diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzphba" "b/data_all_eng_slimpj/shuffled/split2/finalzzphba" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzphba" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe discovery of the Higgs boson at the LHC has marked a major step for our understanding of particle physics, and for the construction of the Higgs sector of new physics scenarios. Direct searches for new particles are currently actively persued at the LHC, in particular in the context of supersymmetry (SUSY). No new physics signal has been discovered so far, implying that new physics should be subtle or heavy. Therefore, indirect constraints are at the moment of utmost importance. The measurements of the properties of the Higgs boson can provide in this respect very strong constraints on new physics scenarios. The measurement of its mass at 125 GeV \\cite{Tanabashi:2018oca} is very constraining for supersymmetry, because the Higgs mass can receive large corrections from the stop sector, and has a large impact on the SUSY parameter space~\\cite{Arbey:2011ab}. In the following, we will discuss the status of the Higgs sector of the phenomenological MSSM.\nTo do so, we perform random scans on the 19 parameters of the pMSSM, following the procedures detailled in~\\cite{Arbey:2011un}. In particular, we use a master program based on SuperIso~\\cite{Mahmoudi:2008tp}, generate the MSSM spectra with SOFTSUSY~\\cite{Allanach:2001kg} and compute the Higgs boson decay widths and couplings with HDECAY~\\cite{Djouadi:1997yw}. We keep only the parameter points where the lightest supersymmetric particle is a neutralino (constituting a dark matter candidate) and a light Higgs mass of $125\\pm3$ GeV.\n\n\n\\section{Higgs coupling measurements and SUSY direct searches}\n\nWe first study the interplay of the measurement of the Higgs boson properties and of the results of the SUSY direct searches. We impose the LEP constraints on the SUSY masses \\cite{Tanabashi:2018oca}. To assess the constraints from SUSY searches at the LHC, we generate events with PYTHIA \\cite{Sjostrand:2014zea}, simulate the detector with Delphes \\cite{deFavereau:2013fsa} and obtain constraints from ATLAS and CMS results with 36 fb$^{-1}$~\\cite{Ventura:2017itv} for gluino and squark, neutralino and chargino, stop and sbottom, and monojet searches. For the Higgs measurements, we consider that there are 6 independent effective Higgs couplings, to the photons $\\kappa_\\gamma$, gluons $\\kappa_g$, vector bosons $\\kappa_V$, tops $\\kappa_t$, bottoms $\\kappa_b$ and taus $\\kappa_\\tau$. We combined the ATLAS and CMS measurements of the Higgs couplings at 7+8 TeV \\cite{Khachatryan:2016vau} and 13 TeV \\cite{Brandstetter:2018eju}, and to check if a point is consistent with these measurements, we use a $\\chi^2$ test and keep only points in agreement at 95\\% C.L. In Figure \\ref{figcoups}, we present the photon, gluon and bottom squared coupling distributions as a function of $M_{A}$, applying different sets of constraints. All the shown couplings are sensitive to $M_A$, in addition to other SUSY parameters which modify the couplings at loop level. In particular, the photon and gluon couplings are sensitive to the stop and sbottom masses. The bottom coupling is modified by the $\\Delta_b$ corrections \\cite{Djouadi:2005gj}. We see that the combination of the direct searches and Higgs measurements strongly restricts the coupling values to be close to 1. Since the different couplings are related to SUSY masses, these results can be used to obtain constraints on the pMSSM parameters.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\hspace*{-0.5cm}\\includegraphics[width=.35\\textwidth]{Kgamma2_MA_2.png}\\includegraphics[width=.35\\textwidth]{Kgluon2_MA_2.png}\\includegraphics[width=.35\\textwidth]{Kb2_MA_2.png}\\vspace*{-0.5cm}\n\\end{center}\n\\caption{Distributions of the squared light scalar Higgs couplings to photons (left), gluons (center) and bottoms (right), as a function of $M_{A}$ in the pMSSM. The grey points correspond to all points with $M_h\\sim125$ GeV, the red ones pass in addition the LEP constraints, the blue points are also consistent with LHC SUSY direct searches and the green points are compatible with Higgs coupling measurements.\\label{figcoups}}\n\\end{figure}\n\n\n\\section{Heavy Higgs direct searches and Higgs coupling measurements}\n\nAnother way to constrain the Higgs sector is through searches for heavier Higgs states. We use HDECAY~\\cite{Djouadi:1997yw} and SusHi~\\cite{Harlander:2012pb} to compute the heavy Higgs decay rates and production cross-sections, respectively, and apply the ATLAS and CMS heavy Higgs search limits \\cite{Schaarschmidt:2018ask}. We compare the exclusion from the Higgs coupling measurements to the one from heavy Higgs searches in Figure~\\ref{figheavy}, which reveals the important interplay between the light Higgs coupling measurements and the heavy Higgs search limits: While $(M_A,\\tan\\beta)$ is very strongly constrained by $H\/A \\to \\tau^+\\tau^-$ searches, the $(M_{\\tilde{b}_1},X_b)$ and $(M_2,\\mu)$ parameter planes are more constrained by the Higgs coupling measurements.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\hspace*{-0.5cm}\\includegraphics[width=.35\\textwidth]{tanb_MA_fractionexcluded_higgscouplings.png}\\includegraphics[width=.35\\textwidth]{MSBOT1_Xb_fractionexcluded_higgscouplings.png}\\includegraphics[width=.35\\textwidth]{M2_Mu_fractionexcluded_higgscouplings.png}\\\\\n\\hspace*{-0.5cm}\\includegraphics[width=.35\\textwidth]{tanb_MA_fractionexcluded_lhcheavyhiggs.png}\\includegraphics[width=.35\\textwidth]{MSBOT1_Xb_fractionexcluded_lhcheavyhiggs.png}\\includegraphics[width=.35\\textwidth]{M2_Mu_fractionexcluded_lhcheavyhiggs.png}\n\\caption{Fraction of excluded points by Higgs coupling measurements (top) and heavy Higgs searches (bottom), in the $(M_A,\\tan\\beta)$ (left), $(M_{\\tilde{b}_1},X_b=A_b-\\mu\\times\\tan\\beta)$ (center) and $(M_2,\\mu)$ (right) parameter planes.\\label{figheavy}}\n\\end{center}\n\\end{figure}\n\n\\section{Prospects for the MSSM Higgs sector}\n\n\\begin{table}[t!]\n\\begin{center}\n\n\\begin{tabular}{|c|cc|cc|cc|}\n \\hline\n & $M_A ({\\rm GeV})$ & $\\tan\\beta$ & $M_A ({\\rm GeV})$ & $\\tan\\beta$ & $M_A ({\\rm GeV})$ & $\\tan\\beta$ \\\\ \\hline\n Original parameters & 334.9 & 9.9 & 427.3 & 5.7 & 657.2 & 12.7\\\\\nHL-LHC recontruction & 394$\\pm$40 & 9.6$\\pm$4.0 & 471$^{+341}_{-56}$ & - & - & -\\\\\nILC recontruction & 351$\\pm$23 & 9.2$\\pm$1.9 & 460$^{+54}_{-45}$ & 10.4$^{+6}_{-4}$ & 747.7$^{+302}_{-97}$ & 10.2$^{+20}_{-4}$\\\\\n\\hline\n\\end{tabular}~\\\\[0.1cm]\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nOriginal $\\mu \\tan\\beta~({\\rm TeV})$ & $-149.9$ & $-86.6$ & 0 & 79.6 & 108.6 \\\\ \\hline\n ILC recontruction & $-76.3^{+28}_{-39}$ & $-124.6^{+46}_{-60}$ & $-2.2\\pm22$ & 67.2$^{+39}_{-22}$& 82.5$^{+40}_{-22}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Reconstruction potential of different pMSSM scenarios with HL-LHC and ILC projections.\\label{tabfuture}}\n\\end{center}\n\\end{table}\n\n\n\nWe now study the prospects for the high-luminosity LHC (HL-LHC) run and ILC \\cite{Moortgat-Picka:2015yla}, by considering the possibility to reconstruct specific scenarios using the Higgs coupling measurements. We test two categories of scenarios: the first one where only $M_A$ and $\\tan\\beta$ are varied, and the second where $\\mu \\tan\\beta$ is modified. We assume the accuracy reached when the ILC collects 1 ab$^{-1}$ of luminosity at energies between 350 and 800 GeV. Table~\\ref{tabfuture} summarises our results for several example scenarios (some at the limit of being excluded by current searches). We can conclude that the HL-LHC alone would allow us to reconstruct CP-odd Higgs masses up to 500 GeV. For higher masses, or for scenarios with modified $\\mu \\tan\\beta$, the ILC will be necessary to identify the underlying parameters of the scenario.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main result}\n\n\nA \\emph{circuit} is defined to be a $2$-regular $2$-connected graph. A \\emph{circuit double cover} (CDC) of a cubic graph $G$ is a set $S$ of circuits of $G$ such that every edge of $G$ is covered by exactly two circuits of $S$. A $2$-regular subgraph $D$ of $G$ is said to be contained in $S$ if every circuit of $D$ is an element of $S$. \n\n\nA cubic graph $G$ with a $2$-factor $F_2$ which consists of two chordless circuits is called a \\emph{cycle permutation graph} and\n$F_2$ is called the \\emph{permutation $2$-factor} of $G$. If $G$ is also a snark, then we say $G$ is a \\emph{permutation snark}.\nThe Petersen graph has been for a long time the only known cyclically $5$-edge connected permutation snark.\nIn \\cite{Br} twelve new cyclically $5$-edge connected permutation snarks have been discovered by computer search. Here, we present the first infinite family of cyclically $5$-edge connected permutation snarks. \n\n\n\nWe state the main theorem, see Theorem \\ref{!!} and Corollary \\ref{wichtig}. \n\n\n\\begin{theorem}\n For every $n \\in \\mathbb N$, there is a cyclically $5$-edge connected permutation snarks $G$ of order $10+24n$. Moreover, $G$ has a permutation $2$-factor which is not contained in any CDC of $G$.\n\\end{theorem}\n\n\n\nApplying the above theorem we obtain infinitely many counterexamples to the following conjectures. \n\n\n\n\n\\begin{conjecture}\\label{c2}\\cite{Z1}\nLet $G$ be a cyclically $5$-edge-connected cycle permutation graph. If $G$ is a snark, then $G$ must be the Petersen graph.\n\\end{conjecture}\n\n\n\\begin{conjecture}\\label{c1}\\cite{F}\nIf $G$ is an essentially $6$-edge-connected $4$-regular\ngraph with a transition system $T$, then $(G, T)$ has no compatible cycle \ndecomposition if and only if $(G, T)$ is the bad loop or the bad $K_5$.\n\\end{conjecture}\n\n\n\n\n\\begin{conjecture}\\label{c4} \\cite{BJ, Z1} \nLet $G$ be a cyclically $5$-edge-connected cubic graph and $D$ be a set of pairwise disjoint circuits of $G$. Then $D$ is a\nsubset of a CDC, unless $G$ is the Petersen graph.\n\\end{conjecture}\n\n\n\\begin{conjecture}\\label{c5} \\cite{Z1} \nLet $G$ be a cycle permutation graph with the cordless circuits $C_1$ and $C_2$ where $C_1 \\cup C_2$ is a 2-factor. If $G$ is cyclically $5$-edge-connected\nand there is no CDC which includes both $C_1$ and $C_2$, then $G$ must be the Petersen graph.\n\\end{conjecture}\n\n\nNote that finitely many counterexamples to the above conjectures were found in \\cite{Br} by computer search.\n\n\n\n\\section{Definitions and proofs}\nWe refer to \\cite {Z1} for the definition of a multitpole and an half-edge.\nMoreover, for terminology not defined here we refer to \\cite{Bo}. We use a more general definition of a CDC than the one stated in the introduction. \n\n\n\\begin{definition}\\label{pcdc}\nWe say a set $S=\\{A_1,A_2,...,A_m\\}$ is a path circuit double cover (PCDC) of a graph $G$ if the the following is true\n\n\n1. $A_i$ is a subgraph of $G$ where every component of $A_i$ is either a circuit or a path with both endvertices being vertices of degree $1$ in $G$, $ \\forall \\,i \\in \\{1,2,...,m\\}$. \n\n\n2. $\\sum_{i=1}^m |e \\cap E(A_i)|=2 \\,\\,\\, \\forall \\,e \\in E(G)$. \n\n\\end{definition}\n\n\n\n\nIf no $A_i$ contains a path as a component, then we call $S$ a \\emph{CDC} of $G$ and if $|S|=k$, then we call $S$ a \\emph{$k$-CDC} of $G$. Obviously, a PCDC is a CDC if $G$ contains no vertex of degree $1$. For a survey on CDC's, see \\cite{Z1,Z2}.