkonen, J; Luukko, P J J; Åberg, S; Räsänen, E (2019-01-21). "Effects of scarring on quantum chaos in disordered quantum wells". Journal of Physics: Condensed Matter. 31 (10): 105301. arXiv:1806.02598. Bibcode:2019JPCM...31j5301K. doi:10.1088/1361-648x/aaf9fb. ISSN 0953-8984. PMID 30566927. S2CID 51693305.
• Shnirelman, A I (1974), Ergodic properties of eigenfunctions, vol. 29(6(180)), Uspekhi Mat. Nauk, Moscow, pp. 181–182
• Zelditch, S (2006), "Quantum ergodicity and mixing of eigenfunctions", in Françoise, Jean-Pierre; Naber, Gregory L.; Tsun, Tsou Sheung (eds.), Encyclopedia of mathematical physics. Vol. 1, 2, 3, 4, 5, Academic Press/Elsevier Science, Oxford, ISBN 9780125126601, MR 2238867
• Sunada, T (1997), "Quantum ergodicity", Trend in Mathematics, Birkhauser Verlag, Basel, pp. 175–196
|
Wikipedia
|
Quaquaversal tiling
The quaquaversal tiling is a nonperiodic tiling of the euclidean 3-space introduced by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern that relies essentially on their previous construct, the pinwheel tiling. The rotations relating these tiles belong to the group G(6,4) generated by two rotations of order 6 and 4 whose axes are perpendicular to each other. These rotations are dense in SO(3).
References
• Conway, John H.; Radin, Charles (1998), "Quaquaversal tilings and rotations", Inventiones Mathematicae, 132 (1): 179–188, Bibcode:1998InMat.132..179C, doi:10.1007/s002220050221, MR 1618635, S2CID 14194250.
• Radin, Charles; Sadun, Lorenzo (1998), "Subgroups of SO(3) associated with tilings", Journal of Algebra, 202 (2): 611–633, doi:10.1006/jabr.1997.7320, MR 1617675.
External links
• A picture of a quaquaversal tiling
• Charles Radin page at the University of Texas
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
|
Wikipedia
|
Quark (hash function)
Quark is a cryptographic hash function (family). It was designed by Jean-Philippe Aumasson, Luca Henzen, Willi Meier and María Naya-Plasencia.
Quark was created because of the expressed need by application designers (notably for implementing RFID protocols) for a lightweight cryptographic hash function. The SHA-3 NIST hash function competition concerned general-purpose designs and focused on software performance.
Quark is a lightweight hash function, based on a single security level and on the sponge construction, to minimize memory requirements. Inspired by the lightweight ciphers Grain and KATAN, the hash function family Quark is composed of the three instances u-Quark, d-Quark, and t-Quark. Hardware benchmarks show that Quark compares well to previous lightweight hashes.
For example, the u-Quark conjecturally instance provides at least 64-bit security against all attacks (collisions, multicollisions, distinguishers, etc.), fits in 1379 gate-equivalents, and consumes in average 2.44 µW at 100 kHz in 0.18 µm ASIC.
External links
• Quark on Jean-Philippe Aumasson's website 131002.net
• Quark on the International Association for Cryptologic Research website iacr.org
|
Wikipedia
|
Quartan prime
In mathematics, a quartan prime is a prime number of the form x4 + y4 where x and y are positive integers. The odd quartan primes are of the form 16n + 1.
For example, 17 is the smallest odd quartan prime: 14 + 24 = 1 + 16 = 17.
With the exception of 2 (x = y = 1), one of x and y will be odd, and the other will be even. If both are odd or even, the resulting integer will be even, and 2 is the only even prime.
The first few quartan primes are
2, 17, 97, 257, 337, 641, 881, … (sequence A002645 in the OEIS).
See also
• Fourth power
• Quartic
References
• Neil Sloane, A Handbook of Integer Sequences, Academic Press, NY, 1973.
Prime number classes
By formula
• Fermat (22n + 1)
• Mersenne (2p − 1)
• Double Mersenne (22p−1 − 1)
• Wagstaff (2p + 1)/3
• Proth (k·2n + 1)
• Factorial (n! ± 1)
• Primorial (pn# ± 1)
• Euclid (pn# + 1)
• Pythagorean (4n + 1)
• Pierpont (2m·3n + 1)
• Quartan (x4 + y4)
• Solinas (2m ± 2n ± 1)
• Cullen (n·2n + 1)
• Woodall (n·2n − 1)
• Cuban (x3 − y3)/(x − y)
• Leyland (xy + yx)
• Thabit (3·2n − 1)
• Williams ((b−1)·bn − 1)
• Mills (⌊A3n⌋)
By integer sequence
• Fibonacci
• Lucas
• Pell
• Newman–Shanks–Williams
• Perrin
• Partitions
• Bell
• Motzkin
By property
• Wieferich (pair)
• Wall–Sun–Sun
• Wolstenholme
• Wilson
• Lucky
• Fortunate
• Ramanujan
• Pillai
• Regular
• Strong
• Stern
• Supersingular (elliptic curve)
• Supersingular (moonshine theory)
• Good
• Super
• Higgs
• Highly cototient
• Unique
Base-dependent
• Palindromic
• Emirp
• Repunit (10n − 1)/9
• Permutable
• Circular
• Truncatable
• Minimal
• Delicate
• Primeval
• Full reptend
• Unique
• Happy
• Self
• Smarandache–Wellin
• Strobogrammatic
• Dihedral
• Tetradic
Patterns
• Twin (p, p + 2)
• Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
• Triplet (p, p + 2 or p + 4, p + 6)
• Quadruplet (p, p + 2, p + 6, p + 8)
• k-tuple
• Cousin (p, p + 4)
• Sexy (p, p + 6)
• Chen
• Sophie Germain/Safe (p, 2p + 1)
• Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
• Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
• Balanced (consecutive p − n, p, p + n)
By size
• Mega (1,000,000+ digits)
• Largest known
• list
Complex numbers
• Eisenstein prime
• Gaussian prime
Composite numbers
• Pseudoprime
• Catalan
• Elliptic
• Euler
• Euler–Jacobi
• Fermat
• Frobenius
• Lucas
• Somer–Lucas
• Strong
• Carmichael number
• Almost prime
• Semiprime
• Sphenic number
• Interprime
• Pernicious
Related topics
• Probable prime
• Industrial-grade prime
• Illegal prime
• Formula for primes
• Prime gap
First 60 primes
• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29
• 31
• 37
• 41
• 43
• 47
• 53
• 59
• 61
• 67
• 71
• 73
• 79
• 83
• 89
• 97
• 101
• 103
• 107
• 109
• 113
• 127
• 131
• 137
• 139
• 149
• 151
• 157
• 163
• 167
• 173
• 179
• 181
• 191
• 193
• 197
• 199
• 211
• 223
• 227
• 229
• 233
• 239
• 241
• 251
• 257
• 263
• 269
• 271
• 277
• 281
List of prime numbers
|
Wikipedia
|
Quarter cubic honeycomb
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.
Quarter cubic honeycomb
TypeUniform honeycomb
FamilyTruncated simplectic honeycomb
Quarter hypercubic honeycomb
Indexing[1] J25,33, A13
W10, G6
Schläfli symbolt0,1{3[4]} or q{4,3,4}
Coxeter-Dynkin diagram = =
Cell types{3,3}
(3.6.6)
Face types{3}, {6}
Vertex figure
(isosceles triangular antiprism)
Space groupFd3m (227)
Coxeter group${\tilde {A}}_{3}$×22, [[3[4]]]
Dualoblate cubille
Cell:
(1/4 of rhombic dodecahedron)
Propertiesvertex-transitive, edge-transitive
It is vertex-transitive with 6 truncated tetrahedra and 2 tetrahedra around each vertex.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
It is one of the 28 convex uniform honeycombs.
The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling.
Its vertex figure is an isosceles antiprism: two equilateral triangles joined by six isosceles triangles.
John Horton Conway calls this honeycomb a truncated tetrahedrille, and its dual oblate cubille.
The vertices and edges represent a Kagome lattice in three dimensions,[2] which is the pyrochlore lattice.
Construction
The quarter cubic honeycomb can be constructed in slab layers of truncated tetrahedra and tetrahedral cells, seen as two trihexagonal tilings. Two tetrahedra are stacked by a vertex and a central inversion. In each trihexagonal tiling, half of the triangles belong to tetrahedra, and half belong to truncated tetrahedra. These slab layers must be stacked with tetrahedra triangles to truncated tetrahedral triangles to construct the uniform quarter cubic honeycomb. Slab layers of hexagonal prisms and triangular prisms can be alternated for elongated honeycombs, but these are also not uniform.
trihexagonal tiling:
Symmetry
Cells can be shown in two different symmetries. The reflection generated form represented by its Coxeter-Dynkin diagram has two colors of truncated cuboctahedra. The symmetry can be doubled by relating the pairs of ringed and unringed nodes of the Coxeter-Dynkin diagram, which can be shown with one colored tetrahedral and truncated tetrahedral cells.
Two uniform colorings
Symmetry ${\tilde {A}}_{3}$, [3[4]] ${\tilde {A}}_{3}$×2, [[3[4]]]
Space group F43m (216) Fd3m (227)
Coloring
Vertex figure
Vertex
figure
symmetry
C3v
[3]
(*33)
order 6
D3d
[2+,6]
(2*3)
order 12
Related polyhedra
The subset of hexagonal faces of this honeycomb contains a regular skew apeirohedron {6,6|3}.
Four sets of parallel planes of trihexagonal tilings exist throughout this honeycomb.
This honeycomb is one of five distinct uniform honeycombs[3] constructed by the ${\tilde {A}}_{3}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A3 honeycombs
Space
group
Fibrifold Square
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
F43m
(216)
1o:2 a1 [3[4]] ${\tilde {A}}_{3}$ (None)
Fm3m
(225)
2−:2 d2 <[3[4]]>
↔ [4,31,1]
↔
${\tilde {A}}_{3}$×21
↔ ${\tilde {B}}_{3}$
1, 2
Fd3m
(227)
2+:2 g2 [[3[4]]]
or [2+[3[4]]]
↔
${\tilde {A}}_{3}$×22 3
Pm3m
(221)
4−:2 d4 <2[3[4]]>
↔ [4,3,4]
↔
${\tilde {A}}_{3}$×41
↔ ${\tilde {C}}_{3}$
4
I3
(204)
8−o r8 [4[3[4]]]+
↔ [[4,3+,4]]
↔
½${\tilde {A}}_{3}$×8
↔ ½${\tilde {C}}_{3}$×2
(*)
Im3m
(229)
8o:2 [4[3[4]]]
↔ [[4,3,4]]
${\tilde {A}}_{3}$×8
↔ ${\tilde {C}}_{3}$×2
5
C3 honeycombs
Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Pm3m
(221)
4−:2 [4,3,4] ×1 1, 2, 3, 4,
5, 6
Fm3m
(225)
2−:2 [1+,4,3,4]
↔ [4,31,1]
↔
Half 7, 11, 12, 13
I43m
(217)
4o:2 [[(4,3,4,2+)]] Half × 2 (7),
Fd3m
(227)
2+:2 [[1+,4,3,4,1+]]
↔ [[3[4]]]
↔
Quarter × 2 10,
Im3m
(229)
8o:2 [[4,3,4]] ×2
(1), 8, 9
The Quarter cubic honeycomb is related to a matrix of 3-dimensional honeycombs: q{2p,4,2q}
Euclidean/hyperbolic(paracompact/noncompact) quarter honeycombs q{p,3,q}
p \ q 4 6 8 ... ∞
4
q{4,3,4}
↔ ↔
q{4,3,6}
↔ ↔
q{4,3,8}
↔
q{4,3,∞}
↔
6 q{6,3,4}
↔ ↔
q{6,3,6}
↔
q{6,3,8}
↔
q{6,3,∞}
↔
8 q{8,3,4}
↔
q{8,3,6}
↔
q{8,3,8}
↔
q{8,3,∞}
↔
... ∞ q{∞,3,4}
↔
q{∞,3,6}
↔
q{∞,3,8}
↔
q{∞,3,∞}
↔
See also
Wikimedia Commons has media related to Quarter cubic honeycomb.
• Truncated simplectic honeycomb
• Triakis truncated tetrahedral honeycomb
• Architectonic and catoptric tessellation
References
1. For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).
2. "Physics Today article on the word kagome".
3. , OEIS sequence A000029 6-1 cases, skipping one with zero marks
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
• Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
• D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
• Klitzing, Richard. "3D Euclidean Honeycombs x3x3o3o3*a - batatoh - O27".
• Uniform Honeycombs in 3-Space: 15-Batatoh
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quarter 5-cubic honeycomb
In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb.[1] Its facets are 5-demicubes and runcinated 5-demicubes.
quarter 5-cubic honeycomb
(No image)
TypeUniform 5-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolq{4,3,3,3,4}
Coxeter-Dynkin diagram =
5-face typeh{4,33},
h4{4,33},
Vertex figure
Rectified 5-cell antiprism
or Stretched birectified 5-simplex
Coxeter group${\tilde {D}}_{5}$×2 = [[31,1,3,31,1]]
Dual
Propertiesvertex-transitive
Related honeycombs
This honeycomb is one of 20 uniform honeycombs constructed by the ${\tilde {D}}_{5}$ Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
D5 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,3,31,1] ${\tilde {D}}_{5}$
<[31,1,3,31,1]>
↔ [31,1,3,3,4]
↔
${\tilde {D}}_{5}$×21 = ${\tilde {B}}_{5}$ , , ,
, , ,
[[31,1,3,31,1]] ${\tilde {D}}_{5}$×22 ,
<2[31,1,3,31,1]>
↔ [4,3,3,3,4]
↔
${\tilde {D}}_{5}$×41 = ${\tilde {C}}_{5}$ , , , , ,
[<2[31,1,3,31,1]>]
↔ [[4,3,3,3,4]]
↔
${\tilde {D}}_{5}$×8 = ${\tilde {C}}_{5}$×2 , ,
See also
Regular and uniform honeycombs in 5-space:
• 5-cube honeycomb
• 5-demicube honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb
Notes
1. Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318
References
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• Klitzing, Richard. "5D Euclidean tesselations#5D". x3o3o x3o3o *b3*e - spaquinoh
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quarter 6-cubic honeycomb
In six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb.[1] Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms.
quarter 6-cubic honeycomb
(No image)
TypeUniform 6-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolq{4,3,3,3,3,4}
Coxeter-Dynkin diagram =
5-face typeh{4,34},
h4{4,34},
{3,3}×{3,3} duoprism
Vertex figure
Coxeter group${\tilde {D}}_{6}$×2 = [[31,1,3,3,31,1]]
Dual
Propertiesvertex-transitive
Related honeycombs
This honeycomb is one of 41 uniform honeycombs constructed by the ${\tilde {D}}_{6}$ Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{6}$ and ${\tilde {C}}_{6}$ constructions:
D6 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,3,3,31,1] ×1 ,
[[31,1,3,3,31,1]] ×2 , , ,
<[31,1,3,3,31,1]>
↔ [31,1,3,3,3,4]
↔
×2 , , , , , , , ,
, , , , , , ,
<2[31,1,3,3,31,1]>
↔ [4,3,3,3,3,4]
↔
×4 ,,
,,
, , , , , , ,
[<2[31,1,3,3,31,1]>]
↔ [[4,3,3,3,3,4]]
↔
×8 , , ,
, , ,
See also
Regular and uniform honeycombs in 5-space:
• 6-cube honeycomb
• 6-demicube honeycomb
• 6-simplex honeycomb
• Truncated 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb
Notes
1. Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318
References
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• Klitzing, Richard. "6D Euclidean tesselations#6D".
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quarter 7-cubic honeycomb
In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb.[1] Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.
quarter 7-cubic honeycomb
(No image)
TypeUniform 7-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolq{4,3,3,3,3,3,4}
Coxeter diagram =
6-face typeh{4,35},
h5{4,35},
{31,1,1}×{3,3} duoprism
Vertex figure
Coxeter group${\tilde {D}}_{7}$×2 = [[31,1,3,3,3,31,1]]
Dual
Propertiesvertex-transitive
Related honeycombs
This honeycomb is one of 77 uniform honeycombs constructed by the ${\tilde {D}}_{7}$ Coxeter group, all but 10 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 77 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{7}$ and ${\tilde {C}}_{7}$ constructions:
D7 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,3,3,3,31,1] ×1 , , , , , ,
[[31,1,3,3,3,31,1]] ×2 , , ,
<[31,1,3,3,3,31,1]>
↔ [31,1,3,3,3,3,4]
↔
×2 ...
<<[31,1,3,3,3,31,1]>>
↔ [4,3,3,3,3,3,4]
↔
×4 ...
[<<[31,1,3,3,3,31,1]>>]
↔ [[4,3,3,3,3,3,4]]
↔
×8 ...
See also
Regular and uniform honeycombs in 7-space:
• 7-cube honeycomb
• 7-demicube honeycomb
• 7-simplex honeycomb
• Truncated 7-simplex honeycomb
• Omnitruncated 7-simplex honeycomb
Notes
1. Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318
References
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• Klitzing, Richard. "7D Euclidean tesselations#7D".
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quarter 8-cubic honeycomb
In seven-dimensional Euclidean geometry, the quarter 8-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 8-demicubic honeycomb, and a quarter of the vertices of a 8-cube honeycomb.[1] Its facets are 8-demicubes h{4,36}, pentic 8-cubes h6{4,36}, {3,3}×{32,1,1} and {31,1,1}×{31,1,1} duoprisms.
quarter 8-cubic honeycomb
(No image)
TypeUniform 8-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolq{4,3,3,3,3,3,3,4}
Coxeter diagram =
7-face typeh{4,36},
h6{4,36},
{3,3}×{32,1,1} duoprism
{31,1,1}×{31,1,1} duoprism
Vertex figure
Coxeter group${\tilde {D}}_{8}$×2 = [[31,1,3,3,3,3,31,1]]
Dual
Propertiesvertex-transitive
See also
Regular and uniform honeycombs in 8-space:
• 8-cube honeycomb
• 8-demicube honeycomb
• 8-simplex honeycomb
• Truncated 8-simplex honeycomb
• Omnitruncated 8-simplex honeycomb
Notes
1. Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318
References
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• Klitzing, Richard. "7D Euclidean tesselations#7D".
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quarter hypercubic honeycomb
In geometry, the quarter hypercubic honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb. It is given a Schläfli symbol q{4,3...3,4} or Coxeter symbol qδ4 representing the regular form with three quarters of the vertices removed and containing the symmetry of Coxeter group ${\tilde {D}}_{n-1}$ for n ≥ 5, with ${\tilde {D}}_{4}$ = ${\tilde {A}}_{4}$ and for quarter n-cubic honeycombs ${\tilde {D}}_{5}$ = ${\tilde {B}}_{5}$.[1]
qδn Name Schläfli
symbol
Coxeter diagrams Facets Vertex figure
qδ3
quarter square tiling
q{4,4} or
or
h{4}={2} { }×{ }
{ }×{ }
qδ4
quarter cubic honeycomb
q{4,3,4} or
or
h{4,3}
h2{4,3}
Elongated
triangular antiprism
qδ5 quarter tesseractic honeycomb q{4,32,4} or
or
h{4,32}
h3{4,32}
{3,4}×{}
qδ6 quarter 5-cubic honeycomb q{4,33,4}
h{4,33}
h4{4,33}
Rectified 5-cell antiprism
qδ7 quarter 6-cubic honeycomb q{4,34,4}
h{4,34}
h5{4,34}
{3,3}×{3,3}
qδ8 quarter 7-cubic honeycomb q{4,35,4}
h{4,35}
h6{4,35}
{3,3}×{3,31,1}
qδ9 quarter 8-cubic honeycomb q{4,36,4}
h{4,36}
h7{4,36}
{3,3}×{3,32,1}
{3,31,1}×{3,31,1}
qδn quarter n-cubic honeycomb q{4,3n-3,4} ... h{4,3n-2} hn-2{4,3n-2} ...
See also
• Hypercubic honeycomb
• Alternated hypercubic honeycomb
• Simplectic honeycomb
• Truncated simplectic honeycomb
• Omnitruncated simplectic honeycomb
References
1. Coxeter, Regular and semi-regular honeycoms, 1988, p.318-319
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
1. pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
2. pp. 154–156: Partial truncation or alternation, represented by q prefix
3. p. 296, Table II: Regular honeycombs, δn+1
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• Klitzing, Richard. "1D-8D Euclidean tesselations".
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quarter order-6 square tiling
In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232 orbifold notation, and can be seen as a half symmetry of *443 and *662, and quarter symmetry of *642.
Quarter order-6 square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex figure3.4.6.6.4
Schläfli symbolq{4,6}
Coxeter diagram = = =
or or
or
Dual ?
PropertiesVertex-transitive
Images
Projections centered on a vertex, triangle and hexagon:
Related polyhedra and tiling
Similar H2 tilings in *3232 symmetry
Coxeter
diagrams
Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image
Dual
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
See also
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
|
Wikipedia
|
Rectified tesseractic honeycomb
In four-dimensional Euclidean geometry, the rectified tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.
quarter cubic honeycomb
(No image)
TypeUniform 4-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolr{4,3,3,4}
r{4,31,1}
r{4,31,1}
q{4,3,3,4}
Coxeter-Dynkin diagram
=
=
=
=
4-face typeh{4,32},
h3{4,32},
Cell type{3,3},
t1{4,3},
Face type{3}
{4}
Edge figure
Square pyramid
Vertex figure
Elongated {3,4}×{}
Coxeter group${\tilde {C}}_{4}$ = [4,3,3,4]
${\tilde {B}}_{4}$ = [4,31,1]
${\tilde {D}}_{4}$ = [31,1,1,1]
Dual
Propertiesvertex-transitive
It is also called a quarter tesseractic honeycomb since it has half the vertices of the 4-demicubic honeycomb, and a quarter of the vertices of a tesseractic honeycomb.[1]
Related honeycombs
The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
C4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1
1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13
[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×6
14, 15, 16, 17
The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.
B4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]: ×1
5, 6, 7, 8
<[4,3,31,1]>:
↔[4,3,3,4]
↔
×2
9, 10, 11, 12, 13, 14,
(10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3]
↔
↔
×3
1, 2, 3, 4
[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×12
20, 21, 22, 23
There are ten uniform honeycombs constructed by the ${\tilde {D}}_{4}$ Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1].
The ten permutations are listed with its highest extended symmetry relation:
D4 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,1,1] ${\tilde {D}}_{4}$ (none)
<[31,1,1,1]>
↔ [31,1,3,4]
↔
${\tilde {D}}_{4}$×2 = ${\tilde {B}}_{4}$ (none)
<2[1,131,1]>
↔ [4,3,3,4]
↔
${\tilde {D}}_{4}$×4 = ${\tilde {C}}_{4}$ 1, 2
[3[3,31,1,1]]
↔ [3,3,4,3]
↔
${\tilde {D}}_{4}$×6 = ${\tilde {F}}_{4}$ 3, 4, 5, 6
[4[1,131,1]]
↔ [[4,3,3,4]]
↔
${\tilde {D}}_{4}$×8 = ${\tilde {C}}_{4}$×2 7, 8, 9
[(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
${\tilde {D}}_{4}$×24 = ${\tilde {F}}_{4}$
[(3,3)[31,1,1,1]]+
↔ [3+,4,3,3]
↔
½${\tilde {D}}_{4}$×24 = ½${\tilde {F}}_{4}$ 10
See also
Regular and uniform honeycombs in 4-space:
• Tesseractic honeycomb
• Demitesseractic honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
Notes
1. Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318
References
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Klitzing, Richard. "4D Euclidean tesselations#4D". o4x3o3o4o, o3o3o *b3x4o, x3o3x *b3o4o, x3o3x *b3o *b3o - rittit - O87
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
|
Wikipedia
|
Quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset $\{1,i,j,k,-1,-i,-j,-k\}$ of the quaternions under multiplication. It is given by the group presentation
$\mathrm {Q} _{8}=\langle {\bar {e}},i,j,k\mid {\bar {e}}^{2}=e,\;i^{2}=j^{2}=k^{2}=ijk={\bar {e}}\rangle ,$
Quaternion group multiplication table (simplified form)
1 i j k
1 1 i j k
i i −1 k −j
j j −k −1 i
k k j −i −1
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
where e is the identity element and e commutes with the other elements of the group.
Another presentation of Q8 is
$\mathrm {Q} _{8}=\langle a,b\mid a^{4}=e,a^{2}=b^{2},ba=a^{-1}b\rangle .$
Compared to dihedral group
The quaternion group Q8 has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs:
Q8 D4
Cayley graph
Red arrows connect g→gi, green connect g→gj.
Cycle graph
In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a subset of the quaternions.
Cayley table
The Cayley table (multiplication table) for Q8 is given by:[1]
×eeiijjkk
e eeiijjkk
e eeiijjkk
i iieekkjj
i iieekkjj
j jjkkeeii
j jjkkeeii
k kkjjiiee
k kkjjiiee
Properties
The elements i, j, and k all have order four in Q8 and any two of them generate the entire group. Another presentation of Q8[2] based in only two elements to skip this redundancy is:
$\left\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\right\rangle .$
One may take, for instance, $i=x,j=y,$ and $k=xy$.
The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal.[3] Every Hamiltonian group contains a copy of Q8.[4]
The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.
The center and the commutator subgroup of Q8 is the subgroup $\{e,{\bar {e}}\}$. The inner automorphism group of Q8 is given by the group modulo its center, i.e. the factor group $\mathrm {Q} _{8}/\{e,{\bar {e}}\},$ which is isomorphic to the Klein four-group V. The full automorphism group of Q8 is isomorphic to S4, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q8 is thus S4/V, which is isomorphic to S3.
The quaternion group Q8 has five conjugacy classes, $\{e\},\{{\bar {e}}\},\{i,{\bar {i}}\},\{j,{\bar {j}}\},\{k,{\bar {k}}\},$ and so five irreducible representations over the complex numbers, with dimensions 1, 1, 1, 1, 2:
Trivial representation.
Sign representations with i, j, k-kernel: Q8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup N, we obtain a one-dimensional representation factoring through the 2-element quotient group G/N. The representation sends elements of N to 1, and elements outside N to −1.
2-dimensional representation: Described below in Matrix representations.
The character table of Q8 turns out to be the same as that of D4:
Representation(ρ)/Conjugacy class{ e }{ e }{ i, i }{ j, j }{ k, k }
Trivial representation11111
Sign representation with i-kernel111−1−1
Sign representation with j-kernel11−11−1
Sign representation with k-kernel11−1−11
2-dimensional representation2−2000
Since the irreducible characters $\chi _{\rho }$ in the rows above have real values, this gives the decomposition of the real group algebra of $G=\mathrm {Q} _{8}$ into minimal two-sided ideals:
$\mathbb {R} [\mathrm {Q} _{8}]=\bigoplus _{\rho }(e_{\rho }),$
where the idempotents $e_{\rho }\in \mathbb {R} [\mathrm {Q} _{8}]$ correspond to the irreducibles:
$e_{\rho }={\frac {\dim(\rho )}{|G|}}\sum _{g\in G}\chi _{\rho }(g^{-1})g,$
so that
${\begin{aligned}e_{\text{triv}}&={\tfrac {1}{8}}(e+{\bar {e}}+i+{\bar {i}}+j+{\bar {j}}+k+{\bar {k}})\\e_{i{\text{-ker}}}&={\tfrac {1}{8}}(e+{\bar {e}}+i+{\bar {i}}-j-{\bar {j}}-k-{\bar {k}})\\e_{j{\text{-ker}}}&={\tfrac {1}{8}}(e+{\bar {e}}-i-{\bar {i}}+j+{\bar {j}}-k-{\bar {k}})\\e_{k{\text{-ker}}}&={\tfrac {1}{8}}(e+{\bar {e}}-i-{\bar {i}}-j-{\bar {j}}+k+{\bar {k}})\\e_{2}&={\tfrac {2}{8}}(2e-2{\bar {e}})={\tfrac {1}{2}}(e-{\bar {e}})\end{aligned}}$
Each of these irreducible ideals is isomorphic to a real central simple algebra, the first four to the real field $\mathbb {R} $. The last ideal $(e_{2})$ is isomorphic to the skew field of quaternions $\mathbb {H} $ by the correspondence:
${\begin{aligned}{\tfrac {1}{2}}(e-{\bar {e}})&\longleftrightarrow 1,\\{\tfrac {1}{2}}(i-{\bar {i}})&\longleftrightarrow i,\\{\tfrac {1}{2}}(j-{\bar {j}})&\longleftrightarrow j,\\{\tfrac {1}{2}}(k-{\bar {k}})&\longleftrightarrow k.\end{aligned}}$
Furthermore, the projection homomorphism $\mathbb {R} [\mathrm {Q} _{8}]\to (e_{2})\cong \mathbb {H} $ given by $r\mapsto re_{2}$ has kernel ideal generated by the idempotent:
$e_{2}^{\perp }=e_{1}+e_{i{\text{-ker}}}+e_{j{\text{-ker}}}+e_{k{\text{-ker}}}={\frac {1}{2}}(e+{\bar {e}}),$
so the quaternions can also be obtained as the quotient ring $\mathbb {R} [\mathrm {Q} _{8}]/(e+{\bar {e}})\cong \mathbb {H} $.
The complex group algebra is thus $\mathbb {C} [\mathrm {Q} _{8}]\cong \mathbb {C} ^{\oplus 4}\oplus M_{2}(\mathbb {C} ),$ where $M_{2}(\mathbb {C} )\cong \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} \cong \mathbb {H} \oplus \mathbb {H} $ is the algebra of biquaternions.
Matrix representations
The two-dimensional irreducible complex representation described above gives the quaternion group Q8 as a subgroup of the general linear group $\operatorname {GL} (2,\mathbb {C} )$. The quaternion group is a multiplicative subgroup of the quaternion algebra:
$\mathbb {H} =\mathbb {R} 1+\mathbb {R} i+\mathbb {R} j+\mathbb {R} k=\mathbb {C} 1+\mathbb {C} j,$
which has a regular representation $\rho :\mathbb {H} \to \operatorname {M} (2,\mathbb {C} )$ :\mathbb {H} \to \operatorname {M} (2,\mathbb {C} )} by left multiplication on itself considered as a complex vector space with basis $\{1,j\},$ so that $z\in \mathbb {H} $ corresponds to the $\mathbb {C} $-linear mapping $\rho _{z}:a+jb\mapsto z\cdot (a+jb).$ The resulting representation
${\begin{cases}\rho :\mathrm {Q} _{8}\to \operatorname {GL} (2,\mathbb {C} )\\g\longmapsto \rho _{g}\end{cases}}$ :\mathrm {Q} _{8}\to \operatorname {GL} (2,\mathbb {C} )\\g\longmapsto \rho _{g}\end{cases}}}
is given by:
${\begin{matrix}e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}&i\mapsto {\begin{pmatrix}i&0\\0&\!\!\!\!-i\end{pmatrix}}&j\mapsto {\begin{pmatrix}0&\!\!\!\!-1\\1&0\end{pmatrix}}&k\mapsto {\begin{pmatrix}0&\!\!\!\!-i\\\!\!\!-i&0\end{pmatrix}}\\{\overline {e}}\mapsto {\begin{pmatrix}\!\!\!-1&0\\0&\!\!\!\!-1\end{pmatrix}}&{\overline {i}}\mapsto {\begin{pmatrix}\!\!\!-i&0\\0&i\end{pmatrix}}&{\overline {j}}\mapsto {\begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}}&{\overline {k}}\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}}.\end{matrix}}$
Since all of the above matrices have unit determinant, this is a representation of Q8 in the special linear group $\operatorname {SL} (2,\mathbb {C} )$.[5]
A variant gives a representation by unitary matrices (table at right). Let $g\in \mathrm {Q} _{8}$ correspond to the linear mapping $\rho _{g}:a+bj\mapsto (a+bj)\cdot jg^{-1}j^{-1},$ so that $\rho :\mathrm {Q} _{8}\to \operatorname {SU} (2)$ :\mathrm {Q} _{8}\to \operatorname {SU} (2)} is given by:
${\begin{matrix}e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}&i\mapsto {\begin{pmatrix}i&0\\0&\!\!\!\!-i\end{pmatrix}}&j\mapsto {\begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}}&k\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}}\\{\overline {e}}\mapsto {\begin{pmatrix}\!\!\!-1&0\\0&\!\!\!\!-1\end{pmatrix}}&{\overline {i}}\mapsto {\begin{pmatrix}\!\!\!-i&0\\0&i\end{pmatrix}}&{\overline {j}}\mapsto {\begin{pmatrix}0&\!\!\!\!-1\\1&0\end{pmatrix}}&{\overline {k}}\mapsto {\begin{pmatrix}0&\!\!\!\!-i\\\!\!\!-i&0\end{pmatrix}}.\end{matrix}}$
It is worth noting that physicists exclusively use a different convention for the $\operatorname {SU} (2)$ matrix representation to make contact with the usual Pauli matrices:
${\begin{matrix}&e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\quad \,1_{2\times 2}&i\mapsto {\begin{pmatrix}0&\!\!\!-i\!\\\!\!-i\!\!&0\end{pmatrix}}=-i\sigma _{x}&j\mapsto {\begin{pmatrix}0&\!\!\!-1\!\\1&0\end{pmatrix}}=-i\sigma _{y}&k\mapsto {\begin{pmatrix}\!\!-i\!\!&0\\0&i\end{pmatrix}}=-i\sigma _{z}\\&{\overline {e}}\mapsto {\begin{pmatrix}\!\!-1\!&0\\0&\!\!\!-1\!\end{pmatrix}}=-1_{2\times 2}&{\overline {i}}\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}}=\,\,\,\,i\sigma _{x}&{\overline {j}}\mapsto {\begin{pmatrix}0&1\\\!\!-1\!\!&0\end{pmatrix}}=\,\,\,\,i\sigma _{y}&{\overline {k}}\mapsto {\begin{pmatrix}i&0\\0&\!\!\!-i\!\end{pmatrix}}=\,\,\,\,i\sigma _{z}.\end{matrix}}$
This particular choice is convenient and elegant when one describes spin-1/2 states in the $({\vec {J}}^{2},J_{z})$ basis and considers angular momentum ladder operators $J_{\pm }=J_{x}\pm iJ_{y}.$
There is also an important action of Q8 on the 2-dimensional vector space over the finite field $\mathbb {F} _{3}=\{0,1,-1\}$ (table at right). A modular representation $\rho :\mathrm {Q} _{8}\to \operatorname {SL} (2,3)$ :\mathrm {Q} _{8}\to \operatorname {SL} (2,3)} is given by
${\begin{matrix}e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}&i\mapsto {\begin{pmatrix}1&1\\1&\!\!\!\!-1\end{pmatrix}}&j\mapsto {\begin{pmatrix}\!\!\!-1&1\\1&1\end{pmatrix}}&k\mapsto {\begin{pmatrix}0&\!\!\!\!-1\\1&0\end{pmatrix}}\\{\overline {e}}\mapsto {\begin{pmatrix}\!\!\!-1&0\\0&\!\!\!\!-1\end{pmatrix}}&{\overline {i}}\mapsto {\begin{pmatrix}\!\!\!-1&\!\!\!\!-1\\\!\!\!-1&1\end{pmatrix}}&{\overline {j}}\mapsto {\begin{pmatrix}1&\!\!\!\!-1\\\!\!\!-1&\!\!\!\!-1\end{pmatrix}}&{\overline {k}}\mapsto {\begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}}.\end{matrix}}$
This representation can be obtained from the extension field:
$\mathbb {F} _{9}=\mathbb {F} _{3}[k]=\mathbb {F} _{3}1+\mathbb {F} _{3}k,$
where $k^{2}=-1$ and the multiplicative group $\mathbb {F} _{9}^{\times }$ has four generators, $\pm (k\pm 1),$ of order 8. For each $z\in \mathbb {F} _{9},$ the two-dimensional $\mathbb {F} _{3}$-vector space $\mathbb {F} _{9}$ admits a linear mapping:
${\begin{cases}\mu _{z}:\mathbb {F} _{9}\to \mathbb {F} _{9}\\\mu _{z}(a+bk)=z\cdot (a+bk)\end{cases}}$
In addition we have the Frobenius automorphism $\phi (a+bk)=(a+bk)^{3}$ satisfying $\phi ^{2}=\mu _{1}$ and $\phi \mu _{z}=\mu _{\phi (z)}\phi .$ Then the above representation matrices are:
${\begin{aligned}\rho ({\bar {e}})&=\mu _{-1},\\\rho (i)&=\mu _{k+1}\phi ,\\\rho (j)&=\mu _{k-1}\phi ,\\\rho (k)&=\mu _{k}.\end{aligned}}$
This representation realizes Q8 as a normal subgroup of GL(2, 3). Thus, for each matrix $m\in \operatorname {GL} (2,3)$, we have a group automorphism
${\begin{cases}\psi _{m}:\mathrm {Q} _{8}\to \mathrm {Q} _{8}\\\psi _{m}(g)=mgm^{-1}\end{cases}}$
with $\psi _{I}=\psi _{-I}=\mathrm {id} _{\mathrm {Q} _{8}}.$ In fact, these give the full automorphism group as:
$\operatorname {Aut} (\mathrm {Q} _{8})\cong \operatorname {PGL} (2,3)=\operatorname {GL} (2,3)/\{\pm I\}\cong S_{4}.$
This is isomorphic to the symmetric group S4 since the linear mappings $m:\mathbb {F} _{3}^{2}\to \mathbb {F} _{3}^{2}$ permute the four one-dimensional subspaces of $\mathbb {F} _{3}^{2},$ i.e., the four points of the projective space $\mathbb {P} ^{1}(\mathbb {F} _{3})=\operatorname {PG} (1,3).$
Also, this representation permutes the eight non-zero vectors of $\mathbb {F} _{3}^{2},$ giving an embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations.
Galois group
As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field over Q of the polynomial
$x^{8}-72x^{6}+180x^{4}-144x^{2}+36$.
The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.[6]
Generalized quaternion group
A generalized quaternion group Q4n of order 4n is defined by the presentation[2]
$\langle x,y\mid x^{2n}=y^{4}=1,x^{n}=y^{2},y^{-1}xy=x^{-1}\rangle $
for an integer n ≥ 2, with the usual quaternion group given by n = 2.[7] Coxeter calls Q4n the dicyclic group $\langle 2,2,n\rangle $, a special case of the binary polyhedral group $\langle \ell ,m,n\rangle $ and related to the polyhedral group $(p,q,r)$ and the dihedral group $(2,2,n)$. The generalized quaternion group can be realized as the subgroup of $\operatorname {GL} _{2}(\mathbb {C} )$ generated by
$\left({\begin{array}{cc}\omega _{n}&0\\0&{\overline {\omega }}_{n}\end{array}}\right){\mbox{ and }}\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)$
where $\omega _{n}=e^{i\pi /n}$.[2] It can also be realized as the subgroup of unit quaternions generated by[8] $x=e^{i\pi /n}$ and $y=j$.
The generalized quaternion groups have the property that every abelian subgroup is cyclic.[9] It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.[10] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group.[11] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL2(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL2(F) is 2n, where n = ord2(p2 − 1) + ord2(r).
The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2,[12] which admits the presentation
$\langle x,y\mid x^{2^{m}}=y^{4}=1,x^{2^{m-1}}=y^{2},y^{-1}xy=x^{-1}\rangle .$
See also
• 16-cell
• Binary tetrahedral group
• Clifford algebra
• Dicyclic group
• Hurwitz integral quaternion
• List of small groups
Notes
1. See also a table from Wolfram Alpha
2. Johnson 1980, pp. 44–45
3. See Hall (1999), p. 190
4. See Kurosh (1979), p. 67
5. Artin 1991
6. Dean, Richard (1981). "A Rational Polynomial whose Group is the Quaternions". The American Mathematical Monthly. 88 (1): 42–45. doi:10.2307/2320711. JSTOR 2320711.
7. Some authors (e.g., Rotman 1995, pp. 87, 351) refer to this group as the dicyclic group, reserving the name generalized quaternion group to the case where n is a power of 2.
8. Brown 1982, p. 98
9. Brown 1982, p. 101, exercise 1
10. Cartan & Eilenberg 1999, Theorem 11.6, p. 262
11. Brown 1982, Theorem 4.3, p. 99
12. Roman, Steven (2011). Fundamentals of Group Theory: An Advanced Approach. Springer. pp. 347–348. ISBN 9780817683016.
References
• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2
• Brown, Kenneth S. (1982), Cohomology of groups (3rd ed.), Springer-Verlag, ISBN 978-0-387-90688-1
• Cartan, Henri; Eilenberg, Samuel (1999), Homological Algebra, Princeton University Press, ISBN 978-0-691-04991-5
• Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.
• Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", American Mathematical Monthly 88:42–5.
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
• Johnson, David L. (1980), Topics in the theory of group presentations, Cambridge University Press, ISBN 978-0-521-23108-4, MR 0695161
• Rotman, Joseph J. (1995), An introduction to the theory of groups (4th ed.), Springer-Verlag, ISBN 978-0-387-94285-8
• P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32.
• Hall, Marshall (1999), The theory of groups (2nd ed.), AMS Bookstore, ISBN 0-8218-1967-4
• Kurosh, Alexander G. (1979), Theory of Groups, AMS Bookstore, ISBN 0-8284-0107-1
External links
• Weisstein, Eric W. "Quaternion group". MathWorld.
• Quaternion groups on GroupNames
• Quaternion group on GroupProps
• Conrad, Keith. "Generalized Quaternions"
|
Wikipedia
|
Quartic equation
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is
$ax^{4}+bx^{3}+cx^{2}+dx+e=0\,$
where a ≠ 0.
The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value).
History
Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately.[1] The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna (1545).
The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois before his death in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result.[2]
Solving a quartic equation, special cases
Consider a quartic equation expressed in the form $a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}=0$:
There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is non-zero. However, since the general method is quite complex and susceptible to errors in execution, it is better to apply one of the special cases listed below if possible.
Degenerate case
If the constant term a4 = 0, then one of the roots is x = 0, and the other roots can be found by dividing by x, and solving the resulting cubic equation,
$a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}=0.\,$
Evident roots: 1 and −1 and −k
Call our quartic polynomial Q(x). Since 1 raised to any power is 1,
$Q(1)=a_{0}+a_{1}+a_{2}+a_{3}+a_{4}\ .$
Thus if $\ a_{0}+a_{1}+a_{2}+a_{3}+a_{4}=0\ ,$ Q(1) = 0 and so x = 1 is a root of Q(x). It can similarly be shown that if $\ a_{0}+a_{2}+a_{4}=a_{1}+a_{3}\ ,$ x = −1 is a root.
In either case the full quartic can then be divided by the factor (x − 1) or (x + 1) respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots.
If $\ a_{1}=a_{0}k\ ,$ $\ a_{2}=0\ $ and $\ a_{4}=a_{3}k\ ,$ then $\ x=-k\ $ is a root of the equation. The full quartic can then be factorized this way:
$\ a_{0}x^{4}+a_{0}kx^{3}+a_{3}x+a_{3}k=a_{0}x^{3}(x+k)+a_{3}(x+k)=(a_{0}x^{3}+a_{3})(x+k)\ .$
Alternatively, if $\ a_{1}=a_{0}k\ ,$ $\ a_{3}=a_{2}k\ ,$ and $\ a_{4}=0\ ,$ then x = 0 and x = −k become two known roots. Q(x) divided by x(x + k) is a quadratic polynomial.
Biquadratic equations
A quartic equation where a3 and a1 are equal to 0 takes the form
$a_{0}x^{4}+a_{2}x^{2}+a_{4}=0\,\!$
and thus is a biquadratic equation, which is easy to solve: let $z=x^{2}$, so our equation turns to
$a_{0}z^{2}+a_{2}z+a_{4}=0\,\!$
which is a simple quadratic equation, whose solutions are easily found using the quadratic formula:
$z={\frac {-a_{2}\pm {\sqrt {a_{2}^{2}-4a_{0}a_{4}}}}{2a_{0}}}\,\!$
When we've solved it (i.e. found these two z values), we can extract x from them
$x_{1}=+{\sqrt {z_{+}}}\,\!$
$x_{2}=-{\sqrt {z_{+}}}\,\!$
$x_{3}=+{\sqrt {z_{-}}}\,\!$
$x_{4}=-{\sqrt {z_{-}}}\,\!$
If either of the z solutions were negative or complex numbers, then some of the x solutions are complex numbers.
Quasi-symmetric equations
$a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}mx+a_{0}m^{2}=0\,$
Steps:
1. Divide by x 2.
2. Use variable change z = x + m/x.
3. So, z 2 = x 2 + (m/x) 2 + 2m.
This leads to:
$a_{0}(x^{2}+m^{2}/x^{2})+a_{1}(x+m/x)+a_{2}=0$,
$a_{0}(z^{2}-2m)+a_{1}(z)+a_{2}=0$,
$z^{2}+(a_{1}/a_{0})z+(a_{2}/a_{0}-2m)=0$ (a quadratic in z = x + m/x)
Multiple roots
If the quartic has a double root, it can be found by taking the polynomial greatest common divisor with its derivative. Then they can be divided out and the resulting quadratic equation solved.
In general, there exist only four possible cases of quartic equations with multiple roots, which are listed below:[3]
1. Multiplicity-4 (M4): when the general quartic equation can be expressed as $a(x-l)^{4}=0$, for some real number $l$. This case can always be reduced to a biquadratic equation.
2. Multiplicity-3 (M3): when the general quartic equation can be expressed as $a(x-l)^{3}(x-m)=0$, where $l$ and $m$ are a couple of two different real numbers. This is the only case that can never be reduced to a biquadratic equation.
3. Double Multiplicity-2 (DM2): when the general quartic equation can be expressed as $a(x-l)^{2}(x-m)^{2}=0$, where $l$ and $m$ are a couple of two different real numbers or a couple of non-real complex conjugate numbers. This case can also always be reduced to a biquadratic equation.
4. Single Multiplicity-2 (SM2): when the general quartic equation can be expressed as $a(x-l)^{2}(x-m)(x-n)=0$, where $l$, $m$, and $n$ are three different real numbers or $l$ is a real number and $m$ and $n$ are a couple of non-real complex conjugate numbers. This case is divided into two subcases, those that can be reduced to a biquadratic equation and those in which this is impossible.
So, if the three non-monic coefficients of the depressed quartic equation in terms of the five coefficients of the general quartic equation are given as follows: $p={\frac {8ac-3b^{2}}{8a^{2}}}$, $q={\frac {b^{3}-4abc+8a^{2}d}{8a^{3}}}$ and $r={\frac {16ab^{2}c-64a^{2}bd-3b^{4}+256a^{3}e}{256a^{4}}}$; then, the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are exposed below.
• M4. The general quartic equation corresponds to this case whenever $p=q=r=0$, so the four roots of this equation are given as follows: $x_{1}=x_{2}=x_{3}=x_{4}=-{\frac {b}{4a}}$.
• M3. The general quartic equation corresponds to this case whenever $p^{2}=-12r>0$ and $27q^{2}=-8p^{3}>0$, so the four roots of this equation are given as follows: $x_{1}=x_{2}=x_{3}={\sqrt {-{\frac {p}{6}}}}-{\frac {b}{4a}}$ and $x_{4}=-{\sqrt {-{\frac {3p}{2}}}}-{\frac {b}{4a}}$ , whether $q>0$; otherwise, $x_{1}=x_{2}=x_{3}=-{\sqrt {-{\frac {p}{6}}}}-{\frac {b}{4a}}$ and $x_{4}={\sqrt {-{\frac {3p}{2}}}}-{\frac {b}{4a}}$.
• DM2. The general quartic equation corresponds to this case whenever $p^{2}=4r>0=q$, so the four roots of this equation are given as follows: $x_{1}=x_{3}={\sqrt {-{\frac {p}{2}}}}-{\frac {b}{4a}}$ and $x_{2}=x_{4}=-{\sqrt {-{\frac {p}{2}}}}-{\frac {b}{4a}}$.
• Biquadratic SM2. The general quartic equation corresponds to this subcase of the SM2 equations whenever $p\neq q=r=0$, so the four roots of this equation are given as follows: $x_{1}=x_{2}=-{\frac {b}{4a}}$, $x_{3}={\sqrt {-p}}-{\frac {b}{4a}}$ and $x_{4}=-{\sqrt {-p}}-{\frac {b}{4a}}$.
• Non-Biquadratic SM2. The general quartic equation corresponds to this subcase of the SM2 equations whenever $(p^{2}+12r)^{3}=[p(p^{2}-36r)+{\frac {27}{2}}q^{2}]^{2}>0\neq {q}$, so the four roots of this equation are given by the following formula:[4] $x={\frac {1}{2}}\left[\xi {\sqrt {s_{1}}}\pm {\sqrt {2{\biggl (}s_{2}-{\frac {\xi q}{\sqrt {s_{1}}}}{\biggr )}}}\right]-{\frac {b}{4a}}$, where $s_{1}={\frac {9q^{2}-32pr}{p^{2}+12r}}>0$, $s_{2}=-{\frac {2p(p^{2}-4r)+9q^{2}}{2(p^{2}+12r)}}\neq 0$ and $\xi =\pm 1$.
The general case
To begin, the quartic must first be converted to a depressed quartic.
Converting to a depressed quartic
Let
$\ Ax^{4}+Bx^{3}+Cx^{2}+Dx+E=0\ $
(1')
be the general quartic equation which it is desired to solve. Divide both sides by A,
$\ x^{4}+{B \over A}x^{3}+{C \over A}x^{2}+{D \over A}x+{E \over A}=0\ .$
The first step, if B is not already zero, should be to eliminate the x3 term. To do this, change variables from x to u, such that
$\ x=u-{B \over 4A}\ .$
Then
$\ \left(u-{B \over 4A}\right)^{4}+{B \over A}\left(u-{B \over 4A}\right)^{3}+{C \over A}\left(u-{B \over 4A}\right)^{2}+{D \over A}\left(u-{B \over 4A}\right)+{E \over A}=0\ .$
Expanding the powers of the binomials produces
$\ \left(u^{4}-{B \over A}u^{3}+{6u^{2}B^{2} \over 16A^{2}}-{4uB^{3} \over 64A^{3}}+{B^{4} \over 256A^{4}}\right)+{B \over A}\left(u^{3}-{3u^{2}B \over 4A}+{3uB^{2} \over 16A^{2}}-{B^{3} \over 64A^{3}}\right)+{C \over A}\left(u^{2}-{uB \over 2A}+{B^{2} \over 16A^{2}}\right)+{D \over A}\left(u-{B \over 4A}\right)+{E \over A}=0\ .$
Collecting the same powers of u yields
$\ u^{4}+\left({-3B^{2} \over 8A^{2}}+{C \over A}\right)u^{2}+\left({B^{3} \over 8A^{3}}-{BC \over 2A^{2}}+{D \over A}\right)u+\left({-3B^{4} \over 256A^{4}}+{CB^{2} \over 16A^{3}}-{BD \over 4A^{2}}+{E \over A}\right)=0\ .$
Now rename the coefficients of u. Let
${\begin{aligned}a&={-3B^{2} \over 8A^{2}}+{C \over A}\ ,\\b&={B^{3} \over 8A^{3}}-{BC \over 2A^{2}}+{D \over A}\ ,\\c&={-3B^{4} \over 256A^{4}}+{CB^{2} \over 16A^{3}}-{BD \over 4A^{2}}+{E \over A}\ .\end{aligned}}$
The resulting equation is
$\ u^{4}+au^{2}+bu+c=0\ $
(1)
which is a depressed quartic equation.
If $\ b=0\ $ then we have the special case of a biquadratic equation, which is easily solved, as explained above. Note that the general solution, given below, will not work for the special case $\ b=0\ .$ The equation must be solved as a biquadratic.
In either case, once the depressed quartic is solved for u, substituting those values into
$\ x=u-{B \over 4A}\ $
produces the values for x that solve the original quartic.
Solving a depressed quartic when b ≠ 0
After converting to a depressed quartic equation
$u^{4}+au^{2}+bu+c=0$
and excluding the special case b = 0, which is solved as a biquadratic, we assume from here on that b ≠ 0 .
We will separate the terms left and right as
$u^{4}=-au^{2}-bu-c$
and add in terms to both sides which make them both into perfect squares.
Let y be any solution of this cubic equation:
$2y^{3}-ay^{2}-2cy+(ac-{\tfrac {1}{4}}b^{2})=(2y-a)(y^{2}-c)-{\tfrac {1}{4}}b^{2}=0\ .$
Then (since b ≠ 0)
$2y-a\neq 0$
so we may divide by it, giving
$y^{2}-c={\frac {b^{2}}{4(2y-a)}}\ .$
Then
$(u^{2}+y)^{2}=u^{4}+2yu^{2}+y^{2}=(2y-a)u^{2}-bu+(y^{2}-c)=(2y-a)u^{2}-bu+{\frac {b^{2}}{\ 4(2y-a)\ }}=\left({\sqrt {2y-a\ }}\,u-{\frac {b}{2{\sqrt {2y-a\ }}}}\right)^{2}\ .$
Subtracting, we get the difference of two squares which is the product of the sum and difference of their roots
$(u^{2}+y)^{2}-\left({\sqrt {2y-a\ }}\,u-{\frac {b}{2{\sqrt {2y-a\ }}}}\right)^{2}=\left(u^{2}+y+{\sqrt {2y-a\ }}\,u-{\frac {b}{2{\sqrt {2y-a\ }}}}\right)\left(u^{2}+y-{\sqrt {2y-a\ }}\,u+{\frac {b}{2{\sqrt {2y-a\ }}}}\right)=0$
which can be solved by applying the quadratic formula to each of the two factors. So the possible values of u are:
$u={\tfrac {1}{2}}\left(-{\sqrt {2y-a\ }}+{\sqrt {-2y-a+{\frac {2b}{\sqrt {2y-a\ }}}\ }}\right)\ ,$
$u={\tfrac {1}{2}}\left(-{\sqrt {2y-a\ }}-{\sqrt {-2y-a+{\frac {2b}{\sqrt {2y-a\ }}}\ }}\right)\ ,$
$u={\tfrac {1}{2}}\left({\sqrt {2y-a\ }}+{\sqrt {-2y-a-{\frac {2b}{\sqrt {2y-a\ }}}\ }}\right)\ ,$ or
$u={\tfrac {1}{2}}\left({\sqrt {2y-a\ }}-{\sqrt {-2y-a-{\frac {2b}{\sqrt {2y-a\ }}}\ }}\right)\ .$
Using another y from among the three roots of the cubic simply causes these same four values of u to appear in a different order. The solutions of the cubic are:
$\ y={\frac {a}{6}}+w-{\frac {p}{3w}}\ $
$\ w={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}\ }}\ }}$
using any one of the three possible cube roots. A wise strategy is to choose the sign of the square-root that makes the absolute value of w as large as possible.
$\ p=-{\frac {a^{2}}{12}}-c\ ,$
$\ q=-{\frac {a^{3}}{108}}+{\frac {ac}{3}}-{\frac {b^{2}}{8}}\ .$
Ferrari's solution
Otherwise, the depressed quartic can be solved by means of a method discovered by Lodovico Ferrari. Once the depressed quartic has been obtained, the next step is to add the valid identity
$\left(u^{2}+\alpha \right)^{2}-u^{4}-2\alpha u^{2}=\alpha ^{2}$
to equation (1), yielding
$\left(u^{2}+\alpha \right)^{2}+\beta u+\gamma =\alpha u^{2}+\alpha ^{2}.$
(2)
The effect has been to fold up the u4 term into a perfect square: (u2 + α)2. The second term, αu2 did not disappear, but its sign has changed and it has been moved to the right side.
The next step is to insert a variable y into the perfect square on the left side of equation (2), and a corresponding 2y into the coefficient of u2 in the right side. To accomplish these insertions, the following valid formulas will be added to equation (2),
${\begin{aligned}(u^{2}+\alpha +y)^{2}-(u^{2}+\alpha )^{2}&=2y(u^{2}+\alpha )+y^{2}\ \ \\&=2yu^{2}+2y\alpha +y^{2},\end{aligned}}$
and
$0=(\alpha +2y)u^{2}-2yu^{2}-\alpha u^{2}\,$
These two formulas, added together, produce
$\left(u^{2}+\alpha +y\right)^{2}-\left(u^{2}+\alpha \right)^{2}=\left(\alpha +2y\right)u^{2}-\alpha u^{2}+2y\alpha +y^{2}\qquad \qquad (y{\hbox{-insertion}})\,$
which added to equation (2) produces
$\left(u^{2}+\alpha +y\right)^{2}+\beta u+\gamma =\left(\alpha +2y\right)u^{2}+\left(2y\alpha +y^{2}+\alpha ^{2}\right).\,$
This is equivalent to
$(u^{2}+\alpha +y)^{2}=(\alpha +2y)u^{2}-\beta u+(y^{2}+2y\alpha +\alpha ^{2}-\gamma ).$
(3)
The objective now is to choose a value for y such that the right side of equation (3) becomes a perfect square. This can be done by letting the discriminant of the quadratic function become zero. To explain this, first expand a perfect square so that it equals a quadratic function:
$\left(su+t\right)^{2}=\left(s^{2}\right)u^{2}+\left(2st\right)u+\left(t^{2}\right).\,$
The quadratic function on the right side has three coefficients. It can be verified that squaring the second coefficient and then subtracting four times the product of the first and third coefficients yields zero:
$\left(2st\right)^{2}-4\left(s^{2}\right)\left(t^{2}\right)=0.\,$
Therefore to make the right side of equation (3) into a perfect square, the following equation must be solved:
$(-\beta )^{2}-4\left(2y+\alpha \right)\left(y^{2}+2y\alpha +\alpha ^{2}-\gamma \right)=0.\,$
Multiply the binomial with the polynomial,
$\beta ^{2}-4\left(2y^{3}+5\alpha y^{2}+\left(4\alpha ^{2}-2\gamma \right)y+\left(\alpha ^{3}-\alpha \gamma \right)\right)=0\,$
Divide both sides by −4, and move the −β2/4 to the right,
$2y^{3}+5\alpha y^{2}+\left(4\alpha ^{2}-2\gamma \right)y+\left(\alpha ^{3}-\alpha \gamma -{\frac {\beta ^{2}}{4}}\right)=0$
Divide both sides by 2,
$y^{3}+{\frac {5}{2}}\alpha y^{2}+\left(2\alpha ^{2}-\gamma \right)y+\left({\alpha ^{3} \over 2}-{\alpha \gamma \over 2}-{\beta ^{2} \over 8}\right)=0.$
(4)
This is a cubic equation in y. Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.
Folding the second perfect square
With the value for y so selected, it is now known that the right side of equation (3) is a perfect square of the form
$\left(s^{2}\right)u^{2}+(2st)u+\left(t^{2}\right)=\left(\left({\sqrt {s^{2}}}\right)u+{(2st) \over 2{\sqrt {s^{2}}}}\right)^{2}$
(This is correct for both signs of square root, as long as the same sign is taken for both square roots. A ± is redundant, as it would be absorbed by another ± a few equations further down this page.)
so that it can be folded:
$(\alpha +2y)u^{2}+(-\beta )u+\left(y^{2}+2y\alpha +\alpha ^{2}-\gamma \right)=\left(\left({\sqrt {\alpha +2y}}\right)u+{(-\beta ) \over 2{\sqrt {\alpha +2y}}}\right)^{2}.$
Note: If β ≠ 0 then α + 2y ≠ 0. If β = 0 then this would be a biquadratic equation, which we solved earlier.
Therefore equation (3) becomes
$\left(u^{2}+\alpha +y\right)^{2}=\left(\left({\sqrt {\alpha +2y}}\right)u-{\beta \over 2{\sqrt {\alpha +2y}}}\right)^{2}.$
(5)
Equation (5) has a pair of folded perfect squares, one on each side of the equation. The two perfect squares balance each other.
If two squares are equal, then the sides of the two squares are also equal, as shown by:
$\left(u^{2}+\alpha +y\right)=\pm \left(\left({\sqrt {\alpha +2y}}\right)u-{\beta \over 2{\sqrt {\alpha +2y}}}\right).$
(5')
Collecting like powers of u produces
$u^{2}+\left(\mp _{s}{\sqrt {\alpha +2y}}\right)u+\left(\alpha +y\pm _{s}{\beta \over 2{\sqrt {\alpha +2y}}}\right)=0.$
(6)
Note: The subscript s of $\pm _{s}$ and $\mp _{s}$ is to note that they are dependent.
Equation (6) is a quadratic equation for u. Its solution is
$u={\frac {\pm _{s}{\sqrt {\alpha +2y}}\pm _{t}{\sqrt {(\alpha +2y)-4\left(\alpha +y\pm _{s}{\beta \over 2{\sqrt {\alpha +2y}}}\right)}}}{2}}.$
Simplifying, one gets
$u={\pm _{s}{\sqrt {\alpha +2y}}\pm _{t}{\sqrt {-\left(3\alpha +2y\pm _{s}{2\beta \over {\sqrt {\alpha +2y}}}\right)}} \over 2}.$
This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are
$x=-{B \over 4A}+{\pm _{s}{\sqrt {\alpha +2y}}\pm _{t}{\sqrt {-\left(3\alpha +2y\pm _{s}{2\beta \over {\sqrt {\alpha +2y}}}\right)}} \over 2}.$
(6')
Remember: The two $\pm _{s}$ come from the same place in equation (5'), and should both have the same sign, while the sign of $\pm _{t}$ is independent.
Summary of Ferrari's method
Given the quartic equation
$Ax^{4}+Bx^{3}+Cx^{2}+Dx+E=0,\,$
its solution can be found by means of the following calculations:
$\alpha =-{3B^{2} \over 8A^{2}}+{C \over A},$
$\beta ={B^{3} \over 8A^{3}}-{BC \over 2A^{2}}+{D \over A},$
$\gamma =-{3B^{4} \over 256A^{4}}+{CB^{2} \over 16A^{3}}-{BD \over 4A^{2}}+{E \over A}.$
If $\,\beta =0,$ then
$x=-{B \over 4A}\pm _{s}{\sqrt {-\alpha \pm _{t}{\sqrt {\alpha ^{2}-4\gamma }} \over 2}}\qquad {\mbox{(for }}\beta =0{\mbox{ only)}}.$
Otherwise, continue with
$P=-{\alpha ^{2} \over 12}-\gamma ,$
$Q=-{\alpha ^{3} \over 108}+{\alpha \gamma \over 3}-{\beta ^{2} \over 8},$
$R=-{Q \over 2}\pm {\sqrt {{Q^{2} \over 4}+{P^{3} \over 27}}},$
(either sign of the square root will do)
$U={\sqrt[{3}]{R}},$
(there are 3 complex roots, any one of them will do)
$y=-{5 \over 6}\alpha +{\begin{cases}U=0&\to -{\sqrt[{3}]{Q}}\\U\neq 0,&\to U-{P \over 3U},\end{cases}}\quad \quad \quad $
$W={\sqrt {\alpha +2y}}$
$x=-{B \over 4A}+{\pm _{s}W\pm _{t}{\sqrt {-\left(3\alpha +2y\pm _{s}{2\beta \over W}\right)}} \over 2}.$
The two ±s must have the same sign, the ±t is independent. To get all roots, compute x for ±s,±t = +,+ and for +,−; and for −,+ and for −,−. This formula handles repeated roots without problem.
Ferrari was the first to discover one of these labyrinthine solutions. The equation which he solved was
$x^{4}+6x^{2}-60x+36=0$
which was already in depressed form. It has a pair of solutions which can be found with the set of formulas shown above.
Ferrari's solution in the special case of real coefficients
If the coefficients of the quartic equation are real then the nested depressed cubic equation (5) also has real coefficients, thus it has at least one real root.
Furthermore the cubic function
$C(v)=v^{3}+Pv+Q,$
where P and Q are given by (5) has the properties that
$C\left({\alpha \over 3}\right)={-\beta ^{2} \over 8}<0$ and
$\lim _{v\to \infty }C(v)=\infty ,$ where α and β are given by (1).
This means that (5) has a real root greater than $\alpha \over 3$, and therefore that (4) has a real root greater than $-\alpha \over 2$.
Using this root the term ${\sqrt {\alpha +2y}}$ in (8) is always real, which ensures that the two quadratic equations (8) have real coefficients.[5]
Obtaining alternative solutions the hard way
It could happen that one only obtained one solution through the formulae above, because not all four sign patterns are tried for four solutions, and the solution obtained is complex. It may also be the case that one is only looking for a real solution. Let x1 denote the complex solution. If all the original coefficients A, B, C, D and E are real—which should be the case when one desires only real solutions – then there is another complex solution x2 which is the complex conjugate of x1. If the other two roots are denoted as x3 and x4 then the quartic equation can be expressed as
$(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})=0,\,$
but this quartic equation is equivalent to the product of two quadratic equations:
$(x-x_{1})(x-x_{2})=0$
(9)
and
$(x-x_{3})(x-x_{4})=0.$
(10)
Since
$x_{2}=x_{1}^{\star }$
then
${\begin{aligned}(x-x_{1})(x-x_{2})&=x^{2}-(x_{1}+x_{1}^{\star })x+x_{1}x_{1}^{\star }\\&=x^{2}-2\operatorname {Re} (x_{1})x+[\operatorname {Re} (x_{1})]^{2}+[\operatorname {Im} (x_{1})]^{2}.\end{aligned}}$
Let
$a=-2\operatorname {Re} (x_{1}),$
$b=\left[\operatorname {Re} (x_{1})\right]^{2}+\left[\operatorname {Im} (x_{1})\right]^{2}$
so that equation (9) becomes
$x^{2}+ax+b=0.$
(11)
Also let there be (unknown) variables w and v such that equation (10) becomes
$x^{2}+wx+v=0.$
(12)
Multiplying equations (11) and (12) produces
$x^{4}+(a+w)x^{3}+(b+wa+v)x^{2}+(wb+va)x+vb=0.$
(13)
Comparing equation (13) to the original quartic equation, it can be seen that
$a+w={B \over A},$
$b+wa+v={C \over A},$
$wb+va={D \over A},$
and
$vb={E \over A}.$
Therefore
$w={B \over A}-a={B \over A}+2\operatorname {Re} (x_{1}),$
$v={E \over Ab}={\frac {E}{A\left(\left[\operatorname {Re} (x_{1})\right]^{2}+\left[\operatorname {Im} (x_{1})\right]^{2}\right)}}.$
Equation (12) can be solved for x yielding
$x_{3}={-w+{\sqrt {w^{2}-4v}} \over 2},$
$x_{4}={-w-{\sqrt {w^{2}-4v}} \over 2}.$
One of these two solutions should be the desired real solution.
Alternative methods
Quick and memorable solution from first principles
Most textbook solutions of the quartic equation require a substitution that is hard to memorize. Here is an approach that makes it easy to understand. The job is done if we can factor the quartic equation into a product of two quadratics. Let
${\begin{aligned}0&=x^{4}+bx^{3}+cx^{2}+dx+e\\&=\left(x^{2}+px+q\right)\left(x^{2}+rx+s\right)\\&=x^{4}+(p+r)x^{3}+(q+s+pr)x^{2}+(ps+qr)x+qs\end{aligned}}$
By equating coefficients, this results in the following set of simultaneous equations:
${\begin{aligned}b&=p+r\\c&=q+s+pr\\d&=ps+qr\\e&=qs\end{aligned}}$
This is harder to solve than it looks, but if we start again with a depressed quartic where $b=0$, which can be obtained by substituting $(x-b/4)$ for $x$, then $r=-p$, and:
${\begin{aligned}c+p^{2}&=s+q\\d/p&=s-q\\e&=sq\end{aligned}}$
It's now easy to eliminate both $s$ and $q$ by doing the following:
${\begin{aligned}\left(c+p^{2}\right)^{2}-(d/p)^{2}&=(s+q)^{2}-(s-q)^{2}\\&=4sq\\&=4e\end{aligned}}$
If we set $P=p^{2}$, then this equation turns into the cubic equation:
$P^{3}+2cP^{2}+\left(c^{2}-4e\right)P-d^{2}=0$
which is solved elsewhere. Once you have $p$, then:
${\begin{aligned}r&=-p\\2s&=c+p^{2}+d/p\\2q&=c+p^{2}-d/p\end{aligned}}$
The symmetries in this solution are easy to see. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of $p$ for the square root of $P$ merely exchanges the two quadratics with one another.
Galois theory and factorization
The symmetric group S4 on four elements has the Klein four-group as a normal subgroup. This suggests using a resolvent whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots. Suppose ri for i from 0 to 3 are roots of
$x^{4}+bx^{3}+cx^{2}+dx+e=0\qquad (1)$
If we now set
${\begin{aligned}s_{0}&={\tfrac {1}{2}}(r_{0}+r_{1}+r_{2}+r_{3}),\\s_{1}&={\tfrac {1}{2}}(r_{0}-r_{1}+r_{2}-r_{3}),\\s_{2}&={\tfrac {1}{2}}(r_{0}+r_{1}-r_{2}-r_{3}),\\s_{3}&={\tfrac {1}{2}}(r_{0}-r_{1}-r_{2}+r_{3}),\end{aligned}}$
then since the transformation is an involution, we may express the roots in terms of the four si in exactly the same way. Since we know the value s0 = −b/2, we really only need the values for s1, s2 and s3. These we may find by expanding the polynomial
$\left(z^{2}-s_{1}^{2}\right)\left(z^{2}-s_{2}^{2}\right)\left(z^{2}-s_{3}^{2}\right)\qquad (2)$
which if we make the simplifying assumption that b = 0, is equal to
$z^{6}+2cz^{4}+\left(c^{2}-4e\right)z^{2}-d^{2}\qquad (3)$
This polynomial is of degree six, but only of degree three in z2, and so the corresponding equation is solvable. By trial we can determine which three roots are the correct ones, and hence find the solutions of the quartic.
We can remove any requirement for trial by using a root of the same resolvent polynomial for factoring; if w is any root of (3), and if
$F_{1}=x^{2}+wx+{\frac {1}{2}}w^{2}+{\frac {1}{2}}c-{\frac {1}{2}}\cdot {\frac {c^{2}w}{d}}-{\frac {1}{2}}\cdot {\frac {w^{5}}{d}}-{\frac {cw^{3}}{d}}+2{\frac {ew}{d}}$
$F_{2}=x^{2}-wx+{\frac {1}{2}}w^{2}+{\frac {1}{2}}c+{\frac {1}{2}}\cdot {\frac {w^{5}}{d}}+{\frac {cw^{3}}{d}}-2{\frac {ew}{d}}+{\frac {1}{2}}\cdot {\frac {c^{2}w}{d}}$
then
$F_{1}F_{2}=x^{4}+cx^{2}+dx+e\qquad \qquad (4)$
We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.
Approximate methods
The methods described above are, in principle, exact root-finding methods. It is also possible to use successive approximation methods which iteratively converge towards the roots, such as the Durand–Kerner method. Iterative methods are the only ones available for quintic and higher-order equations, beyond trivial or special cases.
See also
• Linear equation
• Quadratic equation
• Cubic equation
• Quintic equation
• Polynomial
• Newton's method
References
• Ferrari's achievement
• Quartic formula as four single equations at PlanetMath.
Notes
1. "Lodovico Ferrari".
2. Stewart, Ian, Galois Theory, Third Edition (Chapman & Hall/CRC Mathematics, 2004)
3. Chávez-Pichardo, Mauricio; Martínez-Cruz, Miguel A.; Trejo-Martínez, Alfredo; Martínez-Carbajal, Daniel; Arenas-Resendiz, Tanya (July 2022). "A Complete Review of the General Quartic Equation with Real Coefficients and Multiple Roots". Mathematics. 10 (14): 2377. doi:10.3390/math10142377. ISSN 2227-7390.
4. Chávez-Pichardo, Mauricio; Martínez-Cruz, Miguel A.; Trejo-Martínez, Alfredo; Vega-Cruz, Ana Beatriz; Arenas-Resendiz, Tanya (March 2023). "On the Practicality of the Analytical Solutions for all Third- and Fourth-Degree Algebraic Equations with Real Coefficients". Mathematics. 11 (6): 1447. doi:10.3390/math11061447. ISSN 2227-7390.
5. Carstensen, Jens, Komplekse tal, First Edition, (Systime 1981), ISBN 87-87454-71-8. (in Danish)
External links
• Calculator for solving Quartics
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
• Univariate
• Bivariate
• Multivariate
• Monomial
• Binomial
• Trinomial
• Irreducible
• Square-free
• Homogeneous
• Quasi-homogeneous
Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
|
Wikipedia
|
Quartic graph
In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph.[1]
Examples
Several well-known graphs are quartic. They include:
• The complete graph K5, a quartic graph with 5 vertices, the smallest possible quartic graph.
• The Chvátal graph, another quartic graph with 12 vertices, the smallest quartic graph that both has no triangles and cannot be colored with three colors.[2]
• The Folkman graph, a quartic graph with 20 vertices, the smallest semi-symmetric graph.[3]
• The Meredith graph, a quartic graph with 70 vertices that is 4-connected but has no Hamiltonian cycle, disproving a conjecture of Crispin Nash-Williams.[4]
Every medial graph is a quartic plane graph, and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs.[5] Knot diagrams and link diagrams are also quartic plane multigraphs, in which the vertices represent the crossings of the diagram and are marked with additional information concerning which of the two branches of the knot crosses the other branch at that point.[6]
Properties
Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour. And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for finding such a matching is possible than for irregular graphs: by selecting every other edge of an Euler tour, one may find a 2-factor, which in this case must be a collection of cycles, each of even length, with each vertex of the graph appearing in exactly one cycle. By selecting every other edge again in these cycles, one obtains a perfect matching in linear time. The same method can also be used to color the edges of the graph with four colors in linear time.[7]
Quartic graphs have an even number of Hamiltonian decompositions.[8]
Open problems
It is an open conjecture whether all quartic Hamiltonian graphs have an even number of Hamiltonian circuits, or have more than one Hamiltonian circuit. The answer is known to be false for quartic multigraphs.[9]
See also
Wikimedia Commons has media related to 4-regular graphs.
• Cubic graph
References
1. Toida, S. (1974), "Construction of quartic graphs", Journal of Combinatorial Theory, Series B, 16: 124–133, doi:10.1016/0095-8956(74)90054-9, MR 0347693.
2. Chvátal, V. (1970), "The smallest triangle-free 4-chromatic 4-regular graph", Journal of Combinatorial Theory, 9 (1): 93–94, doi:10.1016/S0021-9800(70)80057-6.
3. Folkman, Jon (1967), "Regular line-symmetric graphs", Journal of Combinatorial Theory, 3: 215–232, doi:10.1016/s0021-9800(67)80069-3, MR 0224498.
4. Meredith, G. H. J. (1973), "Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs", Journal of Combinatorial Theory, Series B, 14: 55–60, doi:10.1016/s0095-8956(73)80006-1, MR 0311503.
5. Bondy, J. A.; Häggkvist, R. (1981), "Edge-disjoint Hamilton cycles in 4-regular planar graphs", Aequationes Mathematicae, 22 (1): 42–45, doi:10.1007/BF02190157, MR 0623315.
6. Welsh, Dominic J. A. (1993), "The complexity of knots", Quo vadis, graph theory?, Annals of Discrete Mathematics, vol. 55, Amsterdam: North-Holland, pp. 159–171, doi:10.1016/S0167-5060(08)70385-6, MR 1217989.
7. Gabow, Harold N. (1976), "Using Euler partitions to edge color bipartite multigraphs", International Journal of Computer and Information Sciences, 5 (4): 345–355, doi:10.1007/bf00998632, MR 0422081.
8. Thomason, A. G. (1978), "Hamiltonian cycles and uniquely edge colourable graphs", Annals of Discrete Mathematics, 3: 259–268, doi:10.1016/s0167-5060(08)70511-9, MR 0499124.
9. Fleischner, Herbert (1994), "Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs", Journal of Graph Theory, 18 (5): 449–459, doi:10.1002/jgt.3190180503, MR 1283310.
External links
• Weisstein, Eric W. "Quartic Graph". MathWorld.
|
Wikipedia
|
Quartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.
More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form
$f(x,y,z)=0\ $
where f is a polynomial of degree 4, such as $f(x,y,z)=x^{4}+y^{4}+xyz+z^{2}-1$. This is a surface in affine space A3.
On the other hand, a projective quartic surface is a surface in projective space P3 of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example $f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}$.
If the base field is $\mathbb {R} $ or $\mathbb {C} $ the surface is said to be real or complex respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over $\mathbb {C} $, and quartic surfaces over $\mathbb {R} $. For instance, the Klein quartic is a real surface given as a quartic curve over $\mathbb {C} $. If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.
Special quartic surfaces
• Dupin cyclides
• The Fermat quartic, given by x4 + y4 + z4 + w4 =0 (an example of a K3 surface).
• More generally, certain K3 surfaces are examples of quartic surfaces.
• Kummer surface
• Plücker surface
• Weddle surface
See also
• Quadric surface (The union of two quadric surfaces is a special case of a quartic surface)
• Cubic surface (The union of a cubic surface and a plane is another particular type of quartic surface)
References
• Hudson, R. W. H. T. (1990), Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, MR 1097176
• Jessop, C. M. (1916), Quartic surfaces with singular points, Cornell University Library, ISBN 978-1-4297-0393-2
|
Wikipedia
|
Quasi-Frobenius Lie algebra
In mathematics, a quasi-Frobenius Lie algebra
$({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )$
over a field $k$ is a Lie algebra
$({\mathfrak {g}},[\,\,\,,\,\,\,])$
equipped with a nondegenerate skew-symmetric bilinear form
$\beta :{\mathfrak {g}}\times {\mathfrak {g}}\to k$ :{\mathfrak {g}}\times {\mathfrak {g}}\to k} , which is a Lie algebra 2-cocycle of ${\mathfrak {g}}$ with values in $k$. In other words,
$\beta \left(\left[X,Y\right],Z\right)+\beta \left(\left[Z,X\right],Y\right)+\beta \left(\left[Y,Z\right],X\right)=0$
for all $X$, $Y$, $Z$ in ${\mathfrak {g}}$.
If $\beta $ is a coboundary, which means that there exists a linear form $f:{\mathfrak {g}}\to k$ such that
$\beta (X,Y)=f(\left[X,Y\right]),$
then
$({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )$
is called a Frobenius Lie algebra.
Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form
If $({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )$ is a quasi-Frobenius Lie algebra, one can define on ${\mathfrak {g}}$ another bilinear product $\triangleleft $ by the formula
$\beta \left(\left[X,Y\right],Z\right)=\beta \left(Z\triangleleft Y,X\right)$.
Then one has $\left[X,Y\right]=X\triangleleft Y-Y\triangleleft X$ and
$({\mathfrak {g}},\triangleleft )$
is a pre-Lie algebra.
See also
• Lie coalgebra
• Lie bialgebra
• Lie algebra cohomology
• Frobenius algebra
• Quasi-Frobenius ring
References
• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
|
Wikipedia
|
Normal extension
In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K which has a root in L, splits into linear factors in L.[1][2] These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension.
Definition
Let $L/K$ be an algebraic extension (i.e., L is an algebraic extension of K), such that $L\subseteq {\overline {K}}$ (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:[3]
• Every embedding of L in ${\overline {K}}$ induces an automorphism of L.
• L is the splitting field of a family of polynomials in $K\left[X\right]$.
• Every irreducible polynomial of $K\left[X\right]$ which has a root in L splits into linear factors in L.
Other properties
Let L be an extension of a field K. Then:
• If L is a normal extension of K and if E is an intermediate extension (that is, L ⊃ E ⊃ K), then L is a normal extension of E.[4]
• If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normal extensions of K.[4]
Equivalent conditions for normality
Let $L/K$ be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.
• The minimal polynomial over K of every element in L splits in L;
• There is a set $S\subseteq K[x]$ of polynomials that simultaneously split over L, such that if $K\subseteq F\subsetneq L$ are fields, then S has a polynomial that does not split in F;
• All homomorphisms $L\to {\bar {K}}$ have the same image;
• The group of automorphisms, ${\text{Aut}}(L/K),$ of L which fixes elements of K, acts transitively on the set of homomorphisms $L\to {\bar {K}}.$
Examples and counterexamples
For example, $\mathbb {Q} ({\sqrt {2}})$ is a normal extension of $\mathbb {Q} ,$ since it is a splitting field of $x^{2}-2.$ On the other hand, $\mathbb {Q} ({\sqrt[{3}]{2}})$ is not a normal extension of $\mathbb {Q} $ since the irreducible polynomial $x^{3}-2$ has one root in it (namely, ${\sqrt[{3}]{2}}$), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field ${\overline {\mathbb {Q} }}$ of algebraic numbers is the algebraic closure of $\mathbb {Q} ,$ that is, it contains $\mathbb {Q} ({\sqrt[{3}]{2}}).$ Since,
$\mathbb {Q} ({\sqrt[{3}]{2}})=\left.\left\{a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\in {\overline {\mathbb {Q} }}\,\,\right|\,\,a,b,c\in \mathbb {Q} \right\}$
and, if $\omega $ is a primitive cubic root of unity, then the map
${\begin{cases}\sigma :\mathbb {Q} ({\sqrt[{3}]{2}})\longrightarrow {\overline {\mathbb {Q} }}\\a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\longmapsto a+b\omega {\sqrt[{3}]{2}}+c\omega ^{2}{\sqrt[{3}]{4}}\end{cases}}$ :\mathbb {Q} ({\sqrt[{3}]{2}})\longrightarrow {\overline {\mathbb {Q} }}\\a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\longmapsto a+b\omega {\sqrt[{3}]{2}}+c\omega ^{2}{\sqrt[{3}]{4}}\end{cases}}}
is an embedding of $\mathbb {Q} ({\sqrt[{3}]{2}})$ in ${\overline {\mathbb {Q} }}$ whose restriction to $\mathbb {Q} $ is the identity. However, $\sigma $ is not an automorphism of $\mathbb {Q} ({\sqrt[{3}]{2}}).$
For any prime $p,$ the extension $\mathbb {Q} ({\sqrt[{p}]{2}},\zeta _{p})$ is normal of degree $p(p-1).$ It is a splitting field of $x^{p}-2.$ Here $\zeta _{p}$ denotes any $p$th primitive root of unity. The field $\mathbb {Q} ({\sqrt[{3}]{2}},\zeta _{3})$ is the normal closure (see below) of $\mathbb {Q} ({\sqrt[{3}]{2}}).$
Normal closure
If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, that is, the only subfield of M which contains L and which is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K.
If L is a finite extension of K, then its normal closure is also a finite extension.
See also
• Galois extension
• Normal basis
Citations
1. Lang 2002, p. 237, Theorem 3.3, NOR 3.
2. Jacobson 1989, p. 489, Section 8.7.
3. Lang 2002, p. 237, Theorem 3.3.
4. Lang 2002, p. 238, Theorem 3.4.
References
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787
|
Wikipedia
|
Quasi-Lie algebra
In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom
$[x,x]=0$
replaced by
$[x,y]=-[y,x]$ (anti-symmetry).
In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers.
In a quasi-Lie algebra,
$2[x,x]=0.$
Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish.
See also
• Whitehead product
References
• Serre, Jean-Pierre (2006). Lie Algebras and Lie Groups. 1964 lectures given at Harvard University. Lecture Notes in Mathematics. Vol. 1500 (Corrected 5th printing of the 2nd (1992) ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-540-70634-2. ISBN 3-540-55008-9. MR 2179691.
|
Wikipedia
|
Quasi-Monte Carlo methods in finance
High-dimensional integrals in hundreds or thousands of variables occur commonly in finance. These integrals have to be computed numerically to within a threshold $\epsilon $. If the integral is of dimension $d$ then in the worst case, where one has a guarantee of error at most $\epsilon $, the computational complexity is typically of order $\epsilon ^{-d}$. That is, the problem suffers the curse of dimensionality. In 1977 P. Boyle, University of Waterloo, proposed using Monte Carlo (MC) to evaluate options.[1] Starting in early 1992, J. F. Traub, Columbia University, and a graduate student at the time, S. Paskov, used quasi-Monte Carlo (QMC) to price a Collateralized mortgage obligation with parameters specified by Goldman Sachs. Even though it was believed by the world's leading experts that QMC should not be used for high-dimensional integration, Paskov and Traub found that QMC beat MC by one to three orders of magnitude and also enjoyed other desirable attributes. Their results were first published[2] in 1995. Today QMC is widely used in the financial sector to value financial derivatives; see list of books below.
QMC is not a panacea for all high-dimensional integrals. A number of explanations have been proposed for why QMC is so good for financial derivatives. This continues to be a very fruitful research area.
Monte Carlo and quasi-Monte Carlo methods
Integrals in hundreds or thousands of variables are common in computational finance. These have to be approximated numerically to within an error threshold $\epsilon $. It is well known that if a worst case guarantee of error at most $\epsilon $ is required then the computational complexity of integration may be exponential in $d$, the dimension of the integrand; See [3] Ch. 3 for details. To break this curse of dimensionality one can use the Monte Carlo (MC) method defined by
$\varphi ^{\mathop {\rm {MC}} }(f)={\frac {1}{n}}\sum _{i=1}^{n}f(x_{i}),$
where the evaluation points $x_{i}$ are randomly chosen. It is well known that the expected error of Monte Carlo is of order $n^{-1/2}$. Thus, the cost of the algorithm that has error $\epsilon $ is of order $\epsilon ^{-2}$ breaking the curse of dimensionality.
Of course in computational practice pseudo-random points are used. Figure 1 shows the distribution of 500 pseudo-random points on the unit square.
Note there are regions where there are no points and other regions where there are clusters of points. It would be desirable to sample the integrand at uniformly distributed points. A rectangular grid would be uniform but even if there were only 2 grid points in each Cartesian direction there would be $2^{d}$ points. So the desideratum should be as few points as possible chosen as uniform as possible.
It turns out there is a well-developed part of number theory which deals exactly with this desideratum. Discrepancy is a measure of deviation from uniformity so what one wants are low discrepancy sequences (LDS).[4] Numerous LDS have been created named after their inventors, e.g.
• Halton
• Hammersley
• Sobol
• Faure
• Niederreiter
Figure 2. gives the distribution of 500 LDS points.
The quasi-Monte Carlo (QMC) method is defined by
$\varphi ^{\mathop {\rm {QMC}} }(f)={\frac {1}{n}}\sum _{i=1}^{n}f(x_{i}),$
where the $x_{i}$ belong to an LDS. The standard terminology quasi-Monte Carlo is somewhat unfortunate since MC is a randomized method whereas QMC is purely deterministic.
The uniform distribution of LDS is desirable. But the worst case error of QMC is of order
${\frac {(\log n)^{d}}{n}},$
where $n$ is the number of sample points. See [4] for the theory of LDS and references to the literature. The rate of convergence of LDS may be contrasted with the expected rate of convergence of MC which is $n^{-1/2}$. For $d$ small the rate of convergence of QMC is faster than MC but for $d$ large the factor $(\log n)^{d}$ is devastating. For example, if $d=360$, then even with $\log n=2$ the QMC error is proportional to $2^{360}$. Thus, it was widely believed by the world's leading experts that QMC should not be used for high-dimensional integration. For example, in 1992 Bratley, Fox and Niederreiter[5] performed extensive testing on certain mathematical problems. They conclude "in high-dimensional problems (say $d>12$), QMC seems to offer no practical advantage over MC". In 1993, Rensburg and Torrie[6] compared QMC with MC for the numerical estimation of high-dimensional integrals which occur in computing virial coefficients for the hard-sphere fluid. They conclude QMC is more effective than MC only if $d<10$. As we shall see, tests on 360-dimensional integrals arising from a collateralized mortgage obligation (CMO) lead to very different conclusions.
Woźniakowski's 1991 paper[7] showing the connection between average case complexity of integration and QMC led to new interest in QMC. Woźniakowski's result received considerable coverage in the scientific press[8] .[9] In early 1992, I. T. Vanderhoof, New York University, became aware of Woźniakowski's result and gave Woźniakowski's colleague J. F. Traub, Columbia University, a CMO with parameters set by Goldman Sachs. This CMO had 10 tranches each requiring the computation of a 360 dimensional integral. Traub asked a Ph.D. student, Spassimir Paskov, to compare QMC with MC for the CMO. In 1992 Paskov built a software system called FinDer and ran extensive tests. To the Columbia's research group's surprise and initial disbelief Paskov reported that QMC was always superior to MC in a number of ways. Details are given below. Preliminary results were presented by Paskov and Traub to a number of Wall Street firms in Fall 1993 and Spring 1994. The firms were initially skeptical of the claim that QMC was superior to MC for pricing financial derivatives. A January 1994 article in Scientific American by Traub and Woźniakowski[9] discussed the theoretical issues and reported that "Preliminary results obtained by testing certain finance problems suggests the superiority of the deterministic methods in practice". In Fall 1994 Paskov wrote a Columbia University Computer Science Report which appeared in slightly modified form in 1997.[10]
In Fall 1995 Paskov and Traub published a paper in the "Journal of Portfolio Management".[2] They compared MC and two QMC methods. The two deterministic methods used Sobol and Halton points. Since better LDS were created later, no comparison will be made between Sobol and Halton sequences. The experiments drew the following conclusions regarding the performance of MC and QMC on the 10 tranche CMO:
• QMC methods converge significantly faster than MC
• MC is sensitive to the initial seed
• The convergence of QMC is smoother than the convergence of MC. This makes automatic termination easier for QMC.
To summarize, QMC beats MC for the CMO on accuracy, confidence level, and speed.
This paper was followed by reports on tests by a number of researchers which also led to the conclusion the QMC is superior to MC for a variety of high-dimensional finance problems. This includes papers by Caflisch and Morokoff (1996),[11] Joy, Boyle, Tan (1996),[12] Ninomiya and Tezuka (1996),[13] Papageorgiou and Traub (1996),[14] Ackworth, Broadie and Glasserman (1997),[15] Kucherenko and co-authors [16] [17]
Further testing of the CMO[14] was carried out by Anargyros Papageorgiou, who developed an improved version of the FinDer software system. The new results include the following:
• Small number of sample points: For the hardest CMO tranche QMC using the generalized Faure LDS due to S. Tezuka[18] achieves accuracy $10^{-2}$ with just 170 points. MC requires 2700 points for the same accuracy. The significance of this is that due to future interest rates and prepayment rates being unknown, financial firms are content with accuracy of $10^{-2}$.
• Large number of sample points: The advantage of QMC over MC is further amplified as the sample size and accuracy demands grow. In particular, QMC is 20 to 50 times faster than MC with moderate sample sizes, and can be up to 1000 times faster than MC[14] when high accuracy is desired QMC.
Currently the highest reported dimension for which QMC outperforms MC is 65536.[19] The software is the Sobol' Sequence generator SobolSeq65536 which generates Sobol' Sequences satisfying Property A for all dimensions and Property A' for the adjacent dimensions. SobolSeq generators outperform all other known generators both in speed and accuracy [20]
Theoretical explanations
The results reported so far in this article are empirical. A number of possible theoretical explanations have been advanced. This has been a very research rich area leading to powerful new concepts but a definite answer has not been obtained.
A possible explanation of why QMC is good for finance is the following. Consider a tranche of the CMO mentioned earlier. The integral gives expected future cash flows from a basket of 30-year mortgages at 360 monthly intervals. Because of the discounted value of money variables representing future times are increasingly less important. In a seminal paper I. Sloan and H. Woźniakowski[21] introduced the idea of weighted spaces. In these spaces the dependence on the successive variables can be moderated by weights. If the weights decrease sufficiently rapidly the curse of dimensionality is broken even with a worst case guarantee. This paper led to a great amount of work on the tractability of integration and other problems.[22] A problem is tractable when its complexity is of order $\epsilon ^{-p}$ and $p$ is independent of the dimension.
On the other hand, effective dimension was proposed by Caflisch, Morokoff and Owen[23] as an indicator of the difficulty of high-dimensional integration. The purpose was to explain the remarkable success of quasi-Monte Carlo (QMC) in approximating the very-high-dimensional integrals in finance. They argued that the integrands are of low effective dimension and that is why QMC is much faster than Monte Carlo (MC). The impact of the arguments of Caflisch et al.[23] was great. A number of papers deal with the relationship between the error of QMC and the effective dimension[24] .[16] [17] [25]
It is known that QMC fails for certain functions that have high effective dimension.[5] However, low effective dimension is not a necessary condition for QMC to beat MC and for high-dimensional integration to be tractable. In 2005, Tezuka[26] exhibited a class of functions of $d$ variables, all with maximum effective dimension equal to $d$. For these functions QMC is very fast since its convergence rate is of order $n^{-1}$, where $n$ is the number of function evaluations.
Isotropic integrals
QMC can also be superior to MC and to other methods for isotropic problems, that is, problems where all variables are equally important. For example, Papageorgiou and Traub[27] reported test results on the model integration problems suggested by the physicist B. D. Keister[28]
$\left({\frac {1}{2\pi }}\right)^{d/2}\int _{\mathbb {R} ^{d}}\cos(\|x\|)e^{-\|x\|^{2}}\,dx,$
where $\|\cdot \|$ denotes the Euclidean norm and $d=25$. Keister reports that using a standard numerical method some 220,000 points were needed to obtain a relative error on the order of $10^{-2}$. A QMC calculation using the generalized Faure low discrepancy sequence[18] (QMC-GF) used only 500 points to obtain the same relative error. The same integral was tested for a range of values of $d$ up to $d=100$. Its error was
$c\cdot n^{-1},$
$c<110$, where $n$ is the number of evaluations of $f$. This may be compared with the MC method whose error was proportional to $n^{-1/2}$.
These are empirical results. In a theoretical investigation Papageorgiou[29] proved that the convergence rate of QMC for a class of $d$-dimensional isotropic integrals which includes the integral defined above is of the order
${\sqrt {\log n}}/n.$
This is with a worst case guarantee compared to the expected convergence rate of $n^{-1/2}$ of Monte Carlo and shows the superiority of QMC for this type of integral.
In another theoretical investigation Papageorgiou[30] presented sufficient conditions for fast QMC convergence. The conditions apply to isotropic and non-isotropic problems and, in particular, to a number of problems in computational finance. He presented classes of functions where even in the worst case the convergence rate of QMC is of order
$n^{-1+p(\log n)^{-1/2}},$
where $p\geq 0$ is a constant that depends on the class of functions.
But this is only a sufficient condition and leaves open the major question we pose in the next section.
Open questions
1. Characterize for which high-dimensional integration problems QMC is superior to MC.
2. Characterize types of financial instruments for which QMC is superior to MC.
See also
• Monte Carlo methods in finance
• Historical simulation (finance)
Resources
Books
• Bruno Dupire (1998). Monte Carlo: methodologies and applications for pricing and risk management. Risk. ISBN 1-899332-91-X.
• Paul Glasserman (2003). Monte Carlo methods in financial engineering. Springer-Verlag. ISBN 0-387-00451-3.
• Peter Jaeckel (2002). Monte Carlo methods in finance. John Wiley and Sons. ISBN 0-471-49741-X.
• Don L. McLeish (2005). Monte Carlo Simulation & Finance. ISBN 0-471-67778-7.
• Christian P. Robert, George Casella (2004). Monte Carlo Statistical Methods. ISBN 0-387-21239-6.
Models
• Spreadsheets available for download, Prof. Marco Dias, PUC-Rio
References
1. Boyle, P. (1977), Options: a Monte Carlo approach, J. Financial Economics, 4, 323-338.
2. Paskov, S. H. and Traub, J. F. (1995), Faster evaluation of financial derivatives, J. Portfolio Management, 22(1), 113-120.
3. Traub, J. F and Werschulz, A. G. (1998), Complexity and Information, Cambridge University Press, Cambridge, UK.
4. Niederreiter, H. (1992), Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
5. Bratley, P., Fox, B. L. and Niederreiter, H. (1992), Implementation and tests of low-discrepancy sequences, ACM Transactions on Modelling and Computer Simulation, Vol. 2, No. 3, 195-213.
6. van Rensburg, E. J. J. and Torrie, G. M. (1993), Estimation of multidimensional integrals: is Monte Carlo the best method? J. Phys. A: Math. Gen., 26(4), 943-953.
7. Woźniakowski, H. (1991), Average case complexity of multivariate integration, Bull. Amer. Math. Soc. (New Ser.), 24(1), 185-194.
8. Cipra, Barry Arthur (1991), Multivariate Integration: It ain't so tough (on average), SIAM NEWS, 28 March.
9. Traub, J. F. and Woźniakowski, H. (1994), Breaking intractability, Scientific American, 270(1), January, 102-107.
10. Paskov, S. H., New methodologies for valuing derivatives, 545-582, in Mathematics of Derivative Securities, S. Pliska and M. Dempster eds., Cambridge University Press, Cambridge.
11. Caflisch, R. E. and Morokoff, W. (1996), Quasi-Monte Carlo computation of a finance problem, 15-30, in Proceedings Workshop on Quasi-Monte Carlo Methods and their Applications, 11 December 1995, K.-T. Fang and F. Hickernell eds., Hong Kong Baptist University.
12. Joy, C., Boyle, P. P. and Tang, K. S. (1996), Quasi-Monte Carlo methods in numerical finance, Management Science, 42(6), 926-938.
13. Ninomiya, S. and Tezuka, S. (1996), Toward real-time pricing of complex financial derivatives, Appl. Math. Finance, 3, 1-20.
14. Papageorgiou, A. and Traub, J. F. (1996), Beating Monte Carlo, Risk, 9(6), 63-65.
15. Ackworth, P., Broadie, M. and Glasserman, P. (1997), A comparison of some Monte Carlo techniques for option pricing, 1-18, in Monte Carlo and Quasi-Monte Carlo Methods '96, H. Hellekalek, P. Larcher and G. Zinterhof eds., Springer Verlag, New York.
16. Kucherenko S., Shah N. The Importance of being Global.Application of Global Sensitivity Analysis in Monte Carlo option Pricing Wilmott, 82-91, July 2007. http://www.broda.co.uk/gsa/wilmott_GSA_SK.pdf
17. Bianchetti M., Kucherenko S., Scoleri S., Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysis, Wilmott, July, pp. 46-70, 2015, http://www.broda.co.uk/doc/PricingRiskManagement_Sobol.pdf
18. Tezuka, S., Uniform Random Numbers:Theory and Practice, Kluwer, Netherlands.
19. BRODA Ltd. http://www.broda.co.uk
20. Sobol’ I., Asotsky D. , Kreinin A. , Kucherenko S. (2012) Construction and Comparison of High-Dimensional Sobol’ Generators, Wilmott, Nov, 64-79
21. Sloan, I. and Woźniakowski, H. (1998), When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity, 14(1), 1-33.
22. Novak, E. and Wozniakowski, H. (2008), Tractability of multivariate problems, European Mathematical Society, Zurich (forthcoming).
23. Caflisch, R. E., Morokoff, W. and Owen, A. B. (1997), Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, Journal of Computational Finance, 1, 27-46.
24. Hickernell, F. J. (1998), Lattice rules: how well do they measure up?, in P. Hellekalek and G. Larcher (Eds.), Random and Quasi-Random Point Sets, Springer, 109-166.
25. Wang, X. and Sloan, I. H. (2005), Why are high-dimensional finance problems often of low effective dimension?, SIAM Journal on Scientific Computing, 27(1), 159-183.
26. Tezuka, S. (2005), On the necessity of low-effective dimension, Journal of Complexity, 21, 710-721.
27. Papageorgiou, A. and Traub, J. F. (1997), Faster evaluation of multidimensional integrals, Computers in Physics, 11(6), 574-578.
28. Keister, B. D. (1996), Multidimensional quadrature algorithms, Computers in Physics, 10(20), 119-122.
29. Papageorgiou, A. (2001), Fast convergence of quasi-Monte Carlo for a class of isotropic integrals, Math. Comp., 70, 297-306.
30. Papageorgiou, A. (2003), Sufficient conditions for fast quasi-Monte Carlo convergence, J. Complexity, 19(3), 332-351.
|
Wikipedia
|
Quasi-abelian category
In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel.
Definition
Let ${\mathcal {A}}$ be a pre-abelian category. A morphism $f$ is a kernel (a cokernel) if there exists a morphism $g$ such that $f$ is a kernel (cokernel) of $g$. The category ${\mathcal {A}}$ is quasi-abelian if for every kernel $f:X\rightarrow Y$ and every morphism $h:X\rightarrow Z$ in the pushout diagram
${\begin{array}{ccc}X&{\xrightarrow {f}}&Y\\\downarrow _{h}&&\downarrow _{h'}\\Z&{\xrightarrow {f'}}&Q\end{array}}$
the morphism $f'$ is again a kernel and, dually, for every cokernel $g:X\rightarrow Y$ and every morphism $h:Z\rightarrow Y$ in the pullback diagram
${\begin{array}{ccc}P&{\xrightarrow {g'}}&Z\\\downarrow _{h'}&&\downarrow _{h}\\X&{\xrightarrow {g}}&Y\end{array}}$
the morphism $g'$ is again a cokernel.
Equivalently, a quasi-abelian category is a pre-abelian category in which the system of all kernel-cokernel pairs forms an exact structure.
Given a pre-abelian category, those kernels, which are stable under arbitrary pushouts, are sometimes called the semi-stable kernels. Dually, cokernels, which are stable under arbitrary pullbacks, are called semi-stable cokernels.[1]
Properties
Let $f$ be a morphism in a quasi-abelian category. Then the induced morphism ${\overline {f}}:\operatorname {cok} \ker f\to \ker \operatorname {cok} f$ is always a bimorphism, i.e., a monomorphism and an epimorphism. A quasi-abelian category is therefore always semi-abelian.
Examples
Every abelian category is quasi-abelian. Typical non-abelian examples arise in functional analysis.[2]
• The category of Banach spaces is quasi-abelian.
• The category of Fréchet spaces is quasi-abelian.
• The category of (Hausdorff) locally convex spaces is quasi-abelian.
History
The concept of quasi-abelian category was developed in the 1960s. The history is involved.[3] This is in particular due to Raikov's conjecture, which stated that the notion of a semi-abelian category is equivalent to that of a quasi-abelian category. Around 2005 it turned out that the conjecture is false.[4]
Left and right quasi-abelian categories
By dividing the two conditions in the definition, one can define left quasi-abelian categories by requiring that cokernels are stable under pullbacks and right quasi-abelian categories by requiring that kernels stable under pushouts.[5]
Citations
1. Richman and Walker, 1977.
2. Prosmans, 2000.
3. Rump, 2008, p. 986f.
4. Rump, 2011, p. 44f.
5. Rump, 2001.
References
• Fabienne Prosmans, Derived categories for functional analysis. Publ. Res. Inst. Math. Sci. 36(5–6), 19–83 (2000).
• Fred Richman and Elbert A. Walker, Ext in pre-Abelian categories. Pac. J. Math. 71(2), 521–535 (1977).
• Wolfgang Rump, A counterexample to Raikov's conjecture, Bull. London Math. Soc. 40, 985–994 (2008).
• Wolfgang Rump, Almost abelian categories, Cahiers Topologie Géom. Différentielle Catég. 42(3), 163–225 (2001).
• Wolfgang Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure and Appl. Algebra 215, 44–52 (2011).
• Jean Pierre Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. Nouv. Sér. 76 (1999).
|
Wikipedia
|
Quasi-algebraically closed field
In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang's advisor Emil Artin.
Formally, if P is a non-constant homogeneous polynomial in variables
X1, ..., XN,
and of degree d satisfying
d < N
then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have
P(x1, ..., xN) = 0.
In geometric language, the hypersurface defined by P, in projective space of degree N − 2, then has a point over F.
Examples
• Any algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.[1]
• Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.[2][3][4]
• Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.[3][5]
• The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.[3]
• A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.[3][6]
• A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.[7]
Properties
• Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.
• The Brauer group of a finite extension of a quasi-algebraically closed field is trivial.[8][9][10]
• A quasi-algebraically closed field has cohomological dimension at most 1.[10]
Ck fields
Quasi-algebraically closed fields are also called C1. A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided
dk < N,
for k ≥ 1.[11] The condition was first introduced and studied by Lang.[10] If a field is Ci then so is a finite extension.[11][12] The C0 fields are precisely the algebraically closed fields.[13][14]
Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree n is Ck+n.[15][16][17] The smallest k such that K is a Ck field ($\infty $ if no such number exists), is called the diophantine dimension dd(K) of K.[13]
C1 fields
Every finite field is C1.[7]
Properties
Suppose that the field k is C2.
• Any skew field D finite over k as centre has the property that the reduced norm D∗ → k∗ is surjective.[16]
• Every quadratic form in 5 or more variables over k is isotropic.[16]
Artin's conjecture
Artin conjectured that p-adic fields were C2, but Guy Terjanian found p-adic counterexamples for all p.[18][19] The Ax–Kochen theorem applied methods from model theory to show that Artin's conjecture was true for Qp with p large enough (depending on d).
Weakly Ck fields
A field K is weakly Ck,d if for every homogeneous polynomial of degree d in N variables satisfying
dk < N
the Zariski closed set V(f) of Pn(K) contains a subvariety which is Zariski closed over K.
A field which is weakly Ck,d for every d is weakly Ck.[2]
Properties
• A Ck field is weakly Ck.[2]
• A perfect PAC weakly Ck field is Ck.[2]
• A field K is weakly Ck,d if and only if every form satisfying the conditions has a point x defined over a field which is a primary extension of K.[20]
• If a field is weakly Ck, then any extension of transcendence degree n is weakly Ck+n.[17]
• Any extension of an algebraically closed field is weakly C1.[21]
• Any field with procyclic absolute Galois group is weakly C1.[21]
• Any field of positive characteristic is weakly C2.[21]
• If the field of rational numbers $\mathbb {Q} $ and the function fields $\mathbb {F} _{p}(t)$ are weakly C1, then every field is weakly C1.[21]
See also
• Brauer's theorem on forms
• Tsen rank
Citations
1. Fried & Jarden (2008) p.455
2. Fried & Jarden (2008) p.456
3. Serre (1979) p.162
4. Gille & Szamuley (2006) p.142
5. Gille & Szamuley (2006) p.143
6. Gille & Szamuley (2006) p.144
7. Fried & Jarden (2008) p.462
8. Lorenz (2008) p.181
9. Serre (1979) p.161
10. Gille & Szamuely (2006) p.141
11. Serre (1997) p.87
12. Lang (1997) p.245
13. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4.
14. Lorenz (2008) p.116
15. Lorenz (2008) p.119
16. Serre (1997) p.88
17. Fried & Jarden (2008) p.459
18. Terjanian, Guy (1966). "Un contre-example à une conjecture d'Artin". Comptes Rendus de l'Académie des Sciences, Série A-B (in French). 262: A612. Zbl 0133.29705.
19. Lang (1997) p.247
20. Fried & Jarden (2008) p.457
21. Fried & Jarden (2008) p.461
References
• Ax, James; Kochen, Simon (1965). "Diophantine problems over local fields I". Amer. J. Math. 87 (3): 605–630. doi:10.2307/2373065. JSTOR 2373065. Zbl 0136.32805.
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
• Greenberg, M.J. (1969). Lectures of forms in many variables. Mathematics Lecture Note Series. New York-Amsterdam: W.A. Benjamin. Zbl 0185.08304.
• Lang, Serge (1952), "On quasi algebraic closure", Annals of Mathematics, 55 (2): 373–390, doi:10.2307/1969785, JSTOR 1969785, Zbl 0046.26202
• Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016.
• Serre, Jean-Pierre (1997). Galois cohomology. Springer-Verlag. ISBN 3-540-61990-9. Zbl 0902.12004.
• Tsen, C. (1936), "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper", J. Chinese Math. Soc., 171: 81–92, Zbl 0015.38803
|
Wikipedia
|
Betrothed numbers
Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (m, n) are a pair of betrothed numbers if s(m) = n + 1 and s(n) = m + 1, where s(n) is the aliquot sum of n: an equivalent condition is that σ(m) = σ(n) = m + n + 1, where σ denotes the sum-of-divisors function.
The first few pairs of betrothed numbers (sequence A005276 in the OEIS) are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128).
All known pairs of betrothed numbers have opposite parity. Any pair of the same parity must exceed 1010.
Quasi-sociable numbers
Quasi-sociable numbers or reduced sociable numbers are numbers whose aliquot sums minus one form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of betrothed numbers and quasiperfect numbers. The first quasi-sociable sequences, or quasi-sociable chains, were discovered by Mitchell Dickerman in 1997:
• 1215571544 = 2^3*11*13813313
• 1270824975 = 3^2*5^2*7*19*42467
• 1467511664 = 2^4*19*599*8059
• 1530808335 = 3^3*5*7*1619903
• 1579407344 = 2^4*31^2*59*1741
• 1638031815 = 3^4*5*7*521*1109
• 1727239544 = 2^3*2671*80833
• 1512587175 = 3*5^2*11*1833439
References
• Hagis, Peter, jr; Lord, Graham (1977). "Quasi-Amicable Numbers". Math. Comput. 31 (138): 608–611. doi:10.1090/s0025-5718-1977-0434939-3. ISSN 0025-5718. Zbl 0355.10010.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of Number Theory I. Dordrecht: Springer-Verlag. p. 113. ISBN 978-1-4020-4215-7. Zbl 1151.11300.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of Number Theory II. Dordrecht: Kluwer Academic. p. 68. ISBN 978-1-4020-2546-4. Zbl 1079.11001.
External links
• Weisstein, Eric W. "Quasiamicable Pair". MathWorld.