\n\n\n\nLater we need the following known lemma \\cite{Z1}.\n\n\\begin{lemma}\\label{bla}\nLet $G$ be a $3$-edge colorable cubic graph and $D$ be a $2$-regular subgraph of $G$. Then $G$ has a $4$-CDC $S$ with $D \\in S$. \n\\end{lemma}\n\n\n\\begin{definition}\nLet $A \\in S $ be given where $S$ is a PCDC of a graph $G$. Let $e$ be an half-edge or edge of $A$, then $[e]$ denotes the \nunique element of $S$ which contains $e$ and which is not $A$. We say $[\\,]$ refers to $A$.\n\\end{definition}\n\n\n\n\\begin{definition}\\label{c5}\nLet $Q^i$ with $i \\in \\{1,2,...,4\\}$ be a cyclically $5$-edge connected permutation snark with a permutation $2$-factor $F^i$ such that $F^i$ is not contained in any CDC of $Q^i$. The two circuits of $F^i$ are denoted by $C^i_1$ and $C^i_2$. We may assume w.l.o.g. that $C^i_1$ ($C^i_2$) contains a subpath which has the following vertices in the following consecutive order: $x^i_1$, $x^i_2$, $z^i_2$, $x^i_6$ ($x^i_4$, $z^i_1$, $x^i_5$), such that $z^i_1z^i_2 \\in E(Q^i)$.\n\\end{definition}\n\n\n\\begin{definition}\nLet $\\widetilde{ Q }^{i}$ be the graph which is obtained from $Q^i$ by removing the edge $x^i_1x^i_2$, the vertices \n$z^i_1$, $z^i_2$ and by adding the vertex $y^i_j$, $j=1,2,...,6$ and the edges \n$e^i_3:=x^i_2y^i_3$, $e^i_s:=x^i_sy^i_s$, $s=1,2,4,5,6$, see Figure 1. \n\\end{definition}\n\n\n\nBy an \\emph{end-edge} of a graph $G$, we mean an edge which is incident with a vertex of $G$ with degree $1$ in $G$. \n\n\\begin{definition}\\label{A}\nThe six end-edges of $\\widetilde{Q}^i$ together with the remaining edges of $F^i$ in $Q^i$ induce the following three paths in $\\widetilde{Q}^i$:\nthe path $A^i_1$ with end-edges $e^i_1$ and $e^i_6$,\nthe path $A^i_2$ with end-edges $e^i_4$ and $e^i_5$ and the path $A^i_3$ with end-edges $e^i_2$ and $e^i_3$.\nMoreover, set $A^i:= A^i_1 \\cup A^i_2 \\cup A^i_3$ and $\\mathbb A^i := \\{A^i_1, A^i_2, A^i_3 \\}$.\n \\end{definition}\n\n\n\\begin{figure}[htpb]\n\\centering\\epsfig{file=Aiii.eps,width=3.2in}\n\\caption{The graph $\\widetilde{Q}^i$, $i \\in \\{1,2,3,4\\}$.} \\label{color1}\n\\end{figure}\n\n\nWe recall that $[\\,]$ refers to the given element of a PCDC or CDC. We need the following propositions and lemmas for proving Theorem \\ref{!!}. \n\n\n\n\\begin{proposition}\\label{qa}\nLet $\\widetilde{Q}^i$ and $A^i$ with $i \\in \\{1,2,...,4\\}$ be defined as above.\nThen every PCDC $S$ of $\\widetilde{Q}^i$ with $A^i \\in S$ satisfies the following\n\n\n$(1)\\,\\,$ If $[e^i_4] \\not= [e^i_5]$, then $[e^i_1] \\not\\in \\{[e^i_2],[e^i_3]\\}$. If $[e^i_1] \\in \\{[e^i_2],[e^i_3]\\}$, then $[e^i_4] = [e^i_5]$.\\\\\n$(2)\\,\\,$ $[e^i_2]\\not=[e^i_3]$.\n\\end{proposition}\n\n\n\n\n\nProof. Suppose by contradiction that one of the two conclusions of $(1)$ is not fulfilled. Then $S$ implies a CDC of $Q^i$ containing $F^i$ which contradicts the definition of $Q^i$. If $(2)$ is not fulfilled, then the edge $e \\not\\in A^i_3$ which is incident with $x^i_2$ cannot be covered \nby $S$ which is impossible. Hence, the proof is finished. \n\n\n\n\\begin{corollary}\\label{ww}\nLet $S$ with $A^i \\in S$ be a PCDC of $\\widetilde{Q}^i$ with $i \\in \\{1,2,...,4\\}$. If $[e^i_4] \\not= [e^i_5]$, then\n\n\n$(1)\\,\\,$ $|\\{[e^i_1],[e^i_2],[e^i_3]\\}|=|\\{[e^i_4],[e^i_5],[e^i_6]\\}|=3$.\\\\\n$(2)\\,\\,$ $|\\{[e^i_1],[e^i_2],...,[e^i_6]\\}|=3$.\n\\end{corollary}\n\n\nProof. Since by Proposition \\ref{qa}, $[e^i_2] \\not= [e^i_3]$ and $[e^i_1] \\not\\in \\{[e^i_2],[e^i_3]\\}$, the corollary follows.\n\n\n\\begin{definition}\nDenote by $P^i$ with $i \\in \\{1,2,3,4\\}$ the connected multipole which is obtained from $\\widetilde{Q}^i$ by transforming every end-edge of $\\widetilde{Q}^i$ \ninto an half-edge except for $i=1$, $e^1_2$; for $i=2$, $e^2_3$; for $i=3$, $e^3_2$; and for $i=4$, $e^4_3$. \n\\end{definition}\n\n\n\\begin{definition}\\label{Hcon}\nDenote by $H(Q^1,Q^2,Q^3,Q^4)$ or in short by $H$ the cubic graph which is constructed from $P^1$, $P^2$, $P^ 3$ and $P^4$, as illustrated in Figure 2, by gluing together half-edges, identifying vertices of degree $1$ and by adding the edge $\\alpha$. \n\\end{definition}\n\n\nNote that we keep in $H$ the edge labels of $P^i$, respectively, of $Q^i$, see Figure 2. \n\n\n\\begin{definition}\nLet $A \\in \\mathbb A^i$ (Def. \\ref{A}), i.e. $A \\subseteq \\widetilde {Q}^i$, $i \\in \\{1,2,3,4\\}$. Then $A \\subseteq H$ \nis defined to be the path in $H$ containing all edges of $H$ which have the same edge-labels as $A \\subseteq \\widetilde {Q}^i$; if an edge $e$ of \n$A \\subseteq \\widetilde {Q}^i$ corresponds to an half-edge of $H$ then the edge of $H$ which contains $e$ is defined to be part of $A \\subseteq H$.\n\\end{definition}\n\n\n\n\\begin{definition}\\label{c1c2}\nDenote by $F$ the permutation $2$-factor of $H$ with $F:=A^1 \\cup A^2 \\cup A^3 \\cup A^4$ and $A_i \\subseteq H$, $i=1,2,3,4$, see Figure 1, 2 and 3.\n\\end{definition}\n\n\n\n\n\n\n\\begin{figure}[htpb]\n\\centering\\epsfig{file=Hnew.eps,width=3.2in}\n\\caption{The graph $H$.} \\label{color1}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\centering\\epsfig{file=Trans2.eps,width=3.2in}\n\\caption{The permutation $2$-factor $F$ of $H$ and $\\alpha \\not\\in E(F)$.} \\label{ff}\n\\end{figure}\n\n\\begin{lemma} \\label{1}\nLet $S$ be a CDC of $H$ with $F \\in S$. Then $[e^i_4] \\not= [e^i_5]$ for some $i \\in \\{1,2,3,4\\}$. \n\\end{lemma}\n\n\nProof by contradiction. Consider $P^3$ in Figure 2. Since $[e^3_4]=[e^3_5]$ we obtain $[e^4_2]=[e^4_1]$. Therefore and since $[e^4_4]= [e^4_5]$\nit follows that $[e^4_3]= [e^4_6]$. Consider $P^1$. By analogous arguments, $[e^2_3]= [e^2_6]$. Since $[e^2_6]= [e^4_6]$ we obtain $[e^4_3]= [e^2_3]$ which is impossible.\n\n\n\n\\begin{lemma} \\label{2} \nLet $S$ be a CDC of $H$ with $F \\in S$ and let $[e^i_4] \\not= [e^i_5]$ for some $i \\in \\{1,2,3,4\\}$. Then $[e^i_4] \\not= [e^i_5]$ $\\forall i \\in \\{1,2,3,4\\}$. \n\\end{lemma}\n\n\n\nProof. Let $[e^1_4] \\not= [e^1_5]$. Then by Proposition \\ref{qa} $(1)$, $[e^1_1] \\not= [e^1_3]$. Hence $[e^4_4] \\not= [e^4_5]$ and by \nProposition \\ref{qa} $(1)$, $[e^4_1] \\not= [e^4_2]$. Hence $[e^3_4] \\not= [e^3_5]$ and thus by Proposition \\ref{qa} $(1)$, $[e^3_1] \\not= [e^3_3]$ implying $[e^2_4] \\not= [e^2_5]$. Each of the three remaining cases to consider, i.e. $[e^i_4] \\not= [e^i_5]$, $i=2,3,4$, can be proven analogously.\n\n\n\nCorollary \\ref{ww}, Lemma \\ref {1} and Lemma \\ref{2} imply the following proposition.\n\n\n\\begin{proposition}\\label{!}\nLet $S$ be a CDC of $H$ with $F \\in S$. Then the following is true for $i=1,2,3,4$. \n\n\n$(1)\\,\\,$ $[e^i_4] \\not= [e^i_5]$.\\\\\n$(2)\\,\\,$ $|\\{[e^i_1],[e^i_2],[e^i_3]\\}|=|\\{[e^i_4],[e^i_5],[e^i_6]\\}|=3$.\\\\\n$(3)\\,\\,$ $|\\{[e^i_1],[e^i_2],...,[e^i_6]\\}|=3$.\n\\end{proposition}\n\n\n\nFor the proof of the next theorem we form a new cubic graph $H'$ from $H$. Consider for this purpose $P^i$, $i=1,2,3,4$ in Figure 2 as a vertex of degree $6$ and split every $P^i$ into two vertices $v^i$ and $w^i$ such that $v^i$ ($w^i$) is incident with $e^i_j$, $j=1,2,3$ ($\\,e^i_j$, $j=4,5,6$) to obtain $H'$. For reasons of convenience we do not use the edge-labels of $H$ for $H'$, see Figure 4. \n\n\n\\begin{figure}[htpb]\n\\centering\\epsfig{file=Hsplit.eps,width=3.3in}\n\\caption{The graph $H'$.} \\label{color1}\n\\end{figure}\n\n\n\\begin{theorem}\\label{!!}\nLet $S$ be CDC of $H$, then $F \\not\\in S$. \n\\end{theorem}\n\n\nProof. Note that for every CDC $S$ of $H$ with $F \\in S$, $\\{[e^1_2],[e^3_2]\\}=\\{[e^2_3],[e^4_3]\\}$. \n\n\n$(2)$ and $(3)$ in Proposition \\ref{!} imply that it suffices to show that there is no proper edge-coloring $f: \\{E(H')- \\alpha\\} \\mapsto \\mathbb N$ such that $v^i$ and $w^i$ in $H'$ are incident with the same colors, $\\forall \\,i \\in \\{0,1,2,3,4\\}$. Let $v \\in V(H')$, then $E_v$ denotes the edge-set containing all edges of $H'$ incident with $v$. We proceed by contradiction. There are two cases to consider. \n\n\nCase 1. $f(a_1)= f(a_3)=1$ and $f(a_2)= f(a_4)=2$.\\\\\nSince $f(a_1)=1$ and $a_1 \\in E_{v^1}$ there is $x \\in E_{w^1}$ with $f(x)=1$. Since $f(a_3)=1$, $f(a_{11})=1$. Since $f(a_2)=2$ and since $a_2 \\in E_{v^3}$ there is $y \\in E_{w^3}$ with $f(y)=2$ which is impossible since $f(a_{11})=1$ and $f(a_4)=2$.\n\n\n\nCase 2. $f(a_1)= f(a_4)=1$ and $f(a_2)= f(a_3)=2$.\\\\\nSince $f(a_3)=2$ and $a_3 \\in E_{v^2}$ there is $x \\in E_{w^2}$ with $f(x)=2$. Since $f(a_2)=2$, $f(a_{12})=2$. \nSince $f(a_4)=1$ and $a_4 \\in E_{v^4}$ there is $y \\in E_{w^4}$ with $f(y)=1$ which is impossible since $f(a_1)=1$ and $f(a_{12})=2$.\n\n\nThe \\emph{cyclic edge-connectivity} of a graph $G$ which contains two vertex-disjoint circuits is denoted by $\\lambda_c(G)$; it is the minimum number of edges one needs to delete from $G$ in order to obtain two components such that each of them contains a circuit. In order to show that $\\lambda_c (H) > 4$, we need several results.\n\n\\begin{definition}\nLet $G$ be a graph with a given $2$-factor $F_2$ consisting of two chordless circuits. We call $e \\in E(G)$ a spoke if $e \\not\\in E(F_2)$.\n\\end{definition}\n\n\n\\begin{proposition}\\label{perch}\nLet $G$ be a cubic graph with a $2$-factor $F_2$ consisting of two chordless circuits $C_1$, $C_2$.\n\n \n(1) Let $|V(G)| \\geq 8$, then $\\lambda_c (G) \\geq 4$.\\\\\n(2) Let $|V(G)| \\geq 10$. Then every cyclic $4$-edge cut $E_0$ of $G$ is a matching and $|E_0 \\cap C_i|=2$, $i=1,2$. \n\\end{proposition}\n\n\nProof. Suppose by contradiction that $M$ is a cyclic $3$-edge cut of $G$. Obviously, $M$ is matching. First we show that $M$ contains no spoke and consider two cases.\n\n\\emph{Case 1.} $M$ contains two or three spokes. \\\\ It is straightforward to see that $G-M$ is connected and thus this is impossible.\n\n\\emph{Case 2.} $M$ contains exactly one spoke. \\\\If $|M \\cap E(C_i)| =1$ for $i=1,2$ then $C_1-M \\subseteq G$ is a path which is connected by more than one spoke to $C_2-M \\subseteq G$. Thus $G-M$ is connected. Hence w.l.o.g. $|M \\cap E(C_1)|=2$. Since both paths of $C_1-M$ contain more than one vertex, both paths in $G$ are connected by more than one spoke to $C_2$ and thus $G-M$ is connected. Hence $M$ contains no spoke. \n\nSuppose $|M \\cap E(C_1)|=2$ and thus $|M \\cap E(C_2)|=1$. Then both paths of $C_1-M \\subseteq G$ are connected by more than one spoke to the path $C_2-M \\subseteq G$. Hence $G-M$ is connected which contradicts the assumption and thus finishes the first part of the proof.\n\n\nSuppose $E_0$ contains a spoke $s$. For every subdivision $Z'$ of a cubic graph $Z$, $\\lambda_c (Z')= \\lambda_c (Z)$. Thus and by the first statement of the Proposition, $G-s$ is cyclically 4-edge connected. Hence $s \\not\\in E_0$. \\\\\nSuppose $E_0$ is not a matching and let $a_1$ be adjacent with $a_2$ where $\\{a_1,a_2\\} \\subseteq E_0 \\cap E(C_1)$. Let $s$ be the unique spoke which is adjacent with $a_1$ and $a_2$. Then $E'_0:= E_0 - a_1 \\cup s$ is a cyclic $4$-edge cut of $G$. This contradicts the previous observation that a spoke is not contained in any cyclic $4$-edge cut. Hence $E_0$ is a matching. Suppose $E_0$ contains no (one) edge of $C_1$ and thus four (three) edges of $C_2$. \n$C_2-E_0$ consists of four (three) paths where each of them is connected by a spoke to $C_1- E_0$ which is connected in both case. Hence $G-E_0$ is connected which contradicts the assumption and thus finishes the proof. \n\n\n\nLet $V'$ be a subset of vertices of a graph $G$, then we denote by $\\langle V'\\rangle $ the \\emph{vertex induced subgraph} of $G$. \n\n\n\\begin{lemma} \\label{4cc} \nLet $G$ with $|V(G)| \\geq 10$ be a cubic graph which contains a $2$-factor $F_2$ consisting of two chordless circuits $C_1$,$C_2$. Then the following is true and the analogous holds for $C_2$.