Divisibility-based sets of integers
Overview
• Integer factorization
• Divisor
• Unitary divisor
• Divisor function
• Prime factor
• Fundamental theorem of arithmetic
Factorization forms
• Prime
• Composite
• Semiprime
• Pronic
• Sphenic
• Square-free
• Powerful
• Perfect power
• Achilles
• Smooth
• Regular
• Rough
• Unusual
Constrained divisor sums
• Perfect
• Almost perfect
• Quasiperfect
• Multiply perfect
• Hemiperfect
• Hyperperfect
• Superperfect
• Unitary perfect
• Semiperfect
• Practical
• Erdős–Nicolas
With many divisors
• Abundant
• Primitive abundant
• Highly abundant
• Superabundant
• Colossally abundant
• Highly composite
• Superior highly composite
• Weird
Aliquot sequence-related
• Untouchable
• Amicable (Triple)
• Sociable
• Betrothed
Base-dependent
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
Other sets
• Arithmetic
• Deficient
• Friendly
• Solitary
• Sublime
• Harmonic divisor
• Descartes
• Refactorable
• Superperfect
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
• Mersenne
• Proth
• Thabit
• Woodall
Other polynomial numbers
• Hilbert
• Idoneal
• Leyland
• Loeschian
• Lucky numbers of Euler
Recursively defined numbers
• Fibonacci
• Jacobsthal
• Leonardo
• Lucas
• Padovan
• Pell
• Perrin
Possessing a specific set of other numbers
• Amenable
• Congruent
• Knödel
• Riesel
• Sierpiński
Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
• Wolstenholme
Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
• Centered hexagonal
• Centered heptagonal
• Centered octagonal
• Centered nonagonal
• Centered decagonal
• Star
non-centered
• Triangular
• Square
• Square triangular
• Pentagonal
• Hexagonal
• Heptagonal
• Octagonal
• Nonagonal
• Decagonal
• Dodecagonal
3-dimensional
centered
• Centered tetrahedral
• Centered cube
• Centered octahedral
• Centered dodecahedral
• Centered icosahedral
non-centered
• Tetrahedral
• Cubic
• Octahedral
• Dodecahedral
• Icosahedral
• Stella octangula
pyramidal
• Square pyramidal
4-dimensional
non-centered
• Pentatope
• Squared triangular
• Tesseractic
Combinatorial numbers
• Bell
• Cake
• Catalan
• Dedekind
• Delannoy
• Euler
• Eulerian
• Fuss–Catalan
• Lah
• Lazy caterer's sequence
• Lobb
• Motzkin
• Narayana
• Ordered Bell
• Schröder
• Schröder–Hipparchus
• Stirling first
• Stirling second
• Telephone number
• Wedderburn–Etherington
Primes
• Wieferich
• Wall–Sun–Sun
• Wolstenholme prime
• Wilson
Pseudoprimes
• Carmichael number
• Catalan pseudoprime
• Elliptic pseudoprime
• Euler pseudoprime
• Euler–Jacobi pseudoprime
• Fermat pseudoprime
• Frobenius pseudoprime
• Lucas pseudoprime
• Lucas–Carmichael number
• Somer–Lucas pseudoprime
• Strong pseudoprime
Arithmetic functions and dynamics
Divisor functions
• Abundant
• Almost perfect
• Arithmetic
• Betrothed
• Colossally abundant
• Deficient
• Descartes
• Hemiperfect
• Highly abundant
• Highly composite
• Hyperperfect
• Multiply perfect
• Perfect
• Practical
• Primitive abundant
• Quasiperfect
• Refactorable
• Semiperfect
• Sublime
• Superabundant
• Superior highly composite
• Superperfect
Prime omega functions
• Almost prime
• Semiprime
Euler's totient function
• Highly cototient
• Highly totient
• Noncototient
• Nontotient
• Perfect totient
• Sparsely totient
Aliquot sequences
• Amicable
• Perfect
• Sociable
• Untouchable
Primorial
• Euclid
• Fortunate
Other prime factor or divisor related numbers
• Blum
• Cyclic
• Erdős–Nicolas
• Erdős–Woods
• Friendly
• Giuga
• Harmonic divisor
• Jordan–Pólya
• Lucas–Carmichael
• Pronic
• Regular
• Rough
• Smooth
• Sphenic
• Størmer
• Super-Poulet
• Zeisel
Numeral system-dependent numbers
Arithmetic functions
and dynamics
• Persistence
• Additive
• Multiplicative
Digit sum
• Digit sum
• Digital root
• Self
• Sum-product
Digit product
• Multiplicative digital root
• Sum-product
Coding-related
• Meertens
Other
• Dudeney
• Factorion
• Kaprekar
• Kaprekar's constant
• Keith
• Lychrel
• Narcissistic
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Happy
P-adic numbers-related
• Automorphic
• Trimorphic
Digit-composition related
• Palindromic
• Pandigital
• Repdigit
• Repunit
• Self-descriptive
• Smarandache–Wellin
• Undulating
Digit-permutation related
• Cyclic
• Digit-reassembly
• Parasitic
• Primeval
• Transposable
Divisor-related
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
• Vampire
Other
• Friedman
Binary numbers
• Evil
• Odious
• Pernicious
Generated via a sieve
• Lucky
• Prime
Sorting related
• Pancake number
• Sorting number
Natural language related
• Aronson's sequence
• Ban
Graphemics related
• Strobogrammatic
• Mathematics portal
|
Wikipedia
|
Quasibarrelled space
In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.
Definition
A subset $B$ of a topological vector space (TVS) $X$ is called bornivorous if it absorbs all bounded subsets of $X$; that is, if for each bounded subset $S$ of $X,$ there exists some scalar $r$ such that $S\subseteq rB.$ A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin.[1][2]
Properties
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.[3] A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.[4] A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.[2]
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.[2]
Characterizations
A Hausdorff topological vector space $X$ is quasibarrelled if and only if every bounded closed linear operator from $X$ into a complete metrizable TVS is continuous.[5] By definition, a linear $F:X\to Y$ operator is called closed if its graph is a closed subset of $X\times Y.$
For a locally convex space $X$ with continuous dual $X^{\prime }$ the following are equivalent:
1. $X$ is quasibarrelled.
2. Every bounded lower semi-continuous semi-norm on $X$ is continuous.
3. Every $\beta (X',X)$-bounded subset of the continuous dual space $X^{\prime }$ is equicontinuous.
If $X$ is a metrizable locally convex TVS then the following are equivalent:
1. The strong dual of $X$ is quasibarrelled.
2. The strong dual of $X$ is barrelled.
3. The strong dual of $X$ is bornological.
Examples and sufficient conditions
Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.[6] Thus, every metrizable TVS is quasibarrelled.
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.[2] There exist Mackey spaces that are not quasibarrelled.[2] There exist distinguished spaces, DF-spaces, and $\sigma $-barrelled spaces that are not quasibarrelled.[2]
The strong dual space $X_{b}^{\prime }$ of a Fréchet space $X$ is distinguished if and only if $X$ is quasibarrelled.[7]
Counter-examples
There exists a DF-space that is not quasibarrelled.[2] There exists a quasibarrelled DF-space that is not bornological.[2] There exists a quasibarrelled space that is not a σ-barrelled space.[2]
See also
• Barrelled space – Type of topological vector space
• Countably barrelled space
• Countably quasibarrelled space
• Infrabarrelled space
• Uniform boundedness principle#Generalisations – A theorem stating that pointwise boundedness implies uniform boundedness
References
1. Jarchow 1981, p. 222.
2. Khaleelulla 1982, pp. 28–63.
3. Khaleelulla 1982, p. 28.
4. Khaleelulla 1982, pp. 35.
5. Adasch, Ernst & Keim 1978, p. 43.
6. Adasch, Ernst & Keim 1978, pp. 70–73.
7. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
Bibliography
• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
• Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
• Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Boundedness and bornology
Basic concepts
• Barrelled space
• Bounded set
• Bornological space
• (Vector) Bornology
Operators
• (Un)Bounded operator
• Uniform boundedness principle
Subsets
• Barrelled set
• Bornivorous set
• Saturated family
Related spaces
• (Countably) Barrelled space
• (Countably) Quasi-barrelled space
• Infrabarrelled space
• (Quasi-) Ultrabarrelled space
• Ultrabornological space
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
• Metrizability
• Operator topologies
• Topological vector space
• Vector space
Main results
• Anderson–Kadec
• Banach–Alaoglu
• Closed graph theorem
• F. Riesz's
• Hahn–Banach (hyperplane separation
• Vector-valued Hahn–Banach)
• Open mapping (Banach–Schauder)
• Bounded inverse
• Uniform boundedness (Banach–Steinhaus)
Maps
• Bilinear operator
• form
• Linear map
• Almost open
• Bounded
• Continuous
• Closed
• Compact
• Densely defined
• Discontinuous
• Topological homomorphism
• Functional
• Linear
• Bilinear
• Sesquilinear
• Norm
• Seminorm
• Sublinear function
• Transpose
Types of sets
• Absolutely convex/disk
• Absorbing/Radial
• Affine
• Balanced/Circled
• Banach disks
• Bounding points
• Bounded
• Complemented subspace
• Convex
• Convex cone (subset)
• Linear cone (subset)
• Extreme point
• Pre-compact/Totally bounded
• Prevalent/Shy
• Radial
• Radially convex/Star-shaped
• Symmetric
Set operations
• Affine hull
• (Relative) Algebraic interior (core)
• Convex hull
• Linear span
• Minkowski addition
• Polar
• (Quasi) Relative interior
Types of TVSs
• Asplund
• B-complete/Ptak
• Banach
• (Countably) Barrelled
• BK-space
• (Ultra-) Bornological
• Brauner
• Complete
• Convenient
• (DF)-space
• Distinguished
• F-space
• FK-AK space
• FK-space
• Fréchet
• tame Fréchet
• Grothendieck
• Hilbert
• Infrabarreled
• Interpolation space
• K-space
• LB-space
• LF-space
• Locally convex space
• Mackey
• (Pseudo)Metrizable
• Montel
• Quasibarrelled
• Quasi-complete
• Quasinormed
• (Polynomially
• Semi-) Reflexive
• Riesz
• Schwartz
• Semi-complete
• Smith
• Stereotype
• (B
• Strictly
• Uniformly) convex
• (Quasi-) Ultrabarrelled
• Uniformly smooth
• Webbed
• With the approximation property
• Mathematics portal
• Category
• Commons
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
|
Wikipedia
|
Quasi-birth–death process
In queueing models, a discipline within the mathematical theory of probability, the quasi-birth–death process describes a generalisation of the birth–death process.[1][2]: 118 As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.
Discrete time
The stochastic matrix describing the Markov chain has block structure[3]
$P={\begin{pmatrix}A_{1}^{\ast }&A_{2}^{\ast }\\A_{0}^{\ast }&A_{1}&A_{2}\\&A_{0}&A_{1}&A_{2}\\&&A_{0}&A_{1}&A_{2}\\&&&\ddots &\ddots &\ddots \end{pmatrix}}$
where each of A0, A1 and A2 are matrices and A*0, A*1 and A*2 are irregular matrices for the first and second levels.[4]
Continuous time
The transition rate matrix for a quasi-birth-death process has a tridiagonal block structure
$Q={\begin{pmatrix}B_{00}&B_{01}\\B_{10}&A_{1}&A_{2}\\&A_{0}&A_{1}&A_{2}\\&&A_{0}&A_{1}&A_{2}\\&&&A_{0}&A_{1}&A_{2}\\&&&&\ddots &\ddots &\ddots \end{pmatrix}}$
where each of B00, B01, B10, A0, A1 and A2 are matrices.[5] The process can be viewed as a two dimensional chain where the block structure are called levels and the intra-block structure phases.[6] When describing the process by both level and phase it is a continuous-time Markov chain, but when considering levels only it is a semi-Markov process (as transition times are then not exponentially distributed).
Usually the blocks have finitely many phases, but models like the Jackson network can be considered as quasi-birth-death processes with infinitely (but countably) many phases.[6][7]
Stationary distribution
The stationary distribution of a quasi-birth-death process can be computed using the matrix geometric method.
References
1. Latouche, G. (2011). "Level-Independent Quasi-Birth-and-Death Processes". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0461. ISBN 9780470400531.
2. Gautam, Natarajan (2012). Analysis of Queues: Methods and Applications. CRC Press. ISBN 9781439806586.
3. Latouche, G.; Pearce, C. E. M.; Taylor, P. G. (1998). "Invariant measures for quasi-birth-and-death processes". Communications in Statistics. Stochastic Models. 14: 443. doi:10.1080/15326349808807481.
4. Palugya, S. N.; Csorba, M. T. J. (2005). "Modeling Access Control Lists with Discrete-Time Quasi Birth-Death Processes". Computer and Information Sciences - ISCIS 2005. Lecture Notes in Computer Science. Vol. 3733. p. 234. doi:10.1007/11569596_26. ISBN 978-3-540-29414-6.
5. Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN 978-0-387-00211-8.
6. Kroese, D. P.; Scheinhardt, W. R. W.; Taylor, P. G. (2004). "Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process". The Annals of Applied Probability. 14 (4): 2057. arXiv:math/0503555. doi:10.1214/105051604000000477.
7. Motyer, A. J.; Taylor, P. G. (2006). "Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators". Advances in Applied Probability. 38 (2): 522. doi:10.1239/aap/1151337083.
|
Wikipedia
|
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.
Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf.
Definitions
A quasi-coherent sheaf on a ringed space $(X,{\mathcal {O}}_{X})$ is a sheaf ${\mathcal {F}}$ of ${\mathcal {O}}_{X}$-modules which has a local presentation, that is, every point in $X$ has an open neighborhood $U$ in which there is an exact sequence
${\mathcal {O}}_{X}^{\oplus I}|_{U}\to {\mathcal {O}}_{X}^{\oplus J}|_{U}\to {\mathcal {F}}|_{U}\to 0$
for some (possibly infinite) sets $I$ and $J$.
A coherent sheaf on a ringed space $(X,{\mathcal {O}}_{X})$ is a sheaf ${\mathcal {F}}$ satisfying the following two properties:
1. ${\mathcal {F}}$ is of finite type over ${\mathcal {O}}_{X}$, that is, every point in $X$ has an open neighborhood $U$ in $X$ such that there is a surjective morphism ${\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}$ for some natural number $n$;
2. for any open set $U\subseteq X$, any natural number $n$, and any morphism $\varphi :{\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}$ :{\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}} of ${\mathcal {O}}_{X}$-modules, the kernel of $\varphi $ is of finite type.
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of ${\mathcal {O}}_{X}$-modules.
The case of schemes
When $X$ is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf ${\mathcal {F}}$ of ${\mathcal {O}}_{X}$-modules is quasi-coherent if and only if over each open affine subscheme $U=\operatorname {Spec} A$ the restriction ${\mathcal {F}}|_{U}$ is isomorphic to the sheaf ${\tilde {M}}$ associated to the module $M=\Gamma (U,{\mathcal {F}})$ over $A$. When $X$ is a locally Noetherian scheme, ${\mathcal {F}}$ is coherent if and only if it is quasi-coherent and the modules $M$ above can be taken to be finitely generated.
On an affine scheme $U=\operatorname {Spec} A$, there is an equivalence of categories from $A$-modules to quasi-coherent sheaves, taking a module $M$ to the associated sheaf ${\tilde {M}}$. The inverse equivalence takes a quasi-coherent sheaf ${\mathcal {F}}$ on $U$ to the $A$-module ${\mathcal {F}}(U)$ of global sections of ${\mathcal {F}}$.
Here are several further characterizations of quasi-coherent sheaves on a scheme.[1]
Theorem — Let $X$ be a scheme and ${\mathcal {F}}$ an ${\mathcal {O}}_{X}$-module on it. Then the following are equivalent.
• ${\mathcal {F}}$ is quasi-coherent.
• For each open affine subscheme $U$ of $X$, ${\mathcal {F}}|_{U}$ is isomorphic as an ${\mathcal {O}}_{U}$-module to the sheaf ${\tilde {M}}$ associated to some ${\mathcal {O}}(U)$-module $M$.
• There is an open affine cover $\{U_{\alpha }\}$ of $X$ such that for each $U_{\alpha }$ of the cover, ${\mathcal {F}}|_{U_{\alpha }}$ is isomorphic to the sheaf associated to some ${\mathcal {O}}(U_{\alpha })$-module.
• For each pair of open affine subschemes $V\subseteq U$ of $X$, the natural homomorphism
${\mathcal {O}}(V)\otimes _{{\mathcal {O}}(U)}{\mathcal {F}}(U)\to {\mathcal {F}}(V),\,f\otimes s\mapsto f\cdot s|_{V}$
is an isomorphism.
• For each open affine subscheme $U=\operatorname {Spec} A$ of $X$ and each $f\in A$, writing $U_{f}$ for the open subscheme of $U$ where $f$ is not zero, the natural homomorphism
${\mathcal {F}}(U){\bigg [}{\frac {1}{f}}{\bigg ]}\to {\mathcal {F}}(U_{f})$
is an isomorphism. The homomorphism comes from the universal property of localization.
Properties
On an arbitrary ringed space quasi-coherent sheaves do not necessarily form an abelian category. On the other hand, the quasi-coherent sheaves on any scheme form an abelian category, and they are extremely useful in that context.[2]
On any ringed space $X$, the coherent sheaves form an abelian category, a full subcategory of the category of ${\mathcal {O}}_{X}$-modules.[3] (Analogously, the category of coherent modules over any ring $A$ is a full abelian subcategory of the category of all $A$-modules.) So the kernel, image, and cokernel of any map of coherent sheaves are coherent. The direct sum of two coherent sheaves is coherent; more generally, an ${\mathcal {O}}_{X}$-module that is an extension of two coherent sheaves is coherent.[4]
A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always an ${\mathcal {O}}_{X}$-module of finite presentation, meaning that each point $x$ in $X$ has an open neighborhood $U$ such that the restriction ${\mathcal {F}}|_{U}$ of ${\mathcal {F}}$ to $U$ is isomorphic to the cokernel of a morphism ${\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {O}}_{X}^{m}|_{U}$ for some natural numbers $n$ and $m$. If ${\mathcal {O}}_{X}$ is coherent, then, conversely, every sheaf of finite presentation over ${\mathcal {O}}_{X}$ is coherent.
The sheaf of rings ${\mathcal {O}}_{X}$ is called coherent if it is coherent considered as a sheaf of modules over itself. In particular, the Oka coherence theorem states that the sheaf of holomorphic functions on a complex analytic space $X$ is a coherent sheaf of rings. The main part of the proof is the case $X=\mathbf {C} ^{n}$. Likewise, on a locally Noetherian scheme $X$, the structure sheaf ${\mathcal {O}}_{X}$ is a coherent sheaf of rings.[5]
Basic constructions of coherent sheaves
• An ${\mathcal {O}}_{X}$-module ${\mathcal {F}}$ on a ringed space $X$ is called locally free of finite rank, or a vector bundle, if every point in $X$ has an open neighborhood $U$ such that the restriction ${\mathcal {F}}|_{U}$ is isomorphic to a finite direct sum of copies of ${\mathcal {O}}_{X}|_{U}$. If ${\mathcal {F}}$ is free of the same rank $n$ near every point of $X$, then the vector bundle ${\mathcal {F}}$ is said to be of rank $n$.
Vector bundles in this sheaf-theoretic sense over a scheme $X$ are equivalent to vector bundles defined in a more geometric way, as a scheme $E$ with a morphism $\pi :E\to X$ and with a covering of $X$ by open sets $U_{\alpha }$ with given isomorphisms $\pi ^{-1}(U_{\alpha })\cong \mathbb {A} ^{n}\times U_{\alpha }$ over $U_{\alpha }$ such that the two isomorphisms over an intersection $U_{\alpha }\cap U_{\beta }$ differ by a linear automorphism.[6] (The analogous equivalence also holds for complex analytic spaces.) For example, given a vector bundle $E$ in this geometric sense, the corresponding sheaf ${\mathcal {F}}$ is defined by: over an open set $U$ of $X$, the ${\mathcal {O}}(U)$-module ${\mathcal {F}}(U)$ is the set of sections of the morphism $\pi ^{-1}(U)\to U$. The sheaf-theoretic interpretation of vector bundles has the advantage that vector bundles (on a locally Noetherian scheme) are included in the abelian category of coherent sheaves.
• Locally free sheaves come equipped with the standard ${\mathcal {O}}_{X}$-module operations, but these give back locally free sheaves.
• Let $X=\operatorname {Spec} (R)$, $R$ a Noetherian ring. Then vector bundles on $X$ are exactly the sheaves associated to finitely generated projective modules over $R$, or (equivalently) to finitely generated flat modules over $R$.[7]
• Let $X=\operatorname {Proj} (R)$, $R$ a Noetherian $\mathbb {N} $-graded ring, be a projective scheme over a Noetherian ring $R_{0}$. Then each $\mathbb {Z} $-graded $R$-module $M$ determines a quasi-coherent sheaf ${\mathcal {F}}$ on $X$ such that ${\mathcal {F}}|_{\{f\neq 0\}}$ is the sheaf associated to the $R[f^{-1}]_{0}$-module $M[f^{-1}]_{0}$, where $f$ is a homogeneous element of $R$ of positive degree and $\{f\neq 0\}=\operatorname {Spec} R[f^{-1}]_{0}$ is the locus where $f$ does not vanish.
• For example, for each integer $n$, let $R(n)$ denote the graded $R$-module given by $R(n)_{l}=R_{n+l}$. Then each $R(n)$ determines the quasi-coherent sheaf ${\mathcal {O}}_{X}(n)$ on $X$. If $R$ is generated as $R_{0}$-algebra by $R_{1}$, then ${\mathcal {O}}_{X}(n)$ is a line bundle (invertible sheaf) on $X$ and ${\mathcal {O}}_{X}(n)$ is the $n$-th tensor power of ${\mathcal {O}}_{X}(1)$. In particular, ${\mathcal {O}}_{\mathbb {P} ^{n}}(-1)$ is called the tautological line bundle on the projective $n$-space.
• A simple example of a coherent sheaf on $\mathbb {P} ^{2}$ which is not a vector bundle is given by the cokernel in the following sequence
${\mathcal {O}}(1){\xrightarrow {\cdot (x^{2}-yz,y^{3}+xy^{2}-xyz)}}{\mathcal {O}}(3)\oplus {\mathcal {O}}(4)\to {\mathcal {E}}\to 0$
this is because ${\mathcal {E}}$ restricted to the vanishing locus of the two polynomials has two-dimensional fibers, and has one-dimensional fibers elsewhere.
• Ideal sheaves: If $Z$ is a closed subscheme of a locally Noetherian scheme $X$, the sheaf ${\mathcal {I}}_{Z/X}$ of all regular functions vanishing on $Z$ is coherent. Likewise, if $Z$ is a closed analytic subspace of a complex analytic space $X$, the ideal sheaf ${\mathcal {I}}_{Z/X}$ is coherent.
• The structure sheaf ${\mathcal {O}}_{Z}$ of a closed subscheme $Z$ of a locally Noetherian scheme $X$ can be viewed as a coherent sheaf on $X$. To be precise, this is the direct image sheaf $i_{*}{\mathcal {O}}_{Z}$, where $i:Z\to X$ is the inclusion. Likewise for a closed analytic subspace of a complex analytic space. The sheaf $i_{*}{\mathcal {O}}_{Z}$ has fiber (defined below) of dimension zero at points in the open set $X-Z$, and fiber of dimension 1 at points in $Z$. There is a short exact sequence of coherent sheaves on $X$:
$0\to {\mathcal {I}}_{Z/X}\to {\mathcal {O}}_{X}\to i_{*}{\mathcal {O}}_{Z}\to 0.$
• Most operations of linear algebra preserve coherent sheaves. In particular, for coherent sheaves ${\mathcal {F}}$ and ${\mathcal {G}}$ on a ringed space $X$, the tensor product sheaf ${\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {G}}$ and the sheaf of homomorphisms ${\mathcal {H}}om_{{\mathcal {O}}_{X}}({\mathcal {F}},{\mathcal {G}})$ are coherent.[8]
• A simple non-example of a quasi-coherent sheaf is given by the extension by zero functor. For example, consider $i_{!}{\mathcal {O}}_{X}$ for
$X=\operatorname {Spec} (\mathbb {C} [x,x^{-1}]){\xrightarrow {i}}\operatorname {Spec} (\mathbb {C} [x])=Y$[9]
Since this sheaf has non-trivial stalks, but zero global sections, this cannot be a quasi-coherent sheaf. This is because quasi-coherent sheaves on an affine scheme are equivalent to the category of modules over the underlying ring, and the adjunction comes from taking global sections.
Functoriality
Let $f:X\to Y$ be a morphism of ringed spaces (for example, a morphism of schemes). If ${\mathcal {F}}$ is a quasi-coherent sheaf on $Y$, then the inverse image ${\mathcal {O}}_{X}$-module (or pullback) $f^{*}{\mathcal {F}}$ is quasi-coherent on $X$.[10] For a morphism of schemes $f:X\to Y$ and a coherent sheaf ${\mathcal {F}}$ on $Y$, the pullback $f^{*}{\mathcal {F}}$ is not coherent in full generality (for example, $f^{*}{\mathcal {O}}_{Y}={\mathcal {O}}_{X}$, which might not be coherent), but pullbacks of coherent sheaves are coherent if $X$ is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle.
If $f:X\to Y$ is a quasi-compact quasi-separated morphism of schemes and ${\mathcal {F}}$ is a quasi-coherent sheaf on $X$, then the direct image sheaf (or pushforward) $f_{*}{\mathcal {F}}$ is quasi-coherent on $Y$.[2]
The direct image of a coherent sheaf is often not coherent. For example, for a field $k$, let $X$ be the affine line over $k$, and consider the morphism $f:X\to \operatorname {Spec} (k)$; then the direct image $f_{*}{\mathcal {O}}_{X}$ is the sheaf on $\operatorname {Spec} (k)$ associated to the polynomial ring $k[x]$, which is not coherent because $k[x]$ has infinite dimension as a $k$-vector space. On the other hand, the direct image of a coherent sheaf under a proper morphism is coherent, by results of Grauert and Grothendieck.
Local behavior of coherent sheaves
An important feature of coherent sheaves ${\mathcal {F}}$ is that the properties of ${\mathcal {F}}$ at a point $x$ control the behavior of ${\mathcal {F}}$ in a neighborhood of $x$, more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if ${\mathcal {F}}$ is a coherent sheaf on a scheme $X$, then the fiber ${\mathcal {F}}_{x}\otimes _{{\mathcal {O}}_{X,x}}k(x)$ of $F$ at a point $x$ (a vector space over the residue field $k(x)$) is zero if and only if the sheaf ${\mathcal {F}}$ is zero on some open neighborhood of $x$. A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous.[11] Thus a coherent sheaf has constant rank on an open set, while the rank can jump up on a lower-dimensional closed subset.
In the same spirit: a coherent sheaf ${\mathcal {F}}$ on a scheme $X$ is a vector bundle if and only if its stalk ${\mathcal {F}}_{x}$ is a free module over the local ring ${\mathcal {O}}_{X,x}$ for every point $x$ in $X$.[12]
On a general scheme, one cannot determine whether a coherent sheaf is a vector bundle just from its fibers (as opposed to its stalks). On a reduced locally Noetherian scheme, however, a coherent sheaf is a vector bundle if and only if its rank is locally constant.[13]
Examples of vector bundles
For a morphism of schemes $X\to Y$, let $\Delta :X\to X\times _{Y}X$ be the diagonal morphism, which is a closed immersion if $X$ is separated over $Y$. Let ${\mathcal {I}}$ be the ideal sheaf of $X$ in $X\times _{Y}X$. Then the sheaf of differentials $\Omega _{X/Y}^{1}$ can be defined as the pullback $\Delta ^{*}{\mathcal {I}}$ of ${\mathcal {I}}$ to $X$. Sections of this sheaf are called 1-forms on $X$ over $Y$, and they can be written locally on $X$ as finite sums $\textstyle \sum f_{j}\,dg_{j}$ for regular functions $f_{j}$ and $g_{j}$. If $X$ is locally of finite type over a field $k$, then $\Omega _{X/k}^{1}$ is a coherent sheaf on $X$.
If $X$ is smooth over $k$, then $\Omega ^{1}$ (meaning $\Omega _{X/k}^{1}$) is a vector bundle over $X$, called the cotangent bundle of $X$. Then the tangent bundle $TX$ is defined to be the dual bundle $(\Omega ^{1})^{*}$. For $X$ smooth over $k$ of dimension $n$ everywhere, the tangent bundle has rank $n$.
If $Y$ is a smooth closed subscheme of a smooth scheme $X$ over $k$, then there is a short exact sequence of vector bundles on $Y$:
$0\to TY\to TX|_{Y}\to N_{Y/X}\to 0,$
which can be used as a definition of the normal bundle $N_{Y/X}$ to $Y$ in $X$.
For a smooth scheme $X$ over a field $k$ and a natural number $i$, the vector bundle $\Omega ^{i}$ of i-forms on $X$ is defined as the $i$-th exterior power of the cotangent bundle, $\Omega ^{i}=\Lambda ^{i}\Omega ^{1}$. For a smooth variety $X$ of dimension $n$ over $k$, the canonical bundle $K_{X}$ means the line bundle $\Omega ^{n}$. Thus sections of the canonical bundle are algebro-geometric analogs of volume forms on $X$. For example, a section of the canonical bundle of affine space $\mathbb {A} ^{n}$ over $k$ can be written as
$f(x_{1},\ldots ,x_{n})\;dx_{1}\wedge \cdots \wedge dx_{n},$
where $f$ is a polynomial with coefficients in $k$.
Let $R$ be a commutative ring and $n$ a natural number. For each integer $j$, there is an important example of a line bundle on projective space $\mathbb {P} ^{n}$ over $R$, called ${\mathcal {O}}(j)$. To define this, consider the morphism of $R$-schemes
$\pi :\mathbb {A} ^{n+1}-0\to \mathbb {P} ^{n}$ :\mathbb {A} ^{n+1}-0\to \mathbb {P} ^{n}}
given in coordinates by $(x_{0},\ldots ,x_{n})\mapsto [x_{0},\ldots ,x_{n}]$. (That is, thinking of projective space as the space of 1-dimensional linear subspaces of affine space, send a nonzero point in affine space to the line that it spans.) Then a section of ${\mathcal {O}}(j)$ over an open subset $U$ of $\mathbb {P} ^{n}$ is defined to be a regular function $f$ on $\pi ^{-1}(U)$ that is homogeneous of degree $j$, meaning that
$f(ax)=a^{j}f(x)$
as regular functions on ($\mathbb {A} ^{1}-0)\times \pi ^{-1}(U)$. For all integers $i$ and $j$, there is an isomorphism ${\mathcal {O}}(i)\otimes {\mathcal {O}}(j)\cong {\mathcal {O}}(i+j)$ of line bundles on $\mathbb {P} ^{n}$.
In particular, every homogeneous polynomial in $x_{0},\ldots ,x_{n}$ of degree $j$ over $R$ can be viewed as a global section of ${\mathcal {O}}(j)$ over $\mathbb {P} ^{n}$. Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles ${\mathcal {O}}(j)$.[14] This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions. The regular functions on projective space $\mathbb {P} ^{n}$ over $R$ are just the "constants" (the ring $R$), and so it is essential to work with the line bundles ${\mathcal {O}}(j)$.
Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let $R$ be a Noetherian ring (for example, a field), and consider the polynomial ring $S=R[x_{0},\ldots ,x_{n}]$ as a graded ring with each $x_{i}$ having degree 1. Then every finitely generated graded $S$-module $M$ has an associated coherent sheaf ${\tilde {M}}$ on $\mathbb {P} ^{n}$ over $R$. Every coherent sheaf on $\mathbb {P} ^{n}$ arises in this way from a finitely generated graded $S$-module $M$. (For example, the line bundle ${\mathcal {O}}(j)$ is the sheaf associated to the $S$-module $S$ with its grading lowered by $j$.) But the $S$-module $M$ that yields a given coherent sheaf on $\mathbb {P} ^{n}$ is not unique; it is only unique up to changing $M$ by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on $\mathbb {P} ^{n}$ is the quotient of the category of finitely generated graded $S$-modules by the Serre subcategory of modules that are nonzero in only finitely many degrees.[15]
The tangent bundle of projective space $\mathbb {P} ^{n}$ over a field $k$ can be described in terms of the line bundle ${\mathcal {O}}(1)$. Namely, there is a short exact sequence, the Euler sequence:
$0\to {\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}(1)^{\oplus \;n+1}\to T\mathbb {P} ^{n}\to 0.$
It follows that the canonical bundle $K_{\mathbb {P} ^{n}}$ (the dual of the determinant line bundle of the tangent bundle) is isomorphic to ${\mathcal {O}}(-n-1)$. This is a fundamental calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle ${\mathcal {O}}(1)$ means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with positive Ricci curvature.
Vector bundles on a hypersurface
Consider a smooth degree-$d$ hypersurface $X\subset \mathbb {P} ^{n}$ defined by the homogeneous polynomial $f$ of degree $d$. Then, there is an exact sequence
$0\to {\mathcal {O}}_{X}(-d)\to i^{*}\Omega _{\mathbb {P} ^{n}}\to \Omega _{X}\to 0$
where the second map is the pullback of differential forms, and the first map sends
$\phi \mapsto d(f\cdot \phi )$
Note that this sequence tells us that ${\mathcal {O}}(-d)$ is the conormal sheaf of $X$ in $\mathbb {P} ^{n}$. Dualizing this yields the exact sequence
$0\to T_{X}\to i^{*}T_{\mathbb {P} ^{n}}\to {\mathcal {O}}(d)\to 0$
hence ${\mathcal {O}}(d)$ is the normal bundle of $X$ in $\mathbb {P} ^{n}$. If we use the fact that given an exact sequence
$0\to {\mathcal {E}}_{1}\to {\mathcal {E}}_{2}\to {\mathcal {E}}_{3}\to 0$
of vector bundles with ranks $r_{1}$,$r_{2}$,$r_{3}$, there is an isomorphism
$\Lambda ^{r_{2}}{\mathcal {E}}_{2}\cong \Lambda ^{r_{1}}{\mathcal {E}}_{1}\otimes \Lambda ^{r_{3}}{\mathcal {E}}_{3}$
of line bundles, then we see that there is the isomorphism
$i^{*}\omega _{\mathbb {P} ^{n}}\cong \omega _{X}\otimes {\mathcal {O}}_{X}(-d)$
showing that
$\omega _{X}\cong {\mathcal {O}}_{X}(d-n-1)$
Serre construction and vector bundles
One useful technique for constructing rank 2 vector bundles is the Serre construction[16][17]pg 3 which establishes a correspondence between rank 2 vector bundles ${\mathcal {E}}$ on a smooth projective variety $X$ and codimension 2 subvarieties $Y$ using a certain ${\text{Ext}}^{1}$-group calculated on $X$. This is given by a cohomological condition on the line bundle $\wedge ^{2}{\mathcal {E}}$ (see below).
The correspondence in one direction is given as follows: for a section $s\in \Gamma (X,{\mathcal {E}})$ we can associated the vanishing locus $V(s)\subset X$. If $V(s)$ is a codimension 2 subvariety, then
1. It is a local complete intersection, meaning if we take an affine chart $U_{i}\subset X$ then $s|_{U_{i}}\in \Gamma (U_{i},{\mathcal {E}})$ can be represented as a function $s_{i}:U_{i}\to \mathbb {A} ^{2}$, where $s_{i}(p)=(s_{i}^{1}(p),s_{i}^{2}(p))$ and $V(s)\cap U_{i}=V(s_{i}^{1},s_{i}^{2})$
2. The line bundle $\omega _{X}\otimes \wedge ^{2}{\mathcal {E}}|_{V(s)}$ is isomorphic to the canonical bundle $\omega _{V(s)}$ on $V(s)$
In the other direction,[18] for a codimension 2 subvariety $Y\subset X$ and a line bundle ${\mathcal {L}}\to X$ such that
1. $H^{1}(X,{\mathcal {L}})=H^{2}(X,{\mathcal {L}})=0$
2. $\omega _{Y}\cong (\omega _{X}\otimes {\mathcal {L}})|_{Y}$
there is a canonical isomorphism
${\text{Hom}}((\omega _{X}\otimes {\mathcal {L}})|_{Y},\omega _{Y})\cong {\text{Ext}}^{1}({\mathcal {I}}_{Y}\otimes {\mathcal {L}},{\mathcal {O}}_{X})$
which is functorial with respect to inclusion of codimension $2$ subvarieties. Moreover, any isomorphism given on the left corresponds to a locally free sheaf in the middle of the extension on the right. That is, for $s\in {\text{Hom}}((\omega _{X}\otimes {\mathcal {L}})|_{Y},\omega _{Y})$ which is an isomorphism there is a corresponding locally free sheaf ${\mathcal {E}}$ of rank 2 which fits into a short exact sequence
$0\to {\mathcal {O}}_{X}\to {\mathcal {E}}\to {\mathcal {I}}_{Y}\otimes {\mathcal {L}}\to 0$
This vector bundle can then be further studied using cohomological invariants to determine if it is stable or not. This forms the basis for studying moduli of stable vector bundles in many specific cases, such as on principally polarized abelian varieties[17] and K3 surfaces.[19]
Chern classes and algebraic K-theory
A vector bundle $E$ on a smooth variety $X$ over a field has Chern classes in the Chow ring of $X$, $c_{i}(E)$ in $CH^{i}(X)$ for $i\geq 0$.[20] These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence
$0\to A\to B\to C\to 0$
of vector bundles on $X$, the Chern classes of $B$ are given by
$c_{i}(B)=c_{i}(A)+c_{1}(A)c_{i-1}(C)+\cdots +c_{i-1}(A)c_{1}(C)+c_{i}(C).$
It follows that the Chern classes of a vector bundle $E$ depend only on the class of $E$ in the Grothendieck group $K_{0}(X)$. By definition, for a scheme $X$, $K_{0}(X)$ is the quotient of the free abelian group on the set of isomorphism classes of vector bundles on $X$ by the relation that $[B]=[A]+[C]$ for any short exact sequence as above. Although $K_{0}(X)$ is hard to compute in general, algebraic K-theory provides many tools for studying it, including a sequence of related groups $K_{i}(X)$ for integers $i>0$.
A variant is the group $G_{0}(X)$ (or $K_{0}'(X)$), the Grothendieck group of coherent sheaves on $X$. (In topological terms, G-theory has the formal properties of a Borel–Moore homology theory for schemes, while K-theory is the corresponding cohomology theory.) The natural homomorphism $K_{0}(X)\to G_{0}(X)$ is an isomorphism if $X$ is a regular separated Noetherian scheme, using that every coherent sheaf has a finite resolution by vector bundles in that case.[21] For example, that gives a definition of the Chern classes of a coherent sheaf on a smooth variety over a field.
More generally, a Noetherian scheme $X$ is said to have the resolution property if every coherent sheaf on $X$ has a surjection from some vector bundle on $X$. For example, every quasi-projective scheme over a Noetherian ring has the resolution property.
Applications of resolution property
Since the resolution property states that a coherent sheaf ${\mathcal {E}}$ on a Noetherian scheme is quasi-isomorphic in the derived category to the complex of vector bundles :${\mathcal {E}}_{k}\to \cdots \to {\mathcal {E}}_{1}\to {\mathcal {E}}_{0}$ we can compute the total Chern class of ${\mathcal {E}}$ with
$c({\mathcal {E}})=c({\mathcal {E}}_{0})c({\mathcal {E}}_{1})^{-1}\cdots c({\mathcal {E}}_{k})^{(-1)^{k}}$
For example, this formula is useful for finding the Chern classes of the sheaf representing a subscheme of $X$. If we take the projective scheme $Z$ associated to the ideal $(xy,xz)\subset \mathbb {C} [x,y,z,w]$, then
$c({\mathcal {O}}_{Z})={\frac {c({\mathcal {O}})c({\mathcal {O}}(-3))}{c({\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2))}}$
since there is the resolution
$0\to {\mathcal {O}}(-3)\to {\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2)\to {\mathcal {O}}\to {\mathcal {O}}_{Z}\to 0$
over $\mathbb {CP} ^{3}$.
Bundle homomorphism vs. sheaf homomorphism
When vector bundles and locally free sheaves of finite constant rank are used interchangeably, care must be given to distinguish between bundle homomorphisms and sheaf homomorphisms. Specifically, given vector bundles $p:E\to X,\,q:F\to X$, by definition, a bundle homomorphism $\varphi :E\to F$ is a scheme morphism over $X$ (i.e., $p=q\circ \varphi $) such that, for each geometric point $x$ in $X$, $\varphi _{x}:p^{-1}(x)\to q^{-1}(x)$ is a linear map of rank independent of $x$. Thus, it induces the sheaf homomorphism ${\widetilde {\varphi }}:{\mathcal {E}}\to {\mathcal {F}}$ of constant rank between the corresponding locally free ${\mathcal {O}}_{X}$-modules (sheaves of dual sections). But there may be an ${\mathcal {O}}_{X}$-module homomorphism that does not arise this way; namely, those not having constant rank.
In particular, a subbundle $E\subset F$ is a subsheaf (i.e., ${\mathcal {E}}$ is a subsheaf of ${\mathcal {F}}$). But the converse can fail; for example, for an effective Cartier divisor $D$ on $X$, ${\mathcal {O}}_{X}(-D)\subset {\mathcal {O}}_{X}$ is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles).
The category of quasi-coherent sheaves
The quasi-coherent sheaves on any fixed scheme form an abelian category. Gabber showed that, in fact, the quasi-coherent sheaves on any scheme form a particularly well-behaved abelian category, a Grothendieck category.[22] A quasi-compact quasi-separated scheme $X$ (such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi-coherent sheaves on $X$, by Rosenberg, generalizing a result of Gabriel.[23]
Coherent cohomology
The fundamental technical tool in algebraic geometry is the cohomology theory of coherent sheaves. Although it was introduced only in the 1950s, many earlier techniques of algebraic geometry are clarified by the language of sheaf cohomology applied to coherent sheaves. Broadly speaking, coherent sheaf cohomology can be viewed as a tool for producing functions with specified properties; sections of line bundles or of more general sheaves can be viewed as generalized functions. In complex analytic geometry, coherent sheaf cohomology also plays a foundational role.
Among the core results of coherent sheaf cohomology are results on finite-dimensionality of cohomology, results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such as Hodge theory, and formulas for Euler characteristics of coherent sheaves such as the Riemann–Roch theorem.
See also
• Picard group
• Divisor (algebraic geometry)
• Reflexive sheaf
• Quot scheme
• Twisted sheaf
• Essentially finite vector bundle
• Bundle of principal parts
• Gabriel–Rosenberg reconstruction theorem
• Pseudo-coherent sheaf
• Quasi-coherent sheaf on an algebraic stack
Notes
1. Mumford 1999, Ch. III, § 1, Theorem-Definition 3.
2. Stacks Project, Tag 01LA.
3. Stacks Project, Tag 01BU.
4. Serre 1955, §13
5. Grothendieck & Dieudonné 1960, Corollaire 1.5.2
6. Hartshorne 1977, Exercise II.5.18
7. Stacks Project, Tag 00NV.
8. Serre 1955, §14
9. Hartshorne 1977
10. Stacks Project, Tag 01BG.
11. Hartshorne 1977, Example III.12.7.2
12. Grothendieck & Dieudonné 1960, Ch. 0, 5.2.7
13. Eisenbud 1995, Exercise 20.13
14. Hartshorne 1977, Corollary II.5.16
15. Stacks Project, Tag 01YR.
16. Serre, Jean-Pierre (1960–1961). "Sur les modules projectifs". Séminaire Dubreil. Algèbre et théorie des nombres (in French). 14 (1): 1–16.
17. Gulbrandsen, Martin G. (2013-05-20). "Vector Bundles and Monads On Abelian Threefolds" (PDF). Communications in Algebra. 41 (5): 1964–1988. arXiv:0907.3597. doi:10.1080/00927872.2011.645977. ISSN 0092-7872.
18. Hartshorne, Robin (1978). "Stable Vector Bundles of Rank 2 on P3". Mathematische Annalen. 238: 229–280.
19. Huybrechts, Daniel; Lehn, Manfred (2010). The Geometry of Moduli Spaces of Sheaves. Cambridge Mathematical Library (2 ed.). Cambridge: Cambridge University Press. pp. 123–128, 238–243. doi:10.1017/cbo9780511711985. ISBN 978-0-521-13420-0.
20. Fulton 1998, §3.2 and Example 8.3.3
21. Fulton 1998, B.8.3
22. Stacks Project, Tag 077K.
23. Antieau 2016, Corollary 4.2
References
• Antieau, Benjamin (2016), "A reconstruction theorem for abelian categories of twisted sheaves", Journal für die reine und angewandte Mathematik, 712: 175–188, arXiv:1305.2541, doi:10.1515/crelle-2013-0119, MR 3466552
• Danilov, V. I. (2001) [1994], "Coherent algebraic sheaf", Encyclopedia of Mathematics, EMS Press
• Grauert, Hans; Remmert, Reinhold (1984), Coherent Analytic Sheaves, Springer-Verlag, doi:10.1007/978-3-642-69582-7, ISBN 3-540-13178-7, MR 0755331
• Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
• Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, ISBN 978-0-387-98549-7, MR 1644323
• Sections 0.5.3 and 0.5.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Mumford, David (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 354063293X. MR 1748380.
• Onishchik, A.L. (2001) [1994], "Coherent analytic sheaf", Encyclopedia of Mathematics, EMS Press
• Onishchik, A.L. (2001) [1994], "Coherent sheaf", Encyclopedia of Mathematics, EMS Press
• Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents", Annals of Mathematics, 61: 197–278, doi:10.2307/1969915, MR 0068874
External links
• The Stacks Project Authors, The Stacks Project
• Part V of Vakil, Ravi, The Rising Sea
|
Wikipedia
|
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.[1] The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers $\mathbb {Q} $ is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers $\mathbb {R} $ is not compact either, because it excludes the two limiting values $+\infty $ and $-\infty $. However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.
One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.[2]
The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.
Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering $\mathbb {R} ^{1}$ (the real number line), the sequence of points 0, 1, 2, 3, ... has no subsequence that converges to any real number.
Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces.[3] In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally — that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character.
The term compact set is sometimes used as a synonym for compact space, but also often refers to a compact subspace of a topological space.
Historical development
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point. Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts — until it closes down on the desired limit point. The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.[4]
In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà.[5] The culmination of their investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt. For a certain class of Green's functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paper[6] which led to the famous 1906 thesis).
However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904). The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
This property was significant because it allowed for the passage from local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearing his name. Ultimately, the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topological space. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called (relative) sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.
Basic examples
Any finite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point among these points in that interval. For instance, the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, ... get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval of the real numbers is not compact. It is also crucial that the interval be bounded, since in the interval [0,∞), one could choose the sequence of points 0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number.
In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary — without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.
Definitions
Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.[3]
In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness — originally called bicompactness — is defined using covers consisting of open sets (see Open cover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally — in a neighbourhood of each point of the space — and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.
Open cover definition
Formally, a topological space X is called compact if every open cover of X has a finite subcover.[7] That is, X is compact if for every collection C of open subsets[8] of X such that
$X=\bigcup _{S\in C}S\ ,$
there is a finite subcollection F ⊆ C such that
$X=\bigcup _{S\in F}S\ .$
Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.
Compactness of subsets
A subset K of a topological space X is said to be compact if it is compact as a subspace (in the subspace topology). That is, K is compact if for every arbitrary collection C of open subsets of X such that
$K\subseteq \bigcup _{S\in C}S\ ,$
there is a finite subcollection F ⊆ C such that
$K\subseteq \bigcup _{S\in F}S\ .$
Compactness is a "topological" property. That is, if $K\subset Z\subset Y$, with subset Z equipped with the subspace topology, then K is compact in Z if and only if K is compact in Y.
Characterization
If X is a topological space then the following are equivalent:
1. X is compact; i.e., every open cover of X has a finite subcover.
2. X has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (Alexander's sub-base theorem).
3. X is Lindelöf and countably compact.[9]
4. Any collection of closed subsets of X with the finite intersection property has nonempty intersection.
5. Every net on X has a convergent subnet (see the article on nets for a proof).
6. Every filter on X has a convergent refinement.
7. Every net on X has a cluster point.
8. Every filter on X has a cluster point.
9. Every ultrafilter on X converges to at least one point.
10. Every infinite subset of X has a complete accumulation point.[10]
11. For every topological space Y, the projection $X\times Y\to Y$ is a closed mapping[11] (see proper map).
Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).[12]
Euclidean space
For any subset A of Euclidean space, A is compact if and only if it is closed and bounded; this is the Heine–Borel theorem.
As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed n-ball.
Metric spaces
For any metric space (X, d), the following are equivalent (assuming countable choice):
1. (X, d) is compact.
2. (X, d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces).[13]
3. (X, d) is sequentially compact; that is, every sequence in X has a convergent subsequence whose limit is in X (this is also equivalent to compactness for first-countable uniform spaces).
4. (X, d) is limit point compact (also called weakly countably compact); that is, every infinite subset of X has at least one limit point in X.
5. (X, d) is countably compact; that is, every countable open cover of X has a finite subcover.
6. (X, d) is an image of a continuous function from the Cantor set.[14]
7. Every decreasing nested sequence of nonempty closed subsets S1 ⊇ S2 ⊇ ... in (X, d) has a nonempty intersection.
8. Every increasing nested sequence of proper open subsets S1 ⊆ S2 ⊆ ... in (X, d) fails to cover X.
A compact metric space (X, d) also satisfies the following properties:
1. Lebesgue's number lemma: For every open cover of X, there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover.
2. (X, d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.
3. X is closed and bounded (as a subset of any metric space whose restricted metric is d). The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.
Ordered Spaces
For an ordered space (X, <) (i.e. a totally ordered set equipped with the order topology), the following are equivalent:
1. (X, <) is compact.
2. Every subset of X has a supremum (i.e. a least upper bound) in X.
3. Every subset of X has an infimum (i.e. a greatest lower bound) in X.
4. Every nonempty closed subset of X has a maximum and a minimum element.
An ordered space satisfying (any one of) these conditions is called a complete lattice.
In addition, the following are equivalent for all ordered spaces (X, <), and (assuming countable choice) are true whenever (X, <) is compact. (The converse in general fails if (X, <) is not also metrizable.):
1. Every sequence in (X, <) has a subsequence that converges in (X, <).
2. Every monotone increasing sequence in X converges to a unique limit in X.
3. Every monotone decreasing sequence in X converges to a unique limit in X.
4. Every decreasing nested sequence of nonempty closed subsets S1 ⊇ S2 ⊇ ... in (X, <) has a nonempty intersection.
5. Every increasing nested sequence of proper open subsets S1 ⊆ S2 ⊆ ... in (X, <) fails to cover X.
Characterization by continuous functions
Let X be a topological space and C(X) the ring of real continuous functions on X. For each p ∈ X, the evaluation map $\operatorname {ev} _{p}\colon C(X)\to \mathbb {R} $ given by evp(f) = f(p) is a ring homomorphism. The kernel of evp is a maximal ideal, since the residue field C(X)/ker evp is the field of real numbers, by the first isomorphism theorem. A topological space X is pseudocompact if and only if every maximal ideal in C(X) has residue field the real numbers. For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.[15] There are pseudocompact spaces that are not compact, though.
In general, for non-pseudocompact spaces there are always maximal ideals m in C(X) such that the residue field C(X)/m is a (non-Archimedean) hyperreal field. The framework of non-standard analysis allows for the following alternative characterization of compactness:[16] a topological space X is compact if and only if every point x of the natural extension *X is infinitely close to a point x0 of X (more precisely, x is contained in the monad of x0).
Hyperreal definition
A space X is compact if its hyperreal extension *X (constructed, for example, by the ultrapower construction) has the property that every point of *X is infinitely close to some point of X ⊂ *X. For example, an open real interval X = (0, 1) is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X.
Sufficient conditions
• A closed subset of a compact space is compact.[17]
• A finite union of compact sets is compact.
• A continuous image of a compact space is compact.[18]
• The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed);
• If X is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).[lower-alpha 1]
• The product of any collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.)
• In a metrizable space, a subset is compact if and only if it is sequentially compact (assuming countable choice)
• A finite set endowed with any topology is compact.
Properties of compact spaces
• A compact subset of a Hausdorff space X is closed.
• If X is not Hausdorff then a compact subset of X may fail to be a closed subset of X (see footnote for example).[lower-alpha 2]
• If X is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).[lower-alpha 3]
• In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are not closed.
• If A and B are disjoint compact subsets of a Hausdorff space X, then there exist disjoint open sets U and V in X such that A ⊆ U and B ⊆ V.
• A continuous bijection from a compact space into a Hausdorff space is a homeomorphism.
• A compact Hausdorff space is normal and regular.
• If a space X is compact and Hausdorff, then no finer topology on X is compact and no coarser topology on X is Hausdorff.
• If a subset of a metric space (X, d) is compact then it is d-bounded.
Functions and compact spaces
Since a continuous image of a compact space is compact, the extreme value theorem holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.[19] (Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a proper map is compact.
Compactifications
Every topological space X is an open dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space X is an open dense subspace of a compact Hausdorff space having at most one point more than X.
Ordered compact spaces
A nonempty compact subset of the real numbers has a greatest element and a least element.
Let X be a simply ordered set endowed with the order topology. Then X is compact if and only if X is a complete lattice (i.e. all subsets have suprema and infima).[20]
Examples
• Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
• Any space carrying the cofinite topology is compact.
• Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point compactification of $\mathbb {R} $ is homeomorphic to the circle S1; the one-point compactification of $\mathbb {R} ^{2}$ is homeomorphic to the sphere S2. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
• The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpiński space is compact.
• No discrete space with an infinite number of points is compact. The collection of all singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
• In $\mathbb {R} $ carrying the lower limit topology, no uncountable set is compact.
• In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not locally compact but is still Lindelöf.
• The closed unit interval [0, 1] is compact. This follows from the Heine–Borel theorem. The open interval (0, 1) is not compact: the open cover $ \left({\frac {1}{n}},1-{\frac {1}{n}}\right)$ for n = 3, 4, ... does not have a finite subcover. Similarly, the set of rational numbers in the closed interval [0,1] is not compact: the sets of rational numbers in the intervals $ \left[0,{\frac {1}{\pi }}-{\frac {1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]$ cover all the rationals in [0, 1] for n = 4, 5, ... but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of $\mathbb {R} $.
• The set $\mathbb {R} $ of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals (n − 1, n + 1) , where n takes all integer values in Z, cover $\mathbb {R} $ but there is no finite subcover.
• On the other hand, the extended real number line carrying the analogous topology is compact; note that the cover described above would never reach the points at infinity and thus would not cover the extended real line. In fact, the set has the homeomorphism to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
• For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
• On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. (Alaoglu's theorem)
• The Cantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set.
• Consider the set K of all functions f : $\mathbb {R} $ → [0, 1] from the real number line to the closed unit interval, and define a topology on K so that a sequence $\{f_{n}\}$ in K converges towards f ∈ K if and only if $\{f_{n}(x)\}$ converges towards f(x) for all real numbers x. There is only one such topology; it is called the topology of pointwise convergence or the product topology. Then K is a compact topological space; this follows from the Tychonoff theorem.
• Consider the set K of all functions f : [0, 1] → [0, 1] satisfying the Lipschitz condition |f(x) − f(y)| ≤ |x − y| for all x, y ∈ [0,1]. Consider on K the metric induced by the uniform distance $d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.$ Then by Arzelà–Ascoli theorem the space K is compact.
• The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers $\mathbb {C} $. Conversely, any compact subset of $\mathbb {C} $ arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space $\ell ^{2}$ may have any compact nonempty subset of $\mathbb {C} $ as spectrum.
Algebraic examples
• Topological groups such as an orthogonal group are compact, while groups such as a general linear group are not.
• Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.
• The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact, but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact schemes, "quasi" referring to the non-Hausdorff nature of the topology.
• The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of profinite groups.
• The structure space of a commutative unital Banach algebra is a compact Hausdorff space.
• The Hilbert cube is compact, again a consequence of Tychonoff's theorem.
• A profinite group (e.g. Galois group) is compact.
See also
• Compactly generated space
• Compactness theorem
• Eberlein compactum
• Exhaustion by compact sets
• Lindelöf space
• Metacompact space
• Noetherian topological space
• Orthocompact space
• Paracompact space
• Quasi-compact morphism
• Precompact set - also called totally bounded
• Relatively compact subspace
• Totally bounded
Notes
1. Let X = {a, b} ∪ $\mathbb {N} $, U = {a} ∪ $\mathbb {N} $, and V = {b} ∪ $\mathbb {N} $. Endow X with the topology generated by the following basic open sets: every subset of $\mathbb {N} $ is open; the only open sets containing a are X and U; and the only open sets containing b are X and V. Then U and V are both compact subsets but their intersection, which is $\mathbb {N} $, is not compact. Note that both U and V are compact open subsets, neither one of which is closed.
2. Let X = {a, b} and endow X with the topology {X, ∅, {a}}. Then {a} is a compact set but it is not closed.
3. Let X be the set of non-negative integers. We endow X with the particular point topology by defining a subset U ⊆ X to be open if and only if 0 ∈ U. Then S := {0} is compact, the closure of S is all of X, but X is not compact since the collection of open subsets {{0, x} : x ∈ X} does not have a finite subcover.
References
1. "Compactness". Encyclopaedia Britannica. mathematics. Retrieved 2019-11-25 – via britannica.com.