\n\n\n(1) $\\lambda_c(G)= 4$ if and only if $C_1$ contains a path $L_1$ with $1 < |V(L_1)| < |V(C_1)|-1 $ such that $L_2:=\\,\\langle N(V(L_1)) \\cap V(C_2) \\rangle $ is a \npath of $C_2$. In particular, the four distinct end-edges of $F_2-E(L_1)-E(L_2)$ form a cyclic $4$-edge cut of $G$. \n\n\n(2) Let $E_0:= \\{a_1,a_2,b_1,b_2\\}$ be a cyclic $4$-edge cut of $G$ where by Prop. \\ref{perch}, w.l.o.g. $\\{a_1,a_2\\} \\subseteq E(C_1)$ and $\\{b_1,b_2\\} \\subseteq E(C_2)$. Denote the two paths of $C_1-a_1-a_2$ ($C_2-b_1-b_2$) by $L'_1$ and $L''_1$ ($L'_2$ and $L''_2$). Then, $\\{\\,\\langle N(V(L'_1)) \\cap V(C_2)\\rangle , \\, \\langle N(V(L''_1)) \\cap V(C_2) \\rangle \\,\\} = \\{L'_2, L''_2\\}$.\n\\end{lemma}\n\n\nProof. We first prove (2). The four paths $L'_1, L''_1, L'_2, L''_2$ decompose the graph $C_1 \\cup C_2 - E_0$. Every neighbor of a vertex of $L'_1$ or $L''_1$ in $C_2$ is thus contained in $L'_2$ or $L''_2$. Hence the equality in (2) does not hold if and only if $L'_1$ or $L''_1$ contains two distinct vertices $x, y$ such that $N(x) \\cap V(C_2) \\in V(L'_2)$ and $N(y) \\cap V(C_2) \\in V(L''_2)$. Let w.l.o.g.\n$\\{x,y\\} \\subseteq V(L'_1)$ and suppose by contradiction that $x$ and $y$ have the before described properties. Then $L'_1 \\subseteq G-E_0$ is connected to $L'_2$ and $L''_2$. Since $L''_1 \\subseteq G-E_0$ is connected to $L'_2$ or $L''_2$, $G-E_0$ is connected which contradicts the assumption and thus finishes this part of the proof.\n \nWe prove (1). Let $\\lambda_c(G)= 4$ and let $E_0$ be a cyclic $4$-edge cut of $G$ as defined in (2). By Proposition \\ref{perch}, $\\{a_1,a_2\\}$ is a matching of $G$. Hence the inequality in (1) is satisfied by setting $L_1:= L_1'$. It is straightforward to check that the four end-edges of \n$F_2-E(L_1)-E(L_2)$ form a cyclic $4$-edge cut of $G$. Hence the proof is finished.\n\n\n\n\\begin{lemma} \\label{kkj} \nLet $A \\subseteq H $ and $A \\in \\mathbb A^i $, $i \\in \\{1,2,3,4\\}$. Then there is no cyclic $4$-edge cut $E_0 $ of $H$ such that $|E_0 \\cap E(A)| =2$. \n\\end{lemma}\n\n\n\n\nProof by contradiction. Set $E_0 \\cap E(A)= \\{a_1,a_2\\}$. By Proposition \\ref{perch}, $E_0$ is a matching. Hence $A \\not= A^i_3$. There are two cases to consider.\n\n\n\\emph{Case 1.} $A= A^i_1$. Set $B:= A^i_2$. $A$ and $B$ belong to different components of $F \\subseteq H$, see Figure 1 and Figure 3. \nDenote by $A^*$ the unique path of $A-a_1-a_2$ which connects one endvertex of $a_1$ with one endvertex of $a_2$.\nSince $E_0$ is a cyclic $4$-edge cut and by Lemma \\ref{4cc} (2) and by the structure of $P^i$, $B^*:=\\langle N(V(A^*)) \\cap V(B) \\rangle $ is a subpath of $B$, see Figure 1. Since $\\{a_1,a_2\\}$ is a matching of $H$, $|V(A^*)| > 1$. Since $|V(A^*)| = |V(B^*)|$, $|V(B^*)| > 1$.\n\nDenote by $w$ the neighbor of $x^i_2$ in $B$. Consider the graph $Q^i$ which was defined for constructing $P^i$, see Figure 1. Then $B^*$ is a path of $C^i_2 \\subseteq Q^i$ (Def. \\ref{c5}) with $1< |V(B^*)|< |V(C^i_2)|-1$ since $\\{z^i_1, w\\} \\cap V(B^*) = \\emptyset $. Moreover, $A^* \\subseteq Q^i$ is a path of $C^i_1$ and every vertex of $A^*$ is adjacent to a vertex of $B^* \\subseteq Q^i$ and vice versa. Therefore and by Lemma \\ref{4cc} (1), $\\lambda_c (Q^i)=4$ which is a contradiction to Def. \\ref{c5}.\n \n \n\\emph{Case 2.} $A= A^i_2$. Set $B:= A^i_1$. Let $A^*$ be the unique path of $A-a_1-a_2$ which connects one endvertex of $a_1$ with one endvertex of $a_2$. Let $C$ denote the component of $F \\subseteq H$ such that $A^* \\not\\subseteq C$. Since $E_0$ is a cyclic $4$-edge cut and by Lemma \\ref{4cc} (2), $B^*:=\\langle N(V(A^*)) \\cap V(C) \\rangle $ is a subpath of $C$. Denote by $w$ the neighbor of $x^i_2$ in $A$. Suppose $w \\in V(A^*)$. Then $x^i_2 \\in V(B^*)$. Since $|V(B^*)| > 1$, $B^*$ contains a vertex $y \\not= x^i_2$. Since $N(V(A^*)-w) \\cap V(C) \\subseteq V(B)$ and since $x^i_2$ is not adjacent to a vertex of $B$, $B^*$ is not a path which is a contradiction. Hence, $w \\not\\in V(A^*)$. Similar to Case 1, $A^* \\subseteq Q^i$ is a path of $C^i_2$ with $\\{z^i_1, w\\} \\cap V(A^*) = \\emptyset$ where every vertex of $A^*$ is adjacent to a vertex of the path $B^* \\subseteq Q^i$ and vice versa. Hence, by applying the same arguments as in Case 1, the proof is finished.\n\n\n\nWe need the following observations and notations for the proof of Theorem \\ref{pppp}.\n\n\\begin{definition}\nLet $X$ be a path with $|V(X)|>2$. Then $oX$ is the subpath of $X$ which contains all vertices of $X$ except the two endvertices of $X$. \n\\end{definition}\n\n\nLet $R$ be a set of subgraphs of $H$, then $E(R)$ denotes the union of the edge-sets of the subgraphs of $R$. \nNote that the vertices of the graph $J$ defined below are not the vertices of $H'$. \n\n\n\\begin{figure}[htpb]\n\\centering\\epsfig{file=pintervall.eps,width=3.6in}\n\\caption{The graph $J$.} \\label{inp}\n\\end{figure}\n\n\n\\begin{definition}\\label{JJ}\nLet $C_1$, $C_2$ denote the two circuits of $F \\subseteq H$ where $e^1_2 \\in E(C_1)$, see Figure 3.\nSet $R:= \\{oA^i_j|\\,i \\in \\{1,2,3,4\\} \\textrm { and } j \\in \\{ 1,2,3 \\} \\}$. Let $J:=H\/R$ be the the graph which is obtained from $H$ by contracting every element of $R$ in $H$ to a distinct vertex and by then replacing every multiple-edge by a single edge, see Figure \\ref{inp}. The edges of $E(F)-E(R)$ induce a $2$-factor of $J$ which consists of two circuits $D_1$, $D_2$, see Figure 5. \n\\end{definition}\n \nNote that only subpaths of $F \\subseteq H$ are contracted in the transformation from $H$ into $J$ and that $C_k \\subseteq H$ is transformed into $D_k \\subseteq J$, $k=1,2$. We keep the labels of the edges respectively half-edges of $E(F)-E(R) \\subseteq E(H)$ for $E(D_1) \\cup E(D_2) \\subseteq E(J)$, see Figure 5. Every $oA^i_j \\subseteq H$ corresponds to a vertex of $J$ and $v_4w_6 \\in E(J)$ corresponds to $\\alpha \\in E(H)$. \nSet $\\mathbb H:= \\{V(oA^i_j) \\,|\\, i=1,2,3,4 \\textrm { and } j=1,2,3\\} \\cup \\{v_4\\} \\cup \\{w_6\\}$. Then $\\mathbb H$ is a vertex partition of \n$V(H)$. Every $v \\in V(J)$ corresponds to an element $\\hat v$ of $\\mathbb H$ and vice versa.\n\n\nLet $h: V(H) \\mapsto V(J)$ be the mapping where $h(x)$ is defined to be the unique $v \\in V(J)$ such that $x \\in \\hat v$.\\\\ \nLet $e \\in E(C_k)-E(R)$, then $h'(e)$ denotes the corresponding edge of $D_k \\subseteq J$, $k =1,2$.\\\\\nLet $X$ be a subpath of $C_k \\subseteq H$, then $X\/R$ denotes the subpath of $D_k \\subseteq J$ with $V(X\/R):= \\{h(v)\\,|\\, v \\in V(X)\\}$ and \n$E(X\/R):= \\{h'(x)\\,|\\,x \\in E(X) \\cap (E(F)-E(R))\\}$. \n\n\n\n\\begin{theorem}\\label{pppp}\n$H$ is a cyclically $5$-edge connected permutation snark.\n\\end{theorem}\n\n\n\nProof. By Theorem \\ref{!!}, $F$ is not contained in a CDC. Hence Lemma \\ref{bla} implies that $H$ is not $3$-edge colorable. \nIt remains to show that $\\lambda_c (H) > 4$.\n\n\nSuppose by contradiction that $E_0:=\\{a_1,a_2,a_3,a_4\\}$ is a cyclic $4$-edge cut of $H$ and let \nby Proposition \\ref{perch} (2), w.l.o.g. $\\{a_1,a_2\\}$ be a matching of $E(C_1)$. Let $X$, $X'$ denote the two components of $C_1-a_1-a_2$. $X\/R$ and $X\/R'$ are edge disjoint paths of $D_1 \\subseteq J$ with $|E(X\/R)| + |E(X'\/R)| \\leq 7$. Let w.l.o.g. $|E(X\/R)| \\leq 3$. Then, $2 \\leq |V(X\/R)| \\leq 4$ since by Lemma \\ref{kkj}, $\\{a_1,a_2\\} \\not\\subseteq E(A^i_j)$ for $i=1,2,3,4$ and $j=1,2,3$. Set $Y:= X\/R$.\n \n \nBy Lemma \\ref{4cc} (2), $X^N := \\langle N(X) \\cap V(C_2) \\rangle $ is a path in $C_2$. Hence $Y^*:=X^N\/R$ is a path of $D_2 \\subseteq J$. Since only subpaths of $F \\subseteq H$ are contracted in the construction of $J$ and since every vertex of $X \\subseteq C_1$ is adjacent to a vertex of $X^N \\subseteq C_2$ and vice versa, every vertex of $Y$ is adjacent to a vertex of $Y^*$ and vice versa. We make the following observation. \n\n\nIf $v$ is an endvertex of $Y$, then $|(N(v) \\cap V(D_2)) \\cap V(Y^*)| \\geq 1$.\\\\\nIf $|V(Y)| > 2$ and $v$ is an inner vertex of $Y$, then $N(v) \\cap V(D_2) \\subseteq V(Y^*)$.\n \n\nSet $U:= N(V(oY)) \\cap V(D_2)$. Denote by $v_s$ and $v_t$ the two distinct endvertices of $Y$. Then, $Y^*$ = $\\langle S \\cup T \\cup U\\rangle $ for some $S \\subseteq N(v_s) \\cap V(D_2)$ and for some $T \\subseteq N(v_t) \\cap V(D_2)$. \n\n\nIf $V(Y)=\\{v_1,v_2\\}$ then $V(Y^*) \\in \\{\\{w_1,w_5\\},\\{w_1,w_3\\},\\{w_1,w_3,w_5\\}\\}$.\nWe abbreviate this conclusion by only writing the indices: $12 \\rightsquigarrow 15,13,135$. We recall that $2 \\leq |V(Y)| \\leq 4$. Analogously, we consider for $Y$ the following cases where $Y \\not= \\,\\langle \\{v_6,v_7,v_1,v_2\\} \\rangle $.\n\n\n$23\\rightsquigarrow 13,15,135$; $34 \\rightsquigarrow 16$; $45 \\rightsquigarrow 46$; $56 \\rightsquigarrow 42,47,247$; $67 \\rightsquigarrow 24,47,247$; $71 \\rightsquigarrow 14$.\n\n\n$123 \\rightsquigarrow 135$; $234 \\rightsquigarrow 136,156,356$; $345 \\rightsquigarrow 146$; $456 \\rightsquigarrow 2467,246,467$; $567 \\rightsquigarrow 247$; \n$671 \\rightsquigarrow 124,147,1247$; $712 \\rightsquigarrow 134,145,1345$.\n\n\n$1234 \\rightsquigarrow 1356$; $2345 \\rightsquigarrow 1346,1456,13456$; $3456 \\rightsquigarrow 1246,1467,12467$; $4567 \\rightsquigarrow 2467$;\n$5671 \\rightsquigarrow1247$; $7123 \\rightsquigarrow 1345$.\n\n\nIn none of the above cases, $V(Y^*)$ can be the vertex set of a path in $D_2$ contradicting the assumption that $Y^*$ is a path.\n\n\nThus it remains to consider $Y:= \\langle \\{v_6,v_7,v_1,v_2 \\} \\rangle $. Since $v_1 \\in V(Y) \\subseteq V(D_1)$ (see the half-edges incident with $v_1$ in Figure 5), the path $X \\subseteq C_1$ contains all vertices of $oA^1_1$ (Figure 1). The vertex $x^1_2 \\in V(H)$ (Figure 1) corresponds to $v_3 \\in V(D_1)$ (Figure 5) with $v_3 \\not\\in V(Y)$. Hence, $x^1_2 \\in V(C_1)$ and $x^1_2 \\not\\in V(X)$. Moreover, $x^1_2$ is neither adjacent to $x^1_4$ nor to $x^1_5$; otherwise $Q^1$ contains a circuit of length $4$ which contradicts Def. \\ref{c5}. Hence, $\\{x^1_4,x^1_5\\} \\subseteq V(X^N)$.\n\n\nBy the structure of $Q_1$ (Figure 1) and since $oA^1_1 \\subseteq X$, $X^N$ contains every vertex of $oA^1_2-w$ where $w$ denotes the neighbor of $x^1_2$ in $oA^1_2$. Thus, and since $X^N$ is a subpath of $C_2$, and since $\\{x^1_4,x^1_5\\} \\subseteq V(X^N)$, $|V(X^N)| = |V(C_2)|-1$. Since $|V(X)| = |V(X^N)|$ and since $\\{a_1,a_2\\}$ in the definition of $X$ is a matching, this is impossible which finishes the proof. \n\n\n\n\nDenote by $P_{10}$ the Petersen graph.\n\n\\begin{definition}\\label{Hmenge}\n Set $\\mathcal H:= \\bigcup_{n=0}^\\infty \\{H_n\\}$ where $H_n:=H(H_{n-1},P_{10},P_{10},P_{10})$ and \n$H_0:=P_{10}$, see Definition \\ref{Hcon}.