2. Engelking, Ryszard (1977). General Topology. Warsaw, PL: PWN. p. 266.
3. "Sequential compactness". www-groups.mcs.st-andrews.ac.uk. MT 4522 course lectures. Retrieved 2019-11-25.
4. Kline 1990, pp. 952–953; Boyer & Merzbach 1991, p. 561
5. Kline 1990, Chapter 46, §2
6. Frechet, M. 1904. Generalisation d'un theorem de Weierstrass. Analyse Mathematique.
7. Weisstein, Eric W. "Compact Space". mathworld.wolfram.com. Retrieved 2019-11-25.
8. Here, "collection" means "set" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset".
9. Howes 1995, pp. xxvi–xxviii.
10. Kelley 1955, p. 163
11. Bourbaki 2007, § 10.2. Theorem 1, Corollary 1.
12. Bourbaki 2007, § 9.1. Definition 1.
13. Arkhangel'skii & Fedorchuk 1990, Theorem 5.3.7
14. Willard 1970 Theorem 30.7.
15. Gillman & Jerison 1976, §5.6
16. Robinson 1996, Theorem 4.1.13
17. Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.3
18. Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.2
19. Arkhangel'skii & Fedorchuk 1990, Corollary 5.2.1
20. Steen & Seebach 1995, p. 67
Bibliography
• Alexandrov, Pavel; Urysohn, Pavel (1929). "Mémoire sur les espaces topologiques compacts". Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Mathematical Sciences. 14.
• Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.). General Topology I. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer. ISBN 978-0-387-18178-3..
• Arkhangel'skii, A.V. (2001) [1994], "Compact space", Encyclopedia of Mathematics, EMS Press.
• Bolzano, Bernard (1817). Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege. Wilhelm Engelmann. (Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation).
• Borel, Émile (1895). "Sur quelques points de la théorie des fonctions". Annales Scientifiques de l'École Normale Supérieure. 3. 12: 9–55. doi:10.24033/asens.406. JFM 26.0429.03.
• Bourbaki, Nicolas (2007). Topologie générale. Chapitres 1 à 4. Berlin: Springer. doi:10.1007/978-3-540-33982-3. ISBN 978-3-540-33982-3.
• Boyer, Carl B. (1959). The history of the calculus and its conceptual development. New York: Dover Publications. MR 0124178.
• Boyer, Carl Benjamin; Merzbach, Uta C (1991). A History of Mathematics (2nd ed.). John Wiley & Sons. ISBN 978-0-471-54397-8.
• Arzelà, Cesare (1895). "Sulle funzioni di linee". Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5): 55–74.
• Arzelà, Cesare (1882–1883). "Un'osservazione intorno alle serie di funzioni". Rend. Dell' Accad. R. Delle Sci. dell'Istituto di Bologna: 142–159.
• Ascoli, G. (1883–1884). "Le curve limiti di una varietà data di curve". Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. 18 (3): 521–586.
• Fréchet, Maurice (1906). "Sur quelques points du calcul fonctionnel". Rendiconti del Circolo Matematico di Palermo. 22 (1): 1–72. doi:10.1007/BF03018603. hdl:10338.dmlcz/100655. S2CID 123251660.
• Gillman, Leonard; Jerison, Meyer (1976). Rings of continuous functions. Springer-Verlag.
• Howes, Norman R. (23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M.
• Kelley, John (1955). General topology. Graduate Texts in Mathematics. Vol. 27. Springer-Verlag.
• Kline, Morris (1990) [1972]. Mathematical thought from ancient to modern times (3rd ed.). Oxford University Press. ISBN 978-0-19-506136-9.
• Lebesgue, Henri (1904). Leçons sur l'intégration et la recherche des fonctions primitives. Gauthier-Villars.
• Robinson, Abraham (1996). Non-standard analysis. Princeton University Press. ISBN 978-0-691-04490-3. MR 0205854.
• Scarborough, C.T.; Stone, A.H. (1966). "Products of nearly compact spaces" (PDF). Transactions of the American Mathematical Society. 124 (1): 131–147. doi:10.2307/1994440. JSTOR 1994440. Archived (PDF) from the original on 2017-08-16..
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover Publications reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.
• Willard, Stephen (1970). General Topology. Dover publications. ISBN 0-486-43479-6.
External links
• Sundström, Manya Raman (2010). "A pedagogical history of compactness". arXiv:1006.4131v1 [math.HO].
This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Topology
Fields
• General (point-set)
• Algebraic
• Combinatorial
• Continuum
• Differential
• Geometric
• low-dimensional
• Homology
• cohomology
• Set-theoretic
• Digital
Key concepts
• Open set / Closed set
• Interior
• Continuity
• Space
• compact
• Connected
• Hausdorff
• metric
• uniform
• Homotopy
• homotopy group
• fundamental group
• Simplicial complex
• CW complex
• Polyhedral complex
• Manifold
• Bundle (mathematics)
• Second-countable space
• Cobordism
Metrics and properties
• Euler characteristic
• Betti number
• Winding number
• Chern number
• Orientability
Key results
• Banach fixed-point theorem
• De Rham cohomology
• Invariance of domain
• Poincaré conjecture
• Tychonoff's theorem
• Urysohn's lemma
• Category
• Mathematics portal
• Wikibook
• Wikiversity
• Topics
• general
• algebraic
• geometric
• Publications
|
Wikipedia
|
Quasi-compact morphism
In algebraic geometry, a morphism $f:X\to Y$ between schemes is said to be quasi-compact if Y can be covered by open affine subschemes $V_{i}$ such that the pre-images $f^{-1}(V_{i})$ are quasi-compact (as topological space).[1] If f is quasi-compact, then the pre-image of a quasi-compact open subscheme (e.g., open affine subscheme) under f is quasi-compact.
It is not enough that Y admits a covering by quasi-compact open subschemes whose pre-images are quasi-compact. To give an example,[2] let A be a ring that does not satisfy the ascending chain conditions on radical ideals, and put $X=\operatorname {Spec} A$. X contains an open subset U that is not quasi-compact. Let Y be the scheme obtained by gluing two X's along U. X, Y are both quasi-compact. If $f:X\to Y$ is the inclusion of one of the copies of X, then the pre-image of the other X, open affine in Y, is U, not quasi-compact. Hence, f is not quasi-compact.
A morphism from a quasi-compact scheme to an affine scheme is quasi-compact.
Let $f:X\to Y$ be a quasi-compact morphism between schemes. Then $f(X)$ is closed if and only if it is stable under specialization.
The composition of quasi-compact morphisms is quasi-compact. The base change of a quasi-compact morphism is quasi-compact.
An affine scheme is quasi-compact. In fact, a scheme is quasi-compact if and only if it is a finite union of open affine subschemes. Serre’s criterion gives a necessary and sufficient condition for a quasi-compact scheme to be affine.
A quasi-compact scheme has at least one closed point.[3]
See also
• fpqc morphism
References
1. This is the definition in Hartshorne.
2. Remark 1.5 in Vistoli
3. Schwede, Karl (2005), "Gluing schemes and a scheme without closed points", Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, pp. 157–172, doi:10.1090/conm/386/07222, MR 2182775. See in particular Proposition 4.1.
• Hartshorne, Algebraic Geometry.
• Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory." arXiv:math/0412512
External links
• When is an irreducible scheme quasi-compact?
|
Wikipedia
|
Quasi-complete space
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete[1] if every closed and bounded subset is complete.[2] This concept is of considerable importance for non-metrizable TVSs.[2]
Properties
• Every quasi-complete TVS is sequentially complete.[2]
• In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact.[3]
• In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.[2]
• If X is a normed space and Y is a quasi-complete locally convex TVS then the set of all compact linear maps of X into Y is a closed vector subspace of $L_{b}(X;Y)$.[4]
• Every quasi-complete infrabarrelled space is barreled.[5]
• If X is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded.[5]
• A quasi-complete nuclear space then X has the Heine–Borel property.[6]
Examples and sufficient conditions
Every complete TVS is quasi-complete.[7] The product of any collection of quasi-complete spaces is again quasi-complete.[2] The projective limit of any collection of quasi-complete spaces is again quasi-complete.[8] Every semi-reflexive space is quasi-complete.[9]
The quotient of a quasi-complete space by a closed vector subspace may fail to be quasi-complete.
Counter-examples
There exists an LB-space that is not quasi-complete.[10]
See also
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Complete uniform space – Topological space with a notion of uniform propertiesPages displaying short descriptions of redirect targets
References
1. Wilansky 2013, p. 73.
2. Schaefer & Wolff 1999, p. 27.
3. Schaefer & Wolff 1999, p. 201.
4. Schaefer & Wolff 1999, p. 110.
5. Schaefer & Wolff 1999, p. 142.
6. Trèves 2006, p. 520.
7. Narici & Beckenstein 2011, pp. 156–175.
8. Schaefer & Wolff 1999, p. 52.
9. Schaefer & Wolff 1999, p. 144.
10. Khaleelulla 1982, pp. 28–63.
Bibliography
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
• Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
• Metrizability
• Operator topologies
• Topological vector space
• Vector space
Main results
• Anderson–Kadec
• Banach–Alaoglu
• Closed graph theorem
• F. Riesz's
• Hahn–Banach (hyperplane separation
• Vector-valued Hahn–Banach)
• Open mapping (Banach–Schauder)
• Bounded inverse
• Uniform boundedness (Banach–Steinhaus)
Maps
• Bilinear operator
• form
• Linear map
• Almost open
• Bounded
• Continuous
• Closed
• Compact
• Densely defined
• Discontinuous
• Topological homomorphism
• Functional
• Linear
• Bilinear
• Sesquilinear
• Norm
• Seminorm
• Sublinear function
• Transpose
Types of sets
• Absolutely convex/disk
• Absorbing/Radial
• Affine
• Balanced/Circled
• Banach disks
• Bounding points
• Bounded
• Complemented subspace
• Convex
• Convex cone (subset)
• Linear cone (subset)
• Extreme point
• Pre-compact/Totally bounded
• Prevalent/Shy
• Radial
• Radially convex/Star-shaped
• Symmetric
Set operations
• Affine hull
• (Relative) Algebraic interior (core)
• Convex hull
• Linear span
• Minkowski addition
• Polar
• (Quasi) Relative interior
Types of TVSs
• Asplund
• B-complete/Ptak
• Banach
• (Countably) Barrelled
• BK-space
• (Ultra-) Bornological
• Brauner
• Complete
• Convenient
• (DF)-space
• Distinguished
• F-space
• FK-AK space
• FK-space
• Fréchet
• tame Fréchet
• Grothendieck
• Hilbert
• Infrabarreled
• Interpolation space
• K-space
• LB-space
• LF-space
• Locally convex space
• Mackey
• (Pseudo)Metrizable
• Montel
• Quasibarrelled
• Quasi-complete
• Quasinormed
• (Polynomially
• Semi-) Reflexive
• Riesz
• Schwartz
• Semi-complete
• Smith
• Stereotype
• (B
• Strictly
• Uniformly) convex
• (Quasi-) Ultrabarrelled
• Uniformly smooth
• Webbed
• With the approximation property
• Mathematics portal
• Category
• Commons
|
Wikipedia
|
Quasi-continuous function
In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general.
Definition
Let $X$ be a topological space. A real-valued function $f:X\rightarrow \mathbb {R} $ is quasi-continuous at a point $x\in X$ if for any $\epsilon >0$ and any open neighborhood $U$ of $x$ there is a non-empty open set $G\subset U$ such that
$|f(x)-f(y)|<\epsilon \;\;\;\;\forall y\in G$
Note that in the above definition, it is not necessary that $x\in G$.
Properties
• If $f:X\rightarrow \mathbb {R} $ is continuous then $f$ is quasi-continuous
• If $f:X\rightarrow \mathbb {R} $ is continuous and $g:X\rightarrow \mathbb {R} $ is quasi-continuous, then $f+g$ is quasi-continuous.
Example
Consider the function $f:\mathbb {R} \rightarrow \mathbb {R} $ defined by $f(x)=0$ whenever $x\leq 0$ and $f(x)=1$ whenever $x>0$. Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set $G\subset U$ such that $y<0\;\forall y\in G$. Clearly this yields $|f(0)-f(y)|=0\;\forall y\in G$ thus f is quasi-continuous.
In contrast, the function $g:\mathbb {R} \rightarrow \mathbb {R} $ defined by $g(x)=0$ whenever $x$ is a rational number and $g(x)=1$ whenever $x$ is an irrational number is nowhere quasi-continuous, since every nonempty open set $G$ contains some $y_{1},y_{2}$ with $|g(y_{1})-g(y_{2})|=1$.
References
• Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange. 33 (2): 339–350.
• T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange. 14 (2): 259–308. JSTOR 44151947.
|
Wikipedia
|
Quasi-derivative
In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative.
Let f : A → F be a continuous function from an open set A in a Banach space E to another Banach space F. Then the quasi-derivative of f at x0 ∈ A is a linear transformation u : E → F with the following property: for every continuous function g : [0,1] → A with g(0)=x0 such that g′(0) ∈ E exists,
$\lim _{t\to 0^{+}}{\frac {f(g(t))-f(x_{0})}{t}}=u(g'(0)).$
If such a linear map u exists, then f is said to be quasi-differentiable at x0.
Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.
References
• Dieudonné, J (1969). Foundations of modern analysis. Academic Press.
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
Analysis in topological vector spaces
Basic concepts
• Abstract Wiener space
• Classical Wiener space
• Bochner space
• Convex series
• Cylinder set measure
• Infinite-dimensional vector function
• Matrix calculus
• Vector calculus
Derivatives
• Differentiable vector–valued functions from Euclidean space
• Differentiation in Fréchet spaces
• Fréchet derivative
• Total
• Functional derivative
• Gateaux derivative
• Directional
• Generalizations of the derivative
• Hadamard derivative
• Holomorphic
• Quasi-derivative
Measurability
• Besov measure
• Cylinder set measure
• Canonical Gaussian
• Classical Wiener measure
• Measure like set functions
• infinite-dimensional Gaussian measure
• Projection-valued
• Vector
• Bochner / Weakly / Strongly measurable function
• Radonifying function
Integrals
• Bochner
• Direct integral
• Dunford
• Gelfand–Pettis/Weak
• Regulated
• Paley–Wiener
Results
• Cameron–Martin theorem
• Inverse function theorem
• Nash–Moser theorem
• Feldman–Hájek theorem
• No infinite-dimensional Lebesgue measure
• Sazonov's theorem
• Structure theorem for Gaussian measures
Related
• Crinkled arc
• Covariance operator
Functional calculus
• Borel functional calculus
• Continuous functional calculus
• Holomorphic functional calculus
Applications
• Banach manifold (bundle)
• Convenient vector space
• Choquet theory
• Fréchet manifold
• Hilbert manifold
|
Wikipedia
|
Quasi-finite morphism
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions:[1]
• Every point x of X is isolated in its fiber f−1(f(x)). In other words, every fiber is a discrete (hence finite) set.
• For every point x of X, the scheme f−1(f(x)) = X ×YSpec κ(f(x)) is a finite κ(f(x)) scheme. (Here κ(p) is the residue field at a point p.)
• For every point x of X, ${\mathcal {O}}_{X,x}\otimes \kappa (f(x))$ is finitely generated over $\kappa (f(x))$.
Quasi-finite morphisms were originally defined by Alexander Grothendieck in SGA 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in EGA II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of stalks.
For a general morphism f : X → Y and a point x in X, f is said to be quasi-finite at x if there exist open affine neighborhoods U of x and V of f(x) such that f(U) is contained in V and such that the restriction f : U → V is quasi-finite. f is locally quasi-finite if it is quasi-finite at every point in X.[2] A quasi-compact locally quasi-finite morphism is quasi-finite.
Properties
For a morphism f, the following properties are true.[3]
• If f is quasi-finite, then the induced map fred between reduced schemes is quasi-finite.
• If f is a closed immersion, then f is quasi-finite.
• If X is noetherian and f is an immersion, then f is quasi-finite.
• If g : Y → Z, and if g ∘ f is quasi-finite, then f is quasi-finite if any of the following are true:
1. g is separated,
2. X is noetherian,
3. X ×Z Y is locally noetherian.
Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite.[3]
If f is unramified at a point x, then f is quasi-finite at x. Conversely, if f is quasi-finite at x, and if also ${\mathcal {O}}_{f^{-1}(f(x)),x}$, the local ring of x in the fiber f−1(f(x)), is a field and a finite separable extension of κ(f(x)), then f is unramified at x.[4]
Finite morphisms are quasi-finite.[5] A quasi-finite proper morphism locally of finite presentation is finite.[6] Indeed, a morphism is finite if and only if it is proper and locally quasi-finite.[7] Since proper morphisms are of finite type and finite type morphisms are quasi-compact[8] one may omit the qualification locally, i.e., a morphism is finite if and only if it is proper and quasi-finite.
A generalized form of Zariski Main Theorem is the following:[9] Suppose Y is quasi-compact and quasi-separated. Let f be quasi-finite, separated and of finite presentation. Then f factors as $X\hookrightarrow X'\to Y$ where the first morphism is an open immersion and the second is finite. (X is open in a finite scheme over Y.)
See also
• The quasi-finite fundamental group scheme
Notes
1. EGA II, Définition 6.2.3
2. EGA III, ErrIII, 20.
3. EGA II, Proposition 6.2.4.
4. EGA IV4, Théorème 17.4.1.
5. EGA II, Corollaire 6.1.7.
6. EGA IV3, Théorème 8.11.1.
7. "Lemma 02LS". The Stacks Project. Retrieved 31 January 2022.
8. "Definition 29.15.1". The Stacks Project. Retrieved 15 August 2023.
9. EGA IV3, Théorème 8.12.6.
References
• Grothendieck, Alexandre; Michèle Raynaud (2003) [1971]. Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3) (in French) (Updated ed.). Société Mathématique de France. xviii+327. ISBN 2-85629-141-4.
• Grothendieck, Alexandre; Jean Dieudonné (1961). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8: 5–222. doi:10.1007/bf02699291.
• Grothendieck, Alexandre; Jean Dieudonné (1966). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28: 5–255.
|
Wikipedia
|
Quasi-free algebra
In abstract algebra, a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative algebra. The notion was introduced by Cuntz and Quillen for the applications to cyclic homology.[1] A quasi-free algebra generalizes a free algebra, as well as the coordinate ring of a smooth affine complex curve. Because of the latter generalization, a quasi-free algebra can be thought of as signifying smoothness on a noncommutative space.[2]
Definition
Let A be an associative algebra over the complex numbers. Then A is said to be quasi-free if the following equivalent conditions are met:[3][4][5]
• Given a square-zero extension $R\to R/I$, each homomorphism $A\to R/I$ lifts to $A\to R$.
• The cohomological dimension of A with respect to Hochschild cohomology is at most one.
Let $(\Omega A,d)$ denotes the differential envelope of A; i.e., the universal differential-graded algebra generated by A.[6][7] Then A is quasi-free if and only if $\Omega ^{1}A$ is projective as a bimodule over A.[3]
There is also a characterization in terms of a connection. Given an A-bimodule E, a right connection on E is a linear map
$\nabla _{r}:E\to E\otimes _{A}\Omega ^{1}A$
that satisfies $\nabla _{r}(as)=a\nabla _{r}(s)$ and $\nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da$.[8] A left connection is defined in the similar way. Then A is quasi-free if and only if $\Omega ^{1}A$ admits a right connection.[9]
Properties and examples
One of basic properties of a quasi-free algebra is that the algebra is left and right hereditary (i.e., a submodule of a projective left or right module is projective or equivalently the left or right global dimension is at most one).[10] This puts a strong restriction for algebras to be quasi-free. For example, a hereditary (commutative) integral domain is precisely a Dedekind domain. In particular, a polynomial ring over a field is quasi-free if and only if the number of variables is at most one.
An analog of the tubular neighborhood theorem, called the formal tubular neighborhood theorem, holds for quasi-free algebras.[11]
References
1. Cuntz & Quillen 1995
2. Cuntz 2013, Introduction
3. Cuntz & Quillen 1995, Proposition 3.3.
4. Vale 2009, Proposotion 7.7.
5. Kontsevich & Rosenberg 2000, 1.1.
6. Cuntz & Quillen 1995, Proposition 1.1.
7. Kontsevich & Rosenberg 2000, 1.1.2.
8. Vale 2009, Definition 8.4.
9. Vale 2009, Remark 7.12.
10. Cuntz & Quillen 1995, Proposition 5.1.
11. Cuntz & Quillen 1995, § 6.
Bibliography
• Cuntz, Joachim (June 2013). "Quillen's work on the foundations of cyclic cohomology". Journal of K-Theory. 11 (3): 559–574. arXiv:1202.5958. doi:10.1017/is012011006jkt201. ISSN 1865-2433.
• Cuntz, Joachim; Quillen, Daniel (1995). "Algebra Extensions and Nonsingularity". Journal of the American Mathematical Society. 8 (2): 251–289. doi:10.2307/2152819. ISSN 0894-0347.
• Kontsevich, Maxim; Rosenberg, Alexander L. (2000). "Noncommutative Smooth Spaces". The Gelfand Mathematical Seminars, 1996–1999. Birkhäuser: 85–108. arXiv:math/9812158. doi:10.1007/978-1-4612-1340-6_5.
• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35
• Vale, R. (2009). "notes on quasi-free algebras" (PDF).
Further reading
• https://ncatlab.org/nlab/show/quasi-free+algebra
|
Wikipedia
|
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have an identity element.
A quasigroup with an identity element is called a loop.
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
Definitions
There are at least two structurally equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup defined with a single binary operation, however, need not be a quasigroup.[1] We begin with the first definition.
Algebra
A quasigroup (Q, ∗) is a non-empty set Q with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy closure property), obeying the Latin square property. This states that, for each a and b in Q, there exist unique elements x and y in Q such that both
a ∗ x = b,
y ∗ a = b
hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, finite group, is a Latin square.) The requirement that x and y be unique can be replaced by the requirement that the magma be cancellative.[2][3]
The unique solutions to these equations are written x = a \ b and y = b / a. The operations '\' and '/' are called, respectively, left division and right division. With regard to the Cayley table, the first equation (left division) means that the b entry in the a row marks the x column while the second equation (right division) means that the b entry in the a column marks the y row.
The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.[4][5]
Universal algebra
Given some algebraic structure, an identity is an equation in which all variables are tacitly universally quantified, and in which all operations are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called a variety. Many standard results in universal algebra hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.
A right-quasigroup (Q, ∗, /) is a type (2, 2) algebra that satisfy both identities:
y = (y / x) ∗ x;
y = (y ∗ x) / x.
A left-quasigroup (Q, ∗, \) is a type (2, 2) algebra that satisfy both identities:
y = x ∗ (x \ y);
y = x \ (x ∗ y).
A quasigroup (Q, ∗, \, /) is a type (2, 2, 2) algebra (i.e., equipped with three binary operations) that satisfy the identities:
y = (y / x) ∗ x,
y = (y ∗ x) / x.
y = x ∗ (x \ y),
y = x \ (x ∗ y),
In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.
Hence if (Q, ∗) is a quasigroup according to the definition of the previous section, then (Q, ∗, \, /) is the same quasigroup in the sense of universal algebra. And vice versa: if (Q, ∗, \, /) is a quasigroup according to the sense of universal algebra, then (Q, ∗) is a quasigroup according to the first definition.
Loops
A loop is a quasigroup with an identity element; that is, an element, e, such that
x ∗ e = x and e ∗ x = x for all x in Q.
It follows that the identity element, e, is unique, and that every element of Q has unique left and right inverses (which need not be the same).
A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed idempotent". (That is, there is a principal isotopy (x, y, z) ↦ (x, −y, z).)
A loop that is associative is a group. A group can have a non-associative pique isotope, but it cannot have a nonassociative loop isotope.
There are weaker associativity properties that have been given special names.
For instance, a Bol loop is a loop that satisfies either:
x ∗ (y ∗ (x ∗ z)) = (x ∗ (y ∗ x)) ∗ z for each x, y and z in Q (a left Bol loop),
or else
((z ∗ x) ∗ y) ∗ x = z ∗ ((x ∗ y) ∗ x) for each x, y and z in Q (a right Bol loop).
A loop that is both a left and right Bol loop is a Moufang loop. This is equivalent to any one of the following single Moufang identities holding for all x, y, z:
x ∗ (y ∗ (x ∗ z)) = ((x ∗ y) ∗ x) ∗ z,
z ∗ (x ∗ (y ∗ x)) = ((z ∗ x) ∗ y) ∗ x,
(x ∗ y) ∗ (z ∗ x) = x ∗ ((y ∗ z) ∗ x), or
(x ∗ y) ∗ (z ∗ x) = (x ∗ (y ∗ z)) ∗ x.
Symmetries
(Smith 2007) names the following important properties and subclasses:
Semisymmetry
A quasigroup is semisymmetric if any of the following equivalent identities hold:[6]
x ∗ y = y / x,
y ∗ x = x \ y,
x = (y ∗ x) ∗ y,
x = y ∗ (x ∗ y).
Although this class may seem special, every quasigroup Q induces a semisymmetric quasigroup QΔ on the direct product cube Q3 via the following operation:
(x1, x2, x3) ⋅ (y1, y2, y3) = (y3 / x2, y1 \ x3, x1 ∗ y2) = (x2 // y3, x3 \\ y1, x1 ∗ y2),
where "//" and "\\" are the conjugate division operations given by y // x = x / y and y \\ x = x \ y.
Triality
Total symmetry
A narrower class is a totally symmetric quasigroup (sometimes abbreviated TS-quasigroup) in which all conjugates coincide as one operation: x ∗ y = x / y = x \ y. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e. x ∗ y = y ∗ x.
Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with) Steiner triples, so such a quasigroup is also called a Steiner quasigroup, and sometimes the latter is even abbreviated as squag. The term sloop refers to an analogue for loops, namely, totally symmetric loops that satisfy x ∗ x = 1 instead of x ∗ x = x. Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).
Total antisymmetry
A quasigroup (Q, ∗) is called weakly totally anti-symmetric if for all c, x, y ∈ Q, the following implication holds.[7]
(c ∗ x) ∗ y = (c ∗ y) ∗ x implies that x = y
A quasigroup (Q, ∗) is called totally anti-symmetric if, in addition, for all x, y ∈ Q, the following implication holds:[7]
x ∗ y = y ∗ x implies that x = y.
This property is required, for example, in the Damm algorithm.
Examples
• Every group is a loop, because a ∗ x = b if and only if x = a−1 ∗ b, and y ∗ a = b if and only if y = b ∗ a−1.
• The integers Z (or the rationals Q or the reals R) with subtraction (−) form a quasigroup. These quasiqroups are not loops because there is no identity element (0 is a right identity because a − 0 = a, but not a left identity because, in general, 0 − a ≠ a).
• The nonzero rationals Q× (or the nonzero reals R×) with division (÷) form a quasigroup.
• Any vector space over a field of characteristic not equal to 2 forms an idempotent, commutative quasigroup under the operation x ∗ y = (x + y) / 2.
• Every Steiner triple system defines an idempotent, commutative quasigroup: a ∗ b is the third element of the triple containing a and b. These quasigroups also satisfy (x ∗ y) ∗ y = x for all x and y in the quasigroup. These quasigroups are known as Steiner quasigroups.[8]
• The set {±1, ±i, ±j, ±k} where ii = jj = kk = +1 and with all other products as in the quaternion group forms a nonassociative loop of order 8. See hyperbolic quaternions for its application. (The hyperbolic quaternions themselves do not form a loop or quasigroup.)
• The nonzero octonions form a nonassociative loop under multiplication. The octonions are a special type of loop known as a Moufang loop.
• An associative quasigroup is either empty or is a group, since if there is at least one element, the invertibility of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements, thus satisfying all three requirements of a group.
• The following construction is due to Hans Zassenhaus. On the underlying set of the four-dimensional vector space F4 over the 3-element Galois field F = Z/3Z define
(x1, x2, x3, x4) ∗ (y1, y2, y3, y4) = (x1, x2, x3, x4) + (y1, y2, y3, y4) + (0, 0, 0, (x3 − y3)(x1y2 − x2y1)).
Then, (F4, ∗) is a commutative Moufang loop that is not a group.[9]
• More generally, the nonzero elements of any division algebra form a quasigroup with the operation of multiplication in the algebra.
Properties
In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.
Quasigroups have the cancellation property: if ab = ac, then b = c. This follows from the uniqueness of left division of ab or ac by a. Similarly, if ba = ca, then b = c.
The Latin square property of quasigroups implies that, given any two of the three variables in xy = z, the third variable is uniquely determined.
Multiplication operators
The definition of a quasigroup can be treated as conditions on the left and right multiplication operators Lx, Rx: Q → Q, defined by
${\begin{aligned}L_{x}(y)&=xy\\R_{x}(y)&=yx\\\end{aligned}}$
The definition says that both mappings are bijections from Q to itself. A magma Q is a quasigroup precisely when all these operators, for every x in Q, are bijective. The inverse mappings are left and right division, that is,
${\begin{aligned}L_{x}^{-1}(y)&=x\backslash y\\R_{x}^{-1}(y)&=y/x\end{aligned}}$
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on universal algebra) are
${\begin{aligned}L_{x}L_{x}^{-1}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad x(x\backslash y)&=y\\L_{x}^{-1}L_{x}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad x\backslash (xy)&=y\\R_{x}R_{x}^{-1}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad (y/x)x&=y\\R_{x}^{-1}R_{x}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad (yx)/x&=y\end{aligned}}$
where id denotes the identity mapping on Q.
Latin squares
Main article: Latin square
The multiplication table of a finite quasigroup is a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. See small Latin squares and quasigroups.
Infinite quasigroups
For a countably infinite quasigroup Q, it is possible to imagine an infinite array in which every row and every column corresponds to some element q of Q, and where the element a ∗ b is in the row corresponding to a and the column responding to b. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.
For an uncountably infinite quasigroup, such as the group of non-zero real numbers under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in a sequence. (This is somewhat misleading however, as the reals can be written in a sequence of length ${\mathfrak {c}}$, assuming the well-ordering theorem.)
Inverse properties
The binary operation of a quasigroup is invertible in the sense that both $L_{x}$ and $R_{x}$, the left and right multiplication operators, are bijective, and hence invertible.
Every loop element has a unique left and right inverse given by
$x^{\lambda }=e/x\qquad x^{\lambda }x=e$
$x^{\rho }=x\backslash e\qquad xx^{\rho }=e$
A loop is said to have (two-sided) inverses if $x^{\lambda }=x^{\rho }$ for all x. In this case the inverse element is usually denoted by $x^{-1}$.
There are some stronger notions of inverses in loops that are often useful:
• A loop has the left inverse property if $x^{\lambda }(xy)=y$ for all $x$ and $y$. Equivalently, $L_{x}^{-1}=L_{x^{\lambda }}$ or $x\backslash y=x^{\lambda }y$.
• A loop has the right inverse property if $(yx)x^{\rho }=y$ for all $x$ and $y$. Equivalently, $R_{x}^{-1}=R_{x^{\rho }}$ or $y/x=yx^{\rho }$.
• A loop has the antiautomorphic inverse property if $(xy)^{\lambda }=y^{\lambda }x^{\lambda }$ or, equivalently, if $(xy)^{\rho }=y^{\rho }x^{\rho }$.
• A loop has the weak inverse property when $(xy)z=e$ if and only if $x(yz)=e$. This may be stated in terms of inverses via $(xy)^{\lambda }x=y^{\lambda }$ or equivalently $x(yx)^{\rho }=y^{\rho }$.
A loop has the inverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop that satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
Any loop that satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
Morphisms
A quasigroup or loop homomorphism is a map f : Q → P between two quasigroups such that f(xy) = f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
Homotopy and isotopy
Let Q and P be quasigroups. A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that
$\alpha (x)\beta (y)=\gamma (xy)\,$
for all x, y in Q. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
An isotopy is a homotopy for which each of the three maps (α, β, γ) is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ.
An autotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with the automorphism group as a subgroup.
Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by (x + y)/2 is isotopic to the additive group (R, +), but is not itself a group as it has no identity element. Every medial quasigroup is isotopic to an abelian group by the Bruck–Toyoda theorem.
Conjugation (parastrophe)
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e., x ∗ y = z) we can form five new operations: x o y := y ∗ x (the opposite operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
Isostrophe (paratopy)
If the set Q has two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be isostrophic to each other. There are also many other names for this relation of "isostrophe", e.g., paratopy.
Generalizations
Polyadic or multiary quasigroups
An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: Qn → Q, such that the equation f(x1,...,xn) = y has a unique solution for any one variable if all the other n variables are specified arbitrarily. Polyadic or multiary means n-ary for some nonnegative integer n.
A 0-ary, or nullary, quasigroup is just a constant element of Q. A 1-ary, or unary, quasigroup is a bijection of Q to itself. A binary, or 2-ary, quasigroup is an ordinary quasigroup.
An example of a multiary quasigroup is an iterated group operation, y = x1 · x2 · ··· · xn; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.
There exist multiary quasigroups that cannot be represented in any of these ways. An n-ary quasigroup is irreducible if its operation cannot be factored into the composition of two operations in the following way:
$f(x_{1},\dots ,x_{n})=g(x_{1},\dots ,x_{i-1},\,h(x_{i},\dots ,x_{j}),\,x_{j+1},\dots ,x_{n}),$
where 1 ≤ i < j ≤ n and (i, j) ≠ (1, n). Finite irreducible n-ary quasigroups exist for all n > 2; see Akivis and Goldberg (2001) for details.
An n-ary quasigroup with an n-ary version of associativity is called an n-ary group.
Number of small quasigroups and loops
Main article: Small Latin squares and quasigroups
The number of isomorphism classes of small quasigroups (sequence A057991 in the OEIS) and loops (sequence A057771 in the OEIS) is given here:[10]
Order Number of quasigroups Number of loops
0 1 0
1 1 1
2 1 1
3 5 1
4 35 2
5 1,411 6
6 1,130,531 109
7 12,198,455,835 23,746
8 2,697,818,331,680,661 106,228,849
9 15,224,734,061,438,247,321,497 9,365,022,303,540
10 2,750,892,211,809,150,446,995,735,533,513 20,890,436,195,945,769,617
11 19,464,657,391,668,924,966,791,023,043,937,578,299,025 1,478,157,455,158,044,452,849,321,016
See also
• Division ring – a ring in which every non-zero element has a multiplicative inverse
• Semigroup – an algebraic structure consisting of a set together with an associative binary operation
• Monoid – a semigroup with an identity element
• Planar ternary ring – has an additive and multiplicative loop structure
• Problems in loop theory and quasigroup theory
• Mathematics of Sudoku
Notes
1. Smith 2007, pp. 3, 26–27
2. H. Rubin; J. E. Rubin (1985). Equivalents of the Axiom of Choice, II. Elsevier. p. 109.
3. For clarity, cancellativity alone is insufficient: the requirement for existence of a solution must be retained.
4. Pflugfelder 1990, p. 2
5. Bruck 1971, p. 1
6. The first two equations are equivalent to the last two by direct application of the cancellation property of quasigroups. The last pair are shown to be equivalent by setting x = ((x ∗ y) ∗ x) ∗ (x ∗ y) = y ∗ (x ∗ y).
7. Damm, H. Michael (2007). "Totally anti-symmetric quasigroups for all orders n ≠ 2, 6". Discrete Mathematics. 307 (6): 715–729. doi:10.1016/j.disc.2006.05.033.
8. Colbourn & Dinitz 2007, p. 497, definition 28.12
9. Romanowska, Anna B.; Smith, Jonathan D. H. (1999), "Example 4.1.3 (Zassenhaus's Commutative Moufang Loop)", Post-modern algebra, Pure and Applied Mathematics, New York: Wiley, p. 93, doi:10.1002/9781118032589, ISBN 978-0-471-12738-3, MR 1673047.
10. McKay, Brendan D.; Meynert, Alison; Myrvold, Wendy (2007). "Small Latin squares, quasigroups, and loops" (PDF). J. Comb. Des. 15 (2): 98–119. CiteSeerX 10.1.1.151.3043. doi:10.1002/jcd.20105. Zbl 1112.05018.
References
• Akivis, M. A.; Goldberg, Vladislav V. (2001). "Solution of Belousov's problem". Discussiones Mathematicae - General Algebra and Applications. 21 (1): 93–103. arXiv:math/0010175. doi:10.7151/dmgaa.1030. S2CID 18421746.
• Belousov, V.D. (1967). Foundations of the Theory of Quasigroups and Loops (in Russian). Moscow: Izdat. "Nauka". OCLC 472241611.
• Belousov, V.D. (1971). Algebraic Nets and Quasigroups (in Russian). Kishinev: Izdat. "Štiinca". OCLC 8292276.
• Belousov, V.D. (1981). Elements of Quasigroup Theory: a Special Course (in Russian). Kishinev: Kishinev State University Printing House. OCLC 318458899.
• Bruck, R.H. (1971) [1958]. A Survey of Binary Systems. Springer. ISBN 978-0-387-03497-3.
• Chein, O.; Pflugfelder, H. O.; Smith, J.D.H., eds. (1990). Quasigroups and Loops: Theory and Applications. Berlin: Heldermann. ISBN 978-3-88538-008-5.
• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), CRC Press, ISBN 978-1-58488-506-1
• Dudek, W.A.; Glazek, K. (2008). "Around the Hosszu-Gluskin Theorem for n-ary groups". Discrete Math. 308 (21): 4861–76. arXiv:math/0510185. doi:10.1016/j.disc.2007.09.005. S2CID 9545943.
• Pflugfelder, H.O. (1990). Quasigroups and Loops: Introduction. Berlin: Heldermann. ISBN 978-3-88538-007-8.
• Smith, J.D.H (2007). An Introduction to Quasigroups and their Representations. CRC Press. ISBN 978-1-58488-537-5.
• Shcherbacov, V.A. (2017). Elements of Quasigroup Theory and Applications. CRC Press. ISBN 978-1-4987-2155-4.
External links
• quasigroups
• "Quasi-group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
|
Wikipedia
|
Hyperelliptic surface
In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.
Invariants
The Kodaira dimension is 0.
Hodge diamond:
1
11
020
11
1
Classification
Any hyperelliptic surface is a quotient (E×F)/G, where E = C/Λ and F are elliptic curves, and G is a subgroup of F (acting on F by translations). There are seven families of hyperelliptic surfaces as in the following table.
order of K Λ G Action of G on E
2 Any Z/2Z e → −e
2 Any Z/2Z ⊕ Z/2Z e → −e, e → e+c, −c=c
3 Z ⊕ Zω Z/3Z e → ωe
3 Z ⊕ Zω Z/3Z ⊕ Z/3Z e → ωe, e → e+c, ωc=c
4 Z ⊕ Zi; Z/4Z e → ie
4 Z ⊕ Zi Z/4Z ⊕ Z/2Z e → ie, e → e+c, ic=c
6 Z ⊕ Zω Z/6Z e → −ωe
Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1.
Quasi hyperelliptic surfaces
A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cusp. They only exist in characteristics 2 or 3. Their second Betti number is 2, the second Chern number vanishes, and the holomorphic Euler characteristic vanishes. They were classified by (Bombieri & Mumford 1976), who found six cases in characteristic 3 (in which case 6K= 0) and eight in characteristic 2 (in which case 6K or 4K vanishes). Any quasi-hyperelliptic surface is a quotient (E×F)/G, where E is a rational curve with one cusp, F is an elliptic curve, and G is a finite subgroup scheme of F (acting on F by translations).
References
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225 - the standard reference book for compact complex surfaces
• Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314, ISBN 978-0-521-49842-5
• Bombieri, Enrico; Mumford, David (1976), "Enriques' classification of surfaces in char. p. III." (PDF), Inventiones Mathematicae, 35: 197–232, doi:10.1007/BF01390138, ISSN 0020-9910, MR 0491720
• Bombieri, Enrico; Mumford, David (1977), "Enriques' classification of surfaces in char. p. II", Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 23–42, MR 0491719
|
Wikipedia
|
Quasi-interior point
In mathematics, specifically in order theory and functional analysis, an element $x$ of an ordered topological vector space $X$ is called a quasi-interior point of the positive cone $C$ of $X$ if $x\geq 0$ and if the order interval $[0,x]:=\{z\in Z:0\leq z{\text{ and }}z\leq x\}$ is a total subset of $X$; that is, if the linear span of $[0,x]$ is a dense subset of $X.$[1]
Properties
If $X$ is a separable metrizable locally convex ordered topological vector space whose positive cone $C$ is a complete and total subset of $X,$ then the set of quasi-interior points of $C$ is dense in $C.$[1]
Examples
If $1\leq p<\infty $ then a point in $L^{p}(\mu )$ is quasi-interior to the positive cone $C$ if and only it is a weak order unit, which happens if and only if the element (which recall is an equivalence class of functions) contains a function that is $>\,0$ almost everywhere (with respect to $\mu $).[1]
A point in $L^{\infty }(\mu )$ is quasi-interior to the positive cone $C$ if and only if it is interior to $C.$[1]
See also
• Weak order unit
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets
References
1. Schaefer & Wolff 1999, pp. 234–242.
Bibliography
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Ordered topological vector spaces
Basic concepts
• Ordered vector space
• Partially ordered space
• Riesz space
• Order topology
• Order unit
• Positive linear operator
• Topological vector lattice
• Vector lattice
Types of orders/spaces
• AL-space
• AM-space
• Archimedean
• Banach lattice
• Fréchet lattice
• Locally convex vector lattice
• Normed lattice
• Order bound dual
• Order dual
• Order complete
• Regularly ordered
Types of elements/subsets
• Band
• Cone-saturated
• Lattice disjoint
• Dual/Polar cone
• Normal cone
• Order complete
• Order summable
• Order unit
• Quasi-interior point
• Solid set
• Weak order unit
Topologies/Convergence
• Order convergence
• Order topology
Operators
• Positive
• State
Main results
• Freudenthal spectral
|
Wikipedia
|
Inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.
Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element such that x * e = x and e * y = y for all x and y for which the left-hand sides are defined.[1])
When the operation ∗ is associative, if an element x has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the inverse element or simply the inverse. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an invertible element, also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).
Inverses are commonly used in groups—where every element is invertible, and rings—where invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism.
The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'. This may take its origin from the case of fractions, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of ${\tfrac {x}{y}}$ is ${\tfrac {y}{x}}$).
In this article, the operations are associative and have identity elements, except when otherwise stated and in section § Generalizations.
Definitions and basic properties
The concepts of inverse element and invertible element are commonly defined for binary operations that are everywhere defined (that is, the operation is defined for any two elements of its domain). However, these concepts are commonly used with partial operations, that is operations that are not defined everywhere. Common examples are matrix multiplication, function composition and composition of morphisms in a category. It follows that the common definitions of associativity and identity element must be extended to partial operations; this is the object of the first subsections.
In this section, X is a set (possibly a proper class) on which a partial operation (possibly total) is defined, which is denoted with $*.$
Associativity
A partial operation is associative if
$x*(y*z)=(x*y)*z$
for every x, y, z in X for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined.
Examples of non-total associative operations are multiplication of matrices of arbitrary size, and function composition.
Identity elements
Let $*$ be a possibly partial associative operation on a set X.
An identity element, or simply an identity is an element e such that
$x*e=x\quad {\text{and}}\quad e*y=y$
for every x and y for which the left-hand sides of the equalities are defined.
If e and f are two identity elements such that $e*f$ is defined, then $e=f.$ (This results immediately from the definition, by $e=e*f=f.$)
It follows that a total operation has at most one identity element, and if e and f are different identities, then $e*f$ is not defined.
For example, in the case of matrix multiplication, there is one n×n identity matrix for every positive integer n, and two identity matrices of different size cannot be multiplied together.
Similarly, identity functions are identity elements for function composition, and the composition of the identity functions of two different sets are not defined.
Left and right inverses
If $x*y=e,$ where e is an identity element, one says that x is a left inverse of y, and y is a right inverse of x.
Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on nonnegative integers, which has 0 as additive identity, and 0 is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.
An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the functions from the integers to the integers. The doubling function $x\mapsto 2x$ has infinitely many left inverses under function composition, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps n to either $2n$ or $2n+1$ is a right inverse of the function $ n\mapsto \left\lfloor {\frac {n}{2}}\right\rfloor ,$ the floor function that maps n to $ {\frac {n}{2}}$ or $ {\frac {n-1}{2}},$ depending whether n is even or odd.
More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is surjective.
In category theory, right inverses are also called sections, and left inverses are called retractions.
Inverses
An element is invertible under an operation if it has a left inverse and a right inverse.
In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if l and r are respectively a left inverse and a right inverse of x, then
$l=l*(x*r)=(l*x)*r=r.$
The inverse of an invertible element is its unique left or right inverse.
If the operation is denoted as an addition, the inverse, or additive inverse, of an element x is denoted $-x.$ Otherwise, the inverse of x is generally denoted $x^{-1},$ or, in the case of a commutative multiplication $ {\frac {1}{x}}.$ When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in $x^{*-1}.$ The notation $f^{\circ -1}$ is not commonly used for function composition, since $ {\frac {1}{f}}$ can be used for the multiplicative inverse.
If x and y are invertible, and $x*y$ is defined, then $x*y$ is invertible, and its inverse is $y^{-1}x^{-1}.$
An invertible homomorphism is called an isomorphism. In category theory, an invertible morphism is also called an isomorphism.
In groups
A group is a set with an associative operation that has an identity element, and for which every element has an inverse.
Thus, the inverse is a function from the group to itself that may also be considered as an operation of arity one. It is also an involution, since the inverse of the inverse of an element is the element itself.
A group may act on a set as transformations of this set. In this case, the inverse $g^{-1}$ of a group element $g$ defines a transformation that is the inverse of the transformation defined by $g,$ that is, the transformation that "undoes" the transformation defined by $g.$
For example, the Rubik's cube group represents the finite sequences of elementary moves. The inverse of such a sequence is obtained by applying the inverse of each move in the reverse order.
In monoids
A monoid is a set with an associative operation that has an identity element.
The invertible elements in a monoid form a group under monoid operation.
A ring is a monoid for ring multiplication. In this case, the invertible elements are also called units and form the group of units of the ring.
If a monoid is not commutative, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible).
For example, the set of the functions from a set to itself is a monoid under function composition. In this monoid, the invertible elements are the bijective functions; the elements that have left inverses are the injective functions, and those that have right inverses are the surjective functions.
Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the cancellation property (an element x has the cancellation property if $xy=xz$ implies $y=z,$ and $yx=zx$ implies $y=z$). This extension of a monoid is allowed by Grothendieck group construction. This is the method that is commonly used for constructing integers from natural numbers, rational numbers from integers and, more generally, the field of fractions of an integral domain, and localizations of commutative rings.
In rings
A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers.
Under addition, a ring is an abelian group, which means that addition is commutative and associative; it has an identity, called the additive identity, and denoted 0; and every element x has an inverse, called its additive inverse and denoted −x. Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.
Under multiplication, a ring is a monoid; this means that multiplication is associative and has an identity called the multiplicative identity and denoted 1. An invertible element for multiplication is called a unit. The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit x is denoted $x^{-1},$ or, when the multiplication is commutative, $ {\frac {1}{x}}.$
The additive identity 0 is never a unit, except when the ring is the zero ring, which has 0 as its unique element.
If 0 is the only non-unit, the ring is a field if the multiplication is commutative, or a division ring otherwise.
In a noncommutative ring (that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the linear functions from a infinite-dimensional vector space to itself.
A commutative ring (that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not zero divisors (that is, their product with a nonzero element cannot be 0). This is the process of localization, which produces, in particular, the field of rational numbers from the ring of integers, and, more generally, the field of fractions of an integral domain. Localization is also used with zero divisors, but, in this case the original ring is not a subring of the localisation; instead, it is mapped non-injectively to the localization.
Matrices
Matrix multiplication is commonly defined for matrices over a field, and straightforwardly extended to matrices over rings, rngs and semirings. However, in this section, only matrices over a commutative ring are considered, because of the use of the concept of rank and determinant.
If A is a m×n matrix (that is, a matrix with m rows and n columns), and B is a p×q matrix, the product AB is defined if n = p, and only in this case. An identity matrix, that is, an identity element for matrix multiplication is a square matrix (same number for rows and columns) whose entries of the main diagonal are all equal to 1, and all other entries are 0.
An invertible matrix is an invertible element under matrix multiplication. A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R (that is, is invertible in R. In this case, its inverse matrix can be computed with Cramer's rule.
If R is a field, the determinant is invertible if and only if it is not zero. As the case of fields is more common, one see often invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings.
In the case of integer matrices (that is, matrices with integer entries), an invertible matrix is a matrix that has an inverse that is also an integer matrix. Such a matrix is called a unimodular matrix for distinguishing it from matrices that are invertible over the real numbers. A square integer matrix is unimodular if and only if its determinant is 1 or −1, since these two numbers are the only units in the ring of integers.
A matrix has a left inverse if and only if its rank equals its number of columns. This left inverse is not unique except for square matrices where the left inverse equal the inverse matrix. Similarly, a right inverse exists if and only if the rank equals the number of rows; it is not unique in the case of a rectangular matrix, and equals the inverse matrix in the case of a square matrix.
Functions, homomorphisms and morphisms
Composition is a partial operation that generalizes to homomorphisms of algebraic structures and morphisms of categories into operations that are also called composition, and share many properties with function composition.
In all the case, composition is associative.
If $f\colon X\to Y$ and $g\colon Y'\to Z,$ the composition $g\circ f$ is defined if and only if $Y'=Y$ or, in the function and homomorphism cases, $Y\subset Y'.$ In the function and homomorphism cases, this means that the codomain of $f$ equals or is included in the domain of g. In the morphism case, this means that the codomain of $f$ equals the domain of g.
There is an identity $\operatorname {id} _{X}\colon X\to X$ for every object X (set, algebraic structure or object), which is called also an identity function in the function case.
A function is invertible if and only if it is a bijection. An invertible homomorphism or morphism is called an isomorphism. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an inverse function. In the other cases, one talks of inverse isomorphisms.
A function has a left inverse or a right inverse if and only it is injective or surjective, respectively. An homomorphism of algebraic structures that has a left inverse or a right inverse is respectively injective or surjective, but the converse is not true in some algebraic structures. For example, the converse is true for vector spaces but not for modules over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a split epimorphism or a split monomorphism. This terminology is also used for morphisms in any category.
Generalizations
In a unital magma
Let $S$ be a unital magma, that is, a set with a binary operation $*$ and an identity element $e\in S$. If, for $a,b\in S$, we have $a*b=e$, then $a$ is called a left inverse of $b$ and $b$ is called a right inverse of $a$. If an element $x$ is both a left inverse and a right inverse of $y$, then $x$ is called a two-sided inverse, or simply an inverse, of $y$. An element with a two-sided inverse in $S$ is called invertible in $S$. An element with an inverse element only on one side is left invertible or right invertible.
Elements of a unital magma $(S,*)$ may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
* 1 2 3
1 1 2 3
2 2 1 1
3 3 1 1
the elements 2 and 3 each have two two-sided inverses.
A unital magma in which all elements are invertible need not be a loop. For example, in the magma $(S,*)$ given by the Cayley table
* 1 2 3
1 1 2 3
2 2 1 2
3 3 2 1
every element has a unique two-sided inverse (namely itself), but $(S,*)$ is not a loop because the Cayley table is not a Latin square.
Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table
* 1 2 3 4 5
1 1 2 3 4 5
2 2 3 1 5 4
3 3 4 5 1 2
4 4 5 2 3 1
5 5 1 4 2 3
the only element with a two-sided inverse is the identity element 1.
If the operation $*$ is associative then if an element has both a left inverse and a right inverse, they are equal. In other words, in a monoid (an associative unital magma) every element has at most one inverse (as defined in this section). In a monoid, the set of invertible elements is a group, called the group of units of $S$, and denoted by $U(S)$ or H1.
In a semigroup
Main article: Regular semigroup
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity; that is, in a semigroup.
In a semigroup S an element x is called (von Neumann) regular if there exists some element z in S such that xzx = x; z is sometimes called a pseudoinverse. An element y is called (simply) an inverse of x if xyx = x and y = yxy. Every regular element has at least one inverse: if x = xzx then it is easy to verify that y = zxz is an inverse of x as defined in this section. Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, that is ee = e and ff = f. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and ex = xf = x, ye = fy = y, and e acts as a left identity on x, while f acts a right identity, and the left/right roles are reversed for y. This simple observation can be generalized using Green's relations: every idempotent e in an arbitrary semigroup is a left identity for Re and right identity for Le.[2] An intuitive description of this fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.
In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given in this section. Only elements in the Green class H1 have an inverse from the unital magma perspective, whereas for any idempotent e, the elements of He have an inverse as defined in this section. Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an inverse semigroup. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an absorbing element 0 because 000 = 0, whereas a group may not.
Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse. This is generally justified because in most applications (for example, all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity (see Generalized inverse).
U-semigroups
A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (a°)° = a for all a in S; this endows S with a type ⟨2,1⟩ algebra. A semigroup endowed with such an operation is called a U-semigroup. Although it may seem that a° will be the inverse of a, this is not necessarily the case. In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classes of U-semigroups have been studied:[3]
• I-semigroups, in which the interaction axiom is aa°a = a
• *-semigroups, in which the interaction axiom is (ab)° = b°a°. Such an operation is called an involution, and typically denoted by a*
Clearly a group is both an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are completely regular semigroups; these are I-semigroups in which one additionally has aa° = a°a; in other words every element has commuting pseudoinverse a°. There are few concrete examples of such semigroups however; most are completely simple semigroups. In contrast, a subclass of *-semigroups, the *-regular semigroups (in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the Moore–Penrose inverse. In this case however the involution a* is not the pseudoinverse. Rather, the pseudoinverse of x is the unique element y such that xyx = x, yxy = y, (xy)* = xy, (yx)* = yx. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the generalized inverse or Moore–Penrose inverse.
Semirings
Main article: Quasiregular element
Examples
All examples in this section involve associative operators.
Galois connections
The lower and upper adjoints in a (monotone) Galois connection, L and G are quasi-inverses of each other; that is, LGL = L and GLG = G and one uniquely determines the other. They are not left or right inverses of each other however.
Generalized inverses of matrices
A square matrix $M$ with entries in a field $K$ is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero. If the determinant of $M$ is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See invertible matrix for more.
More generally, a square matrix over a commutative ring $R$ is invertible if and only if its determinant is invertible in $R$.
Non-square matrices of full rank have several one-sided inverses:[4]
• For $A:m\times n\mid m>n$ we have left inverses; for example, $\underbrace {\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}} _{A_{\text{left}}^{-1}}A=I_{n}$
• For $A:m\times n\mid m<n$ we have right inverses; for example, $A\underbrace {A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}} _{A_{\text{right}}^{-1}}=I_{m}$
The left inverse can be used to determine the least norm solution of $Ax=b$, which is also the least squares formula for regression and is given by $x=\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}b.$
No rank deficient matrix has any (even one-sided) inverse. However, the Moore–Penrose inverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists.
As an example of matrix inverses, consider:
$A:2\times 3={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}$
So, as m < n, we have a right inverse, $A_{\text{right}}^{-1}=A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}.$ By components it is computed as
${\begin{aligned}AA^{\text{T}}&={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}{\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}={\begin{bmatrix}14&32\\32&77\end{bmatrix}}\\[3pt]\left(AA^{\text{T}}\right)^{-1}&={\begin{bmatrix}14&32\\32&77\end{bmatrix}}^{-1}={\frac {1}{54}}{\begin{bmatrix}77&-32\\-32&14\end{bmatrix}}\\[3pt]A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}&={\frac {1}{54}}{\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}{\begin{bmatrix}77&-32\\-32&14\end{bmatrix}}={\frac {1}{18}}{\begin{bmatrix}-17&8\\-2&2\\13&-4\end{bmatrix}}=A_{\text{right}}^{-1}\end{aligned}}$
The left inverse doesn't exist, because
$A^{\text{T}}A={\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}{\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}={\begin{bmatrix}17&22&27\\22&29&36\\27&36&45\end{bmatrix}}$
which is a singular matrix, and cannot be inverted.
See also
• Division ring
• Latin square property
• Loop (algebra)
• Unit (ring theory)
Notes
1. The usual definition of an identity element has been generalized for including the identity functions as identity elements for function composition, and identity matrices as identity elements for matrix multiplication.
2. Howie, prop. 2.3.3, p. 51
3. Howie p. 102
4. "MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse".
References
• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 15 (def in unital magma) and p. 33 (def in semigroup)
• Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9. contains all of the semigroup material herein except *-regular semigroups.
• Drazin, M.P., Regular semigroups with involution, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
• Miyuki Yamada, P-systems in regular semigroups, Semigroup Forum, 24(1), December 1982, pp. 173–187
• Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377.
|
Wikipedia
|
Quasi-isometry
In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.
The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov.[1]
Definition
Suppose that $f$ is a (not necessarily continuous) function from one metric space $(M_{1},d_{1})$ to a second metric space $(M_{2},d_{2})$. Then $f$ is called a quasi-isometry from $(M_{1},d_{1})$ to $(M_{2},d_{2})$ if there exist constants $A\geq 1$, $B\geq 0$, and $C\geq 0$ such that the following two properties both hold:[2]
1. For every two points $x$ and $y$ in $M_{1}$, the distance between their images is up to the additive constant $B$ within a factor of $A$ of their original distance. More formally:
$\forall x,y\in M_{1}:{\frac {1}{A}}\;d_{1}(x,y)-B\leq d_{2}(f(x),f(y))\leq A\;d_{1}(x,y)+B.$
2. Every point of $M_{2}$ is within the constant distance $C$ of an image point. More formally:
$\forall z\in M_{2}:\exists x\in M_{1}:d_{2}(z,f(x))\leq C.$
The two metric spaces $(M_{1},d_{1})$ and $(M_{2},d_{2})$ are called quasi-isometric if there exists a quasi-isometry $f$ from $(M_{1},d_{1})$ to $(M_{2},d_{2})$.
A map is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, $(M_{1},d_{1})$ is quasi-isometric to a subspace of $(M_{2},d_{2})$.
Two metric spaces M1 and M2 are said to be quasi-isometric, denoted $M_{1}{\underset {q.i.}{\sim }}M_{2}$, if there exists a quasi-isometry $f:M_{1}\to M_{2}$.
Examples
The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most ${\sqrt {2}}$. Note that there can be no isometry, since, for example, the points $(1,0),(-1,0),(0,1),(0,-1)$ are of equal distance to each other in Manhattan distance, but in the Euclidean plane, there are no 4 points that are of equal distance to each other.
The map $f:\mathbb {Z} ^{n}\to \mathbb {R} ^{n}$ (both with the Euclidean metric) that sends every $n$-tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance ${\sqrt {n/4}}$ of an integer tuple. In the other direction, the discontinuous function that rounds every tuple of real numbers to the nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance ${\sqrt {n/4}}$ of it, so rounding changes the distance between pairs of points by adding or subtracting at most $2{\sqrt {n/4}}$.
Every pair of finite or bounded metric spaces is quasi-isometric. In this case, every function from one space to the other is a quasi-isometry.
Equivalence relation
If $f:M_{1}\mapsto M_{2}$ is a quasi-isometry, then there exists a quasi-isometry $g:M_{2}\mapsto M_{1}$. Indeed, $g(x)$ may be defined by letting $y$ be any point in the image of $f$ that is within distance $C$ of $x$, and letting $g(x)$ be any point in $f^{-1}(y)$.
Since the identity map is a quasi-isometry, and the composition of two quasi-isometries is a quasi-isometry, it follows that the property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.
Use in geometric group theory
Given a finite generating set S of a finitely generated group G, we can form the corresponding Cayley graph of S and G. This graph becomes a metric space if we declare the length of each edge to be 1. Taking a different finite generating set T results in a different graph and a different metric space, however the two spaces are quasi-isometric.[3] This quasi-isometry class is thus an invariant of the group G. Any property of metric spaces that only depends on a space's quasi-isometry class immediately yields another invariant of groups, opening the field of group theory to geometric methods.
More generally, the Švarc–Milnor lemma states that if a group G acts properly discontinuously with compact quotient on a proper geodesic space X then G is quasi-isometric to X (meaning that any Cayley graph for G is). This gives new examples of groups quasi-isometric to each other:
• If G' is a subgroup of finite index in G then G' is quasi-isometric to G;
• If G and H are the fundamental groups of two compact hyperbolic manifolds of the same dimension d then they are both quasi-isometric to the hyperbolic space Hd and hence to each other; on the other hand there are infinitely many quasi-isometry classes of fundamental groups of finite-volume.[4]
Quasigeodesics and the Morse lemma
A quasi-geodesic in a metric space $(X,d)$ is a quasi-isometric embedding of $\mathbb {R} $ into $X$. More precisely a map $\phi :\mathbb {R} \to X$ :\mathbb {R} \to X} such that there exists $C,K>0$ so that
$\forall s,t\in \mathbb {R} :C^{-1}|s-t|-K\leq d(\phi (t),\phi (s))\leq C|s-t|+K$
is called a $(C,K)$-quasi-geodesic. Obviously geodesics (parametrised by arclength) are quasi-geodesics. The fact that in some spaces the converse is coarsely true, i.e. that every quasi-geodesic stays within bounded distance of a true geodesic, is called the Morse Lemma (not to be confused with the Morse lemma in differential topology). Formally the statement is:
Let $\delta ,C,K>0$ and $X$ a proper δ-hyperbolic space. There exists $M$ such that for any $(C,K)$-quasi-geodesic $\phi $ there exists a geodesic $L$ in $X$ such that $d(\phi (t),L)\leq M$ for all $t\in \mathbb {R} $.
It is an important tool in geometric group theory. An immediate application is that any quasi-isometry between proper hyperbolic spaces induces a homeomorphism between their boundaries. This result is the first step in the proof of the Mostow rigidity theorem.
Furthermore, this result has found utility in analyzing user interaction design in applications similar to Google Maps.[5]
Examples of quasi-isometry invariants of groups
The following are some examples of properties of group Cayley graphs that are invariant under quasi-isometry:[2]
Hyperbolicity
Main article: Hyperbolic group
A group is called hyperbolic if one of its Cayley graphs is a δ-hyperbolic space for some δ. When translating between different definitions of hyperbolicity, the particular value of δ may change, but the resulting notions of a hyperbolic group turn out to be equivalent.
Hyperbolic groups have a solvable word problem. They are biautomatic and automatic.:[6] indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words.
Growth
The growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n.
According to Gromov's theorem, a group of polynomial growth is virtually nilpotent, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth $k_{0}$ has to be a natural number and in fact $\#(n)\sim n^{k_{0}}$.
If $\#(n)$ grows more slowly than any exponential function, G has a subexponential growth rate. Any such group is amenable.
Ends
Main article: End (topology)
The ends of a topological space are, roughly speaking, the connected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification.
The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is independent of the choice of a finite generating set. Every finitely-generated infinite group has either 0,1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
If two connected locally finite graphs are quasi-isometric then they have the same number of ends.[7] In particular, two quasi-isometric finitely generated groups have the same number of ends.
Amenability
Main article: Amenable group
An amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.[8]
In discrete group theory, where G has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proportion of G any given subset takes up.
If a group has a Følner sequence then it is automatically amenable.
Asymptotic cone
Main article: Ultralimit § Asymptotic cones
An ultralimit is a geometric construction that assigns to a sequence of metric spaces Xn a limiting metric space. An important class of ultralimits are the so-called asymptotic cones of metric spaces. Let (X,d) be a metric space, let ω be a non-principal ultrafilter on $\mathbb {N} $ and let pn ∈ X be a sequence of base-points. Then the ω–ultralimit of the sequence $(X,{\frac {d}{n}},p_{n})$ is called the asymptotic cone of X with respect to ω and $(p_{n})_{n}\,$ and is denoted $Cone_{\omega }(X,d,(p_{n})_{n})\,$. One often takes the base-point sequence to be constant, pn = p for some p ∈ X; in this case the asymptotic cone does not depend on the choice of p ∈ X and is denoted by $Cone_{\omega }(X,d)\,$ or just $Cone_{\omega }(X)\,$.
The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular.[9] Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.[10]
See also
• Isometry
• Coarse structure
References
1. Bridson, Martin R. (2008), "Geometric and combinatorial group theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), The Princeton Companion to Mathematics, Princeton University Press, pp. 431–448, ISBN 978-0-691-11880-2
2. P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6
3. R. B. Sher and R. J. Daverman (2002), Handbook of Geometric Topology, North-Holland. ISBN 0-444-82432-4.
4. Schwartz, Richard (1995). "The Quasi-Isometry Classification of Rank One Lattices". I.H.É.S. Publications Mathématiques. 82: 133–168. doi:10.1007/BF02698639. S2CID 67824718.
5. Baryshnikov, Yuliy; Ghrist, Robert (2023-05-08). "Navigating the Negative Curvature of Google Maps". The Mathematical Intelligencer. doi:10.1007/s00283-023-10270-w. ISSN 0343-6993.
6. Charney, Ruth (1992), "Artin groups of finite type are biautomatic", Mathematische Annalen, 292: 671–683, doi:10.1007/BF01444642, S2CID 120654588
7. Stephen G.Brick (1993). "Quasi-isometries and ends of groups". Journal of Pure and Applied Algebra. 86 (1): 23–33. doi:10.1016/0022-4049(93)90150-R.
8. Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups and groups, Bull. A.M.S. 55 (1949) 1054–1055. Many text books on amenability, such as Volker Runde's, suggest that Day chose the word as a pun.
9. John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2
10. Cornelia Druţu and Mark Sapir (with an Appendix by Denis Osin and Mark Sapir), Tree-graded spaces and asymptotic cones of groups. Topology, Volume 44 (2005), no. 5, pp. 959–1058.
|
Wikipedia
|
Quasiprobability distribution
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution. However, they can violate the σ-additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states. Indeed, quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis,[1] and elsewhere.
Introduction
In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ${\widehat {\rho }}$) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove[2] that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function.
The coherent states, i.e. right eigenstates of the annihilation operator ${\widehat {a}}$ serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
${\begin{aligned}{\widehat {a}}|\alpha \rangle &=\alpha |\alpha \rangle \\\langle \alpha |{\widehat {a}}^{\dagger }&=\langle \alpha |\alpha ^{*}.\end{aligned}}$
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
$\langle \beta \mid \alpha \rangle =e^{-{1 \over 2}(|\beta |^{2}+|\alpha |^{2}-2\beta ^{*}\alpha )}\neq \delta (\alpha -\beta ).$
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete.[3] Click to show an informal proof.
Proof of the overcompleteness of the coherent states
Integration over the complex plane can be written in terms of polar coordinates with $d^{2}\alpha =r\,dr\,d\theta $. Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
${\begin{aligned}\int |\alpha \rangle \langle \alpha |\,d^{2}\alpha &=\int \sum _{n=0}^{\infty }\sum _{k=0}^{\infty }e^{-{|\alpha |^{2}}}\cdot {\frac {\alpha ^{n}(\alpha ^{*})^{k}}{\sqrt {n!k!}}}|n\rangle \langle k|\,d^{2}\alpha \\&=\int _{0}^{\infty }\int _{0}^{2\pi }\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }e^{-{r^{2}}}\cdot {\frac {r^{n+k+1}e^{i(n-k)\theta }}{\sqrt {n!k!}}}|n\rangle \langle k|\,d\theta \,dr\\&=\sum _{n=0}^{\infty }\int _{0}^{\infty }\sum _{k=0}^{\infty }\int _{0}^{2\pi }e^{-{r^{2}}}\cdot {\frac {r^{n+k+1}e^{i(n-k)\theta }}{\sqrt {n!k!}}}|n\rangle \langle k|\,d\theta \,dr\\&=2\pi \sum _{n=0}^{\infty }\int _{0}^{\infty }\sum _{k=0}^{\infty }e^{-{r^{2}}}\cdot {\frac {r^{n+k+1}\delta (n-k)}{\sqrt {n!k!}}}|n\rangle \langle k|\,dr\\&=2\pi \sum _{n=0}^{\infty }\int e^{-{r^{2}}}\cdot {\frac {r^{2n+1}}{n!}}|n\rangle \langle n|\,dr\\&=\pi \sum _{n=0}^{\infty }\int e^{-u}\cdot {\frac {u^{n}}{n!}}|n\rangle \langle n|\,du\\&=\pi \sum _{n=0}^{\infty }|n\rangle \langle n|\\&=\pi {\widehat {I}}.\end{aligned}}$
Clearly, one can span the Hilbert space by writing a state as
$|\psi \rangle ={\frac {1}{\pi }}\int |\alpha \rangle \langle \alpha |\psi \rangle \,d^{2}\alpha .$
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete.
In the coherent states basis, however, it is always possible[2] to express the density operator in the diagonal form
${\widehat {\rho }}=\int f(\alpha ,\alpha ^{*})|\alpha \rangle \langle \alpha |\,d^{2}\alpha $
where f is a representation of the phase space distribution. This function f is considered a quasiprobability density because it has the following properties:
• $\int f(\alpha ,\alpha ^{*})\,d^{2}\alpha =\operatorname {tr} ({\widehat {\rho }})=1$ (normalization)
• If $g_{\Omega }({\widehat {a}},{\widehat {a}}^{\dagger })$ is an operator that can be expressed as a power series of the creation and annihilation operators in an ordering Ω, then its expectation value is
$\langle g_{\Omega }({\widehat {a}},{\widehat {a}}^{\dagger })\rangle =\int f(\alpha ,\alpha ^{*})g_{\Omega }(\alpha ,\alpha ^{*})\,d\alpha \,d\alpha ^{*}$ (optical equivalence theorem).
The function f is not unique. There exists a family of different representations, each connected to a different ordering Ω. The most popular in the general physics literature and historically first of these is the Wigner quasiprobability distribution,[4] which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation.[5] The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:[6]
If the quantum system has a classical analog, e.g. a coherent state or thermal radiation, then P is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent Fock state or entangled system, then P is negative somewhere or more singular than a delta function.
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog.[7][8]
In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation,[9] which is useful when operators are in anti-normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
Characteristic functions
Analogous to probability theory, quantum quasiprobability distributions can be written in terms of characteristic functions, from which all operator expectation values can be derived. The characteristic functions for the Wigner, Glauber P and Q distributions of an N mode system are as follows:
• $\chi _{W}(\mathbf {z} ,\mathbf {z} ^{*})=\operatorname {tr} (\rho e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}+i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }})$
• $\chi _{P}(\mathbf {z} ,\mathbf {z} ^{*})=\operatorname {tr} (\rho e^{i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }}e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}})$
• $\chi _{Q}(\mathbf {z} ,\mathbf {z} ^{*})=\operatorname {tr} (\rho e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}}e^{i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }})$
Here ${\widehat {\mathbf {a} }}$ and ${\widehat {\mathbf {a} }}^{\dagger }$ are vectors containing the annihilation and creation operators for each mode of the system. These characteristic functions can be used to directly evaluate expectation values of operator moments. The ordering of the annihilation and creation operators in these moments is specific to the particular characteristic function. For instance, normally ordered (creation operators preceding annihilation operators) moments can be evaluated in the following way from $\chi _{P}\,$:
$\langle {\widehat {a}}_{j}^{\dagger m}{\widehat {a}}_{k}^{n}\rangle ={\frac {\partial ^{m+n}}{\partial (iz_{j}^{*})^{m}\partial (iz_{k})^{n}}}\chi _{P}(\mathbf {z} ,\mathbf {z} ^{*}){\Big |}_{\mathbf {z} =\mathbf {z} ^{*}=0}$
In the same way, expectation values of anti-normally ordered and symmetrically ordered combinations of annihilation and creation operators can be evaluated from the characteristic functions for the Q and Wigner distributions, respectively. The quasiprobability functions themselves are defined as Fourier transforms of the above characteristic functions. That is,
$\{W\mid P\mid Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})={\frac {1}{\pi ^{2N}}}\int \chi _{\{W\mid P\mid Q\}}(\mathbf {z} ,\mathbf {z} ^{*})e^{-i\mathbf {z} ^{*}\cdot \mathbf {\alpha } ^{*}}e^{-i\mathbf {z} \cdot \mathbf {\alpha } }\,d^{2N}\mathbf {z} .$
Here $\alpha _{j}\,$ and $\alpha _{k}^{*}$ may be identified as coherent state amplitudes in the case of the Glauber P and Q distributions, but simply c-numbers for the Wigner function. Since differentiation in normal space becomes multiplication in Fourier space, moments can be calculated from these functions in the following way:
• $\langle {\widehat {\mathbf {a} }}_{j}^{\dagger m}{\widehat {\mathbf {a} }}_{k}^{n}\rangle =\int P(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})\alpha _{j}^{n}\alpha _{k}^{*m}\,d^{2N}\mathbf {\alpha } $
• $\langle {\widehat {\mathbf {a} }}_{j}^{m}{\widehat {\mathbf {a} }}_{k}^{\dagger n}\rangle =\int Q(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})\alpha _{j}^{m}\alpha _{k}^{*n}\,d^{2N}\mathbf {\alpha } $
• $\langle ({\widehat {\mathbf {a} }}_{j}^{\dagger m}{\widehat {\mathbf {a} }}_{k}^{n})_{S}\rangle =\int W(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})\alpha _{j}^{m}\alpha _{k}^{*n}\,d^{2N}\mathbf {\alpha } $
Here $(\cdots )_{S}$ denotes symmetric ordering.
These representations are all interrelated through convolution by Gaussian functions, Weierstrass transforms,
• $W(\alpha ,\alpha ^{*})={\frac {2}{\pi }}\int P(\beta ,\beta ^{*})e^{-2|\alpha -\beta |^{2}}\,d^{2}\beta $
• $Q(\alpha ,\alpha ^{*})={\frac {2}{\pi }}\int W(\beta ,\beta ^{*})e^{-2|\alpha -\beta |^{2}}\,d^{2}\beta $
or, using the property that convolution is associative,
• $Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\int P(\beta ,\beta ^{*})e^{-|\alpha -\beta |^{2}}\,d^{2}\beta ~.$
It follows that
• $P(\alpha ,\alpha ^{*})={\frac {1}{\pi ^{2}}}\int Q(\beta ,\beta ^{*})e^{|\lambda |^{2}+\lambda ^{*}(\alpha -\beta )-\lambda (\alpha -\beta )^{*}}\,d^{2}\beta ~d^{2}\lambda ,$
an often divergent integral, indicating P is often a distribution. Q is always broader than P for the same density matrix. [10]
For example, for a thermal state,
${\hat {\rho }}={\frac {1}{{\bar {n}}+1}}\sum _{n=0}^{\infty }\left({\frac {\bar {n}}{1+{\bar {n}}}}\right)^{n}|n\rangle \langle n|~~,$
one has
$P(\alpha )={\frac {1}{\pi {\bar {n}}}}e^{-{\frac {|\alpha |^{2}}{\bar {n}}}},\qquad Q(\alpha )={\frac {1}{\pi (1+{\bar {n}})}}e^{-{\frac {|\alpha |^{2}}{1+{\bar {n}}}}}~~~.$
Time evolution and operator correspondences
Since each of the above transformations from ρ to the distribution functions is linear, the equation of motion for each distribution can be obtained by performing the same transformations to ${\dot {\rho }}$. Furthermore, as any master equation which can be expressed in Lindblad form is completely described by the action of combinations of annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions.[11] [12]
For instance, consider the annihilation operator ${\widehat {a}}_{j}\,$ acting on ρ. For the characteristic function of the P distribution we have
$\operatorname {tr} ({\widehat {a}}_{j}\rho e^{i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }}e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}})={\frac {\partial }{\partial (iz_{j})}}\chi _{P}(\mathbf {z} ,\mathbf {z} ^{*}).$
Taking the Fourier transform with respect to $\mathbf {z} \,$ to find the action corresponding action on the Glauber P function, we find
${\widehat {a}}_{j}\rho \rightarrow \alpha _{j}P(\mathbf {\alpha } ,\mathbf {\alpha } ^{*}).$
By following this procedure for each of the above distributions, the following operator correspondences can be identified:
• ${\widehat {a}}_{j}\rho \rightarrow \left(\alpha _{j}+\kappa {\frac {\partial }{\partial \alpha _{j}^{*}}}\right)\{W\mid P\mid Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
• $\rho {\widehat {a}}_{j}^{\dagger }\rightarrow \left(\alpha _{j}^{*}+\kappa {\frac {\partial }{\partial \alpha _{j}}}\right)\{W\mid P\mid Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
• ${\widehat {a}}_{j}^{\dagger }\rho \rightarrow \left(\alpha _{j}^{*}-(1-\kappa ){\frac {\partial }{\partial \alpha _{j}}}\right)\{W\mid P\mid Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
• $\rho {\widehat {a}}_{j}\rightarrow \left(\alpha _{j}-(1-\kappa ){\frac {\partial }{\partial \alpha _{j}^{*}}}\right)\{W\mid P\mid Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
Here κ = 0, 1/2 or 1 for P, Wigner, and Q distributions, respectively. In this way, master equations can be expressed as an equations of motion of quasiprobability functions.
Examples
Coherent state
By construction, P for a coherent state $|\alpha _{0}\rangle $ is simply a delta function:
$P(\alpha ,\alpha ^{*})=\delta ^{2}(\alpha -\alpha _{0}).$
The Wigner and Q representations follows immediately from the Gaussian convolution formulas above,
$W(\alpha ,\alpha ^{*})={\frac {2}{\pi }}\int \delta ^{2}(\beta -\alpha _{0})e^{-2|\alpha -\beta |^{2}}\,d^{2}\beta ={\frac {2}{\pi }}e^{-2|\alpha -\alpha _{0}|^{2}}$
$Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\int \delta ^{2}(\beta -\alpha _{0})e^{-|\alpha -\beta |^{2}}\,d^{2}\beta ={\frac {1}{\pi }}e^{-|\alpha -\alpha _{0}|^{2}}.$
The Husimi representation can also be found using the formula above for the inner product of two coherent states,
$Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\langle \alpha |{\widehat {\rho }}|\alpha \rangle ={\frac {1}{\pi }}|\langle \alpha _{0}|\alpha \rangle |^{2}={\frac {1}{\pi }}e^{-|\alpha -\alpha _{0}|^{2}}$
Fock state
The P representation of a Fock state $|n\rangle $ is
$P(\alpha ,\alpha ^{*})={\frac {e^{|\alpha |^{2}}}{n!}}{\frac {\partial ^{2n}}{\partial \alpha ^{*n}\,\partial \alpha ^{n}}}\delta ^{2}(\alpha ).$
Since for n>0 this is more singular than a delta function, a Fock state has no classical analog. The non-classicality is less transparent as one proceeds with the Gaussian convolutions. If Ln is the nth Laguerre polynomial, W is
$W(\alpha ,\alpha ^{*})=(-1)^{n}{\frac {2}{\pi }}e^{-2|\alpha |^{2}}L_{n}\left(4|\alpha |^{2}\right)~,$
which can go negative but is bounded.
Q, by contrast, always remains positive and bounded,
$Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\langle \alpha |{\widehat {\rho }}|\alpha \rangle ={\frac {1}{\pi }}|\langle n|\alpha \rangle |^{2}={\frac {1}{\pi n!}}|\langle 0|{\widehat {a}}^{n}|\alpha \rangle |^{2}={\frac {|\alpha |^{2n}}{\pi n!}}|\langle 0|\alpha \rangle |^{2}~.$
Damped quantum harmonic oscillator
Consider the damped quantum harmonic oscillator with the following master equation,
${\frac {d{\widehat {\rho }}}{dt}}=i\omega _{0}[{\widehat {\rho }},{\widehat {a}}^{\dagger }{\widehat {a}}]+{\frac {\gamma }{2}}(2{\widehat {a}}{\widehat {\rho }}{\widehat {a}}^{\dagger }-{\widehat {a}}^{\dagger }{\widehat {a}}{\widehat {\rho }}-\rho {\widehat {a}}^{\dagger }{\widehat {a}})+\gamma \langle n\rangle ({\widehat {a}}{\widehat {\rho }}{\widehat {a}}^{\dagger }+{\widehat {a}}^{\dagger }{\widehat {\rho }}{\widehat {a}}-{\widehat {a}}^{\dagger }{\widehat {a}}{\widehat {\rho }}-{\widehat {\rho }}{\widehat {a}}{\widehat {a}}^{\dagger }).$
This results in the Fokker–Planck equation,
${\frac {\partial }{\partial t}}\{W\mid P\mid Q\}(\alpha ,\alpha ^{*},t)=\left[(\gamma +i\omega _{0}){\frac {\partial }{\partial \alpha }}\alpha +(\gamma -i\omega _{0}){\frac {\partial }{\partial \alpha ^{*}}}\alpha ^{*}+{\frac {\gamma }{2}}(\langle n\rangle +\kappa ){\frac {\partial ^{2}}{\partial \alpha \,\partial \alpha ^{*}}}\right]\{W\mid P\mid Q\}(\alpha ,\alpha ^{*},t),$
where κ = 0, 1/2, 1 for the P, W, and Q representations, respectively.
If the system is initially in the coherent state $|\alpha _{0}\rangle $, then this equation has the solution
$\{W\mid P\mid Q\}(\alpha ,\alpha ^{*},t)={\frac {1}{\pi \left[\kappa +\langle n\rangle \left(1-e^{-2\gamma t}\right)\right]}}\exp {\left(-{\frac {\left|\alpha -\alpha _{0}e^{-(\gamma +i\omega _{0})t}\right|^{2}}{\kappa +\langle n\rangle \left(1-e^{-2\gamma t}\right)}}\right)}~~.$
References
1. L. Cohen (1995), Time-frequency analysis: theory and applications, Prentice-Hall, Upper Saddle River, NJ, ISBN 0-13-594532-1
2. Sudarshan, E. C. G. (1963-04-01). "Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams". Physical Review Letters. American Physical Society (APS). 10 (7): 277–279. Bibcode:1963PhRvL..10..277S. doi:10.1103/physrevlett.10.277. ISSN 0031-9007.
3. Klauder, John R (1960). "The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers". Annals of Physics. Elsevier BV. 11 (2): 123–168. Bibcode:1960AnPhy..11..123K. doi:10.1016/0003-4916(60)90131-7. ISSN 0003-4916.
4. Wigner, E. (1932-06-01). "On the Quantum Correction For Thermodynamic Equilibrium". Physical Review. American Physical Society (APS). 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/physrev.40.749. ISSN 0031-899X.
5. Glauber, Roy J. (1963-09-15). "Coherent and Incoherent States of the Radiation Field". Physical Review. American Physical Society (APS). 131 (6): 2766–2788. Bibcode:1963PhRv..131.2766G. doi:10.1103/physrev.131.2766. ISSN 0031-899X.
6. Mandel, L.; Wolf, E. (1995), Optical Coherence and Quantum Optics, Cambridge UK: Cambridge University Press, ISBN 0-521-41711-2
7. Cohen, O. (1997-11-01). "Nonlocality of the original Einstein-Podolsky-Rosen state". Physical Review A. American Physical Society (APS). 56 (5): 3484–3492. Bibcode:1997PhRvA..56.3484C. doi:10.1103/physreva.56.3484. ISSN 1050-2947.
8. Banaszek, Konrad; Wódkiewicz, Krzysztof (1998-12-01). "Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation". Physical Review A. 58 (6): 4345–4347. arXiv:quant-ph/9806069. Bibcode:1998PhRvA..58.4345B. doi:10.1103/physreva.58.4345. ISSN 1050-2947. S2CID 119341663.
9. Husimi, Kôdi. Some Formal Properties of the Density Matrix. Proceedings of the Physico-Mathematical Society of Japan. Vol. 22. The Mathematical Society of Japan. pp. 264–314. doi:10.11429/ppmsj1919.22.4_264. ISSN 0370-1239.
10. Wolfgang Schleich, Quantum Optics in Phase Space, (Wiley-VCH, 2001) ISBN 978-3527294350
11. H. J. Carmichael, Statistical Methods in Quantum Optics I: Master Equations and Fokker–Planck Equations, Springer-Verlag (2002).
12. C. W. Gardiner, Quantum Noise, Springer-Verlag (1991).
|
Wikipedia
|
Quasi-projective variety
In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subscheme of some projective space.[1]
Relationship to affine varieties
An affine space is a Zariski-open subset of a projective space, and since any closed affine subset $U$ can be expressed as an intersection of the projective completion ${\bar {U}}$ and the affine space embedded in the projective space, this implies that any affine variety is quasiprojective. There are locally closed subsets of projective space that are not affine, so that quasi-projective is more general than affine. Taking the complement of a single point in projective space of dimension at least 2 gives a non-affine quasi-projective variety. This is also an example of a quasi-projective variety that is neither affine nor projective.
Examples
Since quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as varieties. Varieties isomorphic to affine algebraic varieties as quasi-projective varieties are called affine varieties; similarly for projective varieties. For example, the complement of a point in the affine line, i.e., $X=\mathbb {A} ^{1}\setminus \{0\}$, is isomorphic to the zero set of the polynomial $xy-1$ in the affine plane. As an affine set $X$ is not closed since any polynomial zero on the complement must be zero on the affine line. For another example, the complement of any conic in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called quasi-affine.
Quasi-projective varieties are locally affine in the same sense that a manifold is locally Euclidean: every point of a quasi-projective variety has a neighborhood which is an affine variety. This yields a basis of affine sets for the Zariski topology on a quasi-projective variety.
See also
• Abstract algebraic variety, often synonymous with "quasi-projective variety".
• divisorial scheme, a generalization of a quasi-projective variety
Citations
1. "Quasi-projective scheme", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
References
• Shafarevich, Igor R. (2013). Basic Algebraic Geometry 1. Springer Science. doi:10.1007/978-3-642-37956-7. ISBN 978-0-387-97716-4.
|
Wikipedia
|
Proximity space
In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that characterize topological spaces.
The concept was described by Frigyes Riesz (1909) but ignored at the time.[1] It was rediscovered and axiomatized by V. A. Efremovič in 1934 under the name of infinitesimal space, but not published until 1951. In the interim, A. D. Wallace (1941) discovered a version of the same concept under the name of separation space.
Definition
A proximity space $(X,\delta )$ is a set $X$ with a relation $\delta $ between subsets of $X$ satisfying the following properties:
For all subsets $A,B,C\subseteq X$
1. $A\;\delta \;B$ implies $B\;\delta \;A$
2. $A\;\delta \;B$ implies $A\neq \varnothing $
3. $A\cap B\neq \varnothing $ implies $A\;\delta \;B$
4. $A\;\delta \;(B\cup C)$ implies ($A\;\delta \;B$ or $A\;\delta \;C$)
5. (For all $E,$ $A\;\delta \;E$ or $B\;\delta \;(X\setminus E)$) implies $A\;\delta \;B$
Proximity without the first axiom is called quasi-proximity (but then Axioms 2 and 4 must be stated in a two-sided fashion).
If $A\;\delta \;B$ we say $A$ is near $B$ or $A$ and $B$ are proximal; otherwise we say $A$ and $B$ are apart. We say $B$ is a proximal- or $\delta $-neighborhood of $A,$ written $A\ll B,$ if and only if $A$ and $X\setminus B$ are apart.
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
For all subsets $A,B,C,D\subseteq X$
1. $X\ll X$
2. $A\ll B$ implies $A\subseteq B$
3. $A\subseteq B\ll C\subseteq D$ implies $A\ll D$
4. ($A\ll B$ and $A\ll C$) implies $A\ll B\cap C$
5. $A\ll B$ implies $X\setminus B\ll X\setminus A$
6. $A\ll B$ implies that there exists some $E$ such that $A\ll E\ll B.$
A proximity space is called separated if $\{x\}\;\delta \;\{y\}$implies $x=y.$
A proximity or proximal map is one that preserves nearness, that is, given $f:(X,\delta )\to \left(X^{*},\delta ^{*}\right),$ if $A\;\delta \;B$ in $X,$ then $f[A]\;\delta ^{*}\;f[B]$ in $X^{*}.$ Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if $C\ll ^{*}D$ holds in $X^{*},$ then $f^{-1}[C]\ll f^{-1}[D]$ holds in $X.$
Properties
Given a proximity space, one can define a topology by letting $A\mapsto \left\{x:\{x\}\;\delta \;A\right\}$ be a Kuratowski closure operator. If the proximity space is separated, the resulting topology is Hausdorff. Proximity maps will be continuous between the induced topologies.
The resulting topology is always completely regular. This can be proven by imitating the usual proofs of Urysohn's lemma, using the last property of proximal neighborhoods to create the infinite increasing chain used in proving the lemma.
Given a compact Hausdorff space, there is a unique proximity whose corresponding topology is the given topology: $A$ is near $B$ if and only if their closures intersect. More generally, proximities classify the compactifications of a completely regular Hausdorff space.
A uniform space $X$ induces a proximity relation by declaring $A$ is near $B$ if and only if $A\times B$ has nonempty intersection with every entourage. Uniformly continuous maps will then be proximally continuous.
See also
• Cauchy space – Concept in general topology and analysis
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Pretopological space – Generalized topological space
References
1. W. J. Thron, Frederic Riesz' contributions to the foundations of general topology, in C.E. Aull and R. Lowen (eds.), Handbook of the History of General Topology, Volume 1, 21-29, Kluwer 1997.
• Efremovič, V. A. (1951), "Infinitesimal spaces", Doklady Akademii Nauk SSSR, New Series (in Russian), 76: 341–343, MR 0040748
• Naimpally, Somashekhar A.; Warrack, Brian D. (1970). Proximity Spaces. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 59. Cambridge: Cambridge University Press. ISBN 0-521-07935-7. Zbl 0206.24601.
• Riesz, F. (1909), "Stetigkeit und abstrakte Mengenlehre", Rom. 4. Math. Kongr. 2: 18–24, JFM 40.0098.07
• Wallace, A. D. (1941), "Separation spaces", Ann. of Math., 2, 42 (3): 687–697, doi:10.2307/1969257, JSTOR 1969257, MR 0004756
• Vita, Luminita; Bridges, Douglas (2001). "A Constructive Theory of Point-Set Nearness". CiteSeerX 10.1.1.15.1415.
External links
• "Proximity space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Topology
Fields
• General (point-set)
• Algebraic
• Combinatorial
• Continuum
• Differential
• Geometric
• low-dimensional
• Homology
• cohomology
• Set-theoretic
• Digital
Key concepts
• Open set / Closed set
• Interior
• Continuity
• Space
• compact
• Connected
• Hausdorff
• metric
• uniform
• Homotopy
• homotopy group
• fundamental group
• Simplicial complex
• CW complex
• Polyhedral complex
• Manifold
• Bundle (mathematics)
• Second-countable space
• Cobordism
Metrics and properties
• Euler characteristic
• Betti number
• Winding number
• Chern number
• Orientability
Key results
• Banach fixed-point theorem
• De Rham cohomology
• Invariance of domain
• Poincaré conjecture
• Tychonoff's theorem
• Urysohn's lemma
• Category
• Mathematics portal
• Wikibook
• Wikiversity
• Topics
• general
• algebraic
• geometric
• Publications
|
Wikipedia
|
Quasi-quotation
Quasi-quotation or Quine quotation is a linguistic device in formal languages that facilitates rigorous and terse formulation of general rules about linguistic expressions while properly observing the use–mention distinction. It was introduced by the philosopher and logician Willard Van Orman Quine in his book Mathematical Logic, originally published in 1940. Put simply, quasi-quotation enables one to introduce symbols that stand for a linguistic expression in a given instance and are used as that linguistic expression in a different instance.
For example, one can use quasi-quotation to illustrate an instance of substitutional quantification, like the following:
"Snow is white" is true if and only if snow is white.
Therefore, there is some sequence of symbols that makes the following sentence true when every instance of φ is replaced by that sequence of symbols: "φ" is true if and only if φ.
Quasi-quotation is used to indicate (usually in more complex formulas) that the φ and "φ" in this sentence are related things, that one is the iteration of the other in a metalanguage. Quine introduced quasiquotes because he wished to avoid the use of variables, and work only with closed sentences (expressions not containing any free variables). However, he still needed to be able to talk about sentences with arbitrary predicates in them, and thus, the quasiquotes provided the mechanism to make such statements. Quine had hoped that, by avoiding variables and schemata, he would minimize confusion for the readers, as well as staying closer to the language that mathematicians actually use.[1]
Quasi-quotation is sometimes denoted using the symbols ⌜ and ⌝ (unicode U+231C, U+231D), or double square brackets, ⟦ ⟧ ("Oxford brackets"), instead of ordinary quotation marks.[2][3][4]
How it works
Quasi-quotation is particularly useful for stating formation rules for formal languages. Suppose, for example, that one wants to define the well-formed formulas (wffs) of a new formal language, L, with only a single logical operation, negation, via the following recursive definition:
1. Any lowercase Roman letter (with or without subscripts) is a well-formed formula (wff) of L.
2. If φ is a well-formed formula (wff) of L, then '~φ' is a well-formed formula (wff) of L.
3. Nothing else is a well-formed formula (wff) of L.
Interpreted literally, rule 2 does not express what is apparently intended. For '~φ' (that is, the result of concatenating '~' and 'φ', in that order, from left to right) is not a well-formed formula (wff) of L, because no Greek letter can occur in well-formed formulas (wffs), according to the apparently intended meaning of the rules. In other words, our second rule says "If some sequence of symbols φ (for example, the sequence of 3 symbols φ = '~~p') is a well-formed formula (wff) of L, then the sequence of 2 symbols '~φ' is a well-formed formula (wff) of L". Rule 2 needs to be changed so that the second occurrence of 'φ' (in quotes) be not taken literally.
Quasi-quotation is introduced as shorthand to capture the fact that what the formula expresses isn't precisely quotation, but instead something about the concatenation of symbols. Our replacement for rule 2 using quasi-quotation looks like this:
2'. If φ is a well-formed formula (wff) of L, then ⌜~φ⌝ is a well-formed formula (wff) of L.
The quasi-quotation marks '⌜' and '⌝' are interpreted as follows. Where 'φ' denotes a well-formed formula (wff) of L, '⌜~φ⌝' denotes the result of concatenating '~' and the well-formed formula (wff) denoted by 'φ' (in that order, from left to right). Thus rule 2' (unlike rule 2) entails, e.g., that if 'p' is a well-formed formula (wff) of L, then '~p' is a well-formed formula (wff) of L.
Similarly, we could not define a language with disjunction by adding this rule:
2.5. If φ and ψ are well-formed formulas (wffs) of L, then '(φ v ψ)' is a well-formed formula (wff) of L.
But instead:
2.5'. If φ and ψ are well-formed formulas (wffs) of L, then ⌜(φ v ψ)⌝ is a well-formed formula (wff) of L.
The quasi-quotation marks here are interpreted just the same. Where 'φ' and 'ψ' denote well-formed formulas (wffs) of L, '⌜(φ v ψ)⌝' denotes the result of concatenating left parenthesis, the well-formed formula (wff) denoted by 'φ', space, 'v', space, the well-formed formula (wff) denoted by 'ψ', and right parenthesis (in that order, from left to right). Just as before, rule 2.5' (unlike rule 2.5) entails, e.g., that if 'p' and 'q' are well-formed formulas (wffs) of L, then '(p v q)' is a well-formed formula (wff) of L.
Scope issues
It does not make sense to quantify into quasi-quoted contexts using variables that range over things other than character strings (e.g. numbers, people, electrons). Suppose, for example, that one wants to express the idea that 's(0)' denotes the successor of 0, 's(1)' denotes the successor of 1, etc. One might be tempted to say:
• If φ is a natural number, then ⌜s(φ)⌝ denotes the successor of φ.
Suppose, for example, φ = 7. What is ⌜s(φ)⌝ in this case? The following tentative interpretations would all be equally absurd:
1. ⌜s(φ)⌝ = 's(7)',
2. ⌜s(φ)⌝ = 's(111)' (in the binary system, '111' denotes the integer 7),
3. ⌜s(φ)⌝ = 's(VII)',
4. ⌜s(φ)⌝ = 's(seven)',
5. ⌜s(φ)⌝ = 's(семь)' ('семь' means 'seven' in Russian),
6. ⌜s(φ)⌝ = 's(the number of days in one week)'.
On the other hand, if φ = '7', then ⌜s(φ)⌝ = 's(7)', and if φ = 'seven', then ⌜s(φ)⌝ = 's(seven)'.
The expanded version of this statement reads as follows:
• If φ is a natural number, then the result of concatenating 's', left parenthesis, φ, and right parenthesis (in that order, from left to right) denotes the successor of φ.
This is a category mistake, because a number is not the sort of thing that can be concatenated (though a numeral is).
The proper way to state the principle is:
• If φ is an Arabic numeral that denotes a natural number, then ⌜s(φ)⌝ denotes the successor of the number denoted by φ.
It is tempting to characterize quasi-quotation as a device that allows quantification into quoted contexts, but this is incorrect: quantifying into quoted contexts is always illegitimate. Rather, quasi-quotation is just a convenient shortcut for formulating ordinary quantified expressions—the kind that can be expressed in first-order logic.
As long as these considerations are taken into account, it is perfectly harmless to "abuse" the corner quote notation and simply use it whenever something like quotation is necessary but ordinary quotation is clearly not appropriate.
See also
• Self-evaluating forms and quoting in Lisp, where "quasi-quotation" has been adopted for metaprogramming
• String interpolation
• Truth-value semantics (substitution interpretation)
• Template processor
References
Notes
1. Preface to the 1981 Revised Edition.
2. What are Denotational Semantics and what are they for?. Allyn and Bacon. 1986.
3. Dowty, D., Wall, R. and Peters, S.: 1981, Introduction to Montague semantics, Springer.
4. Scott, D. and Strachey, C.: 1971, Toward a mathematical semantics for computer languages, Oxford University Computing Laboratory, Programming Research Group.
Bibliography
• Quine, W. V. (2003) [1940]. Mathematical Logic (Revised ed.). Cambridge, MA: Harvard University Press. ISBN 0-674-55451-5.
External links
• Stanford Encyclopedia of Philosophy entry on quotation
|
Wikipedia
|
Quasi-set theory
Quasi-set theory is a formal mathematical theory for dealing with collections of objects, some of which may be indistinguishable from one another. Quasi-set theory is mainly motivated by the assumption that certain objects treated in quantum physics are indistinguishable and don't have individuality.
Motivation
The American Mathematical Society sponsored a 1974 meeting to evaluate the resolution and consequences of the 23 problems Hilbert proposed in 1900. An outcome of that meeting was a new list of mathematical problems, the first of which, due to Manin (1976, p. 36), questioned whether classical set theory was an adequate paradigm for treating collections of indistinguishable elementary particles in quantum mechanics. He suggested that such collections cannot be sets in the usual sense, and that the study of such collections required a "new language".
The use of the term quasi-set follows a suggestion in da Costa's 1980 monograph Ensaio sobre os Fundamentos da Lógica (see da Costa and Krause 1994), in which he explored possible semantics for what he called "Schrödinger Logics". In these logics, the concept of identity is restricted to some objects of the domain, and has motivation in Schrödinger's claim that the concept of identity does not make sense for elementary particles (Schrödinger 1952). Thus in order to provide a semantics that fits the logic, da Costa submitted that "a theory of quasi-sets should be developed", encompassing "standard sets" as particular cases, yet da Costa did not develop this theory in any concrete way. To the same end and independently of da Costa, Dalla Chiara and di Francia (1993) proposed a theory of quasets to enable a semantic treatment of the language of microphysics. The first quasi-set theory was proposed by D. Krause in his PhD thesis, in 1990 (see Krause 1992). A related physics theory, based on the logic of adding fundamental indistinguishability to equality and inequality, was developed and elaborated independently in the book The Theory of Indistinguishables by A. F. Parker-Rhodes.[1]
Outline of the theory
We now expound Krause's (1992) axiomatic theory ${\mathfrak {Q}}$, the first quasi-set theory; other formulations and improvements have since appeared. For an updated paper on the subject, see French and Krause (2010). Krause builds on the set theory ZFU, consisting of Zermelo-Fraenkel set theory with an ontology extended to include two kinds of urelements:
• m-atoms, whose intended interpretation is elementary quantum particles;
• M-atoms, macroscopic objects to which classical logic is assumed to apply.
Quasi-sets (q-sets) are collections resulting from applying axioms, very similar to those for ZFU, to a basic domain composed of m-atoms, M-atoms, and aggregates of these. The axioms of ${\mathfrak {Q}}$ include equivalents of extensionality, but in a weaker form, termed "weak extensionality axiom"; axioms asserting the existence of the empty set, unordered pair, union set, and power set; the axiom of separation; an axiom stating the image of a q-set under a q-function is also a q-set; q-set equivalents of the axioms of infinity, regularity, and choice. Q-set theories based on other set-theoretical frameworks are, of course, possible.
${\mathfrak {Q}}$ has a primitive concept of quasi-cardinal, governed by eight additional axioms, intuitively standing for the quantity of objects in a collection. The quasi-cardinal of a quasi-set is not defined in the usual sense (by means of ordinals) because the m-atoms are assumed (absolutely) indistinguishable. Furthermore, it is possible to define a translation from the language of ZFU into the language of ${\mathfrak {Q}}$ in such a way so that there is a 'copy' of ZFU in ${\mathfrak {Q}}$. In this copy, all the usual mathematical concepts can be defined, and the 'sets' (in reality, the '${\mathfrak {Q}}$-sets') turn out to be those q-sets whose transitive closure contains no m-atoms.
In ${\mathfrak {Q}}$ there may exist q-sets, called "pure" q-sets, whose elements are all m-atoms, and the axiomatics of ${\mathfrak {Q}}$ provides the grounds for saying that nothing in ${\mathfrak {Q}}$ distinguishes the elements of a pure q-set from one another, for certain pure q-sets. Within the theory, the idea that there is more than one entity in x is expressed by an axiom stating that the quasi-cardinal of the power quasi-set of x has quasi-cardinal 2qc(x), where qc(x) is the quasi-cardinal of x (which is a cardinal obtained in the 'copy' of ZFU just mentioned).
What exactly does this mean? Consider the level 2p of a sodium atom, in which there are six indiscernible electrons. Even so, physicists reason as if there are in fact six entities in that level, and not only one. In this way, by saying that the quasi-cardinal of the power quasi-set of x is 2qc(x) (suppose that qc(x) = 6 to follow the example), we are not excluding the hypothesis that there can exist six subquasi-sets of x that are 'singletons', although we cannot distinguish among them. Whether there are or not six elements in x is something that cannot be ascribed by the theory (although the notion is compatible with the theory). If the theory could answer this question, the elements of x would be individualized and hence counted, contradicting the basic assumption that they cannot be distinguished.
In other words, we may consistently (within the axiomatics of ${\mathfrak {Q}}$) reason as if there are six entities in x, but x must be regarded as a collection whose elements cannot be discerned as individuals. Using quasi-set theory, we can express some facts of quantum physics without introducing symmetry conditions (Krause et al. 1999, 2005). As is well known, in order to express indistinguishability, the particles are deemed to be individuals, say by attaching them to coordinates or to adequate functions/vectors like |ψ>. Thus, given two quantum systems labeled |ψ1⟩ and |ψ2⟩ at the outset, we need to consider a function like |ψ12⟩ = |ψ1⟩|ψ2⟩ ± |ψ2⟩|ψ1⟩ (except for certain constants), which keep the quanta indistinguishable by permutations; the probability density of the joint system independs on which is quanta #1 and which is quanta #2. (Note that precision requires that we talk of "two" quanta without distinguishing them, which is impossible in conventional set theories.) In ${\mathfrak {Q}}$, we can dispense with this "identification" of the quanta; for details, see Krause et al. (1999, 2005) and French and Krause (2006).
Quasi-set theory is a way to operationalize Heinz Post's (1963) claim that quanta should be deemed indistinguishable "right from the start."
See also
• Multisets
• Quantum physics
• Quantum logic
References
1. A. F. Parker-Rhodes, The Theory of Indistinguishables: A Search for Explanatory Principles below the level of Physics, Reidel (Springer), Dordecht (1981). ISBN 90-277-1214-X
• French, S, and Krause, D. "Remarks on the theory of quasi-sets", Studia Logica 95 (1–2), 2010, pp. 101–124.
• Newton da Costa (1980) Ensaio sobre os Fundamentos da Lógica. São Paulo: Hucitec.
• da Costa, N. C. A. and Krause, D. (1994) "Schrödinger logics," Studia Logica 53: 533–550.
• ------ (1997) "An Intensional Schrödinger Logic," Notre Dame Journal of Formal Logic 38: 179–94.
• Dalla Chiara, M. L. and Toraldo di Francia, G. (1993) "Individuals, kinds and names in physics" in Corsi, G. et al., eds., Bridging the gap: philosophy, mathematics, physics. Kluwer: 261-83.
• Domenech, G. and Holik, F. (2007), 'A Discussion on Particle Number and Quantum Indistinguishability', "Foundations of Physics" vol. 37, no. 6, pp 855–878.
• Domenech, G., Holik, F. and Krause, D., "Q-spaces and the foundations of quantum mechanics", Foundations of Physics 38 (11) Nov. 2008, 969–994.
• Falkenburg, B.: 2007, "Particle Metaphysics: A Critical Account of Subatomic Reality", Springer.
• French, Steven (2006) "Identity and Individuality in Quantum Theory," The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (ed.).
• French, S. and Krause, D. (2006) Identity in Physics: A Historical, Philosophical, and Formal Analysis. Oxford Univ. Press.
• French, S. and Rickles, D. P. (2003), 'Understanding Permutation Symmetry', in K. Brading and E. Castellani, "Symmetries in Physics: New Reflectio, Cambridge University Press, pp. 212–238.
• Krause, Decio (1992) "On a quasi-set theory," Notre Dame Journal of Formal Logic 33: 402–11.
• Krause, D., Sant'Anna, A. S. and Volkov, A. G. (1999) "Quasi-set theory for bosons and fermions: quantum distributions," Foundations of Physics Letters 12: 51–66.
• Krause, D., Sant'Anna, A. S., and Sartorelli, A. (2005) "On the concept of identity in Zermelo-Fraenkel-like axioms and its relationship with quantum statistics," Logique et Analyse: 189–192, 231–260.
• Manin, Yuri (1976) "Problems in Present Day Mathematics: Foundations," in Felix Browder, ed., Proceedings of Symposia in Pure Mathematics, Vol. XXVIII. Providence RI: American Mathematical Society.
• Post, Heinz (1963) "Individuality in physics," The Listener, 10 October 1963: 534–537. Reprinted in (1973) Vedanta for East and West: 14–22.
• Erwin Schrödinger (1952) Science and Humanism. Cambridge Un. Press.
|
Wikipedia
|
Quasisimple group
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence
$1\to Z(E)\to E\to S\to 1$
such that $E=[E,E]$, where $Z(E)$ denotes the center of E and [ , ] denotes the commutator.[1]
Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple.
The subnormal quasisimple subgroups of a group control the structure of a finite insoluble group in much the same way as the minimal normal subgroups of a finite soluble group do, and so are given a name, component.
The subgroup generated by the subnormal quasisimple subgroups is called the layer, and along with the minimal normal soluble subgroups generates a subgroup called the generalized Fitting subgroup.
The quasisimple groups are often studied alongside the simple groups and groups related to their automorphism groups, the almost simple groups. The representation theory of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.
Examples
The covering groups of the alternating groups are quasisimple but not simple, for $n\geq 5.$
See also
• Almost simple group
• Schur multiplier
• Semisimple group
References
• Aschbacher, Michael (2000). Finite Group Theory. Cambridge University Press. ISBN 0-521-78675-4. Zbl 0997.20001.
External links
• http://mathworld.wolfram.com/QuasisimpleGroup.html
Notes
1. I. Martin Isaacs, Finite group theory (2008), p. 272.
|
Wikipedia
|
Quasi-sphere
In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.
Notation and terminology
This article uses the following notation and terminology:
• A pseudo-Euclidean vector space, denoted Rs,t, is a real vector space with a nondegenerate quadratic form with signature (s, t). The quadratic form is permitted to be definite (where s = 0 or t = 0), making this a generalization of a Euclidean vector space.[lower-alpha 1]
• A pseudo-Euclidean space, denoted Es,t, is a real affine space in which displacement vectors are the elements of the space Rs,t. It is distinguished from the vector space.
• The quadratic form Q acting on a vector x ∈ Rs,t, denoted Q(x), is a generalization of the squared Euclidean distance in a Euclidean space. Élie Cartan calls Q(x) the scalar square of x.[1]
• The symmetric bilinear form B acting on two vectors x, y ∈ Rs,t is denoted B(x, y) or x ⋅ y.[lower-alpha 2] This is associated with the quadratic form Q.[lower-alpha 3]
• Two vectors x, y ∈ Rs,t are orthogonal if x ⋅ y = 0.
• A normal vector at a point of a quasi-sphere is a nonzero vector that is orthogonal to each vector in the tangent space at that point.
Definition
A quasi-sphere is a submanifold of a pseudo-Euclidean space Es,t consisting of the points u for which the displacement vector x = u − o from a reference point o satisfies the equation
a x ⋅ x + b ⋅ x + c = 0,
where a, c ∈ R and b, x ∈ Rs,t.[2][lower-alpha 4]
Since a = 0 in permitted, this definition includes hyperplanes; it is thus a generalization of generalized circles and their analogues in any number of dimensions. This inclusion provides a more regular structure under conformal transformations than if they are omitted.
This definition has been generalized to affine spaces over complex numbers and quaternions by replacing the quadratic form with a Hermitian form.[3]
A quasi-sphere P = {x ∈ X : Q(x) = k} in a quadratic space (X, Q) has a counter-sphere N = {x ∈ X : Q(x) = −k}.[lower-alpha 5] Furthermore, if k ≠ 0 and L is an isotropic line in X through x = 0, then L ∩ (P ∪ N) = ∅, puncturing the union of quasi-sphere and counter-sphere. One example is the unit hyperbola that forms a quasi-sphere of the hyperbolic plane, and its conjugate hyperbola, which is its counter-sphere.
Geometric characterizations
Centre and radial scalar square
The centre of a quasi-sphere is a point that has equal scalar square from every point of the quasi-sphere, the point at which the pencil of lines normal to the tangent hyperplanes meet. If the quasi-sphere is a hyperplane, the centre is the point at infinity defined by this pencil.
When a ≠ 0, the displacement vector p of the centre from the reference point and the radial scalar square r may be found as follows. We put Q(x − p) = r, and comparing to the defining equation above for a quasi-sphere, we get
$p=-{\frac {b}{2a}},$
$r=p\cdot p-{\frac {c}{a}}.$
The case of a = 0 may be interpreted as the centre p being a well-defined point at infinity with either infinite or zero radial scalar square (the latter for the case of a null hyperplane). Knowing p (and r) in this case does not determine the hyperplane's position, though, only its orientation in space.
The radial scalar square may take on a positive, zero or negative value. When the quadratic form is definite, even though p and r may be determined from the above expressions, the set of vectors x satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radial scalar square.
Diameter and radius
Any pair of points, which need not be distinct, (including the option of up to one of these being a point at infinity) defines a diameter of a quasi-sphere. The quasi-sphere is the set of points for which the two displacement vectors from these two points are orthogonal.
Any point may be selected as a centre (including a point at infinity), and any other point on the quasi-sphere (other than a point at infinity) define a radius of a quasi-sphere, and thus specifies the quasi-sphere.