\n\\end{definition}\n\n \nNote that in the above definition we use the graph $H_n$, respectively, $P_{10}$ as $Q^i$ and thus suppose that two subpaths in the known \npermutation $2$-factors of $H_n$ and $P_{10}$ are chosen as the paths specified in Definition \\ref{c5}. \n\n\n\n\n\\begin{corollary}\\label{wichtig}\n$\\mathcal H$ is an infinite set of cyclically $5$-edge connected permutation snarks where $H_n \\in \\mathcal H$ has $10+24n$ vertices.\n\\end{corollary}\n\n\n\nHence and by Theorem \\ref{!!}, we obtain the following corollary.\n\n\\begin{corollary}\nFor every $n \\in \\mathbb N$, there is a counterexample of order $10+24n$ to Conjecture \\ref{c2}, Conjecture \\ref{c4} and Conjecture \\ref{c5}.\n\\end{corollary}\n\n\n\\begin{corollary}\nFor every $n \\in \\mathbb N$, there is a counterexample $G$ of order $5+12n$ to Conjecture \\ref{c1}.\n\\end{corollary}\n\n\n\nProof. Set $H:= H_n$, see Definition \\ref{Hmenge}. Contract every spoke of $H$ with respect to $F$ to obtain a $4$-regular graph $G$ of order $5+12n$. Then $G= C'_1 \\cup C'_2$ where $C'_1$ and $C'_2$ are two edge-disjoint hamiltonian circuits of $G$ which correspond to $C_1$ and $C_2$ in $H$. Hence, every edge-cut $E_0$ of $G$ has even seize. Suppose that $E_0$ is an essential $4$-edge cut of $G$. Since $G$ is $4$-regular, every component of $G-E_0$ has more than $2$ vertices. It is straightforward to see that then $E_0$ corresponds to a cyclic $4$-edge cut of $H$ which contradicts $\\lambda_c(H)=5$. Thus and since $E_0$ is of even size, $G$ is essentially $6$-edge connected. By defining that every pair of two edges which are adjacent and part of $C'_i$ for some $i \\in \\{1,2\\}$ from a transition, we obtain a transition system $T(G)$ of $G$. Since every compatible cycle decomposition of $T(G)$ would imply a CDC $S$ of $H$ which contains $F$ and thus would contradict Theorem \\ref{!!}, the proof is finished.\n\n\n\\textbf{Acknowledgement.}\nA. Hoffmann-Ostenhof thanks R. H\\\"aggkvist for the invitation to the University of Umea where part of the work has been done. \n\n\n\\footnotesize\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{General outline}\n\n \n \n \n \n\n\n\\section {Introduction}\n\\label{sec:intro}\nColloidal semiconductor nanocrystals (NCs) offer an idealized test bed to explore the behavior of excitons and multiexcitons from the discrete, molecular limit to the continous, bulk limit.\\cite{Alivisatos1996,Scholes2006NatMater,Klimov2014,EfrosBrus2021,PGS2021,Kagan2021} At low excitation energies, NCs have discrete spectra due to quantum confinement effects, which resemble those of atoms and molecules, while at higher excitation energies, due to increasingly large densities of states, their spectra converge to the bulk continuum limit. Understanding the interplay of degeneracy, size, shape, and material composition on NC electronic structure has been the subject of numerous studies over the past several decades.\\cite{Efros1982,Murray1993,Wang1994,Dabbousi1997,Efros2000,Scholes2006NatMater,Rabani2010,Boles2016,Weiss2021}\n\nFrom a theoretical perspective, the description of excitons and multiexcitons in semiconductor NCs poses several challenges. Because NCs contain several hundreds to thousands of atoms and valence electrons, quantum chemistry techniques that were developed to study the excited states of molecules are far too computationally expensive to be applicable to NCs. Thus, early work focused on the development of continuum approaches starting from the bulk limit, within a family of effective mass models, to describe the quantum confinement of excitons and dielectric screening.\\cite{Efros1982,Rossetti1983} The most popular single-parabolic band approximation provides a \\textit{qualitative} description of the optical properties of NCs, but it does not account for non-parabolic effects and valence-band degeneracies that are important in NCs. A more \\textit{quantitative} description based on the multi-band effective mass model revealed rich behavior and provided accurate predictions of the exciton fine structure and band-edge exciton splittings as well as their dependence on the size, shape, and crystal structure of the NC.\\cite{Ekimov1993,Norris1994,Norris1996a,Efros2000} In addition, the inclusion of many-body exchange interactions of electrons and holes resulted in optically forbidden dark excitons and explained the non-monotonic temperature dependence of the radiative lifetime in NCs.\\cite{Nirmal1994,Nirmal1995,Norris1996b,Efros1996}\n\nDespite the significant progress made based on the effective mass model, the lack of an underlying atomistic description has limited the application primarily to the description of optical properties, which are less sensitive to the atomistic detail of the NC, particularly in the strong confinement limit ($R < a_{\\text{B}}$, where $R$ is the NC radius and $a_{\\text{B}}$ is the bulk exciton Bohr radius). To account for inhomogeneities in the NC structure, semiempirical pseudopotential models, which became popular in the 1970s and 1980s to describe the electronic and optical properties of bulk semiconductors and surfaces,\\cite{Cohen1966} were employed and further developed to study excitons in a variety of semiconductor NCs,\\cite{Friesner1991,Wang1994,Wang1996,Rabani1999b} demonstrating remarkable success in postdicting and predicting the exciton fine structure\\cite{Norris1996b} as well as the roles of defects,\\cite{Califano2005,Jasrasaria2020} stress, and strain on the electronic structure.\\cite{Wang1999, Mattila1999}\n\nIn recent years, growing interest in the dynamics of excitons inspired by novel experimental observations,\\cite{GS1999,Klimov2000,GS2005,Oron2007,Pandey2008,Jones2009,Sukhovatkin2009,McArthur2010,Ulbricht2011,Knowles2011,Bae2013,Qin2014,Kambhampati2015,Wu2016,Li2017c,Kaledin2018,Li2019} has shifted the focus for theory to address issues related to the transients of these nonequilibrium species.\\cite{Oleg2009} Understanding the radiative and nonradiative decay channels depicted in Fig.~\\ref{fig:AllProcesses} as well as the dephasing and energy transfer mechanisms of confined excitons, which are dictated by the exciton-phonon and exciton-exciton couplings, is key to the rational design of NC-based technologies with reduced thermal losses and increased quantum yields. Two central decay channels are the main focus the current perspective.\n\n\\begin{figure*}[ht]\n\\includegraphics[width=\\textwidth]{figures\/Fig1_Processes.pdf}\n\\caption{\\label{fig:AllProcesses}Photoexcitation of a nanocrystal can create multiple electron-hole pairs, which quickly relax to the band edge in a process called cooling. From the band edge, the multi-excitonic state can undergo Auger recombination, which nonradiatively annihilates one exciton and forms a hot electron-hole pair. The hot carriers then cool back to the band edge, from which they can radiatively recombine.}\n\\end{figure*}\n\nThe first decay channel is that of nonradiative relaxation of hot excitons, or cooling, illustrated in Fig.~\\ref{fig:AllProcesses}. The study of this process is motivated by conflicting results for the relaxation times of hot excitons to the band edge in confined structures relative to those of excitons in bulk.\\cite{Gfroerer1996,Haiping1996,Heitz1997,Sosnowski1998,Mukai1998,GS1999,Klimov1999,Klimov2000a,Harbold2005,GS2005} Due to the discrete nature of the excitonic levels in confined NCs, exciton cooling \\textit{via} phonon emission, especially near the band edge, has been thought to require multi-phonon processes and would, therefore, be inefficient, a phenomenon known as the phonon bottleneck.\\cite{Nozik2001} One mechanism for breaking the phonon bottleneck that allows for fast cooling is the Auger process.\\cite{Efros2003} In many NCs, holes relax rapidly to the band edge \\textit{via} phonon emission because valence band degeneracies and a larger hole effective mass lead to a higher density of hole states with smaller energy spacings that are on the order of the phonon frequencies. An electron, then, can relax to the band edge by nonradiatively transferring its energy to a hole \\textit{via} an Auger-like process, and the re-excited hole can quickly relax back to the valence band edge. The Auger-assisted cooling mechanism~\\cite{Wang2003} has been supported by experimental observations~\\cite{Klimov1999,Klimov2000a,Hendry2006} but, as far as we know, the exciton cooling mechanism has not been confirmed or validated by atomistic calculations,\\cite{Kilina2009} mainly because of the significant computational challenges of describing excitons and their coupling to phonons in systems containing thousands of atoms and valence electrons.\n\nThe second decay channel involves the nonradiative decay of multiexcitonic states and is motivated by the observation of a ``universal volume scaling law\" for Auger recombination (AR) lifetimes in NC.\\cite{Klimov2000,Robel2009} At high photo-carrier densities, which are typical of most optoelectronic devices, all semiconductor materials suffer from enhanced exciton-exciton annihilation that occurs primarily \\textit{via} AR processes, shown in Fig.~\\ref{fig:AllProcesses}, in which one exciton recombines by transferring its energy to another exciton.\\cite{Efros2003} This nonradiative process leads to reduced photoluminescence quantum yields and decreases maximum device efficiencies. Thus, understanding the properties of multiexcitonic states and their decay channels is central to improving and further developing many light-induced NC applications. From a theoretical\/computational perspective, calculating AR lifetimes within Fermi's golden rule requires a description of the initial biexcitonic (or higher multiexcitonic) state and all possible final electron-hole pair states, a challenging task for NCs of experimentally relevant sizes. Thus, previous theoretical works have relied on effective mass continuum models, which ignore electron-hole correlations in the biexcitonic state,\\cite{Li2019a} resulting in a much steeper scaling of the AR lifetimes with the NC volume.\\cite{Chepic1990,Vaxenburg2015,Vaxenburg2016} The discrepancy between theory and experiments had been a mystery for several decades. \n\nIn this perspective, we summarize our recent efforts to develop a unified model that address both problems. In Sec.~\\ref{sec:methods} we describe the atomistic approach we have adopted to calculate quasiparticle excitations and neutral excitations in semiconductor NCs. First principles approaches, such as time-dependent density functional theory (DFT)~\\cite{Chelikowsky2003,Shulenberger2021,Song2022} or many-body perturbation approximations,\\cite{Degoli2009} are limited to describing excitons in relatively small clusters, typically those with fewer than 100 atoms, due to their steep computational scaling.\\cite{Friesner2005,Voznyy2016} To make meaningful contact with experimental results on NCs that contain thousands of atoms and tens of thousands of electrons, we rely on the semiempirical pseudopotential model~\\cite{Wang1994,Wang1996,Rabani1999b,Williamson2000} to describe quasiparticle excitations. We use a converged real-space grid method to represent the single-particle states combined with the filter diagonalization method~\\cite{Wall1995,Toledo2002} to compute the single-particle states near the band edge and at higher excitation energies. We then use a subset of converged quasiparticle eigenstates to solve the Bethe-Salpeter equation~\\cite{Rohlfing2000} within the static screening approximation to account for electron-hole correlations in neutral optical excitations.