Partitioning
Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. Q(x − p)) as the radial scalar square, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial scalar square, those with negative radial scalar square, those with zero radial scalar square.[lower-alpha 6]
In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial scalar square is the empty set, one with zero radial scalar square consists of a single point, one with positive radial scalar square is a standard n-sphere, and one with zero curvature is a hyperplane that is partitioned with the n-spheres.
See also
• Anti-de Sitter space
• de Sitter space
• Hyperboloid § Relation to the sphere
• Lie sphere geometry
• Quadratic set
Notes
1. Some authors exclude the definite cases, but in the context of this article, the qualifier indefinite will be used where this exclusion is intended.
2. The symmetric bilinear form applied to the two vectors is also called their scalar product.
3. The associated symmetric bilinear form of a (real) quadratic form Q is defined such that Q(x) = B(x, x), and may be determined as B(x, y) = 1/4(Q(x + y) − Q(x − y)). See Polarization identity for variations of this identity.
4. Though not mentioned in the source, we must exclude the combination b = 0 and a = 0.
5. There are caveats when Q is definite. Also, when k = 0, it follows that N = P.
6. A hyperplane (a quasi-sphere with infinite radial scalar square or zero curvature) is partitioned with quasi-spheres to which it is tangent. The three sets may be defined according to whether the quadratic form applied to a vector that is a normal of the tangent hypersurface is positive, zero or negative. The three sets of objects are preserved under conformal transformations of the space.
References
1. Élie Cartan (1981) [First published in 1937 in French, and in 1966 in English], The Theory of Spinors, Dover Publications, p. 3, ISBN 0486640701
2. Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
3. Ian R. Porteous (1995), Clifford Algebras and the Classical Groups, Cambridge University Press
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
|
Wikipedia
|
Quasi-split group
In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.
Examples
All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial.
Lang (1956) showed that all simple algebraic groups over finite fields are quasi-split.
Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups On,n+2, the unitary groups SUn,n and SUn,n+1, and the form of E6 with signature 2.
References
• Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics, 78: 555–563, doi:10.2307/2372673, ISSN 0002-9327, JSTOR 2372673, MR 0086367
|
Wikipedia
|
Quasisymmetric function
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).
For quasisymmetric functions in the theory of metric spaces or complex analysis, see quasisymmetric map.
Definitions
The ring of quasisymmetric functions, denoted QSym, can be defined over any commutative ring R such as the integers. Quasisymmetric functions are power series of bounded degree in variables $x_{1},x_{2},x_{3},\dots $ with coefficients in R, which are shift invariant in the sense that the coefficient of the monomial $x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{k}^{\alpha _{k}}$ is equal to the coefficient of the monomial $x_{i_{1}}^{\alpha _{1}}x_{i_{2}}^{\alpha _{2}}\cdots x_{i_{k}}^{\alpha _{k}}$ for any strictly increasing sequence of positive integers $i_{1}<i_{2}<\cdots <i_{k}$ indexing the variables and any positive integer sequence $(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})$ of exponents.[1] Much of the study of quasisymmetric functions is based on that of symmetric functions.
A quasisymmetric function in finitely many variables is a quasisymmetric polynomial. Both symmetric and quasisymmetric polynomials may be characterized in terms of actions of the symmetric group $S_{n}$ on a polynomial ring in $n$ variables $x_{1},\dots ,x_{n}$. One such action of $S_{n}$ permutes variables, changing a polynomial $p(x_{1},\dots ,x_{n})$ by iteratively swapping pairs $(x_{i},x_{i+1})$ of variables having consecutive indices. Those polynomials unchanged by all such swaps form the subring of symmetric polynomials. A second action of $S_{n}$ conditionally permutes variables, changing a polynomial $p(x_{1},\ldots ,x_{n})$ by swapping pairs $(x_{i},x_{i+1})$ of variables except in monomials containing both variables. Those polynomials unchanged by all such conditional swaps form the subring of quasisymmetric polynomials. One quasisymmetric function in four variables $x_{1},x_{2},x_{3},x_{4}$ is the polynomial
$x_{1}^{2}x_{2}x_{3}+x_{1}^{2}x_{2}x_{4}+x_{1}^{2}x_{3}x_{4}+x_{2}^{2}x_{3}x_{4}.\,$
The simplest symmetric function containing these monomials is
${\begin{aligned}x_{1}^{2}x_{2}x_{3}+x_{1}^{2}x_{2}x_{4}+x_{1}^{2}x_{3}x_{4}+x_{2}^{2}x_{3}x_{4}+x_{1}x_{2}^{2}x_{3}+x_{1}x_{2}^{2}x_{4}+x_{1}x_{3}^{2}x_{4}+x_{2}x_{3}^{2}x_{4}\\{}+x_{1}x_{2}x_{3}^{2}+x_{1}x_{2}x_{4}^{2}+x_{1}x_{3}x_{4}^{2}+x_{2}x_{3}x_{4}^{2}.\,\end{aligned}}$
Important bases
QSym is a graded R-algebra, decomposing as
$\operatorname {QSym} =\bigoplus _{n\geq 0}\operatorname {QSym} _{n},\,$
where $\operatorname {QSym} _{n}$ is the $R$-span of all quasisymmetric functions that are homogeneous of degree $n$. Two natural bases for $\operatorname {QSym} _{n}$ are the monomial basis $\{M_{\alpha }\}$ and the fundamental basis $\{F_{\alpha }\}$ indexed by compositions $\alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})$ of $n$, denoted $\alpha \vDash n$. The monomial basis consists of $M_{0}=1$ and all formal power series
$M_{\alpha }=\sum _{i_{1}<i_{2}<\cdots <i_{k}}x_{i_{1}}^{\alpha _{1}}x_{i_{2}}^{\alpha _{2}}\cdots x_{i_{k}}^{\alpha _{k}}.\,$
The fundamental basis consists $F_{0}=1$ and all formal power series
$F_{\alpha }=\sum _{\alpha \succeq \beta }M_{\beta },\,$
where $\alpha \succeq \beta $ means we can obtain $\alpha $ by adding together adjacent parts of $\beta $, for example, (3,2,4,2) $\succeq $ (3,1,1,1,2,1,2). Thus, when the ring $R$ is the ring of rational numbers, one has
$\operatorname {QSym} _{n}=\operatorname {span} _{\mathbb {Q} }\{M_{\alpha }\mid \alpha \vDash n\}=\operatorname {span} _{\mathbb {Q} }\{F_{\alpha }\mid \alpha \vDash n\}.\,$
Then one can define the algebra of symmetric functions $\Lambda =\Lambda _{0}\oplus \Lambda _{1}\oplus \cdots $ as the subalgebra of QSym spanned by the monomial symmetric functions $m_{0}=1$ and all formal power series $m_{\lambda }=\sum M_{\alpha },$ where the sum is over all compositions $\alpha $ which rearrange to the partition $\lambda $. Moreover, we have $\Lambda _{n}=\Lambda \cap \operatorname {QSym} _{n}$. For example, $F_{(1,2)}=M_{(1,2)}+M_{(1,1,1)}$ and $m_{(2,1)}=M_{(2,1)}+M_{(1,2)}.$
Other important bases for quasisymmetric functions include the basis of quasisymmetric Schur functions,[2] the "type I" and "type II" quasisymmetric power sums,[3] and bases related to enumeration in matroids.[4][5]
Applications
Quasisymmetric functions have been applied in enumerative combinatorics, symmetric function theory, representation theory, and number theory. Applications of quasisymmetric functions include enumeration of P-partitions,[6][7] permutations,[8][9][10][11] tableaux,[12] chains of posets,[12][13] reduced decompositions in finite Coxeter groups (via Stanley symmetric functions),[12] and parking functions.[14] In symmetric function theory and representation theory, applications include the study of Schubert polynomials,[15][16] Macdonald polynomials,[17] Hecke algebras,[18] and Kazhdan–Lusztig polynomials.[19] Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions.
Related algebras
As a graded Hopf algebra, the dual of the ring of quasisymmetric functions is the ring of noncommutative symmetric functions. Every symmetric function is also a quasisymmetric function, and hence the ring of symmetric functions is a subalgebra of the ring of quasisymmetric functions.
The ring of quasisymmetric functions is the terminal object in category of graded Hopf algebras with a single character.[20] Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions.
One example of this is the peak algebra.[21]
Other related algebras
The Malvenuto–Reutenauer algebra[22] is a Hopf algebra based on permutations that relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions, (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram. The duality between QSym and NSym mentioned above is reflected in the main diagonal of this diagram.
Many related Hopf algebras were constructed from Hopf monoids in the category of species by Aguiar and Majahan.[23]
One can also construct the ring of quasisymmetric functions in noncommuting variables.[24][25]
References
1. Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).
2. Haglund, J.; Luoto, K.; Mason, S.; van Willigenburg, S. (2011), "Quasisymmetric Schur functions", Journal of Combinatorial Theory, Series A, 118 (2): 463–490, arXiv:0810.2489, doi:10.1016/j.jcta.2009.11.002
3. Ballantine, Christina; Daughtery, Zajj; Hicks, Angela; Mason, Sarah; Niese, Elizabeth (2020), "On Quasisymmetric Power Sums", Journal of Combinatorial Theory, Series A, 175: 105273, arXiv:1710.11613, doi:10.1016/j.jcta.2020.105273, S2CID 51775423
4. Luoto, K. (2008), "A matroid-friendly basis for the quasisymmetric functions", Journal of Combinatorial Theory, Series A, 115 (5): 777–798, arXiv:0704.0836, Bibcode:2007arXiv0704.0836L, doi:10.1016/j.jcta.2007.10.003
5. Billera, L.; Jia, N.; Reiner, V. (2009), "A quasisymmetric function for matroids", European Journal of Combinatorics, 30 (8): 1727–1757, arXiv:math/0606646, Bibcode:2006math......6646B, doi:10.1016/j.ejc.2008.12.007
6. Stanley, Richard P. Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, 1972.
7. Gessel, Ira. Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), 289–317, Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984.
8. Gessel, Ira; Reutenauer, Christophe (1993), "Counting permutations with given cycle structure and descent set", Journal of Combinatorial Theory, Series A, 64 (2): 189–215, doi:10.1016/0097-3165(93)90095-P
9. Shareshian, John; Wachs, Michelle L. (2007), "$q$-Eulerian polynomials: excedance number and major index", Electron. Res. Announc. Amer. Math. Soc., 13 (4): 33–45, arXiv:math/0608274, doi:10.1090/S1079-6762-07-00172-2, S2CID 15394306
10. Shareshian, John; Wachs, Michelle L. (2010), "Eulerian quasisymmetric functions", Advances in Mathematics, 225 (6): 2921–2966, arXiv:0812.0764, doi:10.1016/j.aim.2010.05.009
11. Hyatt, Matthew (2012), "Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups", Advances in Applied Mathematics, 48 (3): 465–505, arXiv:1007.0459, Bibcode:2010arXiv1007.0459H, doi:10.1016/j.aam.2011.11.005, S2CID 119118644
12. Stanley, Richard P. (1984), "On the number of reduced decompositions of elements of Coxeter groups", European Journal of Combinatorics, 5 (4): 359–372, doi:10.1016/s0195-6698(84)80039-6
13. Ehrenborg, Richard (1996), "On posets and Hopf algebras", Advances in Mathematics, 119 (1): 1–25, doi:10.1006/aima.1996.0026
14. Haglund, James; The q,t-Catalan numbers and the space of diagonal harmonics. University Lecture Series, 41. American Mathematical Society, Providence, RI, 2008. viii+167 pp. ISBN 978-0-8218-4411-3; 0-8218-4411-3
15. Billey, Sara C.; Jockusch, William; Stanley, Richard P. (1993), "Some combinatorial properties of Schubert polynomials" (PDF), Journal of Algebraic Combinatorics, 2 (4): 345–374, doi:10.1023/A:1022419800503
16. Fomin, Sergey; Stanley, Richard P. (1994), "Schubert polynomials and the nil-Coxeter algebra", Advances in Mathematics, 103 (2): 196–207, doi:10.1006/aima.1994.1009
17. Assaf, Sami (2010), Dual Equivalence Graphs I: A combinatorial proof of LLT and Macdonald positivity, arXiv:1005.3759, Bibcode:2010arXiv1005.3759A
18. Duchamp, Gérard; Krob, Daniel; Leclerc, Bernard; Thibon, Jean-Yves (1996), "Fonctions quasi-symétriques, fonctions symétriques non commutatives et algèbres de Hecke à $q=0$", C. R. Acad. Sci. Paris, Sér. I Math., 322 (2): 107–112
19. Billera, Louis J.; Brenti, Francesco (2011), "Quasisymmetric functions and Kazhdan–Lusztig polynomials", Israel Journal of Mathematics, 184: 317–348, arXiv:0710.3965, doi:10.1007/s11856-011-0070-0
20. Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank (2006), "Combinatorial Hopf algebras and generalized Dehn–Sommerville relations", Compositio Mathematica, 142 (1): 1–30, arXiv:math/0310016, Bibcode:2003math.....10016A, doi:10.1112/S0010437X0500165X, S2CID 2635356
21. Stembridge, John R. (1997), "Enriched P-partitions", Trans. Amer. Math. Soc., 349 (2): 763–788, doi:10.1090/S0002-9947-97-01804-7
22. Malvenuto, Clauda; Reutenauer, Christophe (1995), "Duality between quasi-symmetric functions and the Solomon descent algebra", Journal of Algebra, 177 (3): 967–982, doi:10.1006/jabr.1995.1336
23. Aguiar, Marcelo; Mahajan, Swapneel Monoidal Functors, Species and Hopf Algebras CRM Monograph Series, no. 29. American Mathematical Society, Providence, RI, 2010.
24. Hivert, Florent, Ph.D. Thesis, Marne-la-Vallée
25. Bergeron, Nantel; Zabrocki, Mike (2009), "The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free", J. Algebra Appl., 8 (4): 581–600, arXiv:math/0509265, doi:10.1142/S0219498809003485, S2CID 18601994
External links
• BIRS Workshop on Quasisymmetric Functions
|
Wikipedia
|
Trivially perfect graph
In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques.[1] Trivially perfect graphs were first studied by (Wolk 1962, 1965) but were named by Golumbic (1978); Golumbic writes that "the name was chosen since it is trivial to show that such a graph is perfect." Trivially perfect graphs are also known as comparability graphs of trees,[2] arborescent comparability graphs,[3] and quasi-threshold graphs.[4]
Equivalent characterizations
Trivially perfect graphs have several other equivalent characterizations:
• They are the comparability graphs of order-theoretic trees. That is, let T be a partial order such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <, and also T possesses a minimum element r. Then the comparability graph of T is trivially perfect, and every trivially perfect graph can be formed in this way.[5]
• They are the graphs that do not have a P4 path graph or a C4 cycle graph as induced subgraphs.[6]
• They are the graphs in which every connected induced subgraph contains a universal vertex.[7]
• They are the graphs that can be represented as the interval graphs for a set of nested intervals. A set of intervals is nested if, for every two intervals in the set, either the two are disjoint or one contains the other.[8]
• They are the graphs that are both chordal and cographs.[9] This follows from the characterization of chordal graphs as the graphs without induced cycles of length greater than three, and of cographs as the graphs without induced paths on four vertices (P4).
• They are the graphs that are both cographs and interval graphs.[9]
• They are the graphs that can be formed, starting from one-vertex graphs, by two operations: disjoint union of two smaller trivially perfect graphs, and the addition of a new vertex adjacent to all the vertices of a smaller trivially perfect graph.[10] These operations correspond, in the underlying forest, to forming a new forest by the disjoint union of two smaller forests and forming a tree by connecting a new root node to the roots of all the trees in a forest.
• They are the graphs in which, for every edge uv, the neighborhoods of u and v (including u and v themselves) are nested: one neighborhood must be a subset of the other.[11]
• They are the permutation graphs defined from stack-sortable permutations.[12]
• They are the graphs with the property that in each of its induced subgraphs the clique cover number equals the number of maximal cliques.[13]
• They are the graphs with the property that in each of its induced subgraphs the clique number equals the pseudo-Grundy number.[13]
• They are the graphs with the property that in each of its induced subgraphs the chromatic number equals the pseudo-Grundy number.[13]
Related classes of graphs
It follows from the equivalent characterizations of trivially perfect graphs that every trivially perfect graph is also a cograph, a chordal graph, a Ptolemaic graph, an interval graph, and a perfect graph.
The threshold graphs are exactly the graphs that are both themselves trivially perfect and the complements of trivially perfect graphs (co-trivially perfect graphs).[14]
Windmill graphs are trivially perfect.
Recognition
Chu (2008) describes a simple linear time algorithm for recognizing trivially perfect graphs, based on lexicographic breadth-first search. Whenever the LexBFS algorithm removes a vertex v from the first set on its queue, the algorithm checks that all remaining neighbors of v belong to the same set; if not, one of the forbidden induced subgraphs can be constructed from v. If this check succeeds for every v, then the graph is trivially perfect. The algorithm can also be modified to test whether a graph is the complement graph of a trivially perfect graph, in linear time.
Determining if a general graph is k edge deletions away from a trivially perfect graph is NP-complete,[15] fixed-parameter tractable[16] and can be solved in O(2.45k(m + n)) time.[17]
Notes
1. Brandstädt, Le & Spinrad (1999), definition 2.6.2, p.34; Golumbic (1978).
2. Wolk (1962); Wolk (1965).
3. Donnelly & Isaak (1999).
4. Yan, Chen & Chang (1996).
5. Brandstädt, Le & Spinrad (1999), theorem 6.6.1, p. 99; Golumbic (1978), corollary 4.
6. Brandstädt, Le & Spinrad (1999), theorem 6.6.1, p. 99; Golumbic (1978), theorem 2. Wolk (1962) and Wolk (1965) proved this for comparability graphs of rooted forests.
7. Wolk (1962).
8. Brandstädt, Le & Spinrad (1999), p. 51.
9. Brandstädt, Le & Spinrad (1999), p. 248; Yan, Chen & Chang (1996), theorem 3.
10. Yan, Chen & Chang (1996); Gurski (2006).
11. Yan, Chen & Chang (1996), theorem 3.
12. Rotem (1981).
13. Rubio-Montiel (2015).
14. Brandstädt, Le & Spinrad (1999), theorem 6.6.3, p. 100; Golumbic (1978), corollary 5.
15. Sharan (2002).
16. Cai (1996).
17. Nastos & Gao (2010).
References
• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X.
• Cai, L. (1996), "Fixed-parameter tractability of graph modification problems for hereditary properties", Information Processing Letters, 58 (4): 171–176, doi:10.1016/0020-0190(96)00050-6.
• Chu, Frank Pok Man (2008), "A simple linear time certifying LBFS-based algorithm for recognizing trivially perfect graphs and their complements", Information Processing Letters, 107 (1): 7–12, doi:10.1016/j.ipl.2007.12.009.
• Donnelly, Sam; Isaak, Garth (1999), "Hamiltonian powers in threshold and arborescent comparability graphs", Discrete Mathematics, 202 (1–3): 33–44, doi:10.1016/S0012-365X(98)00346-X
• Golumbic, Martin Charles (1978), "Trivially perfect graphs", Discrete Mathematics, 24 (1): 105–107, doi:10.1016/0012-365X(78)90178-4.
• Gurski, Frank (2006), "Characterizations for co-graphs defined by restricted NLC-width or clique-width operations", Discrete Mathematics, 306 (2): 271–277, doi:10.1016/j.disc.2005.11.014.
• Nastos, James; Gao, Yong (2010), "A novel branching strategy for parameterized graph modification problems", in Wu, Weili; Daescu, Ovidiu (eds.), Combinatorial Optimization and Applications – 4th International Conference, COCOA 2010, Kailua-Kona, HI, USA, December 18–20, 2010, Proceedings, Part II, Lecture Notes in Computer Science, vol. 6509, Springer, pp. 332–346, arXiv:1006.3020, doi:10.1007/978-3-642-17461-2_27
• Rotem, D. (1981), "Stack sortable permutations", Discrete Mathematics, 33 (2): 185–196, doi:10.1016/0012-365X(81)90165-5, MR 0599081.
• Rubio-Montiel, C. (2015), "A new characterization of trivially perfect graphs", Electronic Journal of Graph Theory and Applications, 3 (1): 22–26, doi:10.5614/ejgta.2015.3.1.3.
• Sharan, Roded (2002), "Graph modification problems and their applications to genomic research", PhD Thesis, Tel Aviv University.
• Wolk, E. S. (1962), "The comparability graph of a tree", Proceedings of the American Mathematical Society (5 ed.), 13: 789–795, doi:10.1090/S0002-9939-1962-0172273-0.
• Wolk, E. S. (1965), "A note on the comparability graph of a tree", Proceedings of the American Mathematical Society (1 ed.), 16: 17–20, doi:10.1090/S0002-9939-1965-0172274-5.
• Yan, Jing-Ho; Chen, Jer-Jeong; Chang, Gerard J. (1996), "Quasi-threshold graphs", Discrete Applied Mathematics, 69 (3): 247–255, doi:10.1016/0166-218X(96)00094-7.
External links
• "Trivially perfect graphs", Information System on Graph Classes and their Inclusions
|
Wikipedia
|
Quasitriangular Hopf algebra
In mathematics, a Hopf algebra, H, is quasitriangular[1] if there exists an invertible element, R, of $H\otimes H$ such that
• $R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)$ for all $x\in H$, where $\Delta $ is the coproduct on H, and the linear map $T:H\otimes H\to H\otimes H$ is given by $T(x\otimes y)=y\otimes x$,
• $(\Delta \otimes 1)(R)=R_{13}\ R_{23}$,
• $(1\otimes \Delta )(R)=R_{13}\ R_{12}$,
where $R_{12}=\phi _{12}(R)$, $R_{13}=\phi _{13}(R)$, and $R_{23}=\phi _{23}(R)$, where $\phi _{12}:H\otimes H\to H\otimes H\otimes H$, $\phi _{13}:H\otimes H\to H\otimes H\otimes H$, and $\phi _{23}:H\otimes H\to H\otimes H\otimes H$, are algebra morphisms determined by
$\phi _{12}(a\otimes b)=a\otimes b\otimes 1,$
$\phi _{13}(a\otimes b)=a\otimes 1\otimes b,$
$\phi _{23}(a\otimes b)=1\otimes a\otimes b.$
R is called the R-matrix.
As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang–Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, $(\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H$; moreover $R^{-1}=(S\otimes 1)(R)$, $R=(1\otimes S)(R^{-1})$, and $(S\otimes S)(R)=R$. One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element: $S^{2}(x)=uxu^{-1}$ where $u:=m(S\otimes 1)R^{21}$ (cf. Ribbon Hopf algebras).
It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.
If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding
$c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)$.
Twisting
The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element $F=\sum _{i}f^{i}\otimes f_{i}\in {\mathcal {A\otimes A}}$ such that $(\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1$ and satisfying the cocycle condition
$(F\otimes 1)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)$
Furthermore, $u=\sum _{i}f^{i}S(f_{i})$ is invertible and the twisted antipode is given by $S'(a)=uS(a)u^{-1}$, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.
See also
• Quasi-triangular quasi-Hopf algebra
• Ribbon Hopf algebra
Notes
1. Montgomery & Schneider (2002), p. 72.
References
• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
• Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Mathematical Sciences Research Institute Publications. Vol. 43. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.
|
Wikipedia
|
Quasi-triangular quasi-Hopf algebra
A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangular Hopf algebra.
A quasi-triangular quasi-Hopf algebra is a set ${\mathcal {H_{A}}}=({\mathcal {A}},R,\Delta ,\varepsilon ,\Phi )$ where ${\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )$ is a quasi-Hopf algebra and $R\in {\mathcal {A\otimes A}}$ known as the R-matrix, is an invertible element such that
$R\Delta (a)=\sigma \circ \Delta (a)R$
for all $a\in {\mathcal {A}}$, where $\sigma \colon {\mathcal {A\otimes A}}\rightarrow {\mathcal {A\otimes A}}$ is the switch map given by $x\otimes y\rightarrow y\otimes x$, and
$(\Delta \otimes \operatorname {id} )R=\Phi _{312}R_{13}\Phi _{132}^{-1}R_{23}\Phi _{123}$
$(\operatorname {id} \otimes \Delta )R=\Phi _{231}^{-1}R_{13}\Phi _{213}R_{12}\Phi _{123}^{-1}$
where $\Phi _{abc}=x_{a}\otimes x_{b}\otimes x_{c}$ and $\Phi _{123}=\Phi =x_{1}\otimes x_{2}\otimes x_{3}\in {\mathcal {A\otimes A\otimes A}}$.
The quasi-Hopf algebra becomes triangular if in addition, $R_{21}R_{12}=1$.
The twisting of ${\mathcal {H_{A}}}$ by $F\in {\mathcal {A\otimes A}}$ is the same as for a quasi-Hopf algebra, with the additional definition of the twisted R-matrix
A quasi-triangular (resp. triangular) quasi-Hopf algebra with $\Phi =1$ is a quasi-triangular (resp. triangular) Hopf algebra as the latter two conditions in the definition reduce the conditions of quasi-triangularity of a Hopf algebra.
Similarly to the twisting properties of the quasi-Hopf algebra, the property of being quasi-triangular or triangular quasi-Hopf algebra is preserved by twisting.
See also
• Ribbon Hopf algebra
References
• Vladimir Drinfeld, "Quasi-Hopf algebras", Leningrad mathematical journal (1989), 1419–1457
• J. M. Maillet and J. Sanchez de Santos, "Drinfeld Twists and Algebraic Bethe Ansatz", American Mathematical Society Translations: Series 2 Vol. 201, 2000
|
Wikipedia
|
Quasi-ultrabarrelled space
In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
Definition
A subset B0 of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence $\left(B_{i}\right)_{i=1}^{\infty }$ of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1, .... In this case, $\left(B_{i}\right)_{i=1}^{\infty }$ is called a defining sequence for B0. A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.[1]
Properties
A locally convex quasi-ultrabarrelled space is quasi-barrelled.[1]
Examples and sufficient conditions
Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.[1]
See also
• Barrelled space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Ultrabarrelled space
• Uniform boundedness principle#Generalisations
References
1. Khaleelulla 1982, pp. 65–76.
• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 65–75.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
Boundedness and bornology
Basic concepts
• Barrelled space
• Bounded set
• Bornological space
• (Vector) Bornology
Operators
• (Un)Bounded operator
• Uniform boundedness principle
Subsets
• Barrelled set
• Bornivorous set
• Saturated family
Related spaces
• (Countably) Barrelled space
• (Countably) Quasi-barrelled space
• Infrabarrelled space
• (Quasi-) Ultrabarrelled space
• Ultrabornological space
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
• Metrizability
• Operator topologies
• Topological vector space
• Vector space
Main results
• Anderson–Kadec
• Banach–Alaoglu
• Closed graph theorem
• F. Riesz's
• Hahn–Banach (hyperplane separation
• Vector-valued Hahn–Banach)
• Open mapping (Banach–Schauder)
• Bounded inverse
• Uniform boundedness (Banach–Steinhaus)
Maps
• Bilinear operator
• form
• Linear map
• Almost open
• Bounded
• Continuous
• Closed
• Compact
• Densely defined
• Discontinuous
• Topological homomorphism
• Functional
• Linear
• Bilinear
• Sesquilinear
• Norm
• Seminorm
• Sublinear function
• Transpose
Types of sets
• Absolutely convex/disk
• Absorbing/Radial
• Affine
• Balanced/Circled
• Banach disks
• Bounding points
• Bounded
• Complemented subspace
• Convex
• Convex cone (subset)
• Linear cone (subset)
• Extreme point
• Pre-compact/Totally bounded
• Prevalent/Shy
• Radial
• Radially convex/Star-shaped
• Symmetric
Set operations
• Affine hull
• (Relative) Algebraic interior (core)
• Convex hull
• Linear span
• Minkowski addition
• Polar
• (Quasi) Relative interior
Types of TVSs
• Asplund
• B-complete/Ptak
• Banach
• (Countably) Barrelled
• BK-space
• (Ultra-) Bornological
• Brauner
• Complete
• Convenient
• (DF)-space
• Distinguished
• F-space
• FK-AK space
• FK-space
• Fréchet
• tame Fréchet
• Grothendieck
• Hilbert
• Infrabarreled
• Interpolation space
• K-space
• LB-space
• LF-space
• Locally convex space
• Mackey
• (Pseudo)Metrizable
• Montel
• Quasibarrelled
• Quasi-complete
• Quasinormed
• (Polynomially
• Semi-) Reflexive
• Riesz
• Schwartz
• Semi-complete
• Smith
• Stereotype
• (B
• Strictly
• Uniformly) convex
• (Quasi-) Ultrabarrelled
• Uniformly smooth
• Webbed
• With the approximation property
• Mathematics portal
• Category
• Commons
|
Wikipedia
|
Unipotent
In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n.
In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix are 1.
The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.
In the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then a group with all elements unipotent.
Definition
Definition with matrices
Consider the group $\mathbb {U} _{n}$ of upper-triangular matrices with $1$'s along the diagonal, so they are the group of matrices[1]
$\mathbb {U} _{n}=\left\{{\begin{bmatrix}1&*&\cdots &*&*\\0&1&\cdots &*&*\\\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&*\\0&0&\cdots &0&1\end{bmatrix}}\right\}.$
Then, a unipotent group can be defined as a subgroup of some $\mathbb {U} _{n}$. Using scheme theory the group $\mathbb {U} _{n}$ can be defined as the group scheme
${\text{Spec}}\left({\frac {\mathbb {C} \!\left[x_{11},x_{12},\ldots ,x_{nn},{\frac {1}{\text{det}}}\right]}{(x_{ii}=1,x_{i>j}=0)}}\right)$
and an affine group scheme is unipotent if it is a closed group scheme of this scheme.
Definition with ring theory
An element x of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affine coordinate ring A[G] of G is locally unipotent as an element of the ring of linear endomorphism of A[G]. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of A[G] is unipotent in the usual ring-theoretic sense.)
An affine algebraic group is called unipotent if all its elements are unipotent. Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GLn(k)).
For example, the standard representation of $\mathbb {U} _{n}$ on $k^{n}$ with standard basis $e_{i}$ has the fixed vector $e_{1}$.
Definition with representation theory
If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite-dimensional vector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups.[1] In particular, this implies there are no non-trivial semisimple representations.
Examples
Un
Of course, the group of matrices $\mathbb {U} _{n}$ is unipotent. Using the lower central series
$\mathbb {U} _{n}=\mathbb {U} _{n}^{(0)}\supset \mathbb {U} _{n}^{(1)}\supset \mathbb {U} _{n}^{(2)}\supset \cdots \supset \mathbb {U} _{n}^{(m)}=e$
where
$\mathbb {U} _{n}^{(1)}=[\mathbb {U} _{n},\mathbb {U} _{n}]$ and $\mathbb {U} _{n}^{(2)}=[\mathbb {U} _{n},\mathbb {U} _{n}^{(1)}]$
there are associated unipotent groups. For example, on $n=4$, the central series are the matrix groups
$\mathbb {U} _{4}=\left\{{\begin{bmatrix}1&*&*&*\\0&1&*&*\\0&0&1&*\\0&0&0&1\end{bmatrix}}\right\}$, $\mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}$, $\mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}$, and $\mathbb {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}$
given some induced examples of unipotent groups.
Gan
The additive group $\mathbb {G} _{a}$ is a unipotent group through the embedding
$a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}$
Notice the matrix multiplication gives
${\begin{bmatrix}1&a\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&b\\0&1\end{bmatrix}}={\begin{bmatrix}1&a+b\\0&1\end{bmatrix}}$
hence this is a group embedding. More generally, there is an embedding $\mathbb {G} _{a}^{n}\to \mathbb {U} _{n+1}$ from the map
$(a_{1},\ldots ,a_{n})\,\mapsto {\begin{bmatrix}1&a_{1}&a_{2}&\cdots &a_{n-1}&a_{n}\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\\0&0&0&\cdots &0&1\end{bmatrix}}$
Using scheme theory, $\mathbb {G} _{a}$ is given by the functor
${\mathcal {O}}:{\textbf {Sch}}^{op}\to {\textbf {Sets}}$
where
$(X,{\mathcal {O}}_{X})\mapsto {\mathcal {O}}_{X}(X)$
Kernel of the Frobenius
Consider the functor ${\mathcal {O}}$ on the subcategory ${\textbf {Sch}}/\mathbb {F} _{p}$, there is the subfunctor $\alpha _{p}$ where
$\alpha _{p}(X)=\{x\in {\mathcal {O}}(X):x^{p}=0\}$
so it is given by the kernel of the Frobenius endomorphism.
Classification of unipotent groups over characteristic 0
Over characteristic 0 there is a nice classification of unipotent algebraic groups with respect to nilpotent Lie algebras. Recall that a nilpotent Lie algebra is a subalgebra of some ${\mathfrak {gl}}_{n}$ such that the iterated adjoint action eventually terminates to the zero-map. In terms of matrices, this means it is a subalgebra ${\mathfrak {g}}$ of ${\mathfrak {n}}_{n}$, the matrices with $a_{ij}=0$ for $i\leq j$.
Then, there is an equivalence of categories of finite-dimensional nilpotent Lie algebras and unipotent algebraic groups.[1]page 261 This can be constructed using the Baker–Campbell–Hausdorff series $H(X,Y)$, where given a finite-dimensional nilpotent Lie algebra, the map
$H:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}{\text{ where }}(X,Y)\mapsto H(X,Y)$
gives a Unipotent algebraic group structure on ${\mathfrak {g}}$.
In the other direction the exponential map takes any nilpotent square matrix to a unipotent matrix. Moreover, if U is a commutative unipotent group, the exponential map induces an isomorphism from the Lie algebra of U to U itself.
Remarks
Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6.
Unipotent radical
The unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. It is a connected unipotent normal subgroup of G, and contains all other such subgroups. A group is called reductive if its unipotent radical is trivial. If G is reductive then its radical is a torus.
Decomposition of algebraic groups
Algebraic groups can be decomposed into unipotent groups, multiplicative groups, and abelian varieties, but the statement of how they decompose depends upon the characteristic of their base field.
Characteristic 0
Over characteristic 0 there is a nice decomposition theorem of an algebraic group $G$ relating its structure to the structure of a linear algebraic group and an Abelian variety. There is a short exact sequence of groups[2]page 8
$0\to M\times U\to G\to A\to 0$
where $A$ is an abelian variety, $M$ is of multiplicative type (meaning, $M$ is, geometrically, a product of tori and algebraic groups of the form $\mu _{n}$) and $U$ is a unipotent group.
Characteristic p
When the characteristic of the base field is p there is an analogous statement[2] for an algebraic group $G$: there exists a smallest subgroup $H$ such that
1. $G/H$ is a unipotent group
2. $H$ is an extension of an abelian variety $A$ by a group $M$ of multiplicative type.
3. $M$ is unique up to commensurability in $G$ and $A$ is unique up to isogeny.
Jordan decomposition
Main article: Jordan–Chevalley decomposition
Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gu gs of commuting unipotent and semisimple elements gu and gs. In the case of the group GLn(C), this essentially says that any invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is (more or less) the multiplicative version of the Jordan–Chevalley decomposition.
There is also a version of the Jordan decomposition for groups: any commutative linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group.
See also
• Reductive group
• Unipotent representation
• Deligne–Lusztig theory
References
1. Milne, J. S. Linear Algebraic Groups (PDF). pp. 252–253, Unipotent algebraic groups.
2. Brion, Michel (2016-09-27). "Commutative algebraic groups up to isogeny". arXiv:1602.00222 [math.AG].
• A. Borel, Linear algebraic groups, ISBN 0-387-97370-2
• Borel, Armand (1956), "Groupes linéaires algébriques", Annals of Mathematics, Second Series, Annals of Mathematics, 64 (1): 20–82, doi:10.2307/1969949, JSTOR 1969949
• Popov, V.L. (2001) [1994], "unipotent element", Encyclopedia of Mathematics, EMS Press
• Popov, V.L. (2001) [1994], "unipotent group", Encyclopedia of Mathematics, EMS Press
• Suprunenko, D.A. (2001) [1994], "unipotent matrix", Encyclopedia of Mathematics, EMS Press
Matrix classes
Explicitly constrained entries
• Alternant
• Anti-diagonal
• Anti-Hermitian
• Anti-symmetric
• Arrowhead
• Band
• Bidiagonal
• Bisymmetric
• Block-diagonal
• Block
• Block tridiagonal
• Boolean
• Cauchy
• Centrosymmetric
• Conference
• Complex Hadamard
• Copositive
• Diagonally dominant
• Diagonal
• Discrete Fourier Transform
• Elementary
• Equivalent
• Frobenius
• Generalized permutation
• Hadamard
• Hankel
• Hermitian
• Hessenberg
• Hollow
• Integer
• Logical
• Matrix unit
• Metzler
• Moore
• Nonnegative
• Pentadiagonal
• Permutation
• Persymmetric
• Polynomial
• Quaternionic
• Signature
• Skew-Hermitian
• Skew-symmetric
• Skyline
• Sparse
• Sylvester
• Symmetric
• Toeplitz
• Triangular
• Tridiagonal
• Vandermonde
• Walsh
• Z
Constant
• Exchange
• Hilbert
• Identity
• Lehmer
• Of ones
• Pascal
• Pauli
• Redheffer
• Shift
• Zero
Conditions on eigenvalues or eigenvectors
• Companion
• Convergent
• Defective
• Definite
• Diagonalizable
• Hurwitz
• Positive-definite
• Stieltjes
Satisfying conditions on products or inverses
• Congruent
• Idempotent or Projection
• Invertible
• Involutory
• Nilpotent
• Normal
• Orthogonal
• Unimodular
• Unipotent
• Unitary
• Totally unimodular
• Weighing
With specific applications
• Adjugate
• Alternating sign
• Augmented
• Bézout
• Carleman
• Cartan
• Circulant
• Cofactor
• Commutation
• Confusion
• Coxeter
• Distance
• Duplication and elimination
• Euclidean distance
• Fundamental (linear differential equation)
• Generator
• Gram
• Hessian
• Householder
• Jacobian
• Moment
• Payoff
• Pick
• Random
• Rotation
• Seifert
• Shear
• Similarity
• Symplectic
• Totally positive
• Transformation
Used in statistics
• Centering
• Correlation
• Covariance
• Design
• Doubly stochastic
• Fisher information
• Hat
• Precision
• Stochastic
• Transition
Used in graph theory
• Adjacency
• Biadjacency
• Degree
• Edmonds
• Incidence
• Laplacian
• Seidel adjacency
• Tutte
Used in science and engineering
• Cabibbo–Kobayashi–Maskawa
• Density
• Fundamental (computer vision)
• Fuzzy associative
• Gamma
• Gell-Mann
• Hamiltonian
• Irregular
• Overlap
• S
• State transition
• Substitution
• Z (chemistry)
Related terms
• Jordan normal form
• Linear independence
• Matrix exponential
• Matrix representation of conic sections
• Perfect matrix
• Pseudoinverse
• Row echelon form
• Wronskian
• Mathematics portal
• List of matrices
• Category:Matrices
|
Wikipedia
|
Quasi-Fuchsian group
In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two. Some authors use "quasi-Fuchsian group" to mean "quasi-Fuchsian group of type 1", in other words the limit set is the whole Jordan curve. This terminology is incompatible with the use of the terms "type 1" and "type 2" for Kleinian groups: all quasi-Fuchsian groups are Kleinian groups of type 2 (even if they are quasi-Fuchsian groups of type 1), as their limit sets are proper subsets of the Riemann sphere. The special case when the Jordan curve is a circle or line is called a Fuchsian group, named after Lazarus Fuchs by Henri Poincaré.
Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations.
The space of quasi-Fuchsian groups of the first kind is described by the simultaneous uniformization theorem of Bers.
References
• Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen. (in German), Leipzig: B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01
• Fricke, Robert; Klein, Felix (1912), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen. (in German), Leipzig: B. G. Teubner., ISBN 978-1-4297-0552-3, JFM 32.0430.01
• Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Berlin, New York: Springer-Verlag, ISBN 978-3-540-17746-3, MR 0959135
External links
• A picture of the limit set of a quasi-Fuchsian group from (Fricke & Klein 1897, p. 418).
• A picture of the limit set of a quasi-Fuchsian group
• A picture of the limit set of a quasi-Fuchsian group
|
Wikipedia
|
Excellent ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal.
The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; Hironaka (1964) showed this in characteristic 0, but the positive characteristic case is (as of 2016) still a major open problem. Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent.
Definitions
The definition of excellent rings is quite involved, so we recall the definitions of the technical conditions it satisfies. Although it seems like a long list of conditions, most rings in practice are excellent, such as fields, polynomial rings, complete Noetherian rings, Dedekind domains over characteristic 0 (such as $\mathbb {Z} $), and quotient and localization rings of these rings.
Recalled definitions
• A ring $R$ containing a field $k$ is called geometrically regular over $k$ if for any finite extension $K$ of $k$ the ring $R\otimes _{k}K$ is regular.
• A homomorphism of rings from $R\to S$ is called regular if it is flat and for every ${\mathfrak {p}}\in {\text{Spec}}(R)$ the fiber $S\otimes _{R}\kappa ({\mathfrak {p}})$ is geometrically regular over the residue field $\kappa ({\mathfrak {p}})$ of ${\mathfrak {p}}$.
• A ring $R$ is called a G-ring[1] (or Grothendieck ring) if it is Noetherian and its formal fibers are geometrically regular; this means that for any ${\mathfrak {p}}\in {\text{Spec}}(R)$, the map from the local ring $R_{\mathfrak {p}}\to {\hat {R_{\mathfrak {p}}}}$ to its completion is regular in the sense above.
Finally, a ring is J-2[2] if any finite type $R$-algebra $S$ is J-1, meaning the regular subscheme ${\text{Reg}}({\text{Spec}}(S))\subset {\text{Spec}}(S)$ is open.
Definition of (quasi-)excellence
A ring $R$ is called quasi-excellent if it is a G-ring and J-2 ring. It is called excellent[3]pg 214 if it is quasi-excellent and universally catenary. In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings.
A scheme is called excellent or quasi-excellent if it has a cover by open affine subschemes with the same property, which implies that every open affine subscheme has this property.
Properties
Because an excellent ring $R$ is a G-ring,[1] it is Noetherian by definition. Because it is universally catenary, every maximal chain of prime ideals has the same length. This is useful for studying the dimension theory of such rings because their dimension can be bounded by a fixed maximal chain. In practice, this means infinite-dimensional Noetherian rings[4] which have an inductive definition of maximal chains of prime ideals, giving an infinite-dimensional ring, cannot be constructed.
Schemes
Given an excellent scheme $X$ and a locally finite type morphism $f:X'\to X$, then $X'$ is excellent[3]pg 217.
Quasi-excellence
Any quasi-excellent ring is a Nagata ring.
Any quasi-excellent reduced local ring is analytically reduced.
Any quasi-excellent normal local ring is analytically normal.
Examples
Excellent rings
Most naturally occurring commutative rings in number theory or algebraic geometry are excellent. In particular:
• All complete Noetherian local rings, for instance all fields and the ring Zp of p-adic integers, are excellent.
• All Dedekind domains of characteristic 0 are excellent. In particular the ring Z of integers is excellent. Dedekind domains over fields of characteristic greater than 0 need not be excellent.
• The rings of convergent power series in a finite number of variables over R or C are excellent.
• Any localization of an excellent ring is excellent.
• Any finitely generated algebra over an excellent ring is excellent. This includes all polynomial algebras $R[x_{1},\ldots ,x_{n}]/(f_{1},\ldots ,f_{k})$ with $R$ excellent. This means most rings considered in algebraic geometry are excellent.
A J-2 ring that is not a G-ring
Here is an example of a discrete valuation ring A of dimension 1 and characteristic p > 0 which is J-2 but not a G-ring and so is not quasi-excellent. If k is any field of characteristic p with [k : kp] = ∞ and A is the ring of power series Σaixi such that [kp(a0, a1, ...) : kp] is finite then the formal fibers of A are not all geometrically regular so A is not a G-ring. It is a J-2 ring as all Noetherian local rings of dimension at most 1 are J-2 rings. It is also universally catenary as it is a Dedekind domain. Here kp denotes the image of k under the Frobenius morphism a → ap.
A G-ring that is not a J-2 ring
Here is an example of a ring that is a G-ring but not a J-2 ring and so not quasi-excellent. If R is the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and S is obtained from R by adjoining inverses to all elements not in any of the ideals generated by some xn, then S is a 1-dimensional Noetherian domain that is not a J-1 ring as S has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring. This ring is also universally catenary, as its localization at every prime ideal is a quotient of a regular ring.
A quasi-excellent ring that is not excellent
Nagata's example of a 2-dimensional Noetherian local ring that is catenary but not universally catenary is a G-ring, and is also a J-2 ring as any local G-ring is a J-2 ring (Matsumura 1980, p.88, 260). So it is a quasi-excellent catenary local ring that is not excellent.
Resolution of singularities
Quasi-excellent rings are closely related to the problem of resolution of singularities, and this seems to have been Grothendieck's motivation[3]pg 218 for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi-excellent rings. Hironaka (1964) proved this for all complete integral Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral finite algebras over a Noetherian ring R then the ring R is quasi-excellent.
See also
• Resolution of singularities
References
1. "Section 15.49 (07GG): G-rings—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
2. "Section 15.46 (07P6): The singular locus—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
3. Grothendieck, Alexander (1965). "Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS. 24: 5–231.
4. "Section 108.14 (02JC): A Noetherian ring of infinite dimension—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
• Alexandre Grothendieck, Jean Dieudonné, Eléments de géométrie algébrique IV Publications Mathématiques de l'IHÉS 24 (1965), section 7
• V.I. Danilov (2001) [1994], "Excellent ring", Encyclopedia of Mathematics, EMS Press
• Hironaka, Heisuke (1964). "Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I". Annals of Mathematics. 79 (1): 109–203. doi:10.2307/1970486. ISSN 0003-486X. JSTOR 1970486.
• Hironaka, Heisuke (1964). "Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II". Annals of Mathematics. 79 (2): 205–326. doi:10.2307/1970547. ISSN 0003-486X. JSTOR 1970547.
• Matsumura, Hideyuki (1980). "Chapter 13". Commutative algebra. Reading, Mass.: Benjamin/Cummings Pub. Co. ISBN 0-8053-7026-9.
|
Wikipedia
|
Quasi-finite field
In mathematics, a quasi-finite field[1] is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite (i.e. non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite.[2]
Formal definition
A quasi-finite field is a perfect field K together with an isomorphism of topological groups
$\phi :{\hat {\mathbb {Z} }}\to \operatorname {Gal} (K_{s}/K),$ :{\hat {\mathbb {Z} }}\to \operatorname {Gal} (K_{s}/K),}
where Ks is an algebraic closure of K (necessarily separable because K is perfect). The field extension Ks/K is infinite, and the Galois group is accordingly given the Krull topology. The group ${\widehat {\mathbb {Z} }}$ is the profinite completion of integers with respect to its subgroups of finite index.
This definition is equivalent to saying that K has a unique (necessarily cyclic) extension Kn of degree n for each integer n ≥ 1, and that the union of these extensions is equal to Ks.[3] Moreover, as part of the structure of the quasi-finite field, there is a generator Fn for each Gal(Kn/K), and the generators must be coherent, in the sense that if n divides m, the restriction of Fm to Kn is equal to Fn.
Examples
The most basic example, which motivates the definition, is the finite field K = GF(q). It has a unique cyclic extension of degree n, namely Kn = GF(qn). The union of the Kn is the algebraic closure Ks. We take Fn to be the Frobenius element; that is, Fn(x) = xq.
Another example is K = C((T)), the ring of formal Laurent series in T over the field C of complex numbers. (These are simply formal power series in which we also allow finitely many terms of negative degree.) Then K has a unique cyclic extension
$K_{n}=\mathbf {C} ((T^{1/n}))$
of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and that a generator of Gal(Kn/K) is given by
$F_{n}(T^{1/n})=e^{2\pi i/n}T^{1/n}.$
This construction works if C is replaced by any algebraically closed field C of characteristic zero.[4]
Notes
1. (Artin & Tate 2009, §XI.3) say that the field satisfies "Moriya's axiom"
2. As shown by Mikao Moriya (Serre 1979, chapter XIII, p. 188)
3. (Serre 1979, §XIII.2 exercise 1, p. 192)
4. (Serre 1979, §XIII.2, p. 191)
References
• Artin, Emil; Tate, John (2009) [1967], Class field theory, American Mathematical Society, ISBN 978-0-8218-4426-7, MR 2467155, Zbl 1179.11040
• Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, vol. 67, translated by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, MR 0554237, Zbl 0423.12016
|
Wikipedia
|
Quasi-open map
In topology a branch of mathematics, a quasi-open map or quasi-interior map is a function which has similar properties to continuous maps. However, continuous maps and quasi-open maps are not related.[1]
Definition
A function f : X → Y between topological spaces X and Y is quasi-open if, for any non-empty open set U ⊆ X, the interior of f ('U) in Y is non-empty.[1][2]
Properties
Let $f:X\to Y$ be a map between topological spaces.
• If $f$ is continuous, it need not be quasi-open. Conversely if $f$ is quasi-open, it need not be continuous.[1]
• If $f$ is open, then $f$ is quasi-open.[1]
• If $f$ is a local homeomorphism, then $f$ is quasi-open.[1]
• The composition of two quasi-open maps is again quasi-open.[note 1][1]
See also
• Almost open map – Map that satisfies a condition similar to that of being an open map.
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed linear operator – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Proper map – Map between topological spaces with the property that the preimage of every compact is compact
• Quotient map (topology) – Topological space constructionPages displaying short descriptions of redirect targets
Notes
1. This means that if $f:X\to Y$ and $g:Y\to Z$ are both quasi-open (such that all spaces are topological), then the function composition $g\circ f:X\to Z$ is quasi-open.
References
1. Kim, Jae Woon (1998). "A Note on Quasi-Open Maps" (PDF). Journal of the Korean Mathematical Society. B: The Pure and Applied Mathematics. 5 (1): 1–3. Archived from the original (PDF) on March 4, 2016. Retrieved October 20, 2011.
2. Blokh, A.; Oversteegen, L.; Tymchatyn, E.D. (2006). "On almost one-to-one maps". Trans. Amer. Math. Soc. 358 (11): 5003–5015. doi:10.1090/s0002-9947-06-03922-5.
|
Wikipedia
|
Minuscule representation
In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl group acts transitively on the weights. Some authors exclude the trivial representation. A quasi-minuscule representation (also called a basic representation) is an irreducible representation such that all non-zero weights are in the same orbit under the Weyl group; each simple Lie algebra has a unique quasi-minuscule representation that is not minuscule, and the multiplicity of the zero weight is the number of short nodes of the Dynkin diagram. (The highest weight of that quasi-minuscule representation is the highest short root, which in the simply-laced case is also the highest long root, making the quasi-minuscule representation be the adjoint representation.)
The minuscule representations are indexed by the weight lattice modulo the root lattice, or equivalently by irreducible representations of the center of the simply connected compact group. For the simple Lie algebras, the dimensions of the minuscule representations are given as follows.
• An (n+1
k
) for 0 ≤ k ≤ n (exterior powers of vector representation). Quasi-minuscule: n2+2n (adjoint)
• Bn 1 (trivial), 2n (spin). Quasi-minuscule: 2n+1 (vector)
• Cn 1 (trivial), 2n (vector). Quasi-minuscule: 2n2–n–1 if n>1
• Dn 1 (trivial), 2n (vector), 2n−1 (half spin), 2n−1 (half spin). Quasi-minuscule: 2n2–n (adjoint)
• E6 1, 27, 27. Quasi-minuscule: 78 (adjoint)
• E7 1, 56. Quasi-minuscule: 133 (adjoint)
• E8 1. Quasi-minuscule: 248 (adjoint)
• F4 1. Quasi-minuscule: 26
• G2 1. Quasi-minuscule: 7
References
• Seshadri, C. S. (1978), "Geometry of G/P. I. Theory of standard monomials for minuscule representations", C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Berlin, New York: Springer-Verlag, pp. 207–239
|
Wikipedia
|
Almost holomorphic modular form
In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. An almost holomorphic modular form is determined by its holomorphic part, so the operation of taking the holomorphic part gives an isomorphism between the spaces of almost holomorphic modular forms and quasimodular forms. The archetypal examples of quasimodular forms are the Eisenstein series E2(τ) (the holomorphic part of the almost holomorphic modular form E2(τ) – 3/πIm(τ)), and derivatives of modular forms.
Not to be confused with Weakly holomorphic modular form.
In terms of representation theory, modular forms correspond roughly to highest weight vectors of certain discrete series representations of SL2(R), while almost holomorphic or quasimodular forms correspond roughly to other (not necessarily highest weight) vectors of these representations.
Definitions
To simplify notation this section treats the level 1 case; the extension to higher levels is straightforward.
A level 1 almost holomorphic modular form is a function f on the upper half plane with the properties:
• f transforms like a modular form: $f((a\tau +b)/(c\tau +d))=(c\tau +d)^{k}f(\tau )$ for some integer k called the weight, for any elements of SL2(Z) (that is: a, b, c, d are integers with ad - bc = 1).
• As a function of q=e2πiτ, f is a polynomial in 1/Im(τ) with coefficients that are holomorphic functions of q.
A level 1 quasimodular form is defined to be the constant term of an almost holomorphic modular form (considered as a polynomial in 1/Im(τ)).
Structure
The ring of almost holomorphic modular forms of level 1 is a polynomial ring over the complex numbers in the three generators $E_{2}(\tau )-3/\pi \Im (\tau ),E_{4}(\tau ),E_{6}(\tau )$. Similarly the ring of quasimodular forms of level 1 is a polynomial ring over the complex numbers in the three generators $E_{2}(\tau ),E_{4}(\tau ),E_{6}(\tau )$.
Quasimodular forms can be interpreted as sections of certain jet bundles.[1]
Derivatives
Ramanujan observed that the derivative of any quasimodular form is another quasimodular form.[2] For example,
${\begin{aligned}{\frac {1}{2\pi i}}{\frac {dE_{2}}{d\tau }}&={\frac {E_{2}^{2}-E_{4}}{12}}\\[6pt]{\frac {1}{2\pi i}}{\frac {dE_{4}}{d\tau }}&={\frac {E_{2}E_{4}-E_{6}}{3}}\\[6pt]{\frac {1}{2\pi i}}{\frac {dE_{6}}{d\tau }}&={\frac {E_{2}E_{6}-E_{4}^{2}}{2}}\end{aligned}}$
As the field generated by quasimodular forms of some level has transcendence degree 3 over C, this implies that any quasimodular form satisfies some nonlinear differential equation of order 3. For example, the Eisenstein series E2 satisfies the Chazy equation (give or take a few constants).
References
1. Movasati (2012, Appendix A)
• Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Camb. Philos. Soc., 22 (9): 159–184, MR 2280861
• Movasati, Hossein (2012), "Quasi-modular forms attached to elliptic curves, I", Ann. Math. Blaise Pascal, 19 (2): 307–377, MR 3025138
• Zagier, Don (2008), "Elliptic modular forms and their applications", in Ranestad, Kristian (ed.), The 1-2-3 of modular forms. Lectures at a summer school in Nordfjordeid, Norway, June 2004, Universitext, with Bruinier, Jan Hendrik; van der Geer, Gerard; Harder, Günter, Berlin: Springer-Verlag, pp. 1–103, doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6, MR 2409678, Zbl 1197.11047
|
Wikipedia
|
Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation.
This article is about binary relations. For the graph vertex ordering, see depth-first search. For purchase orders for unreleased products, see pre-order. For other uses, see Preorder (disambiguation).
"Quasiorder" redirects here. For irreflexive transitive relations, see strict order.
Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗
Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗
Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗
Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗
Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗
Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗
Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗
Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗
Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all $a,b$ and $S\neq \varnothing :$ :} ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.
The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol $\,\leq \,$ can be used as the notational device for the relation. However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol $\,\leq \,$ may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.
In words, when $a\leq b,$ one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or → or $\,\lesssim \,$ is used instead of $\,\leq .$
To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components.
Formal definition
Consider a homogeneous relation $\,\leq \,$ on some given set $P,$ so that by definition, $\,\leq \,$ is some subset of $P\times P$ and the notation $a\leq b$ is used in place of $(a,b)\in \,\leq .$ Then $\,\leq \,$ is called a preorder or quasiorder if it is reflexive and transitive; that is, if it satisfies:
1. Reflexivity: $a\leq a$ for all $a\in P,$ and
2. Transitivity: if $a\leq b{\text{ and }}b\leq c{\text{ then }}a\leq c$ for all $a,b,c\in P.$
A set that is equipped with a preorder is called a preordered set (or proset).[2] For emphasis or contrast to strict preorders (defined next), a preorder may also be referred to as a non-strict preorder.
If reflexivity is replaced with irreflexivity (while keeping transitivity) then the result is called a strict preorder; explicitly, a strict preorder on $P$ is a homogeneous binary relation $\,<\,$ on $P$ that satisfies the following conditions:
1. Irreflexivity or Anti-reflexivity: not $a<a$ for all $a\in P;$ that is, $\,a<a$ is false for all $a\in P,$ and
2. Transitivity: if $a<b{\text{ and }}b<c{\text{ then }}a<c$ for all $a,b,c\in P.$
A binary relation is a strict preorder if and only if it is a strict partial order. By definition, a strict partial order is an asymmetric strict preorder, where $\,<\,$ is called asymmetric if $a<b{\text{ implies }}{\textit {not}}\ b<a$ for all $a,b.$ Conversely, every strict preorder is a strict partial order because every transitive irreflexive relation is necessarily asymmetric. Although they are equivalent, the term "strict partial order" is typically preferred over "strict preorder" and readers are referred to the article on strict partial orders for details about such relations. In contrast to strict preorders, there are many (non-strict) preorders that are not (non-strict) partial orders.
Related definitions
If a preorder is also antisymmetric, that is, $a\leq b$ and $b\leq a$ implies $a=b,$ then it is a partial order.
On the other hand, if it is symmetric, that is, if $a\leq b$ implies $b\leq a,$ then it is an equivalence relation.
A preorder is total if $a\leq b$ or $b\leq a$ for all $a,b\in P.$
The notion of a preordered set $P$ can be formulated in a categorical framework as a thin category; that is, as a category with at most one morphism from an object to another. Here the objects correspond to the elements of $P,$ and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can be understood as an enriched category, enriched over the category $2=(0\to 1).$
A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class.
Examples
Graph theory
• (see figure above) By x//4 is meant the greatest integer that is less than or equal to x divided by 4, thus 1//4 is 0, which is certainly less than or equal to 0, which is itself the same as 0//4.
• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where $x\leq y$ in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for every pair (x, y) with $x\leq y.$ However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partially ordered sets (preorders satisfying an additional antisymmetry property).
• The graph-minor relation in graph theory.
Computer science
In computer science, one can find examples of the following preorders.
• Asymptotic order causes a preorder over functions $f:\mathbb {N} \to \mathbb {N} $. The corresponding equivalence relation is called asymptotic equivalence.
• Polynomial-time, many-one (mapping) and Turing reductions are preorders on complexity classes.
• Subtyping relations are usually preorders.[3]
• Simulation preorders are preorders (hence the name).
• Reduction relations in abstract rewriting systems.
• The encompassment preorder on the set of terms, defined by $s\leq t$ if a subterm of t is a substitution instance of s.
• Theta-subsumption,[4] which is when the literals in a disjunctive first-order formula are contained by another, after applying a substitution to the former.
Other
Further examples:
• Every finite topological space gives rise to a preorder on its points by defining $x\leq y$ if and only if x belongs to every neighborhood of y. Every finite preorder can be formed as the specialization preorder of a topological space in this way. That is, there is a one-to-one correspondence between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not one-to-one.
• A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without losing important features.
• The relation defined by $x\leq y$ if $f(x)\leq f(y),$ where f is a function into some preorder.
• The relation defined by $x\leq y$ if there exists some injection from x to y. Injection may be replaced by surjection, or any type of structure-preserving function, such as ring homomorphism, or permutation.
• The embedding relation for countable total orderings.
• A category with at most one morphism from any object x to any other object y is a preorder. Such categories are called thin. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.
Example of a total preorder:
• Preference, according to common models.
Uses
Preorders play a pivotal role in several situations:
• Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
• Preorders may be used to define interior algebras.
• Preorders provide the Kripke semantics for certain types of modal logic.
• Preorders are used in forcing in set theory to prove consistency and independence results.[5]
Constructions
Every binary relation $R$ on a set $S$ can be extended to a preorder on $S$ by taking the transitive closure and reflexive closure, $R^{+=}.$ The transitive closure indicates path connection in $R:xR^{+}y$ if and only if there is an $R$-path from $x$ to $y.$
Left residual preorder induced by a binary relation
Given a binary relation $R,$ the complemented composition $R\backslash R={\overline {R^{\textsf {T}}\circ {\overline {R}}}}$ forms a preorder called the left residual,[6] where $R^{\textsf {T}}$ denotes the converse relation of $R,$ and ${\overline {R}}$ denotes the complement relation of $R,$ while $\circ $ denotes relation composition.
Preorders and partial orders on partitions
Given a preorder $\,\lesssim \,$ on $S$ one may define an equivalence relation $\,\sim \,$ on $S$ such that
$a\sim b\quad {\text{ if and only if }}\quad a\lesssim b\;{\text{ and }}\;b\lesssim a.$
The resulting relation $\,\sim \,$ is reflexive since the preorder $\,\lesssim \,$ is reflexive; transitive by applying the transitivity of $\,\lesssim \,$ twice; and symmetric by definition.
Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, $S/\sim ,$ which is the set of all equivalence classes of $\,\sim .$ If the preorder is denoted by $R^{+=},$ then $S/\sim $ is the set of $R$-cycle equivalence classes: $x\in [y]$ if and only if $x=y$ or $x$ is in an $R$-cycle with $y$ In any case, on $S/\sim $ it is possible to define $[x]\leq [y]$ if and only if $x\lesssim y.$ That this is well-defined, meaning that its defining condition does not depend on which representatives of $[x]$ and $[y]$ are chosen, follows from the definition of $\,\sim .\,$ It is readily verified that this yields a partially ordered set.
Conversely, from any partial order on a partition of a set $S,$ it is possible to construct a preorder on $S$ itself. There is a one-to-one correspondence between preorders and pairs (partition, partial order).
Example: Let $S$ be a formal theory, which is a set of sentences with certain properties (details of which can be found in the article on the subject). For instance, $S$ could be a first-order theory (like Zermelo–Fraenkel set theory) or a simpler zeroth-order theory. One of the many properties of $S$ is that it is closed under logical consequences so that, for instance, if a sentence $A\in S$ logically implies some sentence $B,$ which will be written as $A\Rightarrow B$ and also as $B\Leftarrow A,$ then necessarily $B\in S$ (by modus ponens). The relation $\,\Leftarrow \,$ is a preorder on $S$ because $A\Leftarrow A$ always holds and whenever $A\Leftarrow B$ and $B\Leftarrow C$ both hold then so does $A\Leftarrow C.$ Furthermore, for any $A,B\in S,$ $A\sim B$ if and only if $A\Leftarrow B{\text{ and }}B\Leftarrow A$; that is, two sentences are equivalent with respect to $\,\Leftarrow \,$ if and only if they are logically equivalent. This particular equivalence relation $A\sim B$ is commonly denoted with its own special symbol $A\iff B,$ and so this symbol $\,\iff \,$ may be used instead of $\,\sim .$ The equivalence class of a sentence $A,$ denoted by $[A],$ consists of all sentences $B\in S$ that are logically equivalent to $A$ (that is, all $B\in S$ such that $A\iff B$). The partial order on $S/\sim $ induced by $\,\Leftarrow ,\,$ which will also be denoted by the same symbol $\,\Leftarrow ,\,$ is characterized by $[A]\Leftarrow [B]$ if and only if $A\Leftarrow B,$ where the right hand side condition is independent of the choice of representatives $A\in [A]$ and $B\in [B]$ of the equivalence classes. All that has been said of $\,\Leftarrow \,$ so far can also be said of its converse relation $\,\Rightarrow .\,$ The preordered set $(S,\Leftarrow )$ is a directed set because if $A,B\in S$ and if $C:=A\wedge B$ denotes the sentence formed by logical conjunction $\,\wedge ,\,$ then $A\Leftarrow C$ and $B\Leftarrow C$ where $C\in S.$ The partially ordered set $\left(S/\sim ,\Leftarrow \right)$ is consequently also a directed set. See Lindenbaum–Tarski algebra for a related example.
Preorders and strict preorders
Strict preorder induced by a preorder
Given a preorder $\,\lesssim ,$ a new relation $\,<\,$ can be defined by declaring that $a<b$ if and only if $a\lesssim b{\text{ and not }}b\lesssim a.$ Using the equivalence relation $\,\sim \,$ introduced above, $a<b$ if and only if $a\lesssim b{\text{ and not }}a\sim b;$ and so the following holds
$a\lesssim b\quad {\text{ if and only if }}\quad a<b\;{\text{ or }}\;a\sim b.$
The relation $\,<\,$ is a strict partial order and every strict partial order can be constructed this way. If the preorder $\,\lesssim \,$ is antisymmetric (and thus a partial order) then the equivalence $\,\sim \,$ is equality (that is, $a\sim b$ if and only if $a=b$) and so in this case, the definition of $\,<\,$ can be restated as:
$a<b\quad {\text{ if and only if }}\quad a\leq b\;{\text{ and }}\;a\neq b\quad \quad ({\text{assuming }}\lesssim {\text{ is antisymmetric}}).$
But importantly, this new condition is not used as (nor is it equivalent to) the general definition of the relation $\,<\,$ (that is, $\,<\,$ is not defined as: $a<b$ if and only if $a\lesssim b{\text{ and }}a\neq b$) because if the preorder $\,\lesssim \,$ is not antisymmetric then the resulting relation $\,<\,$ would not be transitive (consider how equivalent non-equal elements relate). This is the reason for using the symbol "$\lesssim $" instead of the "less than or equal to" symbol "$\leq $", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that $a\leq b$ implies $a<b{\text{ or }}a=b.$
Preorders induced by a strict preorder
Using the construction above, multiple non-strict preorders can produce the same strict preorder $\,<,\,$ so without more information about how $\,<\,$ was constructed (such knowledge of the equivalence relation $\,\sim \,$ for instance), it might not be possible to reconstruct the original non-strict preorder from $\,<.\,$ Possible (non-strict) preorders that induce the given strict preorder $\,<\,$ include the following:
• Define $a\leq b$ as $a<b{\text{ or }}a=b$ (that is, take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "$<$" through reflexive closure; in this case the equivalence is equality $\,=,$ so the symbols $\,\lesssim \,$ and $\,\sim \,$ are not needed.
• Define $a\lesssim b$ as "${\text{ not }}b<a$" (that is, take the inverse complement of the relation), which corresponds to defining $a\sim b$ as "neither $a<b{\text{ nor }}b<a$"; these relations $\,\lesssim \,$ and $\,\sim \,$ are in general not transitive; however, if they are then $\,\sim \,$ is an equivalence; in that case "$<$" is a strict weak order. The resulting preorder is connected (formerly called total); that is, a total preorder.
If $a\leq b$ then $a\lesssim b.$ The converse holds (that is, $\,\lesssim \;\;=\;\;\leq \,$) if and only if whenever $a\neq b$ then $a<b$ or $b<a.$
Number of preorders
Number of n-element binary relations of different types
Elements Any Transitive Reflexive Symmetric Preorder Partial order Total preorder Total order Equivalence relation
0111111111
1221211111
216134843322
3512171646429191365
465,5363,9944,0961,024355219752415
n 2n2 2n2−n 2n(n+1)/2 $ \sum _{k=0}^{n}k!S(n,k)$ n! $ \sum _{k=0}^{n}S(n,k)$
OEIS A002416 A006905 A053763 A006125 A000798 A001035 A000670 A000142 A000110
Note that S(n, k) refers to Stirling numbers of the second kind.
As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
• for $n=3:$
• 1 partition of 3, giving 1 preorder
• 3 partitions of 2 + 1, giving $3\times 3=9$ preorders
• 1 partition of 1 + 1 + 1, giving 19 preorders
I.e., together, 29 preorders.
• for $n=4:$
• 1 partition of 4, giving 1 preorder
• 7 partitions with two classes (4 of 3 + 1 and 3 of 2 + 2), giving $7\times 3=21$ preorders
• 6 partitions of 2 + 1 + 1, giving $6\times 19=114$ preorders
• 1 partition of 1 + 1 + 1 + 1, giving 219 preorders
I.e., together, 355 preorders.
Interval
For $a\lesssim b,$ the interval $[a,b]$ is the set of points x satisfying $a\lesssim x$ and $x\lesssim b,$ also written $a\lesssim x\lesssim b.$ It contains at least the points a and b. One may choose to extend the definition to all pairs $(a,b)$ The extra intervals are all empty.
Using the corresponding strict relation "$<$", one can also define the interval $(a,b)$ as the set of points x satisfying $a<x$ and $x<b,$ also written $a<x<b.$ An open interval may be empty even if $a<b.$
Also $[a,b)$ and $(a,b]$ can be defined similarly.
See also
• Partial order – preorder that is antisymmetric
• Equivalence relation – preorder that is symmetric
• Total preorder – preorder that is total
• Total order – preorder that is antisymmetric and total
• Directed set
• Category of preordered sets
• Prewellordering
• Well-quasi-ordering
Notes
1. on the set of numbers divisible by 4
2. For "proset", see e.g. Eklund, Patrik; Gähler, Werner (1990), "Generalized Cauchy spaces", Mathematische Nachrichten, 147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.
3. Pierce, Benjamin C. (2002). Types and Programming Languages. Cambridge, Massachusetts/London, England: The MIT Press. pp. 182ff. ISBN 0-262-16209-1.
4. Robinson, J. A. (1965). "A machine-oriented logic based on the resolution principle". ACM. 12 (1): 23–41. doi:10.1145/321250.321253. S2CID 14389185.
5. Kunen, Kenneth (1980), Set Theory, An Introduction to Independence Proofs, Studies in logic and the foundation of mathematics, vol. 102, Amsterdam, the Netherlands: Elsevier.
6. In this context, "$\backslash $" does not mean "set difference".
References
• Schmidt, Gunther, "Relational Mathematics", Encyclopedia of Mathematics and its Applications, vol. 132, Cambridge University Press, 2011, ISBN 978-0-521-76268-7
• Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9
Order theory
• Topics
• Glossary
• Category
Key concepts
• Binary relation
• Boolean algebra
• Cyclic order
• Lattice
• Partial order
• Preorder
• Total order
• Weak ordering
Results
• Boolean prime ideal theorem
• Cantor–Bernstein theorem
• Cantor's isomorphism theorem
• Dilworth's theorem
• Dushnik–Miller theorem
• Hausdorff maximal principle
• Knaster–Tarski theorem
• Kruskal's tree theorem
• Laver's theorem
• Mirsky's theorem
• Szpilrajn extension theorem
• Zorn's lemma
Properties & Types (list)
• Antisymmetric
• Asymmetric
• Boolean algebra
• topics
• Completeness
• Connected
• Covering
• Dense
• Directed
• (Partial) Equivalence
• Foundational
• Heyting algebra
• Homogeneous
• Idempotent
• Lattice
• Bounded
• Complemented
• Complete
• Distributive
• Join and meet
• Reflexive
• Partial order
• Chain-complete
• Graded
• Eulerian
• Strict
• Prefix order
• Preorder
• Total
• Semilattice
• Semiorder
• Symmetric
• Total
• Tolerance
• Transitive
• Well-founded
• Well-quasi-ordering (Better)
• (Pre) Well-order
Constructions
• Composition
• Converse/Transpose
• Lexicographic order
• Linear extension
• Product order
• Reflexive closure
• Series-parallel partial order
• Star product
• Symmetric closure
• Transitive closure
Topology & Orders
• Alexandrov topology & Specialization preorder
• Ordered topological vector space
• Normal cone
• Order topology
• Order topology
• Topological vector lattice
• Banach
• Fréchet
• Locally convex
• Normed
Related
• Antichain
• Cofinal
• Cofinality
• Comparability
• Graph
• Duality
• Filter
• Hasse diagram
• Ideal
• Net
• Subnet
• Order morphism
• Embedding
• Isomorphism
• Order type
• Ordered field
• Ordered vector space
• Partially ordered
• Positive cone
• Riesz space
• Upper set
• Young's lattice
|
Wikipedia
|
Low-discrepancy sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of an equidistributed sequence. Specific definitions of discrepancy differ regarding the choice of B (hyperspheres, hypercubes, etc.) and how the discrepancy for every B is computed (usually normalized) and combined (usually by taking the worst value).
Low-discrepancy sequences are also called quasirandom sequences, due to their common use as a replacement of uniformly distributed random numbers. The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither random nor pseudorandom, but such sequences share some properties of random variables and in certain applications such as the quasi-Monte Carlo method their lower discrepancy is an important advantage.
Applications
Quasirandom numbers have an advantage over pure random numbers in that they cover the domain of interest quickly and evenly.
Two useful applications are in finding the characteristic function of a probability density function, and in finding the derivative function of a deterministic function with a small amount of noise. Quasirandom numbers allow higher-order moments to be calculated to high accuracy very quickly.
Applications that don't involve sorting would be in finding the mean, standard deviation, skewness and kurtosis of a statistical distribution, and in finding the integral and global maxima and minima of difficult deterministic functions. Quasirandom numbers can also be used for providing starting points for deterministic algorithms that only work locally, such as Newton–Raphson iteration.
Quasirandom numbers can also be combined with search algorithms. A binary tree Quicksort-style algorithm ought to work exceptionally well because quasirandom numbers flatten the tree far better than random numbers, and the flatter the tree the faster the sorting. With a search algorithm, quasirandom numbers can be used to find the mode, median, confidence intervals and cumulative distribution of a statistical distribution, and all local minima and all solutions of deterministic functions.
Low-discrepancy sequences in numerical integration
Various methods of numerical integration can be phrased as approximating the integral of a function f in some interval, e.g. [0,1], as the average of the function evaluated at a set {x1, ..., xN} in that interval:
$\int _{0}^{1}f(u)\,du\approx {\frac {1}{N}}\,\sum _{i=1}^{N}f(x_{i}).$
If the points are chosen as xi = i/N, this is the rectangle rule. If the points are chosen to be randomly (or pseudorandomly) distributed, this is the Monte Carlo method. If the points are chosen as elements of a low-discrepancy sequence, this is the quasi-Monte Carlo method. A remarkable result, the Koksma–Hlawka inequality (stated below), shows that the error of such a method can be bounded by the product of two terms, one of which depends only on f, and the other one is the discrepancy of the set {x1, ..., xN}.
It is convenient to construct the set {x1, ..., xN} in such a way that if a set with N+1 elements is constructed, the previous N elements need not be recomputed. The rectangle rule uses points set which have low discrepancy, but in general the elements must be recomputed if N is increased. Elements need not be recomputed in the random Monte Carlo method if N is increased, but the point sets do not have minimal discrepancy. By using low-discrepancy sequences we aim for low discrepancy and no need for recomputations, but actually low-discrepancy sequences can only be incrementally good on discrepancy if we allow no recomputation.
Definition of discrepancy
The discrepancy of a set P = {x1, ..., xN} is defined, using Niederreiter's notation, as
$D_{N}(P)=\sup _{B\in J}\left|{\frac {A(B;P)}{N}}-\lambda _{s}(B)\right|$
where λs is the s-dimensional Lebesgue measure, A(B;P) is the number of points in P that fall into B, and J is the set of s-dimensional intervals or boxes of the form
$\prod _{i=1}^{s}[a_{i},b_{i})=\{\mathbf {x} \in \mathbf {R} ^{s}:a_{i}\leq x_{i}<b_{i}\}\,$
where $0\leq a_{i}<b_{i}\leq 1$.
The star-discrepancy D*N(P) is defined similarly, except that the supremum is taken over the set J* of rectangular boxes of the form
$\prod _{i=1}^{s}[0,u_{i})$
where ui is in the half-open interval [0, 1).
The two are related by
$D_{N}^{*}\leq D_{N}\leq 2^{s}D_{N}^{*}.\,$
Note: With these definitions, discrepancy represents the worst-case or maximum point density deviation of a uniform set. However, also other error measures are meaningful, leading to other definitions and variation measures. For instance, L2 discrepancy or modified centered L2 discrepancy are also used intensively to compare the quality of uniform point sets. Both are much easier to calculate for large N and s.
The Koksma–Hlawka inequality
Let Īs be the s-dimensional unit cube, Īs = [0, 1] × ... × [0, 1]. Let f have bounded variation V(f) on Īs in the sense of Hardy and Krause. Then for any x1, ..., xN in Is = [0, 1) × ... × [0, 1),
$\left|{\frac {1}{N}}\sum _{i=1}^{N}f(x_{i})-\int _{{\bar {I}}^{s}}f(u)\,du\right|\leq V(f)\,D_{N}^{*}(x_{1},\ldots ,x_{N}).$
The Koksma–Hlawka inequality is sharp in the following sense: For any point set {x1,...,xN} in Is and any $\varepsilon >0$, there is a function f with bounded variation and V(f) = 1 such that
$\left|{\frac {1}{N}}\sum _{i=1}^{N}f(x_{i})-\int _{{\bar {I}}^{s}}f(u)\,du\right|>D_{N}^{*}(x_{1},\ldots ,x_{N})-\varepsilon .$
Therefore, the quality of a numerical integration rule depends only on the discrepancy D*N(x1,...,xN).
The formula of Hlawka–Zaremba
Let $D=\{1,2,\ldots ,d\}$. For $\emptyset \neq u\subseteq D$ we write
$dx_{u}:=\prod _{j\in u}dx_{j}$
and denote by $(x_{u},1)$ the point obtained from x by replacing the coordinates not in u by $1$. Then
${\frac {1}{N}}\sum _{i=1}^{N}f(x_{i})-\int _{{\bar {I}}^{s}}f(u)\,du=\sum _{\emptyset \neq u\subseteq D}(-1)^{|u|}\int _{[0,1]^{|u|}}\operatorname {disc} (x_{u},1){\frac {\partial ^{|u|}}{\partial x_{u}}}f(x_{u},1)\,dx_{u},$
where $\operatorname {disc} (z)={\frac {1}{N}}\sum _{i=1}^{N}\prod _{j=1}^{d}1_{[0,z_{j})}(x_{i,j})-\prod _{j=1}^{d}z_{i}$ is the discrepancy function.
The L² version of the Koksma–Hlawka inequality
Applying the Cauchy–Schwarz inequality for integrals and sums to the Hlawka–Zaremba identity, we obtain an $L^{2}$ version of the Koksma–Hlawka inequality:
$\left|{\frac {1}{N}}\sum _{i=1}^{N}f(x_{i})-\int _{{\bar {I}}^{s}}f(u)\,du\right|\leq \|f\|_{d}\operatorname {disc} _{d}(\{t_{i}\}),$
where
$\operatorname {disc} _{d}(\{t_{i}\})=\left(\sum _{\emptyset \neq u\subseteq D}\int _{[0,1]^{|u|}}\operatorname {disc} (x_{u},1)^{2}\,dx_{u}\right)^{1/2}$
and
$\|f\|_{d}=\left(\sum _{u\subseteq D}\int _{[0,1]^{|u|}}\left|{\frac {\partial ^{|u|}}{\partial x_{u}}}f(x_{u},1)\right|^{2}dx_{u}\right)^{1/2}.$
$L^{2}$ discrepancy has a high practical importance because fast explicit calculations are possible for a given point set. This way it is easy to create point set optimizers using $L^{2}$ discrepancy as criteria.
The Erdős–Turán–Koksma inequality
It is computationally hard to find the exact value of the discrepancy of large point sets. The Erdős–Turán–Koksma inequality provides an upper bound.
Let x1,...,xN be points in Is and H be an arbitrary positive integer. Then
$D_{N}^{*}(x_{1},\ldots ,x_{N})\leq \left({\frac {3}{2}}\right)^{s}\left({\frac {2}{H+1}}+\sum _{0<\|h\|_{\infty }\leq H}{\frac {1}{r(h)}}\left|{\frac {1}{N}}\sum _{n=1}^{N}e^{2\pi i\langle h,x_{n}\rangle }\right|\right)$
where
$r(h)=\prod _{i=1}^{s}\max\{1,|h_{i}|\}\quad {\mbox{for}}\quad h=(h_{1},\ldots ,h_{s})\in \mathbb {Z} ^{s}.$
The main conjectures
Conjecture 1. There is a constant cs depending only on the dimension s, such that
$D_{N}^{*}(x_{1},\ldots ,x_{N})\geq c_{s}{\frac {(\ln N)^{s-1}}{N}}$
for any finite point set {x1,...,xN}.
Conjecture 2. There is a constant c's depending only on s, such that
$D_{N}^{*}(x_{1},\ldots ,x_{N})\geq c'_{s}{\frac {(\ln N)^{s}}{N}}$
for infinite number of N for any infinite sequence x1,x2,x3,....
These conjectures are equivalent. They have been proved for s ≤ 2 by W. M. Schmidt. In higher dimensions, the corresponding problem is still open. The best-known lower bounds are due to Michael Lacey and collaborators.
Lower bounds
Let s = 1. Then
$D_{N}^{*}(x_{1},\ldots ,x_{N})\geq {\frac {1}{2N}}$
for any finite point set {x1, ..., xN}.
Let s = 2. W. M. Schmidt proved that for any finite point set {x1, ..., xN},
$D_{N}^{*}(x_{1},\ldots ,x_{N})\geq C{\frac {\log N}{N}}$
where
$C=\max _{a\geq 3}{\frac {1}{16}}{\frac {a-2}{a\log a}}=0.023335\dots .$
For arbitrary dimensions s > 1, K. F. Roth proved that
$D_{N}^{*}(x_{1},\ldots ,x_{N})\geq {\frac {1}{2^{4s}}}{\frac {1}{((s-1)\log 2)^{\frac {s-1}{2}}}}{\frac {\log ^{\frac {s-1}{2}}N}{N}}$
for any finite point set {x1, ..., xN}. Jozef Beck [1] established a double log improvement of this result in three dimensions. This was improved by D. Bilyk and M. T. Lacey to a power of a single logarithm. The best known bound for s > 2 is due D. Bilyk and M. T. Lacey and A. Vagharshakyan.[2] For s > 2 there is a t > 0 so that
$D_{N}^{*}(x_{1},\ldots ,x_{N})\geq t{\frac {\log ^{{\frac {s-1}{2}}+t}N}{N}}$
for any finite point set {x1, ..., xN}.
Construction of low-discrepancy sequences
Because any distribution of random numbers can be mapped onto a uniform distribution, and quasirandom numbers are mapped in the same way, this article only concerns generation of quasirandom numbers on a multidimensional uniform distribution.
There are constructions of sequences known such that
$D_{N}^{*}(x_{1},\ldots ,x_{N})\leq C{\frac {(\ln N)^{s}}{N}}.$
where C is a certain constant, depending on the sequence. After Conjecture 2, these sequences are believed to have the best possible order of convergence. Examples below are the van der Corput sequence, the Halton sequences, and the Sobol’ sequences. One general limitation is that construction methods can usually only guarantee the order of convergence. Practically, low discrepancy can be only achieved if N is large enough, and for large given s this minimum N can be very large. This means running a Monte-Carlo analysis with e.g. s=20 variables and N=1000 points from a low-discrepancy sequence generator may offer only a very minor accuracy improvement .
Random numbers
Sequences of quasirandom numbers can be generated from random numbers by imposing a negative correlation on those random numbers. One way to do this is to start with a set of random numbers $r_{i}$ on $[0,0.5)$ and construct quasirandom numbers $s_{i}$ which are uniform on $[0,1)$ using:
$s_{i}=r_{i}$ for $i$ odd and $s_{i}=0.5+r_{i}$ for $i$ even.
A second way to do it with the starting random numbers is to construct a random walk with offset 0.5 as in:
$s_{i}=s_{i-1}+0.5+r_{i}{\pmod {1}}.\,$
That is, take the previous quasirandom number, add 0.5 and the random number, and take the result modulo 1.
For more than one dimension, Latin squares of the appropriate dimension can be used to provide offsets to ensure that the whole domain is covered evenly.
Additive recurrence
For any irrational $\alpha $, the sequence
$s_{n}=\{s_{0}+n\alpha \}$
has discrepancy tending to $1/N$. Note that the sequence can be defined recursively by
$s_{n+1}=(s_{n}+\alpha ){\bmod {1}}\;.$
A good value of $\alpha $ gives lower discrepancy than a sequence of independent uniform random numbers.
The discrepancy can be bounded by the approximation exponent of $\alpha $. If the approximation exponent is $\mu $, then for any $\varepsilon >0$, the following bound holds:[3]
$D_{N}((s_{n}))=O_{\varepsilon }(N^{-1/(\mu -1)+\varepsilon }).$
By the Thue–Siegel–Roth theorem, the approximation exponent of any irrational algebraic number is 2, giving a bound of $N^{-1+\varepsilon }$ above.
The recurrence relation above is similar to the recurrence relation used by a linear congruential generator, a poor-quality pseudorandom number generator:[4]
$r_{i}=(ar_{i-1}+c){\bmod {m}}$
For the low discrepancy additive recurrence above, a and m are chosen to be 1. Note, however, that this will not generate independent random numbers, so should not be used for purposes requiring independence.
The value of $c$ with lowest discrepancy is the fractional part of the golden ratio:[5]
$c={\frac {{\sqrt {5}}-1}{2}}=\varphi -1\approx 0.618034.$
Another value that is nearly as good is the fractional part of the silver ratio, which is the fractional part of the square root of 2:
$c={\sqrt {2}}-1\approx 0.414214.\,$
In more than one dimension, separate quasirandom numbers are needed for each dimension. A convenient set of values that are used, is the square roots of primes from two up, all taken modulo 1:
$c={\sqrt {2}},{\sqrt {3}},{\sqrt {5}},{\sqrt {7}},{\sqrt {11}},\ldots \,$
However, a set of values based on the generalised golden ratio has been shown to produce more evenly distributed points. [6]
The list of pseudorandom number generators lists methods for generating independent pseudorandom numbers. Note: In few dimensions, recursive recurrence leads to uniform sets of good quality, but for larger s (like s>8) other point set generators can offer much lower discrepancies.
van der Corput sequence
Let
$n=\sum _{k=0}^{L-1}d_{k}(n)b^{k}$
be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ dk(n) < b. Set
$g_{b}(n)=\sum _{k=0}^{L-1}d_{k}(n)b^{-k-1}.$
Then there is a constant C depending only on b such that (gb(n))n ≥ 1satisfies
$D_{N}^{*}(g_{b}(1),\dots ,g_{b}(N))\leq C{\frac {\log N}{N}},$
where D*N is the star discrepancy.
Halton sequence
The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let s be an arbitrary dimension and b1, ..., bs be arbitrary coprime integers greater than 1. Define
$x(n)=(g_{b_{1}}(n),\dots ,g_{b_{s}}(n)).$
Then there is a constant C depending only on b1, ..., bs, such that sequence {x(n)}n≥1 is a s-dimensional sequence with
$D_{N}^{*}(x(1),\dots ,x(N))\leq C'{\frac {(\log N)^{s}}{N}}.$
Hammersley set
Let b1,...,bs−1 be coprime positive integers greater than 1. For given s and N, the s-dimensional Hammersley set of size N is defined by[7]
$x(n)=\left(g_{b_{1}}(n),\dots ,g_{b_{s-1}}(n),{\frac {n}{N}}\right)$
for n = 1, ..., N. Then
$D_{N}^{*}(x(1),\dots ,x(N))\leq C{\frac {(\log N)^{s-1}}{N}}$
where C is a constant depending only on b1, ..., bs−1. Note: The formulas show that the Hammersley set is actually the Halton sequence, but we get one more dimension for free by adding a linear sweep. This is only possible if N is known upfront. A linear set is also the set with lowest possible one-dimensional discrepancy in general. Unfortunately, for higher dimensions, no such "discrepancy record sets" are known. For s = 2, most low-discrepancy point set generators deliver at least near-optimum discrepancies.
Sobol’ sequence
The Antonov–Saleev variant of the Sobol’ sequence generates numbers between zero and one directly as binary fractions of length $w$, from a set of $w$ special binary fractions, $V_{i},i=1,2,\dots ,w$ called direction numbers. The bits of the Gray code of $i$, $G(i)$, are used to select direction numbers. To get the Sobol’ sequence value $s_{i}$ take the exclusive or of the binary value of the Gray code of $i$ with the appropriate direction number. The number of dimensions required affects the choice of $V_{i}$.
Poisson disk sampling
Poisson disk sampling is popular in video games to rapidly place objects in a way that appears random-looking but guarantees that every two points are separated by at least the specified minimum distance.[8] This does not guarantee low discrepancy (as e. g. Sobol’), but at least a significantly lower discrepancy than pure random sampling. The goal of these sampling patterns is based on frequency analysis rather than discrepancy, a type of so-called "blue noise" patterns.
Graphical examples
The points plotted below are the first 100, 1000, and 10000 elements in a sequence of the Sobol' type. For comparison, 10000 elements of a sequence of pseudorandom points are also shown. The low-discrepancy sequence was generated by TOMS algorithm 659.[9] An implementation of the algorithm in Fortran is available from Netlib.
See also
• Discrepancy theory
• Markov chain Monte Carlo
• Quasi-Monte Carlo method
• Sparse grid
• Systematic sampling
Notes
1. Beck, József (1989). "A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution". Compositio Mathematica. 72 (3): 269–339. MR 1032337. S2CID 125940424. Zbl 0691.10041.
2. Bilyk, Dmitriy; Lacey, Michael T.; Vagharshakyan, Armen (2008). "On the Small Ball Inequality in all dimensions". Journal of Functional Analysis. 254 (9): 2470–2502. doi:10.1016/j.jfa.2007.09.010. S2CID 14234006.
3. Kuipers & Niederreiter 2005, p. 123
4. Knuth, Donald E. "Chapter 3 – Random Numbers". The Art of Computer Programming. Vol. 2.
5. Skarupke, Malte (16 June 2018). "Fibonacci Hashing: The Optimization that the World Forgot". One property of the Golden Ratio is that you can use it to subdivide any range roughly evenly ... if you don't know ahead of time how many steps you're going to take
6. Roberts, Martin (2018). "The Unreasonable Effectiveness of Quasirandom Sequences".
7. Hammersley, J. M.; Handscomb, D. C. (1964). Monte Carlo Methods. doi:10.1007/978-94-009-5819-7. ISBN 978-94-009-5821-0.
8. Herman Tulleken. Tulleken, Herman (March 2008). "Poisson Disk Sampling". Dev.Mag. No. 21. pp. 21–25.
9. Bratley, Paul; Fox, Bennett L. (1988). "Algorithm 659". ACM Transactions on Mathematical Software. 14: 88–100. doi:10.1145/42288.214372. S2CID 17325779.
References
• Dick, Josef; Pillichshammer, Friedrich (2010). Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. ISBN 978-0-521-19159-3.
• Kuipers, L.; Niederreiter, H. (2005), Uniform distribution of sequences, Dover Publications, ISBN 0-486-45019-8
• Harald Niederreiter (1992). Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics. ISBN 0-89871-295-5.
• Drmota, Michael; Tichy, Robert F. (1997). Sequences, Discrepancies and Applications. Lecture Notes in Math. Vol. 1651. Springer. ISBN 3-540-62606-9.
• Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992). Numerical Recipes in C (2nd ed.). Cambridge University Press. see Section 7.7 for a less technical discussion of low-discrepancy sequences. ISBN 0-521-43108-5.
External links
• Collected Algorithms of the ACM (See algorithms 647, 659, and 738.)
• Quasi-Random Sequences from the GNU Scientific Library
• Quasi-random sampling subject to constraints at FinancialMathematics.Com
• C++ generator of Sobol’ sequence
• SciPy QMC API Reference: scipy.stats.qmc
|
Wikipedia
|
Quasi-relative interior
In topology, a branch of mathematics, the quasi-relative interior of a subset of a vector space is a refinement of the concept of the interior. Formally, if $X$ is a linear space then the quasi-relative interior of $A\subseteq X$ is
$\operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}$
where $\operatorname {\overline {cone}} (\cdot )$ denotes the closure of the conic hull.[1]
Let $X$ is a normed vector space, if $C\subseteq X$ is a convex finite-dimensional set then $\operatorname {qri} (C)=\operatorname {ri} (C)$ such that $\operatorname {ri} $ is the relative interior.[2]
See also
• Interior (topology) – Largest open subset of some given set
• Relative interior – Generalization of topological interior
• Algebraic interior – Generalization of topological interior
References
1. Zălinescu 2002, pp. 2–3.
2. Borwein, J.M.; Lewis, A.S. (1992). "Partially finite convex programming, Part I: Quasi relative interiors and duality theory" (pdf). Mathematical Programming. 57: 15–48. doi:10.1007/bf01581072. Retrieved October 19, 2011.
• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
Convex analysis and variational analysis
Basic concepts
• Convex combination
• Convex function
• Convex set
Topics (list)
• Choquet theory
• Convex geometry
• Convex metric space
• Convex optimization
• Duality
• Lagrange multiplier
• Legendre transformation
• Locally convex topological vector space
• Simplex
Maps
• Convex conjugate
• Concave
• (Closed
• K-
• Logarithmically
• Proper
• Pseudo-
• Quasi-) Convex function
• Invex function
• Legendre transformation
• Semi-continuity
• Subderivative
Main results (list)
• Carathéodory's theorem
• Ekeland's variational principle
• Fenchel–Moreau theorem
• Fenchel-Young inequality
• Jensen's inequality
• Hermite–Hadamard inequality
• Krein–Milman theorem
• Mazur's lemma
• Shapley–Folkman lemma
• Robinson-Ursescu
• Simons
• Ursescu
Sets
• Convex hull
• (Orthogonally, Pseudo-) Convex set
• Effective domain
• Epigraph
• Hypograph
• John ellipsoid
• Lens
• Radial set/Algebraic interior
• Zonotope
Series
• Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality
• Dual system
• Duality gap
• Strong duality
• Weak duality
Applications and related
• Convexity in economics
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
• Metrizability
• Operator topologies
• Topological vector space
• Vector space
Main results
• Anderson–Kadec
• Banach–Alaoglu
• Closed graph theorem
• F. Riesz's
• Hahn–Banach (hyperplane separation
• Vector-valued Hahn–Banach)
• Open mapping (Banach–Schauder)
• Bounded inverse
• Uniform boundedness (Banach–Steinhaus)
Maps
• Bilinear operator
• form
• Linear map
• Almost open
• Bounded
• Continuous
• Closed
• Compact
• Densely defined
• Discontinuous
• Topological homomorphism
• Functional
• Linear
• Bilinear
• Sesquilinear
• Norm
• Seminorm
• Sublinear function
• Transpose
Types of sets
• Absolutely convex/disk
• Absorbing/Radial
• Affine
• Balanced/Circled
• Banach disks
• Bounding points
• Bounded
• Complemented subspace
• Convex
• Convex cone (subset)
• Linear cone (subset)
• Extreme point
• Pre-compact/Totally bounded
• Prevalent/Shy
• Radial
• Radially convex/Star-shaped
• Symmetric
Set operations
• Affine hull
• (Relative) Algebraic interior (core)
• Convex hull
• Linear span
• Minkowski addition
• Polar
• (Quasi) Relative interior
Types of TVSs
• Asplund
• B-complete/Ptak
• Banach
• (Countably) Barrelled
• BK-space
• (Ultra-) Bornological
• Brauner
• Complete
• Convenient
• (DF)-space
• Distinguished
• F-space
• FK-AK space
• FK-space
• Fréchet
• tame Fréchet
• Grothendieck
• Hilbert
• Infrabarreled
• Interpolation space
• K-space
• LB-space
• LF-space
• Locally convex space
• Mackey
• (Pseudo)Metrizable
• Montel
• Quasibarrelled
• Quasi-complete
• Quasinormed
• (Polynomially
• Semi-) Reflexive
• Riesz
• Schwartz
• Semi-complete
• Smith
• Stereotype
• (B
• Strictly
• Uniformly) convex
• (Quasi-) Ultrabarrelled
• Uniformly smooth
• Webbed
• With the approximation property
• Mathematics portal
• Category
• Commons
|
Wikipedia
|
Quasireversibility
In queueing theory, a discipline within the mathematical theory of probability, quasireversibility (sometimes QR) is a property of some queues. The concept was first identified by Richard R. Muntz[1] and further developed by Frank Kelly.[2][3] Quasireversibility differs from reversibility in that a stronger condition is imposed on arrival rates and a weaker condition is applied on probability fluxes. For example, an M/M/1 queue with state-dependent arrival rates and state-dependent service times is reversible, but not quasireversible.[4]
A network of queues, such that each individual queue when considered in isolation is quasireversible, always has a product form stationary distribution.[5] Quasireversibility had been conjectured to be a necessary condition for a product form solution in a queueing network, but this was shown not to be the case. Chao et al. exhibited a product form network where quasireversibility was not satisfied.[6]
Definition
A queue with stationary distribution $\pi $ is quasireversible if its state at time t, x(t) is independent of
• the arrival times for each class of customer subsequent to time t,
• the departure times for each class of customer prior to time t
for all classes of customer.[7]
Partial balance formulation
Quasireversibility is equivalent to a particular form of partial balance. First, define the reversed rates q'(x,x') by
$\pi (\mathbf {x} )q'(\mathbf {x} ,\mathbf {x'} )=\pi (\mathbf {x'} )q(\mathbf {x'} ,\mathbf {x} )$
then considering just customers of a particular class, the arrival and departure processes are the same Poisson process (with parameter $\alpha $), so
$\alpha =\sum _{\mathbf {x'} \in M_{\mathbf {x} }}q(\mathbf {x} ,\mathbf {x'} )=\sum _{\mathbf {x'} \in M_{\mathbf {x} }}q'(\mathbf {x} ,\mathbf {x'} )$
where Mx is a set such that $\scriptstyle {\mathbf {x'} \in M_{\mathbf {x} }}$ means the state x' represents a single arrival of the particular class of customer to state x.
Examples
• Burke's theorem shows that an M/M/m queueing system is quasireversible.[8][9][10]
• Kelly showed that each station of a BCMP network is quasireversible when viewed in isolation.[11]
• G-queues in G-networks are quasireversible.[12]
See also
• Time reversibility
References
1. Muntz, R.R. (1972). Poisson departure process and queueing networks (IBM Research Report RC 4145) (Technical report). Yorktown Heights, N.Y.: IBM Thomas J. Watson Research Center.
2. Kelly, F. P. (1975). "Networks of Queues with Customers of Different Types". Journal of Applied Probability. 12 (3): 542–554. doi:10.2307/3212869. JSTOR 3212869. S2CID 51917794.
3. Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability. 8 (2): 416–432. doi:10.2307/1425912. JSTOR 1425912. S2CID 204177645.
4. Harrison, Peter G.; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. p. 288. ISBN 0-201-54419-9.
5. Kelly, F.P. (1982). Networks of quasireversible nodes. In Applied Probability and Computer Science: The Interface (Ralph L. Disney and Teunis J. Ott, editors.) 1 3-29. Birkhäuser, Boston
6. Chao, X.; Miyazawa, M.; Serfozo, R. F.; Takada, H. (1998). "Markov network processes with product form stationary distributions". Queueing Systems. 28 (4): 377. doi:10.1023/A:1019115626557. S2CID 14471818.
7. Kelly, F.P., Reversibility and Stochastic Networks, 1978 pages 66-67
8. Burke, P. J. (1956). "The Output of a Queuing System". Operations Research. 4 (6): 699–704. doi:10.1287/opre.4.6.699. S2CID 55089958.
9. Burke, P. J. (1968). "The Output Process of a Stationary M/M/s Queueing System". The Annals of Mathematical Statistics. 39 (4): 1144–1152. doi:10.1214/aoms/1177698238.
10. O'Connell, N.; Yor, M. (December 2001). "Brownian analogues of Burke's theorem". Stochastic Processes and Their Applications. 96 (2): 285–298. doi:10.1016/S0304-4149(01)00119-3.
11. Kelly, F.P. (1979). Reversibility and Stochastic Networks. New York: Wiley.
12. Dao-Thi, T. H.; Mairesse, J. (2005). "Zero-Automatic Queues". Formal Techniques for Computer Systems and Business Processes. Lecture Notes in Computer Science. Vol. 3670. p. 64. doi:10.1007/11549970_6. ISBN 978-3-540-28701-8.
Queueing theory
Single queueing nodes
• D/M/1 queue
• M/D/1 queue
• M/D/c queue
• M/M/1 queue
• Burke's theorem
• M/M/c queue
• M/M/∞ queue
• M/G/1 queue
• Pollaczek–Khinchine formula
• Matrix analytic method
• M/G/k queue
• G/M/1 queue
• G/G/1 queue
• Kingman's formula
• Lindley equation
• Fork–join queue
• Bulk queue
Arrival processes
• Poisson point process
• Markovian arrival process
• Rational arrival process
Queueing networks
• Jackson network
• Traffic equations
• Gordon–Newell theorem
• Mean value analysis
• Buzen's algorithm
• Kelly network
• G-network
• BCMP network
Service policies
• FIFO
• LIFO
• Processor sharing
• Round-robin
• Shortest job next
• Shortest remaining time
Key concepts
• Continuous-time Markov chain
• Kendall's notation
• Little's law
• Product-form solution
• Balance equation
• Quasireversibility
• Flow-equivalent server method
• Arrival theorem
• Decomposition method
• Beneš method
Limit theorems
• Fluid limit
• Mean-field theory
• Heavy traffic approximation
• Reflected Brownian motion
Extensions
• Fluid queue
• Layered queueing network
• Polling system
• Adversarial queueing network
• Loss network
• Retrial queue
Information systems
• Data buffer
• Erlang (unit)
• Erlang distribution
• Flow control (data)
• Message queue
• Network congestion
• Network scheduler
• Pipeline (software)
• Quality of service
• Scheduling (computing)
• Teletraffic engineering
Category
|
Wikipedia
|
Nonconvex great rhombicosidodecahedron
In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices.[1] It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol rr{5⁄3,3}. Its vertex figure is a crossed quadrilateral.
Nonconvex great rhombicosidodecahedron
TypeUniform star polyhedron
ElementsF = 62, E = 120
V = 60 (χ = 2)
Faces by sides20{3}+30{4}+12{5/2}
Coxeter diagram
Wythoff symbol5/3 3 | 2
5/2 3/2 | 2
Symmetry groupIh, [5,3], *532
Index referencesU67, C84, W105
Dual polyhedronGreat deltoidal hexecontahedron
Vertex figure
3.4.5/3.4
Bowers acronymQrid
This model shares the name with the convex great rhombicosidodecahedron, also known as the truncated icosidodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a nonconvex great rhombicosidodecahedron are all the even permutations of
(±1/τ2, 0, ±(2−1/τ))
(±1, ±1/τ3, ±1)
(±1/τ, ±1/τ2, ±2/τ)
where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
Related polyhedra
It shares its vertex arrangement with the truncated great dodecahedron, and with the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the great dodecicosidodecahedron (having the triangular and pentagrammic faces in common), and the great rhombidodecahedron (having the square faces in common).
Nonconvex great rhombicosidodecahedron
Great dodecicosidodecahedron
Great rhombidodecahedron
Truncated great dodecahedron
Compound of six pentagonal prisms
Compound of twelve pentagonal prisms
Great deltoidal hexecontahedron
Great deltoidal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 62 (χ = 2)
Symmetry groupIh, [5,3], *532
Index referencesDU67
dual polyhedronNonconvex great rhombicosidodecahedron
The great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices.
It is also called a great strombic hexecontahedron.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "67: great rhombicosidodecahedron". MathConsult.
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Great rhombicosidodecahedron". MathWorld.
• Weisstein, Eric W. "Great deltoidal hexecontahedron". MathWorld.
• Uniform polyhedra and duals
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
|
Wikipedia
|
Quasi-bialgebra
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element $\Phi $ which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category.
Definition
A quasi-bialgebra ${\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi ,l,r)$ is an algebra ${\mathcal {A}}$ over a field $\mathbb {F} $ equipped with morphisms of algebras
$\Delta :{\mathcal {A}}\rightarrow {\mathcal {A\otimes A}}$ :{\mathcal {A}}\rightarrow {\mathcal {A\otimes A}}}
$\varepsilon :{\mathcal {A}}\rightarrow \mathbb {F} $ :{\mathcal {A}}\rightarrow \mathbb {F} }
along with invertible elements $\Phi \in {\mathcal {A\otimes A\otimes A}}$, and $r,l\in A$ such that the following identities hold:
$(id\otimes \Delta )\circ \Delta (a)=\Phi \lbrack (\Delta \otimes id)\circ \Delta (a)\rbrack \Phi ^{-1},\quad \forall a\in {\mathcal {A}}$
$\lbrack (id\otimes id\otimes \Delta )(\Phi )\rbrack \ \lbrack (\Delta \otimes id\otimes id)(\Phi )\rbrack =(1\otimes \Phi )\ \lbrack (id\otimes \Delta \otimes id)(\Phi )\rbrack \ (\Phi \otimes 1)$
$(\varepsilon \otimes id)(\Delta a)=l^{-1}al,\qquad (id\otimes \varepsilon )\circ \Delta =r^{-1}ar,\quad \forall a\in {\mathcal {A}}$
$(id\otimes \varepsilon \otimes id)(\Phi )=r\otimes l^{-1}.$
Where $\Delta $ and $\epsilon $ are called the comultiplication and counit, $r$ and $l$ are called the right and left unit constraints (resp.), and $\Phi $ is sometimes called the Drinfeld associator.[1]: 369–376 This definition is constructed so that the category ${\mathcal {A}}-Mod$ is a tensor category under the usual vector space tensor product, and in fact this can be taken as the definition instead of the list of above identities.[1]: 368 Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. $l=r=1$ the definition may sometimes be given with this assumed.[1]: 370 Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: $l=r=1$ and $\Phi =1\otimes 1\otimes 1$.
Braided quasi-bialgebras
A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category ${\mathcal {A}}-Mod$ is braided. Equivalently, by analogy with braided bialgebras, we can construct a notion of a universal R-matrix which controls the non-cocommutativity of a quasi-bialgebra. The definition is the same as in the braided bialgebra case except for additional complications in the formulas caused by adding in the associator.
Proposition: A quasi-bialgebra $({\mathcal {A}},\Delta ,\epsilon ,\Phi ,l,r)$ is braided if it has a universal R-matrix, ie an invertible element $R\in {\mathcal {A\otimes A}}$ such that the following 3 identities hold:
$(\Delta ^{op})(a)=R\Delta (a)R^{-1}$
$(id\otimes \Delta )(R)=(\Phi _{231})^{-1}R_{13}\Phi _{213}R_{12}(\Phi _{213})^{-1}$
$(\Delta \otimes id)(R)=(\Phi _{321})R_{13}(\Phi _{213})^{-1}R_{23}\Phi _{123}$
Where, for every $a_{1}\otimes ...\otimes a_{k}\in {\mathcal {A}}^{\otimes k}$, $a_{i_{1}i_{2}...i_{n}}$ is the monomial with $a_{j}$ in the $i_{j}$th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of ${\mathcal {A}}^{\otimes k}$.[1]: 371
Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang–Baxter equation:
$R_{12}\Phi _{321}R_{13}(\Phi _{132})^{-1}R_{23}\Phi _{123}=\Phi _{321}R_{23}(\Phi _{231})^{-1}R_{13}\Phi _{213}R_{12}$[1]: 372
Twisting
Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume $r=l=1$) .
If ${\mathcal {B_{A}}}$ is a quasi-bialgebra and $F\in {\mathcal {A\otimes A}}$ is an invertible element such that $(\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1$, set
$\Delta '(a)=F\Delta (a)F^{-1},\quad \forall a\in {\mathcal {A}}$
$\Phi '=(1\otimes F)\ ((id\otimes \Delta )F)\ \Phi \ ((\Delta \otimes id)F^{-1})\ (F^{-1}\otimes 1).$
Then, the set $({\mathcal {A}},\Delta ',\varepsilon ,\Phi ')$ is also a quasi-bialgebra obtained by twisting ${\mathcal {B_{A}}}$ by F, which is called a twist or gauge transformation.[1]: 373 If $({\mathcal {A}},\Delta ,\varepsilon ,\Phi )$ was a braided quasi-bialgebra with universal R-matrix $R$ , then so is $({\mathcal {A}},\Delta ',\varepsilon ,\Phi ')$ with universal R-matrix $F_{21}RF^{-1}$ (using the notation from the above section).[1]: 376 However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by $F_{1}$ and then $F_{2}$ is equivalent to twisting by $F_{2}F_{1}$, and twisting by $F$ then $F^{-1}$ recovers the original quasi-bialgebra.
Twistings have the important property that they induce categorical equivalences on the tensor category of modules:
Theorem: Let ${\mathcal {B_{A}}}$, ${\mathcal {B_{A'}}}$ be quasi-bialgebras, let ${\mathcal {B'_{A'}}}$ be the twisting of ${\mathcal {B_{A'}}}$ by $F$, and let there exist an isomorphism: $\alpha :{\mathcal {B_{A}}}\to {\mathcal {B'_{A'}}}$ :{\mathcal {B_{A}}}\to {\mathcal {B'_{A'}}}} . Then the induced tensor functor $(\alpha ^{*},id,\phi _{2}^{F})$ is a tensor category equivalence between ${\mathcal {A'}}-mod$ and ${\mathcal {A}}-mod$. Where $\phi _{2}^{F}(v\otimes w)=F^{-1}(v\otimes w)$. Moreover, if $\alpha $ is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.[1]: 375–376
Usage
Quasi-bialgebras form the basis of the study of quasi-Hopf algebras and further to the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang–Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the XXZ in the framework of the Algebraic Bethe ansatz.
See also
• Bialgebra
• Hopf algebra
• Quasi-Hopf algebra
References
1. C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Springer-Verlag. ISBN 0387943706
Further reading
• Vladimir Drinfeld, Quasi-Hopf algebras, Leningrad Math J. 1 (1989), 1419-1457
• J.M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000
|
Wikipedia
|
Circumcenter of mass
In geometry, the circumcenter of mass is a center associated with a polygon which shares many of the properties of the center of mass. More generally, the circumcenter of mass may be defined for simplicial polytopes and also in the spherical and hyperbolic geometries.
In the special case when the polytope is a quadrilateral or hexagon, the circumcenter of mass has been called the "quasicircumcenter" and has been used to define an Euler line of a quadrilateral.[1][2] The circumcenter of mass allows us to define an Euler line for simplicial polytopes.
Definition in the plane
Let $P$ be an oriented polygon (with vertices counted countercyclically) in the plane with vertices $V_{1},V_{2},\ldots ,V_{n}$ and let $O$ be an arbitrary point not lying on the sides (or their extensions). Consider the triangulation of $P$ by the oriented triangles $OV_{i}V_{i+1}$ (the index $i$ is viewed modulo $n$). Associate with each of these triangles its circumcenter $C_{i}$ with weight equal to its oriented area (positive if its sequence of vertices is countercyclical; negative otherwise). The circumcenter of mass of $P$ is the center of mass of these weighted circumcenters. The result is independent of the choice of point $O$.[3]
Properties
In the special case when the polygon is cyclic, the circumcenter of mass coincides with the circumcenter.