\\cite{Philbin2018} Sec.~\\ref{sec:methods} also provides validation of the approach for the quasiparticle and optical gaps and the exciton binding energies for II-VI and III-V semiconductor NCs in both the strongly ($Ra_{\\text{B}}$) confined regimes.\n\nIn Sec.~\\ref{sec:exph} we present and assess the accuracy of our approach for determining the exciton-phonon couplings in semiconductors NCs, and we analyze the contributions of acoustic, optical, and surface modes to the overall magnitude of the exciton-phonon couplings. The standard model Hamiltonian that describes a manifold of excitonic states and phonons that are coupled to first order in the atomic displacements is given by:\\cite{Giustino2017}\n\\begin{align}\nH = &\\sum_{n}E_{n}\\left|\\psi_{n}\\right\\rangle \\left\\langle \\psi_{n}\\right|+\\sum_{\\alpha}\\hbar\\omega_{\\alpha}b_{\\alpha}^{\\dagger}b_{\\alpha} \\nonumber \\\\ \n&+\\sum_{\\alpha nm}V_{n,m}^{\\alpha}\\left|\\psi_{n}\\right\\rangle \\left\\langle \\psi_{m}\\right|q_{\\alpha}\\,,\\label{eq:hamiltonian}\n\\end{align}\nwhere $\\left|\\psi_{n}\\right\\rangle $ describes exciton $n$ with energy $E_{n}$, and $b_{\\alpha}^{\\dagger}$ and $b_{\\alpha}$ are the Boson creation and annihilation operators, respectively, of phonon mode $\\alpha$ with frequency $\\omega_{\\alpha}$ and displacement $q_{\\alpha}=\\sqrt{\\frac{\\hbar}{2\\omega_{\\alpha}}}\\left(b_{\\alpha}^{\\dagger}+b_{\\alpha}\\right)$. Describing the nonequilibrium dynamics of excitons requires knowledge of the excitonic transition energies $E_{n}$, the phonon modes and their corresponding frequencies, $\\omega_\\alpha$, and the exciton-phonon couplings, $V_{n,m}^{\\alpha}$. Sec.~\\ref{sec:exph} provides the details for obtaining both the phonon modes using an atomistic force field~\\cite{Zhou2013} and the exciton-phonon couplings directly from the atomistic pseudopotential model described in Sec.~\\ref{sec:methods}. We compare the predictions for the reorganization energies (\\textit{i.e.}, polaron shifts) computed from $V_{n,n}^{\\alpha}$ to experimentally measured Stokes shifts and demonstrate that \\textit{acoustic} modes that are delocalized across the NC contribute more significantly than optical modes to the reorganization energy in all NC systems and sizes. Excitons in smaller NCs are more strongly coupled to modes localized near the surface of the NC, while excitons in larger NCs are more strongly coupled to modes in the interior of the NC. The assessment of the exciton-phonon couplings is essential for addressing the dynamics and mechanism for exciton cooling. This topic is further discussed in Sec.~\\ref{sec:conclusions}.\n\nNext, in Sec.~\\ref{sec:auger} we turn to the ``universal volume scaling law'' for AR lifetimes and present our recent developments for calculating AR lifetimes in NCs that have thousands to tens of thousands of electrons.\\cite{Philbin2018} We demonstrate that the inclusion of electron-hole correlations in the initial biexcitonic state (but not in the final electron-hole state) is imperative to capturing the experimentally observed scaling of AR lifetimes with the size and shape of the NC. In addition, we find that electron-hole correlations are essential for obtaining quantitatively accurate lifetimes and that neglecting such correlations can result in AR lifetimes that are orders of magnitude too long. We demonstrate the strength of our approach for 0D spherical quantum dots, 1D nanorods, and 2D nanoplatelets of varying diameters and lengths. To perform these calculations and compute AR lifetimes, we developed a low-scaling approach~\\cite{Philbin2020a} based on the stochastic resolution of identity,\\cite{Takeshita2017,Dou2019} which is briefly summarized in Sec.~\\ref{sec:auger}.\n\nThe role of interfaces on reorganization energies and AR lifetimes is the central topic of Sec.~\\ref{sec:interfaces}. We focus on core-shell quantum dots and elucidate the shell thickness and band alignment dependencies for quasi-type II CdSe\/CdS and type I CdSe\/ZnS systems. The introduction of interfaces in these heterostructures allows for wave function engineering that affects electron-hole correlations, exciton-phonon couplings, and exciton-exciton interactions, which impacts both the magnitudes of reorganization energies and AR lifetimes. These insights serve as a starting point for realizing NC systems that readily control both exciton-phonon and exciton-exciton interactions, enabling unique, emergent phenomena, such as room-temperature superfluorescence, fast exciton transport, and near-unity photoluminescence quantum yields. Finally, in Sec.~\\ref{sec:conclusions} we summarize the main conclusions and provide an outlook for future directions.\n\n\n\\section{Model Hamiltonian}\n\\label{sec:methods}\nThe diversity of dynamic processes in NCs requires a comprehensive model that captures a wide spectrum of physics. The finite size of NCs modifies the electronic structure relative to the bulk material. The continuous conduction and valence bands of the bulk are split into discrete states for finite crystals, and the quantum confinement of carriers gives rise to NCs' hallmark size-dependent optical properties. To properly describe these optical properties, a model must go beyond the ground state to describe the excited electronic configurations. While these excited states are generally well understood in the bulk, quantum confinement complicates our understanding by significantly enhancing the electron-hole interactions.\\cite{Williamson2000} The small size of NCs compared to the exciton Bohr radius forces the electron and hole closer to each other than they would be in bulk, increasing the strength of their Coulomb interactions. Additionally, dielectric screening is reduced at the nanoscale as quantum confinement widens the band gap and increases the energy required to polarize the medium. This effect leads to a size-dependent reduction in screening, further contributing to size-dependent modifications of excited states in NCs. These enhanced interactions must be properly considered in order to describe the correlations between electrons and holes and in order to achieve agreement with experimental measurements. \n\n\\begin{figure*}[ht]\n\\includegraphics[width=\\textwidth]{figures\/Fig2_Pseudopotentials.pdf}\n\\caption{\\label{fig:fitting}(a) The bulk band structures of wurtzite CdSe (right) and CdS (left) obtained from the pseudopotential Hamiltonian (red points) are compared to literature values\\cite{Bergstresser1967} (black lines). The resulting band structures show excellent agreement both around the band gap and across the entire Brilouin zone. (b) The corresponding real-space pseudopotentials for Cd, Se, and S. The inset illustrates a cross-section of the pseudoptential for a wurtzite 3.9~nm CdSe NC as constructed from these atom-centered functions.}\n\\end{figure*}\n\nExperimentally relevant NCs are highly crystalline, and, in the interior of the structure, they closely resemble the corresponding bulk materials. The atomic configuration aligns closely with the bulk crystalline lattice across the majority of unit cells, suggesting that a description based on bulk bands would be a valid starting point. However, NCs possess additional features that distinguish them from bulk. The NC surface truncates the lattice symmetry, which gives rise to quantum confinement. Core-shell structures also form a nanoscale heterojunction that can introduce significant amounts of strain into the crystal structure.\\cite{CaoBanin200, ReissSmall2009} Both these internal interfaces and surfaces cause deformations from crystallinity on the scale of individual atoms, so accurate modeling of NCs must include this atomistic detail. For example, localized trap states at surfaces or interfaces due to atomic defects are ubiquitous in experimental studies of NCs, where they are observed to rapidly quench photoluminescence and result in significantly lower quantum yields.\\cite{Wuister2004, Guzelturk2021} An atomistic description of the NC structure allows for the introduction of site-specific defects or alloying to understand their roles in trap formation and to determine the dynamics of trapping in NC systems.\\cite{Jasrasaria2020,Enright2022} In addition to the static deformation of the crystal lattice, the effects of lattice fluctuations (\\textit{i.e.}, phonons) play a key role in the physics of NCs and must be properly incorporated.\\cite{Jasrasaria2021} Finally, in order to make meaningful contact with experimental measurements on NCs that contain thousands of atoms and tens of thousands of electrons, computational evaluation of the model must scale moderately with system size in comparison to first principles approaches. Because NC systems have important size dependent properties, such as optical gaps,\\cite{Ekimov1985} radiative lifetimes, and AR lifetimes,\\cite{Philbin2020a} and the scaling of these properties with system size is often an important question, the ability to access experimentally relevant sizes with volumes ranging across multiple orders of magnitude is crucial. \n\n\\begin{figure*}[ht]\n\\includegraphics[width=0.85\\textwidth]{figures\/Fig3_States.pdf}\n\\caption{\\label{fig:states} (a) Densities of the quasi-electron (red) and quasi-hole (blue) wave functions reveal that they are periodic across several unit cells in the interior of the NC. The electron states are labeled based on the symmetry of the envelope function in analogy to effective mass descriptions. (b) The densities of single-particle states (DOS) for wurtzite CdSe NCs of different sizes shows the effects of quantum confinement and the larger density of hole states in these II-VI systems. The inset illustrates the density of states across a larger energy range (that is normalized to the fundamental gap, $E_g$, of each NC), where the continuum of high energy states can be seen.}\n\\end{figure*}\n\nThese considerations have informed our development of the semiempirical pseudopotential model as a sufficiently detailed description of NCs that can also tackle calculations of experimentally relevant systems. For example, a CdSe quantum dot only 4~nm in diameter has over $\\sim$1000 atoms and $\\sim$4000 valence electrons, so the conventional workhorses of quantum chemistry, such as DFT and related methods for excited states, despite making significant progress,\\cite{Shulenberger2021,Song2022} are still far from being able to tackle this problem. On the other hand, continuum models based on the effective mass approximation have produced successful predictions for simple, linear spectroscopic observables~\\cite{Efros2000} but are unable to capture many of the more complicated dynamic processes that determine the timescales of process like nonradiative exciton relaxation and AR. Furthermore, these continuum models are, by nature, blind to atomistic detail, such as defects, strain at heterostructure interfaces, and facet-dependent properties.\\cite{Ondry2019, Cui2019}\n\nOur approach is based on the semiempirical pseudopotential method,\\cite{Wang1994,Rabani1999b,Williamson2000} which was first developed to characterize the band structures of simple bulk materials\\cite{Cohen1966} and was later extended to describe the role of surfaces~\\cite{Zunger1980} and confinement.