The circumcenter of mass satisfies an analog of Archimedes' Lemma, which states that if a polygon is decomposed into two smaller polygons, then the circumcenter of mass of that polygon is a weighted sum of the circumcenters of mass of the two smaller polygons. As a consequence, any triangulation with nondegenerate triangles may be used to define the circumcenter of mass.
For an equilateral polygon, the circumcenter of mass and center of mass coincide. More generally, the circumcenter of mass and center of mass coincide for a simplicial polytope for which each face has the sum of squares of its edges a constant.[4]
The circumcenter of mass is invariant under the operation of "recutting" of polygons.[5] and the discrete bicycle (Darboux) transformation; in other words, the image of a polygon under these operations has the same circumcenter of mass as the original polygon. The generalized Euler line makes other appearances in the theory of integrable systems.[6]
Let $V_{i}=(x_{i},y_{i})$ be the vertices of $P$ and let $A$ denote its area. The circumcenter of mass $CCM(P)$ of the polygon $P$ is given by the formula
$CCM(P)={\frac {1}{4A}}(\sum _{i=0}^{n-1}-y_{i}y_{i+1}^{2}+y_{i}^{2}y_{i+1}+x_{i}^{2}y_{i+1}-x_{i+1}^{2}y_{i},\sum _{i=0}^{n-1}-x_{i+1}y_{i}^{2}+x_{i}y_{i+1}^{2}+x_{i}x_{i+1}^{2}-x_{i}^{2}x_{i+1}).$
The circumcenter of mass can be extended to smooth curves via a limiting procedure. This continuous limit coincides with the center of mass of the homogeneous lamina bounded by the curve.
Under natural assumptions, the centers of polygons which satisfy Archimedes' Lemma are precisely the points of its Euler line. In other words, the only "well-behaved" centers which satisfy Archimedes' Lemma are the affine combinations of the circumcenter of mass and center of mass.
Generalized Euler line
The circumcenter of mass allows an Euler line to be defined for any polygon (and more generally, for a simplicial polytope). This generalized Euler line is defined as the affine span of the center of mass and circumcenter of mass of the polytope.
See also
• Circumcenter
• Circumscribed sphere
References
1. Myakishev, Alexei (2006), "On Two Remarkable Lines Related to a Quadrilateral" (PDF), Forum Geometricorum, 6: 289–295.
2. de Villiers, Michael (2014), "Quasi-circumcenters and a generalization of the quasi-Euler line to a hexagon" (PDF), Forum Geometricorum, 14: 233–236
3. Tabachnikov, Serge; Tsukerman, Emmanuel (May 2014), "Circumcenter of Mass and Generalized Euler Line", Discrete and Computational Geometry, 51 (4): 815–836, arXiv:1301.0496, doi:10.1007/s00454-014-9597-2, S2CID 12307207
4. Akopyan, Arseniy (May 2014), "Some Remarks on the Circumcenter of Mass", Discrete and Computational Geometry, 51 (4): 837–841, arXiv:1512.08655, doi:10.1007/s00454-014-9596-3, S2CID 3464833
5. Adler, V. (1993), "Cutting of polygons", Funct. Anal. Appl., 27 (2): 141–143, doi:10.1007/BF01085984, S2CID 122179363
6. Schief, W. K. (2014), "Integrable structure in discrete shell membrane theory", Proceedings of the Royal Society of London A, 470 (2165): 22, doi:10.1098/rspa.2013.0757, PMC 3973394, PMID 24808755
|
Wikipedia
|
Quasi-commutative property
In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions.
Applied to matrices
Two matrices $p$ and $q$ are said to have the commutative property whenever
$pq=qp$
The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices $x$ and $y$
$xy-yx=z$
satisfy the quasi-commutative property whenever $z$ satisfies the following properties:
${\begin{aligned}xz&=zx\\yz&=zy\end{aligned}}$
An example is found in the matrix mechanics introduced by Heisenberg as a version of quantum mechanics. In this mechanics, p and q are infinite matrices corresponding respectively to the momentum and position variables of a particle.[1] These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite unit matrix, where ħ is the reduced Planck constant.
Applied to functions
A function $f:X\times Y\to X$ is said to be quasi-commutative[2] if
$f\left(f\left(x,y_{1}\right),y_{2}\right)=f\left(f\left(x,y_{2}\right),y_{1}\right)\qquad {\text{ for all }}x\in X,\;y_{1},y_{2}\in Y.$
If $f(x,y)$ is instead denoted by $x\ast y$ then this can be rewritten as:
$(x\ast y)\ast y_{2}=\left(x\ast y_{2}\right)\ast y\qquad {\text{ for all }}x\in X,\;y,y_{2}\in Y.$
See also
• Commutative property – Property of some mathematical operations
• Accumulator (cryptography)
References
1. Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.
2. Benaloh, J., & De Mare, M. (1994, January). One-way accumulators: A decentralized alternative to digital signatures. In Advances in Cryptology – EUROCRYPT’93 (pp. 274–285). Springer Berlin Heidelberg.
|
Wikipedia
|
Quasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.
Intuitively, let f : D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K.
Definition
Suppose f : D → D′ where D and D′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of f. If f is assumed to have continuous partial derivatives, then f is quasiconformal provided it satisfies the Beltrami equation
${\frac {\partial f}{\partial {\bar {z}}}}=\mu (z){\frac {\partial f}{\partial z}},$
(1)
for some complex valued Lebesgue measurable μ satisfying sup |μ| < 1 (Bers 1977). This equation admits a geometrical interpretation. Equip D with the metric tensor
$ds^{2}=\Omega (z)^{2}\left|\,dz+\mu (z)\,d{\bar {z}}\right|^{2},$
where Ω(z) > 0. Then f satisfies (1) precisely when it is a conformal transformation from D equipped with this metric to the domain D′ equipped with the standard Euclidean metric. The function f is then called μ-conformal. More generally, the continuous differentiability of f can be replaced by the weaker condition that f be in the Sobolev space W1,2(D) of functions whose first-order distributional derivatives are in L2(D). In this case, f is required to be a weak solution of (1). When μ is zero almost everywhere, any homeomorphism in W1,2(D) that is a weak solution of (1) is conformal.
Without appeal to an auxiliary metric, consider the effect of the pullback under f of the usual Euclidean metric. The resulting metric is then given by
$\left|{\frac {\partial f}{\partial z}}\right|^{2}\left|\,dz+\mu (z)\,d{\bar {z}}\right|^{2}$
which, relative to the background Euclidean metric $dzd{\bar {z}}$, has eigenvalues
$(1+|\mu |)^{2}\textstyle {\left|{\frac {\partial f}{\partial z}}\right|^{2}},\qquad (1-|\mu |)^{2}\textstyle {\left|{\frac {\partial f}{\partial z}}\right|^{2}}.$
The eigenvalues represent, respectively, the squared length of the major and minor axis of the ellipse obtained by pulling back along f the unit circle in the tangent plane.
Accordingly, the dilatation of f at a point z is defined by
$K(z)={\frac {1+|\mu (z)|}{1-|\mu (z)|}}.$
The (essential) supremum of K(z) is given by
$K=\sup _{z\in D}|K(z)|={\frac {1+\|\mu \|_{\infty }}{1-\|\mu \|_{\infty }}}$
and is called the dilatation of f.
A definition based on the notion of extremal length is as follows. If there is a finite K such that for every collection Γ of curves in D the extremal length of Γ is at most K times the extremal length of {f o γ : γ ∈ Γ}. Then f is K-quasiconformal.
If f is K-quasiconformal for some finite K, then f is quasiconformal.
A few facts about quasiconformal mappings
If K > 1 then the maps x + iy ↦ Kx + iy and x + iy ↦ x + iKy are both quasiconformal and have constant dilatation K.
If s > −1 then the map $z\mapsto z\,|z|^{s}$ is quasiconformal (here z is a complex number) and has constant dilatation $\max(1+s,{\frac {1}{1+s}})$. When s ≠ 0, this is an example of a quasiconformal homeomorphism that is not smooth. If s = 0, this is simply the identity map.
A homeomorphism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal. If f : D → D′ is K-quasiconformal and g : D′ → D′′ is K′-quasiconformal, then g o f is KK′-quasiconformal. The inverse of a K-quasiconformal homeomorphism is K-quasiconformal. The set of 1-quasiconformal maps forms a group under composition.
The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact.
Measurable Riemann mapping theorem
Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable Riemann mapping theorem, proved by Lars Ahlfors and Lipman Bers. The theorem generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that D is a simply connected domain in C that is not equal to C, and suppose that μ : D → C is Lebesgue measurable and satisfies $\|\mu \|_{\infty }<1$. Then there is a quasiconformal homeomorphism f from D to the unit disk which is in the Sobolev space W1,2(D) and satisfies the corresponding Beltrami equation (1) in the distributional sense. As with Riemann's mapping theorem, this f is unique up to 3 real parameters.
Computational quasi-conformal geometry
Recently, quasi-conformal geometry has attracted attention from different fields, such as applied mathematics, computer vision and medical imaging. Computational quasi-conformal geometry has been developed, which extends the quasi-conformal theory into a discrete setting. It has found various important applications in medical image analysis, computer vision and graphics.
See also
• Isothermal coordinates
• Quasiregular map
• Pseudoanalytic function
• Teichmüller space
• Tissot's indicatrix
References
• Ahlfors, Lars (1935), "Zur Theorie der Überlagerungsflächen", Acta Mathematica (in German), 65 (1): 157–194, doi:10.1007/BF02420945, ISSN 0001-5962, JFM 61.0365.03, Zbl 0012.17204.
• Ahlfors, Lars V. (2006) [1966], Lectures on quasiconformal mappings, University Lecture Series, vol. 38 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3644-6, MR 2241787, Zbl 1103.30001, (reviews of the first edition: MR0200442, Zbl 1103.30001).
• Bers, Lipman (1977), "Quasiconformal mappings, with applications to differential equations, function theory and topology", Bull. Amer. Math. Soc., 83 (6): 1083–1100, doi:10.1090/S0002-9904-1977-14390-5, MR 0463433.
• Caraman, Petru (1974) [1968], n–Dimensional Quasiconformal (QCf) Mappings (revised ed.), București / Tunbridge Wells, Kent: Editura Academiei / Abacus Press, p. 553, ISBN 0-85626-005-3, MR 0357782, Zbl 0342.30015.
• Grötzsch, Herbert (1928), "Über einige Extremalprobleme der konformen Abbildung. I, II.", Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe (in German), 80: 367–376, 497–502, JFM 54.0378.01.
• Heinonen, Juha (December 2006), "What Is ... a Quasiconformal Mapping?" (PDF), Notices of the American Mathematical Society, 53 (11): 1334–1335, MR 2268390, Zbl 1142.30322.
• Jones, Gareth Wyn; Mahadevan, L. (2013-05-08). "Planar morphometry, shear and optimal quasi-conformal mappings". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 469 (2153): 20120653. Bibcode:2013RSPSA.46920653J. doi:10.1098/rspa.2012.0653. ISSN 1364-5021. S2CID 123826235.
• Lehto, O.; Virtanen, K.I. (1973), Quasiconformal mappings in the plane, Die Grundlehren der mathematischen Wissenschaften, vol. 126 (2nd ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. VIII+258, ISBN 3-540-03303-3, MR 0344463, Zbl 0267.30016 (also available as ISBN 0-387-03303-3).
• Morrey, Charles B. Jr. (1938), "On the solutions of quasi-linear elliptic partial differential equations", Transactions of the American Mathematical Society, 43 (1): 126–166, doi:10.2307/1989904, JFM 62.0565.02, JSTOR 1989904, MR 1501936, Zbl 0018.40501.
• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR2284826.
• Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN 978-3-03719-055-5, MR2524085.
• Zorich, V. A. (2001) [1994], "Quasi-conformal mapping", Encyclopedia of Mathematics, EMS Press.
|
Wikipedia
|
Quasicircle
In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by Pfluger (1961) and Tienari (1962), in the older literature (in German) they were referred to as quasiconformal curves, a terminology which also applied to arcs.[1][2] In complex analysis and geometric function theory, quasicircles play a fundamental role in the description of the universal Teichmüller space, through quasisymmetric homeomorphisms of the circle. Quasicircles also play an important role in complex dynamical systems.
Definitions
A quasicircle is defined as the image of a circle under a quasiconformal mapping of the extended complex plane. It is called a K-quasicircle if the quasiconformal mapping has dilatation K. The definition of quasicircle generalizes the characterization of a Jordan curve as the image of a circle under a homeomorphism of the plane. In particular a quasicircle is a Jordan curve. The interior of a quasicircle is called a quasidisk.[3]
As shown in Lehto & Virtanen (1973), where the older term "quasiconformal curve" is used, if a Jordan curve is the image of a circle under a quasiconformal map in a neighbourhood of the curve, then it is also the image of a circle under a quasiconformal mapping of the extended plane and thus a quasicircle. The same is true for "quasiconformal arcs" which can be defined as quasiconformal images of a circular arc either in an open set or equivalently in the extended plane.[4]
Geometric characterizations
Ahlfors (1963) gave a geometric characterization of quasicircles as those Jordan curves for which the absolute value of the cross-ratio of any four points, taken in cyclic order, is bounded below by a positive constant.
Ahlfors also proved that quasicircles can be characterized in terms of a reverse triangle inequality for three points: there should be a constant C such that if two points z1 and z2 are chosen on the curve and z3 lies on the shorter of the resulting arcs, then[5]
$|z_{1}-z_{3}|+|z_{2}-z_{3}|\leq C|z_{1}-z_{2}|.$
This property is also called bounded turning[6] or the arc condition.[7]
For Jordan curves in the extended plane passing through ∞, Ahlfors (1966) gave a simpler necessary and sufficient condition to be a quasicircle.[8][9] There is a constant C > 0 such that if z1, z2 are any points on the curve and z3 lies on the segment between them, then
$\displaystyle {\left|z_{3}-{z_{1}+z_{2} \over 2}\right|\leq C|z_{1}-z_{2}|.}$
These metric characterizations imply that an arc or closed curve is quasiconformal whenever it arises as the image of an interval or the circle under a bi-Lipschitz map f, i.e. satisfying
$C_{1}|s-t|\leq |f(s)-f(t)|\leq C_{2}|s-t|$
for positive constants Ci.[10]
Quasicircles and quasisymmetric homeomorphisms
If φ is a quasisymmetric homeomorphism of the circle, then there are conformal maps f of [z| < 1 and g of |z|>1 into disjoint regions such that the complement of the images of f and g is a Jordan curve. The maps f and g extend continuously to the circle |z| = 1 and the sewing equation
$\varphi =g^{-1}\circ f$
holds. The image of the circle is a quasicircle.
Conversely, using the Riemann mapping theorem, the conformal maps f and g uniformizing the outside of a quasicircle give rise to a quasisymmetric homeomorphism through the above equation.
The quotient space of the group of quasisymmetric homeomorphisms by the subgroup of Möbius transformations provides a model of universal Teichmüller space. The above correspondence shows that the space of quasicircles can also be taken as a model.[11]
Quasiconformal reflection
A quasiconformal reflection in a Jordan curve is an orientation-reversing quasiconformal map of period 2 which switches the inside and the outside of the curve fixing points on the curve. Since the map
$\displaystyle {R_{0}(z)={1 \over {\overline {z}}}}$
provides such a reflection for the unit circle, any quasicircle admits a quasiconformal reflection. Ahlfors (1963) proved that this property characterizes quasicircles.
Ahlfors noted that this result can be applied to uniformly bounded holomorphic univalent functions f(z) on the unit disk D. Let Ω = f(D). As Carathéodory had proved using his theory of prime ends, f extends continuously to the unit circle if and only if ∂Ω is locally connected, i.e. admits a covering by finitely many compact connected sets of arbitrarily small diameter. The extension to the circle is 1-1 if and only if ∂Ω has no cut points, i.e. points which when removed from ∂Ω yield a disconnected set. Carathéodory's theorem shows that a locally set without cut points is just a Jordan curve and that in precisely this case is the extension of f to the closed unit disk a homeomorphism.[12] If f extends to a quasiconformal mapping of the extended complex plane then ∂Ω is by definition a quasicircle. Conversely Ahlfors (1963) observed that if ∂Ω is a quasicircle and R1 denotes the quasiconformal reflection in ∂Ω then the assignment
$\displaystyle {f(z)=R_{1}fR_{0}(z)}$
for |z| > 1 defines a quasiconformal extension of f to the extended complex plane.
Complex dynamical systems
Quasicircles were known to arise as the Julia sets of rational maps R(z). Sullivan (1985) proved that if the Fatou set of R has two components and the action of R on the Julia set is "hyperbolic", i.e. there are constants c > 0 and A > 1 such that
$|\partial _{z}R^{n}(z)|\geq cA^{n}$
on the Julia set, then the Julia set is a quasicircle.[5]
There are many examples:[13][14]
• quadratic polynomials R(z) = z2 + c with an attracting fixed point
• the Douady rabbit (c = –0.122561 + 0.744862i, where c3 + 2 c2 + c + 1 = 0)
• quadratic polynomials z2 + λz with |λ| < 1
• the Koch snowflake
Quasi-Fuchsian groups
Quasi-Fuchsian groups are obtained as quasiconformal deformations of Fuchsian groups. By definition their limit sets are quasicircles.[15][16][17][18][19]
Let Γ be a Fuchsian group of the first kind: a discrete subgroup of the Möbius group preserving the unit circle. acting properly discontinuously on the unit disk D and with limit set the unit circle.
Let μ(z) be a measurable function on D with
$\|\mu \|_{\infty }<1$
such that μ is Γ-invariant, i.e.
$\mu (g(z)){{\overline {\partial _{z}g(z)}} \over \partial _{z}g(z)}=\mu (z)$
for every g in Γ. (μ is thus a "Beltrami differential" on the Riemann surface D / Γ.)
Extend μ to a function on C by setting μ(z) = 0 off D.
The Beltrami equation
$\partial _{\overline {z}}f(z)=\mu (z)\partial _{z}f(z)$
admits a solution unique up to composition with a Möbius transformation.
It is a quasiconformal homeomorphism of the extended complex plane.
If g is an element of Γ, then f(g(z)) gives another solution of the Beltrami equation, so that
$\alpha (g)=f\circ g\circ f^{-1}$
is a Möbius transformation.
The group α(Γ) is a quasi-Fuchsian group with limit set the quasicircle given by the image of the unit circle under f.
Hausdorff dimension
It is known that there are quasicircles for which no segment has finite length.[21] The Hausdorff dimension of quasicircles was first investigated by Gehring & Väisälä (1973), who proved that it can take all values in the interval [1,2).[22] Astala (1993), using the new technique of "holomorphic motions" was able to estimate the change in the Hausdorff dimension of any planar set under a quasiconformal map with dilatation K. For quasicircles C, there was a crude estimate for the Hausdorff dimension[23]
$d_{H}(C)\leq 1+k$
where
$k={K-1 \over K+1}.$
On the other hand, the Hausdorff dimension for the Julia sets Jc of the iterates of the rational maps
$R(z)=z^{2}+c$
had been estimated as result of the work of Rufus Bowen and David Ruelle, who showed that
$1<d_{H}(J_{c})<1+{|c|^{2} \over 4\log 2}+o(|c|^{2}).$
Since these are quasicircles corresponding to a dilatation
$K={\sqrt {1+t \over 1-t}}$
where
$t=|1-{\sqrt {1-4c}}|,$
this led Becker & Pommerenke (1987) to show that for k small
$1+0.36k^{2}\leq d_{H}(C)\leq 1+37k^{2}.$
Having improved the lower bound following calculations for the Koch snowflake with Steffen Rohde and Oded Schramm, Astala (1994) conjectured that
$d_{H}(C)\leq 1+k^{2}.$
This conjecture was proved by Smirnov (2010); a complete account of his proof, prior to publication, was already given in Astala, Iwaniec & Martin (2009).
For a quasi-Fuchsian group Bowen (1979) and Sullivan (1982) showed that the Hausdorff dimension d of the limit set is always greater than 1. When d < 2, the quantity
$\lambda =d(2-d)\,\in (0,1)$
is the lowest eigenvalue of the Laplacian of the corresponding hyperbolic 3-manifold.[24][25]
Notes
1. Lehto & Virtanen 1973
2. Lehto 1983, p. 49 harvnb error: no target: CITEREFLehto1983 (help)
3. Lehto 1987, p. 38
4. Lehto & Virtanen 1973, pp. 97–98
5. Carleson & Gamelin 1993, p. 102
6. Lehto & Virtanen 1973, pp. 100–102
7. Lehto 1983, p. 45 harvnb error: no target: CITEREFLehto1983 (help)
8. Ahlfors 1966, p. 81
9. Lehto 1983, pp. 48–49 harvnb error: no target: CITEREFLehto1983 (help)
10. Lehto & Virtanen 1973, pp. 104–105
11. Lehto 1983 harvnb error: no target: CITEREFLehto1983 (help)
12. Pommerenke 1975, pp. 271–281
13. Carleson & Gamelin 1993, pp. 123–126
14. Rohde 1991
15. Bers 1961
16. Bowen 1979
17. Mumford, Series & Wright 2002
18. Imayoshi & Taniguchi 1992, p. 147
19. Marden 2007, pp. 79–80, 134
20. Carleson & Gamelin 1993, p. 122
21. Lehto & Virtanen 1973, p. 104
22. Lehto 1982, p. 38 harvnb error: no target: CITEREFLehto1982 (help)
23. Astala, Iwaniec & Martin 2009
24. Astala & Zinsmeister 1994
25. Marden 2007, p. 284
References
• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Ahlfors, L. (1963), "Quasiconformal reflections", Acta Mathematica, 109: 291–301, doi:10.1007/bf02391816, Zbl 0121.06403
• Astala, K. (1993), "Distortion of area and dimension under quasiconformal mappings in the plane", Proc. Natl. Acad. Sci. U.S.A., 90 (24): 11958–11959, Bibcode:1993PNAS...9011958A, doi:10.1073/pnas.90.24.11958, PMC 48104, PMID 11607447
• Astala, K.; Zinsmeister, M. (1994), "Holomorphic families of quasi-Fuchsian groups", Ergodic Theory Dynam. Systems, 14 (2): 207–212, doi:10.1017/s0143385700007847, S2CID 121209816
• Astala, K. (1994), "Area distortion of quasiconformal mappings", Acta Math., 173: 37–60, doi:10.1007/bf02392568
• Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven (2009), Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton mathematical series, vol. 48, Princeton University Press, pp. 332–342, ISBN 978-0-691-13777-3, Section 13.2, Dimension of quasicircles.
• Becker, J.; Pommerenke, C. (1987), "On the Hausdorff dimension of quasicircles", Ann. Acad. Sci. Fenn. Ser. A I Math., 12: 329–333, doi:10.5186/aasfm.1987.1206
• Bers, Lipman (August 1961). "Uniformization by Beltrami equations". Communications on Pure and Applied Mathematics. 14 (3): 215–228. doi:10.1002/cpa.3160140304.
• Bowen, R. (1979), "Hausdorff dimension of quasicircles", Inst. Hautes Études Sci. Publ. Math., 50: 11–25, doi:10.1007/BF02684767, S2CID 55631433
• Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 978-0-387-97942-7
• Gehring, F. W.; Väisälä, J. (1973), "Hausdorff dimension and quasiconformal mappings", Journal of the London Mathematical Society, 6 (3): 504–512, CiteSeerX 10.1.1.125.2374, doi:10.1112/jlms/s2-6.3.504
• Gehring, F. W. (1982), Characteristic properties of quasidisks, Séminaire de Mathématiques Supérieures, vol. 84, Presses de l'Université de Montréal, ISBN 978-2-7606-0601-2
• Imayoshi, Y.; Taniguchi, M. (1992), An Introduction to Teichmüller spaces, Springer-Verlag, ISBN 978-0-387-70088-5 +
• Lehto, O. (1987), Univalent functions and Teichmüller spaces, Springer-Verlag, pp. 50–59, 111–118, 196–205, ISBN 978-0-387-96310-5
• Lehto, O.; Virtanen, K. I. (1973), Quasiconformal mappings in the plane, Die Grundlehren der mathematischen Wissenschaften, vol. 126 (Second ed.), Springer-Verlag
• Marden, A. (2007), Outer circles. An introduction to hyperbolic 3-manifolds, Cambridge University Press, ISBN 978-0-521-83974-7
• Mumford, D.; Series, C.; Wright, David (2002), Indra's pearls. The vision of Felix Klein, Cambridge University Press, ISBN 978-0-521-35253-6
• Pfluger, A. (1961), "Ueber die Konstruktion Riemannscher Flächen durch Verheftung", J. Indian Math. Soc., 24: 401–412
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
• Rohde, S. (1991), "On conformal welding and quasicircles", Michigan Math. J., 38: 111–116, doi:10.1307/mmj/1029004266
• Sullivan, D. (1982), "Discrete conformal groups and measurable dynamics", Bull. Amer. Math. Soc., 6: 57–73, doi:10.1090/s0273-0979-1982-14966-7
• Sullivan, D. (1985), "Quasiconformal homeomorphisms and dynamics, I, Solution of the Fatou-Julia problem on wandering domains", Annals of Mathematics, 122 (2): 401–418, doi:10.2307/1971308, JSTOR 1971308
• Tienari, M. (1962), "Fortsetzung einer quasikonformen Abbildung über einen Jordanbogen", Ann. Acad. Sci. Fenn. Ser. A, 321
• Smirnov, S. (2010), "Dimension of quasicircles", Acta Mathematica, 205: 189–197, arXiv:0904.1237, doi:10.1007/s11511-010-0053-8, MR 2736155, S2CID 17945998
|
Wikipedia
|
Quasicontraction semigroup
In mathematical analysis, a C0-semigroup Γ(t), t ≥ 0, is called a quasicontraction semigroup if there is a constant ω such that ||Γ(t)|| ≤ exp(ωt) for all t ≥ 0. Γ(t) is called a contraction semigroup if ||Γ(t)|| ≤ 1 for all t ≥ 0.
See also
• Contraction (operator theory)
• Hille–Yosida theorem
• Lumer–Phillips theorem
References
• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. xiv+434. ISBN 0-387-00444-0. MR2028503
|
Wikipedia
|
Quasiconvexity (calculus of variations)
In the calculus of variations, a subfield of mathematics, quasiconvexity is a generalisation of the notion of convexity. It is used to characterise the integrand of a functional and related to the existence of minimisers. Under some natural conditions, quasiconvexity of the integrand is a necessary and sufficient condition for a functional
${\mathcal {F}}:W^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R} \qquad u\mapsto \int _{\Omega }f(x,u(x),\nabla u(x))dx$
to be lower semi-continuous in the weak topology, for a sufficient regular domain $ \Omega \subset \mathbb {R} ^{d}$. By compactness arguments (Banach–Alaoglu theorem) the existence of minimisers of weakly lower semicontinuous functionals may then follow from the direct method.[1] This concept was introduced by Morrey in 1952.[2] This generalisation should not be confused with the same concept of a quasiconvex function which has the same name.
Definition
A locally bounded Borel-measurable function $ f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R} $ is called quasiconvex if
$\int _{B(0,1)}{\bigl (}f(A+\nabla \psi (x))-f(A){\bigr )}dx\geq 0$
for all $A\in \mathbb {R} $ and all $\psi \in W_{0}^{1,\infty }(B(0,1),\mathbb {R} ^{m})$ , where B(0,1) is the unit ball and $W_{0}^{1,\infty }$ is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.[3]
Properties of quasiconvex functions
• The domain B(0,1) can be replaced by any other bounded Lipschitz domain.[4]
• Quasiconvex functions are locally Lipschitz-continuous.[5]
• In the definition the space $W_{0}^{1,\infty }$ can be replaced by periodic Sobolev functions.[6]
Relations to other notions of convexity
Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let $A\in \mathbb {R} ^{m\times d}$ and $V\in L^{1}(B(0,1),\mathbb {R} ^{m})$ with $\int _{B(0,1)}V(x)dx=0$. The Riesz-Markov-Kakutani representation theorem states that the dual space of $C_{0}(\mathbb {R} ^{m\times d})$ can be identified with the space of signed, finite Radon measures on it. We define a Radon measure $\mu $ by
$\langle h,\mu \rangle ={\frac {1}{|B(0,1)|}}\int _{B(0,1)}h(A+V(x))dx$
for $h\in C_{0}(\mathbb {R} ^{m\times d})$. It can be verfied that $\mu $ is a probability measure and its barycenter is given
$[\mu ]=\langle \operatorname {id} ,\mu \rangle =A+\int _{B(0,1)}V(x)dx=A.$
If h is a convex function, then Jensens' Inequality gives
$h(A)=h([\mu ])\leq \langle h,\mu \rangle ={\frac {1}{|B(0,1)|}}\int _{B(0,1)}h(A+V(x))dx.$
This holds in particular if V(x) is the derivative of $\psi \in W_{0}^{1,\infty }(B(0,1),\mathbb {R} ^{m\times d})$ by the generalised Stokes' Theorem.[7]
The determinant $\det \mathbb {R} ^{d\times d}\rightarrow \mathbb {R} $ is an example of a quasiconvex function, which is not convex.[8] To see that the determinant is not convex, consider
$A={\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}\quad {\text{and}}\quad B={\begin{pmatrix}0&0\\0&1\\\end{pmatrix}}.$
It then holds $\det A=\det B=0$ but for $\lambda \in (0,1)$ we have $\det(\lambda A+(1-\lambda )B)=\lambda (1-\lambda )>0=\max(\det A,\det B)$. This shows that the determinant is not a quasiconvex function like in Game Theory and thus a distinct notion of convexity.
In the vectorial case of the Calculus of Variations there are other notions of convexity. For a function $f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R} $ it holds that [9]
$f{\text{ convex}}\Rightarrow f{\text{ polyconvex}}\Rightarrow f{\text{ quasiconvex}}\Rightarrow f{\text{ rank-1-convex}}.$
These notions are all equivalent if $d=1$ or $m=1$. Already in 1952, Morrey conjectured that rank-1-convexity does not imply quasiconvexity.[10] This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case $d\geq 2$ and $m\geq 3$.[11] The case $d=2$ or $m=2$ is still an open problem, known as Morrey's conjecture.[12]
Relation to weak lower semi-continuity
Under certain growth condition of the integrand, the sequential weakly lower semi-continuity (swlsc) of an integral functional in an appropriate Sobolev space is equivalent to the quasiconvexity of the integrand. Acerbi and Fusco proved the following theorem:
Theorem: If $f:\mathbb {R} ^{d}\times \mathbb {R} ^{m}\times \mathbb {R} ^{d\times m}\rightarrow \mathbb {R} ,(x,v,A)\mapsto f(x,v,A)$ is Carathéodory function and it holds $0\leq f(x,v,A)\leq a(x)+C(|v|^{p}+|A|^{p})$. Then the functional
${\mathcal {F}}[u]=\int _{\Omega }f(x,u(x),\nabla u(x))dx$
is swlsc in the Sobolev Space $W^{1,p}(\Omega ,\mathbb {R} ^{m})$ with $p>1$ if and only if $f$ is quasiconvex. Here $C$ is a positive constant and $a(x)$ an integrable function.[13]
Other authors use different growth conditions and different proof conditions.[14][15] The first proof of it was due to Morrey in his paper, but he required additional assumptions.[16]
References
1. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 125. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
2. Morrey, Charles B. (1952). "Quasiconvexity and the Lower Semicontinuity of Multiple Integrals". Pacific Journal of Mathematics. Mathematical Sciences Publishers. 2 (1): 25–53. doi:10.2140/pjm.1952.2.25. Retrieved 2022-06-30.
3. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 106. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
4. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 108. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
5. Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. Vol. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 159. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
6. Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. Vol. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 173. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
7. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 107. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
8. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 105. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
9. Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. Vol. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 159. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
10. Morrey, Charles B. (1952). "Quasiconvexity and the Lower Semicontinuity of Multiple Integrals". Pacific Journal of Mathematics. Mathematical Sciences Publishers. 2 (1): 25–53. doi:10.2140/pjm.1952.2.25. Retrieved 2022-06-30.
11. Šverák, Vladimir (1993). "Rank-one convexity does not imply quasiconvexity". Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge University Press, Cambridge; RSE Scotland Foundation. 120 (1–2): 185–189. doi:10.1017/S0308210500015080. S2CID 120192116. Retrieved 2022-06-30.
12. Voss, Jendrik; Martin, Robert J.; Sander, Oliver; Kumar, Siddhant; Kochmann, Dennis M.; Neff, Patrizio (2022-01-17). "Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey's Conjecture". Journal of Nonlinear Science. 32 (6). arXiv:2201.06392. doi:10.1007/s00332-022-09820-x. S2CID 246016000.
13. Acerbi, Emilio; Fusco, Nicola (1984). "Semicontinuity problems in the calculus of variations". Archive for Rational Mechanics and Analysis. Springer, Berlin/Heidelberg. 86 (1–2): 125–145. Bibcode:1984ArRMA..86..125A. doi:10.1007/BF00275731. S2CID 121494852. Retrieved 2022-06-30.
14. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 128. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
15. Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. Vol. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 368. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
16. Morrey, Charles B. (1952). "Quasiconvexity and the Lower Semicontinuity of Multiple Integrals". Pacific Journal of Mathematics. Mathematical Sciences Publishers. 2 (1): 25–53. doi:10.2140/pjm.1952.2.25. Retrieved 2022-06-30.
|
Wikipedia
|
Quasideterminant
In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
$\left|{\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}}\right|_{11}=a_{11}-a_{12}{a_{22}}^{-1}a_{21}\qquad \left|{\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}}\right|_{12}=a_{12}-a_{11}{a_{21}}^{-1}a_{22}.$
In general, there are n2 quasideterminants defined for an n × n matrix (one for each position in the matrix), but the presence of the inverted terms above should give the reader pause: they are not always defined, and even when they are defined, they do not reduce to determinants when the entries commute. Rather,
$\left|A\right|_{ij}=(-1)^{i+j}{\frac {\det A}{\det A^{ij}}},$
where $A^{ij}$ means delete the ith row and jth column from A.
The $2\times 2$ examples above were introduced between 1926 and 1928 by Richardson[1][2] and Heyting,[3] but they were marginalized at the time because they were not polynomials in the entries of $A$. These examples were rediscovered and given new life in 1991 by Israel Gelfand and Vladimir Retakh.[4][5] There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if $B$ is built from $A$ by rescaling its $i$-th row (on the left) by $\left.\rho \right.$, then $\left|B\right|_{ij}=\rho \left|A\right|_{ij}$. Similarly, if $B$ is built from $A$ by adding a (left) multiple of the $k$-th row to another row, then $\left|B\right|_{ij}=\left|A\right|_{ij}\,\,(\forall j;\forall k\neq i)$. They even develop a quasideterminantal version of Cramer's rule.
Definition
Let $A$ be an $n\times n$ matrix over a (not necessarily commutative) ring $R$ and fix $1\leq i,j\leq n$. Let $a_{ij}$ denote the ($i,j$)-entry of $A$, let $r_{i}^{j}$ denote the $i$-th row of $A$ with column $j$ deleted, and let $c_{j}^{i}$ denote the $j$-th column of $A$ with row $i$ deleted. The ($i,j$)-quasideterminant of $A$ is defined if the submatrix $A^{ij}$ is invertible over $R$. In this case,
$\left|A\right|_{ij}=a_{ij}-r_{i}^{j}\,{\bigl (}A^{ij}{\bigr )}^{-1}\,c_{j}^{i}.$
Recall the formula (for commutative rings) relating $A^{-1}$ to the determinant, namely $(A^{-1})_{ji}=(-1)^{i+j}{\frac {\det A^{ij}}{\det A}}$. The above definition is a generalization in that (even for noncommutative rings) one has
${\bigl (}A^{-1}{\bigr )}_{\!ji}=\left|A\right|_{ij}^{\,-1}$
whenever the two sides makes sense.
Identities
One of the most important properties of the quasideterminant is what Gelfand and Retakh call the "heredity principle". It allows one to take a quasideterminant in stages (and has no commutative counterpart). To illustrate, suppose
$\left({\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}}\right)$
is a block matrix decomposition of an $n\times n$ matrix $A$ with $A_{11}$ a $k\times k$ matrix. If the ($i,j$)-entry of $A$ lies within $A_{11}$, it says that
$\left|A\right|_{ij}=\left|A_{11}-A_{12}\,{A_{22}}^{-1}\,A_{21}\right|_{ij}.$
That is, the quasideterminant of a quasideterminant is a quasideterminant. To put it less succinctly: UNLIKE determinants, quasideterminants treat matrices with block-matrix entries no differently than ordinary matrices (something determinants cannot do since block-matrices generally don't commute with one another). That is, while the precise form of the above identity is quite surprising, the existence of some such identity is less so. Other identities from the papers [4][5] are (i) the so-called "homological relations", stating that two quasideterminants in a common row or column are closely related to one another, and (ii) the Sylvester formula.
(i) Two quasideterminants sharing a common row or column satisfy
$\left|A\right|_{ij}|A^{il}|_{kj}^{\,-1}=-\left|A\right|_{il}|A^{ij}|_{kl}^{\,-1}$
or
$|A^{kj}|_{il}^{\,-1}\left|A\right|_{ij}=-|A^{ij}|_{kl}^{\,-1}\left|A\right|_{kj},$
respectively, for all choices $i\neq k$, $j\neq l$ so that the quasideterminants involved are defined.
(ii) Like the heredity principle, the Sylvester identity is a way to recursively compute a quasideterminant. To ease notation, we display a special case. Let $A_{0}$ be the upper-left $k\times k$ submatrix of an $n\times n$ matrix $A$ and fix a coordinate ($i,j$) in $A_{0}$. Let $B=(b_{pq})$ be the $(n-k)\times (n-k)$ matrix, with $b_{pq}$ defined as the ($p,q$)-quasideterminant of the $(k+1)\times (k+1)$ matrix formed by adjoining to $A_{0}$ the first $k$ columns of row $p$, the first $k$ rows of column $q$, and the entry $a$$pq$. Then one has
$\left|B\right|_{ij}=\left|A\right|_{ij}.$
Many more identities have appeared since the first articles of Gelfand and Retakh on the subject, most of them being analogs of classical determinantal identities. An important source is Krob and Leclerc's 1995 article.[6] To highlight one, we consider the row/column expansion identities. Fix a row $i$ to expand along. Recall the determinantal formula $\det A=\sum _{l}(-1)^{i+l}a_{il}\cdot \det A^{il}$. Well, it happens that quasideterminants satisfy
$\left|A\right|_{ij}=a_{ij}-\sum _{l\neq j}a_{il}\cdot |A^{ij}|_{kl}^{\,-1}|A^{il}|_{kj}$
(expansion along column $j$), and
$\left|A\right|_{ij}=a_{ij}-\sum _{k\neq i}|A^{kj}|_{il}|A^{ij}|_{kl}^{\,-1}\cdot a_{kj}$
(expansion along row $i$).
Connections to other determinants
The quasideterminant is certainly not the only existing determinant analog for noncommutative settings—perhaps the most famous examples are the Dieudonné determinant and quantum determinant. However, these are related to the quasideterminant in some way. For example,
${\det }_{q}A={\bigl |}A{\bigr |}_{11}\,\left|A^{11}\right|_{22}\,\left|A^{12,12}\right|_{33}\,\cdots \,|a_{nn}|_{nn},$
with the factors on the right-hand side commuting with each other. Other famous examples, such as Berezinians, Moore and Study determinants, Capelli determinants, and Cartier-Foata-type determinants are also expressible in terms of quasideterminants. Gelfand has been known to define a (noncommutative) determinant as "good" if it may be expressed as products of quasiminors.
Applications
Paraphrasing their 2005 survey article with Sergei Gelfand and Robert Wilson ,[7] Israel Gelfand and Vladimir Retakh advocate for the adoption of quasideterminants as "a main organizing tool in noncommutative algebra, giving them the same role determinants play in commutative algebra." Substantive use has been made of the quasideterminant in such fields of mathematics as integrable systems,[8][9] representation theory,[10][11] algebraic combinatorics,[12] the theory of noncommutative symmetric functions,[13] the theory of polynomials over division rings,[14] and noncommutative geometry.[15][16][17]
Several of the applications above make use of quasi-Plücker coordinates, which parametrize noncommutative Grassmannians and flags in much the same way as Plücker coordinates do Grassmannians and flags over commutative fields. More information on these can be found in the survey article.[7]
See also
• MacMahon Master theorem
References
1. Richardson, Archibald Read (1926). "Hypercomplex determinants". Messenger of Mathematics. 55: 145–152.
2. Richardson, Archibald Read (1928). "Simultaneous linear equations over a division algebra". Proceedings of the London Mathematical Society. 28: 395–420. doi:10.1112/plms/s2-28.1.395.
3. Heyting, Arend (1928). "Die theorie der linearen gleichungen in einer zahlenspezies mit nichtkommutativer multiplikation". Mathematische Annalen. 98: 465–490. doi:10.1007/BF01451604.
4. Gelfand, Israel; Retakh, Vladimir (1991). "Determinants of matrices over noncommutative rings". Functional Analysis and its Applications. 25: 91–102. doi:10.1007/BF01079588.
5. Gelfand, Israel; Retakh, Vladimir (1992). "A theory of noncommutative determinants, and characteristic functions of graphs". Functional Analysis and its Applications. 26: 231–246. doi:10.1007/BF01075044.
6. Krob, Daniel; Leclerc, Bernard (1995). "Minor identities for quasi-determinants and quantum determinants". Communications in Mathematical Physics. 169: 1–23. arXiv:hep-th/9411194. doi:10.1007/BF02101594.
7. Gelfand, Israel; Gelfand, Sergei; Retakh, Vladimir; Wilson, Robert Lee (2005). "Quasideterminants". Advances in Mathematics. 193: 56–141. arXiv:math/0208146. doi:10.1016/j.aim.2004.03.018.
8. Etingof, Pavel; Gelfand, Israel; Retakh, Vladimir (1998). "Nonabelian integrable systems, quasideterminants, and Marchenko lemma". Mathematical Research Letters. 5: 1–12. doi:10.4310/MRL.1998.v5.n1.a1.
9. Gilson, Claire R.; Nimmo, Jonathan J.C.; Sooman, C.M. (2008). "On a direct approach to quasideterminant solutions of a noncommutative modified KP equation". Journal of Physics A: Mathematical and Theoretical. 41: 085202. arXiv:0711.3733. doi:10.1088/1751-8113/41/8/085202.
10. A. Molev, Yangians and their applications, in Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003. (eprint)
11. Brundan, Jonathan; Kleshchev, Alexander (2005). "Parabolic presentations of the Yangian $Y(gl_{n})^{\star }$". Communications in Mathematical Physics. 254. arXiv:math/0407011. doi:10.1007/s00220-004-1249-6.
12. Konvalinka, Matjaž; Pak, Igor (2007). "Non-commutative extensions of the MacMahon Master Theorem". Advances in Mathematics. 216: 29–61. arXiv:math/0607737. doi:10.1016/j.aim.2007.05.020.
13. Gelfand, Israel; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir; Thibon, Jean-Yves (1995). "Noncommutative Symmetric Functions". Advances in Mathematics. 112: 218–348. arXiv:hep-th/9407124. doi:10.1006/aima.1995.1032.
14. Israel Gelfand, Vladimir Retakh, Noncommutative Vieta theorem and symmetric functions. The Gelfand Mathematical Seminars, 1993–1995.
15. Zoran Škoda, Noncommutative localization in noncommutative geometry, in "Non-commutative localization in algebra and topology", London Math. Soc. Lecture Note Ser., 330, Cambridge Univ. Press, Cambridge, 2006. (eprint)
16. Lauve, Aaron (2006). "Quantum and quasi-Plücker coordinates". Journal of Algebra. 296: 440–461. arXiv:math/0406062. doi:10.1016/j.jalgebra.2005.12.004.
17. Berenstein, Arkady; Retakh, Vladimir (2005). "Noncommutative double Bruhat cells and their factorizations". International Mathematics Research Notices. 2005: 477–516. arXiv:math/0407010. doi:10.1155/IMRN.2005.477.
|
Wikipedia
|
Quasi-fibration
In algebraic topology, a quasifibration is a generalisation of fibre bundles and fibrations introduced by Albrecht Dold and René Thom. Roughly speaking, it is a continuous map p: E → B having the same behaviour as a fibration regarding the (relative) homotopy groups of E, B and p−1(x). Equivalently, one can define a quasifibration to be a continuous map such that the inclusion of each fibre into its homotopy fibre is a weak equivalence. One of the main applications of quasifibrations lies in proving the Dold-Thom theorem.
Definition
A continuous surjective map of topological spaces p: E → B is called a quasifibration if it induces isomorphisms
$p_{*}\colon \pi _{i}(E,p^{-1}(x),y)\to \pi _{i}(B,x)$
for all x ∈ B, y ∈ p−1(x) and i ≥ 0. For i = 0,1 one can only speak of bijections between the two sets.
By definition, quasifibrations share a key property of fibrations, namely that a quasifibration p: E → B induces a long exact sequence of homotopy groups
${\begin{aligned}\dots \to \pi _{i+1}(B,x)\to \pi _{i}(p^{-1}(x),y)\to \pi _{i}(E,y)&\to \pi _{i}(B,x)\to \dots \\&\to \pi _{0}(B,x)\to 0\end{aligned}}$
as follows directly from the long exact sequence for the pair (E, p−1(x)).
This long exact sequence is also functorial in the following sense: Any fibrewise map f: E → E′ induces a morphism between the exact sequences of the pairs (E, p−1(x)) and (E′, p′−1(x)) and therefore a morphism between the exact sequences of a quasifibration. Hence, the diagram
commutes with f0 being the restriction of f to p−1(x) and x′ being an element of the form p′(f(e)) for an e ∈ p−1(x).
An equivalent definition is saying that a surjective map p: E → B is a quasifibration if the inclusion of the fibre p−1(b) into the homotopy fibre Fb of p over b is a weak equivalence for all b ∈ B. To see this, recall that Fb is the fibre of q under b where q: Ep → B is the usual path fibration construction. Thus, one has
$E_{p}=\{(e,\gamma )\in E\times B^{I}:\gamma (0)=p(e)\}$
and q is given by q(e, γ) = γ(1). Now consider the natural homotopy equivalence φ : E → Ep, given by φ(e) = (e, p(e)), where p(e) denotes the corresponding constant path. By definition, p factors through Ep such that one gets a commutative diagram
Applying πn yields the alternative definition.
Examples
• Every Serre fibration is a quasifibration. This follows from the Homotopy lifting property.
• The projection of the letter L onto its base interval is a quasifibration, but not a fibration. More generally, the projection Mf → I of the mapping cylinder of a map f: X → Y between connected CW complexes onto the unit interval is a quasifibration if and only if πi(Mf, p−1(b)) = 0 = πi(I, b) holds for all i ∈ I and b ∈ B. But by the long exact sequence of the pair (Mf, p−1(b)) and by Whitehead's theorem, this is equivalent to f being a homotopy equivalence. For topological spaces X and Y in general, it is equivalent to f being a weak homotopy equivalence. Furthermore, if f is not surjective, non-constant paths in I starting at 0 cannot be lifted to paths starting at a point of Y outside the image of f in Mf. This means that the projection is not a fibration in this case.
• The map SP(p) : SP(X) → SP(X/A) induced by the projection p: X → X/A is a quasifibration for a CW pair (X, A) consisting of two connected spaces. This is one of the main statements used in the proof of the Dold-Thom theorem. In general, this map also fails to be a fibration.
Properties
The following is a direct consequence of the alternative definition of a fibration using the homotopy fibre:
Theorem. Every quasifibration p: E → B factors through a fibration whose fibres are weakly homotopy equivalent to the ones of p.
A corollary of this theorem is that all fibres of a quasifibration are weakly homotopy equivalent if the base space is path-connected, as this is the case for fibrations.
Checking whether a given map is a quasifibration tends to be quite tedious. The following two theorems are designed to make this problem easier. They will make use of the following notion: Let p: E → B be a continuous map. A subset U ⊂ p(E) is called distinguished (with respect to p) if p: p−1(U) → U is a quasifibration.
Theorem. If the open subsets U,V and U ∩ V are distinguished with respect to the continuous map p: E → B, then so is U ∪ V.[1]
Theorem. Let p: E → B be a continuous map where B is the inductive limit of a sequence B1 ⊂ B2 ⊂ ... All Bn are moreover assumed to satisfy the first separation axiom. If all the Bn are distinguished, then p is a quasifibration.
To see that the latter statement holds, one only needs to bear in mind that continuous images of compact sets in B already lie in some Bn. That way, one can reduce it to the case where the assertion is known. These two theorems mean that it suffices to show that a given map is a quasifibration on certain subsets. Then one can patch these together in order to see that it holds on bigger subsets and finally, using a limiting argument, one sees that the map is a quasifibration on the whole space. This procedure has e.g. been used in the proof of the Dold-Thom theorem.
Notes
1. Dold and Thom (1958), Satz 2.2
References
• Aguilar, Marcelo; Gitler, Samuel; Prieto, Carlos (2008). Algebraic Topology from a Homotopical Viewpoint. Springer Science & Business Media. ISBN 978-0-387-22489-3.
• Dold, Albrecht; Lashof, Richard (1959), "Principal Quasifibrations and Fibre Homotopy Equivalence of Bundles", Illinois Journal of Mathematics, 2 (2): 285–305
• Dold, Albrecht; Thom, René (1958), "Quasifaserungen und unendliche symmetrische Produkte", Annals of Mathematics, Second Series, 67 (2): 239–281, doi:10.2307/1970005, ISSN 0003-486X, JSTOR 1970005, MR 0097062
• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 978-0-521-79540-1.
• May, J. Peter (1990), "Weak Equivalences and Quasifibrations", Springer Lecture Notes, 1425: 91–101
• Piccinini, Renzo A. (1992). Lectures on Homotopy Theory. Elsevier. ISBN 9780080872827.
External Links
• Quasifibrations and homotopy pullbacks on MathOverflow
• Quasifibrations from the Lehigh University
|
Wikipedia
|
Quasi-homogeneous polynomial
In algebra, a multivariate polynomial
$f(x)=\sum _{\alpha }a_{\alpha }x^{\alpha }{\text{, where }}\alpha =(i_{1},\dots ,i_{r})\in \mathbb {N} ^{r}{\text{, and }}x^{\alpha }=x_{1}^{i_{1}}\cdots x_{r}^{i_{r}},$
is quasi-homogeneous or weighted homogeneous, if there exist r integers $w_{1},\ldots ,w_{r}$, called weights of the variables, such that the sum $w=w_{1}i_{1}+\cdots +w_{r}i_{r}$ is the same for all nonzero terms of f. This sum w is the weight or the degree of the polynomial.
The term quasi-homogeneous comes from the fact that a polynomial f is quasi-homogeneous if and only if
$f(\lambda ^{w_{1}}x_{1},\ldots ,\lambda ^{w_{r}}x_{r})=\lambda ^{w}f(x_{1},\ldots ,x_{r})$
for every $\lambda $ in any field containing the coefficients.
A polynomial $f(x_{1},\ldots ,x_{n})$ is quasi-homogeneous with weights $w_{1},\ldots ,w_{r}$ if and only if
$f(y_{1}^{w_{1}},\ldots ,y_{n}^{w_{n}})$
is a homogeneous polynomial in the $y_{i}$. In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1.
A polynomial is quasi-homogeneous if and only if all the $\alpha $ belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set $\{\alpha \mid a_{\alpha }\neq 0\},$ the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polytope (here "degenerate" means "contained in some affine hyperplane").
Introduction
Consider the polynomial $f(x,y)=5x^{3}y^{3}+xy^{9}-2y^{12}$, which is not homogeneous. However, if instead of considering $f(\lambda x,\lambda y)$ we use the pair $(\lambda ^{3},\lambda )$ to test homogeneity, then
$f(\lambda ^{3}x,\lambda y)=5(\lambda ^{3}x)^{3}(\lambda y)^{3}+(\lambda ^{3}x)(\lambda y)^{9}-2(\lambda y)^{12}=\lambda ^{12}f(x,y).$
We say that $f(x,y)$ is a quasi-homogeneous polynomial of type (3,1), because its three pairs (i1, i2) of exponents (3,3), (1,9) and (0,12) all satisfy the linear equation $3i_{1}+1i_{2}=12$. In particular, this says that the Newton polytope of $f(x,y)$ lies in the affine space with equation $3x+y=12$ inside $\mathbb {R} ^{2}$.
The above equation is equivalent to this new one: ${\tfrac {1}{4}}x+{\tfrac {1}{12}}y=1$. Some authors[1] prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type $({\tfrac {1}{4}},{\tfrac {1}{12}})$.
As noted above, a homogeneous polynomial $g(x,y)$ of degree d is just a quasi-homogeneous polynomial of type (1,1); in this case all its pairs of exponents will satisfy the equation $1i_{1}+1i_{2}=d$.
Definition
Let $f(x)$ be a polynomial in r variables $x=x_{1}\ldots x_{r}$ with coefficients in a commutative ring R. We express it as a finite sum
$f(x)=\sum _{\alpha \in \mathbb {N} ^{r}}a_{\alpha }x^{\alpha },\alpha =(i_{1},\ldots ,i_{r}),a_{\alpha }\in \mathbb {R} .$
We say that f is quasi-homogeneous of type $\varphi =(\varphi _{1},\ldots ,\varphi _{r})$, $\varphi _{i}\in \mathbb {N} $, if there exists some $a\in \mathbb {R} $ such that
$\langle \alpha ,\varphi \rangle =\sum _{k}^{r}i_{k}\varphi _{k}=a$
whenever $a_{\alpha }\neq 0$.
References
1. Steenbrink, J. (1977). "Intersection form for quasi-homogeneous singularities" (PDF). Compositio Mathematica. 34 (2): 211–223 See p. 211. ISSN 0010-437X.
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
• Univariate
• Bivariate
• Multivariate
• Monomial
• Binomial
• Trinomial
• Irreducible
• Square-free
• Homogeneous
• Quasi-homogeneous
Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
|
Wikipedia
|
Compound of five stellated truncated hexahedra
This uniform polyhedron compound is a composition of 5 stellated truncated hexahedra, formed by star-truncating each of the cubes in the compound of 5 cubes.
Compound of five stellated truncated hexahedra
TypeUniform compound
IndexUC58
Polyhedra5 stellated truncated hexahedra
Faces40 triangles, 30 octagrams
Edges180
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
(±(2−√2), ±√2, ±(2−√2))
(±φ, ±(φ−1−φ−1√2), ±(2φ−1−φ√2))
(±1, ±(φ−2+φ−1√2), ±(φ2−φ√2))
(±(1−√2), ±(−φ−2+√2), ±(φ2−√2))
(±(φ−φ√2), ±(−φ−1), ±(2φ−1−φ−1√2))
where φ = (1+√5)/2 is the golden ratio.
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
|
Wikipedia
|
Quasi-identity
In universal algebra, a quasi-identity is an implication of the form
s1 = t1 ∧ … ∧ sn = tn → s = t
where s1, ..., sn, t1, ..., tn, s, and t are terms built up from variables using the operation symbols of the specified signature.
A quasi-identity amounts to a conditional equation for which the conditions themselves are equations. Alternatively, it can be seen as a disjunction of inequations and one equation s1 ≠ t1 ∨ ... ∨ sn ≠ tn ∨ s = t—that is, as a definite Horn clause. A quasi-identity with n = 0 is an ordinary identity or equation, so quasi-identities are a generalization of identities.
See also
• Quasivariety
References
• Burris, Stanley N.; H.P. Sankappanavar (1981). A Course in Universal Algebra. Springer. ISBN 3-540-90578-2. Free online edition.
|
Wikipedia
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.