\\cite{Friesner1991,Wang1994} The basic assumption made is that the bulk band structure can be described by a simple, non-interacting model Hamiltonian\n\\begin{equation}\n \\hat{h}_{\\text{qp}}=\\hat{t}+\\hat{v}(\\bm{r})=\\hat{t}+\\sum_\\mu \\hat{v}_\\mu(\\bm{r})\\,,\n \\label{eqn:Hamil}\n\\end{equation} \nwhere $\\hat{t}$ is the single-particle kinetic energy operator, and $\\hat{v}(\\bm{r})$ is the local (or non-local) pseudopotential, which is given by a sum over all atoms $\\mu$ of a pseudopotential $\\hat{v}_\\mu(\\bm{r})$ centered at the location of each atom $\\bm{R}_\\mu$. The parameters used to describe the pseudopotential of each atom are obtained by fitting the form factors to the bulk band structure obtained either from experimental measurements or high-accuracy electronic structure calculations, such as DFT+GW.\\cite{Cohen1966,Hybertsen1986} Within the fitting procedure, we describe the real-space atomistic pseudopotential $\\hat{v}_\\mu(\\bm{r})$ by its reciprocal-space counterpart, $\\hat{\\tilde{v}}_\\mu(\\bm{q})$. For example, one popular form of a local reciprocal-space pseudopotential is given by:\\cite{Wang1999}\n\\begin{equation}\n \\hat{\\tilde{v}}_\\mu(\\bm{q})=\\left[1+a_4\\operatorname{Tr}\\epsilon_\\mu\\right]\\frac{a_{0}\\left(q^{2}-a_{1}\\right)}{a_{2}e^{a_{3}q^{2}}-1}\\,,\n \\label{eqn:LocalPotq}\n\\end{equation}\nwhere $\\epsilon_\\mu$ is the local strain tensor around atom $\\mu$, the parameters $\\{a_0, a_1, a_2, a_3\\}$ are used to fit the band structure at the equilibrium configuration, and $a_4$ is fit to match the absolute hydrostatic deformation potentials of the valence and conduction bands.\\cite{Wei2006PRB_DefPot} The trace of the local strain tensor at each atom is approximated by the ratio of the volume of the tetrahedron formed by the nearest neighbors in the strained structure to that volume in the equilibrium bulk structure. For NCs with significant strain, such as core-shell QDs or other heterostructures,\\cite{Grunwald2013} additional fitting parameters multiplying higher orders of the strain tensor can be included. Furthermore, note that this formalism accounts for hydrostratic strain that occurs due to isotropic compression or expansion of a material, such as the core in spherical core-shell QDs. In anisotropic core-shell nanoplatelets, however, the core experiences biaxial strain,\\cite{Talapin2019} which may need to be incorporated into the model using additional terms.\n\n\\begin{figure*}[ht]\n\\includegraphics[width=\\textwidth]{figures\/Fig4_Validation.png}\n\\caption{\\label{fig:validation} Gaps for wurtzite CdSe quantum dots of different sizes (left). The optical gaps computed by our semiempirical pseudopotential method agree with experimental measurements of the optical gap by Fan \\textit{et al.}\\cite{Zhang2015} (black squares) and Yu \\textit{et al.}\\cite{Peng2003} (black triangles). The inset shows the exciton binding energy, $E_{\\text{B}}$, computed by our method and compared to values computed by Franceschetti and Zunger\\cite{Zunger1997} (black asterisks). Gaps for zincblende CdSe-CdS core-shell nanoplatelets with different thicknesses of CdS shell (middle). The optical gaps calculated by our method compare favorably with those measured experimentally by Hazarika \\textit{et al.}\\cite{Talapin2019} (black squares). Gaps for zincblende InAs quantum dots of different sizes (right). The fundamental gaps calculated are in excellent agreement with those measured by Banin \\textit{et al.}\\cite{Millo1999} using scanning tunneling microscopy (black squares), and the optical gaps compare well with those measured by Guzelian \\textit{et al.}\\cite{Alivisatos1997} (black triangles) and computed by Franceschetti and Zunger\\cite{Zunger2000} (black asterisks).}\n\\end{figure*} \n\nThe fitting of parameters $\\{a_0, a_1, a_2, a_3\\}$ proceeds by comparing the generated band structure to the expected band structure with special care taken to correctly capture the band gaps and effective masses. As shown in Fig.~ \\ref{fig:fitting}a, the model captures all band features and describes the band structure across the entire Brillouin zone. The real-space forms of the corresponding pseudopotentials are illustrated in Fig.~\\ref{fig:fitting}b, where the psuedopotentials have been simultaneously fit to generate the correct band structures for both wurtzite and zincblende CdSe and CdS. The effects of strain are then incorporated through the $a_4$ parameter (and any necessary higher-order terms) to fit the absolute deformation potentials of both the conduction band minimum and valence band maximum. This fitting procedure ensures that hydrostatic deformation of the crystal alters the energies of the electron and hole levels in the correct manner. \n\nOnce the pseudopotentials have been fit to describe bulk systems (the fits are not unique and often other physical measures are used to choose the best set of parameters~\\cite{Wang1996}), they are used to construct the NC Hamiltonian. The central assumption made here is that the pseudopotentials that describe single particle properties in the bulk are adequate also when applied to quantum confined nanostructures. While this might seem a large leap, the error introduced by this assumption is relatively small compared to the fundamental band gap.\\cite{Ogut1997} A cross-section of the resulting pseudopotential for a wurtzite 3.9~nm CdSe NC is shown in the inset of Fig.~\\ref{fig:fitting}b, illustrating both the near-periodic potential in the interior of the NC and the manner by which it is modified at the surface. The NC atomic configurations are obtained by first pruning the correctly sized NC from bulk. The atomic positions are then relaxed using molecular dynamics-based geometry optimization with previously-parameterized force fields,\\cite{Rabani2002,Zhou2013} which includes two- and three-body terms to enforce tetrahedral bonding geometries, to produce NC configurations that are relatively crystalline in agreement with experiment.\\cite{KelleyJCP2016} In the case of core-shell structures, the core is cut from bulk, and the shell material is grown on the surface using the lattice constant of the core material. The subsequent geometry optimization allows the shell to relax and results in compressive strain on the core to minimize the stress along the core-shell interface.\\cite{Grunwald2013, Talapin2019}\n\nThe description of the surface of the NC presents a challenge, as simply terminating the NC will result in dangling bonds. These dangling bonds can give rise to localized electronic states within the band gap, which act as traps. For the II-VI and III-V families of semiconductors, we have found that dangling bonds from the non-metal atoms result in hole traps slightly above the valence band maximum, but metal dangling bonds do not result in electron traps due to the light electron effective mass relative to the hole effective mass.\\cite{Jasrasaria2020,Enright2022} To passivate the surface of the NC, the outermost layer of atoms is replaced with passivation potentials that mimic the effect of organic ligands that terminate the surfaces of experimentally synthesized NCs, pushing the mid-gap electronic states out of the band gap.\\cite{Wang1994} This procedure for building NC structures can be easily adapted to produce more complicated NCs, such as the core-shell NCs, nanorods, and nanoplatelets. Further modification, such as alloying, multi-layered NCs, dimer NC assemblies, and structural defects can also be modeled with atomistic detail. \n\nWhile a NC of experimentally relevant size will have many single-particle states (see Fig.~\\ref{fig:states}b), only a few highest-energy, occupied and lowest-energy, unoccupied states are relevant to describing the optical properties near the band edge. These single-particle states are obtained using the filter diagonalization method,\\cite{Wall1995,Toledo2002} which provides a framework to extract all the eigen-solutions within a specific energy window. This process can be done with nearly linear scaling with the system size due to the locality of the single-particle Hamiltonian, making feasible the calculation for NCs with volumes spanning several orders of magnitude. As the pseudopotentials are fit to reproduce quasiparticle band structures, the eigenstates of the pseudopotential Hamiltonian are assumed to describe the quasi-electron and quasi-hole wave functions of the NC. Examples of the quasi-electron and quasi-hole densities are shown in Fig.~\\ref{fig:states}a. We see that both the electron and hole states show Bloch-like oscillations, which are significantly more pronounced for the hole, and the electron states show a progression of envelope functions with $s$- then $p$-type characteristics, in line with effective mass descriptions of NC electronic states.\\cite{Wang1998,Efros2000} \n\nAs previously stated, connection to experiments also requires an accurate description of the neutral excited states probed by optical spectroscopy. To account for electron-hole correlations, we use the single-particle eigenstates as the basis to solve the Bethe-Salpeter equation (BSE)~\\cite{Rohlfing2000} for the correlated excitonic states using the static screening approximation.\\cite{Eshet2013} This approach describes electron-hole correlations beyond the standard perturbation technique and is essential to describe excitons even in the strongly confined limit. We take the excitonic states to be a linear combination of noninteracting, electron-hole pair states:\n\\begin{equation}\n \\vert \\psi_n \\rangle = \\sum_{ai} c_{a,i}^n a_a^\\dagger a_i \\vert 0 \\rangle\\,,\n \\label{eqn:PsiExciton}\n\\end{equation}\nwhere $a_a^\\dagger$ and $a_i$ are electron creation and annihilation operators in quasiparticle states $a$ and $i$, respectively. The indexes $a,b,c,\\dots$ refer to quasi-electron (unoccupied) states while the indexes $i,j,k,\\dots$ refer to quasi-hole (occupied) states. The expansion coefficients $c_{a,i}$ are determined by solving the eigenvalue equation~\\cite{Rohlfing2000}\n\\begin{equation}\n (E_n-\\Delta \\varepsilon_{ai})c^n_{a,i}=\\sum_{bj}\\left(K^d_{ai;bj}+K^x_{ai;bj}\\right)c^n_{b,j}\\,,\n \\label{eqn:BSEEig}\n\\end{equation}\nwhich also determines the energy of exciton $n$, $E_n$, in terms of the direct and exchange parts of the electron-hole interaction kernel,\\cite{Rohlfing2000} $K^d_{ai;bj}$ and $K^x_{ai;bj}$, respectively, and the quasiparticle energy difference $\\Delta \\varepsilon_{ai}=\\varepsilon_a-\\varepsilon_i$. The direct part of the kernel describes the main attractive interaction between quasi-electrons and quasi-holes while the exchange part controls details of the excitation spectrum, such as the singlet-triplet splittings. Importantly, the direct term is mediated by a screened Coulomb interaction,\\cite{Rohlfing2000} which we approximate using the static screening limit with a dielectric constant that is obtained directly from the quasiparticle Hamiltonian\\cite{Williamson2000} and that depends on the size and shape of the NC. The binding energy of the $n$th excitonic state, $E^n_\\text{B}$, is calculated as\n\\begin{equation}\n E^n_\\text{B}=\\sum_{abij}\\left(c^{n}_{a,i}\\right)^{*}\\left( K^d_{ai;bj}+K^x_{ai;bj}\\right)c^{n}_{b,j} .\n \\label{eqn:ExBind}\n\\end{equation}\n\nAs this model was built on semiempirical foundations, it is necessary to validate the resulting calculations on well-known NC properties before using the model to explore more complex phenomena. Furthermore, the fitting was carried out on pure bulk materials, so it is important to assess the performance of the model on different NCs across a range of sizes and compositions. One of the most fundamental properties we need to capture is the optical gap. As shown in Fig.~\\ref{fig:validation}, we obtain results that compare favorably with experiments with respect to the magnitude of the gap and the scaling with NC size for several different NC compositions and geometries. We additionally validate properties, such as exciton binding energies,\\cite{Philbin2018,Brumberg2019}, exciton fine structure effects on polarized emission,\\cite{Hadar2017,Brumberg2019} radiative and AR lifetimes,\\cite{Philbin2018,Philbin2020a,Philbin2020b} and optical signals of trapped carriers.\\cite{Jasrasaria2020} The strong agreement we obtain between theoretical predictions and experimental observations across a variety of system sizes, compositions, and dimensionalities demonstrates that our approach is suitable for understanding and rationalizing trends across a wide range of nanomaterial systems. Additionally, as we will discuss in the following sections, this model is extremely versatile and lends itself to new development and expansion.\n\n\\section{Phonons and exciton-phonon couplings}\n\\label{sec:exph}\nElectronic degrees of freedom couple to phonons in semiconductors, resulting in a diverse set of processes that affect electronic properties and dynamics. These electron-phonon interactions in bulk semiconductors tend to be weaker than electron-vibrational interactions in molecular systems because the relatively large dielectric screening in bulk semiconductors leads to delocalized, Wannier-Mott excitons, which do not depend as sensitively on the nuclear configuration as do the localized, Frenkel excitons in molecular systems.\\cite{Scholes2006NatMater} Additionally, phonons in bulk semiconductors are also delocalized over the material, unlike localized vibrations in molecules.\\cite{Mahan2000}\n\nSemiconductor NCs lie somewhere in between the bulk and molecular limits. Excitons in NCs are delocalized over multiple atoms, but they are confined to the extent of the NC. Similarly, lattice vibrations resemble phonons in bulk semiconductors, but they are finite in number and spatially confined to the NC. The effects of confinement on the magnitude of exciton-phonon coupling (EXPC) are still poorly understood, and the challenges associated with studying EXPC, both experimentally and theoretically, have led to a set of outstanding questions regarding EXPC in semiconductor NCs. A detailed description of EXPC in NCs is essential for understanding the temperature dependence of excitonic properties\\cite{Balan2017} and phenomena, such as exciton dephasing (\\textit{i.e.}, homogeneous emission linewidths),\\cite{Bawendi2016, Mack2017} phonon-mediated carrier relaxation,\\cite{Nozik2001, Kilina2009,Peterson2014} and charge transfer.\\cite{Kamat2011,Harris2016} \n\nBefore delving into the role of confinement on EXPC, we will examine the phonon states in NCs. Phonon confinement to the spatial extent of the NC results in quantization of the phonon frequencies. This confinement introduces additional complicating factors, such as the role of the NC surface,\\cite{Wood2016Nature, Mack2019} that motivate the need for an atomistic description of phonon modes.\\cite{KelleyJCP2016} While DFT-based frozen phonon approaches have been used to compute phonon modes and frequencies,\\cite{Chou1992, Bester2012, Bester2019} their computational expense restricts these methods to small NCs or to the computation of specific modes which are known \\textit{a priori} to be relevant for the properties or processes of interest. Therefore, we model phonons using classical, atomic force fields, which allows for the computation of all phonon modes and frequencies in NC systems of experimentally relevant sizes. The dynamical matrix, or mass-weighted Hessian, can be computed at the equilibrium configuration of a NC:\\cite{Kong2011}\n\\begin{equation}\n D_{\\mu k, \\mu^\\prime k^\\prime} = \\frac{1}{\\sqrt{m_\\mu m_{\\mu^\\prime}}} \\bigg( \\frac{\\partial^2 U(\\boldsymbol{R})}{\\partial u_{\\mu k} \\partial u_{\\mu^\\prime k^\\prime}} \\bigg)_{\\boldsymbol{R}_0}\\,,\n \\label{eqn:dynamical}\n\\end{equation}\nwhere $U(\\boldsymbol{R})$ is the potential energy given by the force field, $u_{\\mu k}=R_{\\mu k} - R_{0,\\mu k}$ is the displacement of nucleus $\\mu$ away from its equilibrium position in the $k\\in\\{x,y,z\\}$ direction, and $m_\\mu$ is the mass of nucleus $\\mu$. This $3N \\times 3N$ dynamical matrix, where $N$ is the number of atoms in the NC, can be diagonalized to obtain the phonon mode frequencies and coordinates.\n\n\\begin{figure*}[ht]\n\\includegraphics[width=0.85\\textwidth]{figures\/Fig5_ExPh.pdf}\n\\caption{\\label{fig:ExPh} (a) The phonon densities of states (PDOS) calculated for wurtzite CdSe NCs of different sizes. The inset shows the phonon gap, $\\omega_\\text{gap}$, which decreases with increasing NC size. (b) Phonon lifetimes calculated for wurtzite CdSe NCs. For all systems, almost all modes have sub-picosecond lifetimes, indicating significant phonon-phonon coupling. (c) Reorganization energies for wurtzite CdSe NCs. Values calculated by our approach are compared to experimental measurements by Liptay \\textit{et~al.}\\cite{Bawendi2007} (black triangles) and Salvador \\textit{et~al.}\\cite{Scholes2006JCP} (black squares) as well as to calculations using an effective mass model by Kelley\\cite{Kelley2011} (black asterisks). The inset schematically depicts the reorganization energy, $\\lambda$, which corresponds to the energy of lattice rearrangement after vertical excitation from the electronic ground state $\\vert g \\rangle$ to the excited state $\\vert e \\rangle$. The ground and excited state minima are displaced along the phonon mode coordinate $q$ by a distance $d$. (d) The spectral densities, $J(\\omega)$, calculated for wurtzite CdSe NCs show significant coupling to lower-frequency acoustic modes and to optical modes around $\\hbar \\omega \\sim 30$~meV.}\n\\end{figure*}\n\nThe phonon densities of states (PDOS) for wurtzite CdSe NCs of different sizes computed using a previously-parameterized Stillinger-Weber interaction potential\\cite{Zhou2013} are illustrated in Fig.~\\ref{fig:ExPh}a. Acoustic modes, which involve in-phase motion of atoms, have lower frequencies (1\\,THz\\,$\\sim$\\,5\\,meV or lower in CdSe NCs) while optical modes, which are made up of out-of-phase movements of atoms, have higher frequencies (4\\,THz\\,$\\sim$\\,16\\,meV or higher).\\cite{Grunwald2012} Modes at intermediate frequencies are difficult to characterize due to the overlap of acoustic and optical branches in the bulk phonon dispersion relation as well as the confounding effects of phonon confinement. Phonon confinement also leads to a gap in the PDOS near zero-frequency since the longest-wavelength (\\textit{i.e.}, lowest-frequency) phonon mode in a NC is dictated by the NC size. As shown in the inset of Fig.~\\ref{fig:ExPh}a, the lowest-frequency phonon mode in the system is inversely proportional to the NC diameter, as observed by Raman spectroscopy measurements.\\cite{Weller2008, Tisdale2016}\n\nThis zero-frequency gap has led to a hypothesized hot phonon bottleneck in NCs, in which phonon-phonon scattering rates are slow because of the lack of low-frequency modes, and phonon thermalization becomes the rate-limiting step in processes, such as hot carrier relaxation.\\cite{Ploog1988,Klimov1995,Li2017} However, we have found that phonons have significant coupling with one another (\\textit{i.e.}, are anharmonic) at room temperature (300\\,K).\\cite{Guzelturk2021} We performed molecular dynamics simulations using a force field that consists of Lennard-Jones and Coulomb terms\\cite{Rabani2002} to compute phonon relaxation lifetimes of CdSe NCs, shown in Fig.~\\ref{fig:ExPh}b. Unlike the Stillinger-Weber potential, this force field includes long-range interactions that are necessary to accurately describe the splitting between acoustic and optical branches at the Brillouin zone edge in bulk polar semiconductors\\cite{Bester2017} and, thus, the phonon lifetimes. Our calculations are within linear response theory, so they assume phonon modes are only weakly excited, but no assumptions are made about the strength of coupling between different phonon modes. The lifetimes, which are dictated by the phonon-phonon interactions, are sub-picosecond for all modes except for the lowest-energy acoustic modes, for which the lifetimes reach $\\sim$4\\,ps. Smaller NCs have shorter lifetimes because surface atoms, which are proportionally larger in number in smaller NCs, have increased anharmonic motion that leads to greater phonon-phonon coupling. For all systems, the phonon dynamics are overdamped, in agreement with experimental observations,\\cite{Lounis2014,Lindenberg2015} and the relaxation timescales are less than the periods of the phonon modes. We expect that phonon modes that are strongly out of equilibrium would thermalize even more quickly, indicating that a hot phonon bottleneck is unlikely in these systems. (Note that the ``hot phonon bottleneck\" is distinct from the ``phonon bottleneck\", which describes slow carrier relaxation due to the mismatch between electronic gaps and phonon frequencies, that is discussed in Secs.~\\ref{sec:intro} and \\ref{sec:conclusions}.) \n\nWith this discussion of NC phonons in mind, we now turn to discuss the EXPC terms, $V_{n,m}^{\\alpha}$, that appear in Eq.~\\eqref{eq:hamiltonian}. Historically, studies of EXPC in NCs have described electronic states within parameterized models and\/or have described phonons as vibrations of an elastically isotropic sphere,\\cite{Flytzanis1990,Kobayashi1992,Kartheuser1994, Takagahara1996, Zorkani2007, Kelley2011} leading to widely varying results, as the magnitude of EXPC is extremely sensitive to the descriptions of both excitons and phonons. Fully atomistic \\textit{ab initio} methods have been used for small NCs of $\\sim$100 atoms or fewer, for which the electron-phonon coupling can be inferred from fluctuations of the adiabatic electronic states that are generated ``on the fly\" within DFT and time-dependent DFT frameworks.\\cite{Prezhdo2005PRL, Prezhdo2013JCTC, Wood2018NL, Wood2021} While these methods have been moderately effective in modeling electron-phonon coupling, they are often limited to small systems due to the computational expense of DFT, and they ignore excitonic effects. One recent study that did include these excitonic effects using \\textit{ab initio} methods was limited to clusters of tens of atoms.\\cite{Zeng2021}\n\nInstead, we rely on the semiempirical pseudopotential model to describe excitonic states and atomic force fields to describe phonons.\\cite{Jasrasaria2021} Within this framework, the standard electron-nuclear matrix element to first order in the atomic displacements is given by:\\cite{Giustino2017}\n\\begin{equation}\n V_{n,m}^{\\mu k} \\equiv \\bigg\\langle \\psi_n \\bigg\\vert \\bigg( \\frac{\\partial \\hat{v}(\\bm{r})}{\\partial R_{\\mu k}} \\bigg)_{\\boldsymbol{R}_0} \\bigg\\vert \\psi_m \\bigg\\rangle\\,,\n\\end{equation}\nwhere $\\vert \\psi_{n} \\rangle$ is the state of exciton $n$ (cf., Eq.~\\eqref{eqn:PsiExciton}), $\\hat{v}(\\bm{r}) = \\sum_\\mu \\hat{v}_\\mu(\\boldsymbol{r})$ is the sum over atomic pseudopotentials given in Eq.~(\\ref{eqn:Hamil}), $R_{\\mu k}$ is the position of atom $\\mu$ in the $k\\in \\{x,y,z\\}$ direction, and $\\boldsymbol{R}_0$ is the equilibrium configuration of the NC. Using the static dielectric BSE approximation for the excitonic wave function, we can reduce the calculation of these matrix elements to a simpler form given by:\\cite{Jasrasaria2021}\n\\begin{equation}\n V_{n,m}^{\\mu k} = \\sum_{abi} c_{a,i}^n c_{b,i}^m v_{ab,\\mu}^\\prime (R_{\\mu k}) - \\sum_{aij} c_{a,i}^n c_{a,j}^m v_{ij,\\mu}^\\prime (R_{\\mu k})\\,,\n\\label{eqn:ExPh_ElHoleChannels}\n\\end{equation}\nwhere\n\\begin{equation}\n v_{rs,\\mu}^\\prime = \\int d\\boldsymbol{r} \\phi^{*}_r (\\boldsymbol{r}) \\frac{\\partial \\hat{v}(\\bm{r})}{\\partial R_{\\mu k}} \\phi_s (\\boldsymbol{r})\\,.\n\\label{eqn:single-ElHoleChannels}\n\\end{equation}\nHere, $c_{a,i}^n$ represent the Bethe-Salpeter coefficients introduced in Eq.~\\eqref{eqn:PsiExciton}, and $\\phi_a(\\boldsymbol{r})$ are the real-space quasiparticle wave functions. The first term in Eq.~\\eqref{eqn:ExPh_ElHoleChannels} represents the electron channel of EXPC, in which excitons comprised of different single-particle electron states are coupled. The second term describes the hole channel, in which excitons comprised of different single-particle hole states are coupled. These matrix elements can be transformed to phonon mode coordinates using the eigenvectors of the dynamical matrix:\n\\begin{equation}\n V_{n,m}^\\alpha = \\sum_{\\mu k} \\frac{1}{\\sqrt{m_\\mu}} e_{\\alpha,\\mu k}^{-1} V_{n,m}^{\\mu k}\\,, \n\\end{equation}\nwhere $e_{\\alpha, \\mu k}$ is the $\\mu k$ element of the $\\alpha$ eigenvector of the dynamical matrix given in Eq.~\\eqref{eqn:dynamical}, and $m_{\\mu}$ is the mass of atom $\\mu$. The diagonal matrix elements $V_{n,n}^\\alpha$ describe the renormalization of the energy of exciton $n$ through its interaction with phonon mode $\\alpha$, and the off-diagonal matrix elements $V_{n,m}^\\alpha$ describe the interaction of excitons $n$ and $m$ through the absorption or emission of a phonon of mode $\\alpha$.\n\nOne measure of the EXPC is the reorganization energy,\\cite{Ekimov1993, Scholes2006JCP} depicted schematically in the inset of Fig.~\\ref{fig:ExPh}c, which is the energy associated with rearrangement of the NC lattice after exciton formation and is relevant for optical Stokes shifts, charge transfer processes, and NC-based device efficiencies. In the harmonic approximation, the total reorganization energy for a NC is the sum of reorganization energies for each mode, $\\lambda = \\sum_\\alpha \\lambda_\\alpha$, where\n\\begin{equation}\n \\lambda_\\alpha = \\frac{2}{Z} \\sum_n e^{-\\beta E_n} \\bigg( \\frac{1}{2\\omega_\\alpha} V_{n,n}^\\alpha \\bigg)^2\\,.\n\\label{eq:renormalization}\n\\end{equation}\nThe above equation includes a Boltzmann-weighted average over excitonic states, where $Z=\\sum_n e^{-\\beta E_n}$ is the partition function, $\\beta = \\frac{1}{k_{\\rm B}T}$, and $T$ is the temperature. For wurtzite CdSe NCs ranging from 3 to 5~nm in diameter, we calculated the reorganization energy to be between 60 and 20~meV, which is in good agreement with experimentally measured values\\cite{Scholes2006JCP, Bawendi2007} and previous effective mass model calculations\\cite{Kelley2011}, as shown in Fig.~\\ref{fig:ExPh}c. The remarkable agreement is an important validation of the semiempirical technique and is essential for describing the nonadiabatic transitions involved in exciton cooling. To investigate the contribution of each mode to the overall reorganization energy, we examine the spectral density, or weighted density of states, ($J(\\omega) = \\sum_\\alpha \\lambda_\\alpha \\delta (\\omega - \\omega_\\alpha)$), which is illustrated in Fig.~\\ref{fig:ExPh}d. Lower-frequency acoustic modes, which tend to involve collective motions of many atoms in the NC, are more significantly coupled to the exciton. Higher-frequency optical modes have weaker EXPC, but they have a large density of phonon states around 25~meV and 30~meV, which are at an energy scale that is more relevant for excitonic transitions. This feature suggests that optical modes may, in fact, be more important than acoustic modes for phonon-mediated exciton dynamics, but further assessment of the model and the EXCP is required for developing a better understanding of the exciton cooling process. The spectral density, however, may explain discrepancies in experimental results, some of which find stronger exciton coupling to acoustic modes\\cite{Scholes2006JCP, Anni2007, Kambhampati2008PRB, Kambhampati2008JPCC} while others observe stronger exciton coupling to optical modes.\\cite{Kobayashi1992, Righini1996, Bimberg1999, Kelley2015ACSNano, Lin2015} Exciton formation causes NC lattice distortion primarily along the phonon coordinates of acoustic modes while optical modes may play a larger role in exciton dynamics.\n\nFurthermore, our calculations show that excitons in all core and core-shell NCs are more strongly coupled to phonons \\textit{via} the hole channel (\\textit{i.e.}, the second term in Eq.~(\\ref{eqn:ExPh_ElHoleChannels})) than through the electron channel.\\cite{Jasrasaria2021} This effect is a result of the heavier effective hole mass, which makes hole states more sensitive to nuclear configuration and decreases the energy spacing between hole states, allowing them to couple more readily \\textit{via} phonon absorption or emission. Moreover, we have found that phonon modes localized to the surface of the NC have significant contributions to the overall reorganization energy in small NCs because of increased surface strain and strong exciton confinement, which causes the exciton wave function to extend to the NC surface. This surface effect decreases drastically as the NC size increases.\n\nOur framework includes electron-hole correlations, exciton-phonon coupling, and phonon-phonon interactions, enabling our ongoing work to address open questions regarding timescales and mechanisms of phonon-mediated exciton dynamics, such as hot exciton cooling. This atomistic theory can simultaneously study both hypothesized mechanisms -- the Auger decay mechanism, which would occur on the order of picoseconds, and the slower phonon-mediated transitions -- allowing a unified mechanism to emerge from the theory to explain experimental results in a range of NC systems.\n\n\\section{Auger recombination}\n\\label{sec:auger}\n\nAuger recombination (AR) is the primary nonradiative, Coulomb-mediated, exciton-exciton decay channel of multiexcitons, in which one exciton recombines and transfers its energy to an additional electron-hole pair, as schematically illustrated in Fig.~\\ref{fig:AllProcesses}, on timescales of a few hundreds of picoseconds.\\cite{Klimov2014} The energy given to the second exciton primarily excites one of the carriers, which then quickly dissipates the excess energy \\textit{via} phonon emission.\\cite{Achermann2006,Harvey2018} While fast AR in NCs is often responsible for decreased photoluminescence quantum yields and device efficiencies, it also makes NCs potentially useful as single photon sources.\\cite{Correa2012,Utzat2019} AR lifetimes are commonly measured using time-resolved photoluminescence and transient absorption experiments.\\cite{Klimov2000,Klimov2008,Robel2009,Baghani2015,Ben-Shahar2016,Li2016,Pelton2017} Recent experiments have demonstrated that for quasi-0D quantum dots (QDs),~\\cite{Garcia-Santamaria2011} quasi-1D nanorods,~\\cite{Stolle2017} and quasi-2D nanoplatelets,~\\cite{She2015, Li2017, Philbin2020b} AR lifetimes increase linearly with the volume of the NC. Understanding the key factor that leads to this so-called ``universal volume scaling\" of AR lifetimes is key to further controlling AR processes. Understanding the dependence of AR lifetimes on NC size, shape, and composition as well as on the number of excitons present in the NC is central to our understanding of this many-body relaxation process and will provide tools to control AR lifetimes, as further discussed below.\n\nPrior to focusing on the microscopic origins of AR, we will discuss the nature of Coulomb-mediated interactions within and between excitons and their particular importance in confined semiconductors. The strength of Coulomb-mediated interactions within an exciton is normally characterized by the exciton binding energy (see Eq. \\eqref{eqn:ExBind}), which is typically $\\sim$10~meV\nin bulk semiconductors. Due the small sizes and reduced dimensionalities of NCs, Coulomb interactions are enhanced, leading to exciton binding energies that are greater than $100$~meV in QDs,~\\cite{Franceschetti1997} nanorods,~\\cite{Baskoutas2005,Rajadell2009,Royo2010} and nanoplatelets.\\cite{Scholes2006NatMater,Scott2016,Rajadell2017,Brumberg2019} As this exciton binding energy is much greater than the thermal energy scale at room temperature ($k_{\\text{B}}T\\sim25$~meV), electrons and holes readily form bound excitons in NCs. The physics of electrons and holes forming bound, correlated electron-hole pairs impacts almost all physical processes, both radiative and nonradiative, in NCs. The enhancement of Coulomb interactions in NCs also affects interactions between excitons. Bulk materials require optical excitation from intense lasers to reach the exciton densities at which Coulomb-mediated exciton-exciton interactions are important. However, the small volumes of NCs lead to these large exciton densities even with just two excitons on a NC. Moreover, two excitons on a single NC will have significant wave function overlap, further increasing Coulomb interactions between excitons in confined systems. These enhanced Coulomb interactions are the primary reason for significant AR in NCs. \n\nWe will now dive into the details of the approach we have developed for computing AR lifetimes, $\\tau_{\\text{AR}}$. AR is a process by which an initial biexcitonic state, $\\left|B\\right\\rangle $, of energy $E_{B}$ decays into a final excitonic state, $\\left|S\\right\\rangle$, of energy $E_{S}$ \\textit{via} Coulomb scattering, $V$. AR lifetimes of a NC can be calculated using Fermi's golden rule, where we average over thermally distributed initial biexcitonic states and sum over all final decay channels into single excitonic states:\n\\begin{eqnarray}\n\\tau_{\\text{AR}}^{-1} & = & \\sum_{B}\\frac{e^{-\\beta E_{B}}}{Z_{B}}\\frac{2\\pi}{\\hbar}\\sum_{S}\\left|\\left\\langle B\\left|V\\right|S\\right\\rangle \\right|^{2}\\delta\\left(E_{B}-E_{S}\\right)\\,.\\label{eq:fermisGoldenRule}\n\\end{eqnarray}\nIn the above, the Dirac delta function ($\\delta\\left(E_{B}-E_{S}\\right)$) enforces energy conservation between the initial and final states and the partition function, $Z_{B}=\\sum_{B}e^{-\\beta E_{B}}$, describes a thermal average of initial biexcitonic states (assuming Boltzmann statistics for biexcitons). Despite the known fact that electron-hole interactions in NCs are significant, AR lifetimes had previously been calculated by approximating the initial biexcitonic state as two quasi-electrons and two quasi-holes, without any correlations between them.\\cite{Chepic1990, Wang2003, Cragg2010, Vaxenburg2015, Vaxenburg2016} Mathematically, this approximation yields the initial biexcitonic state as \n\\begin{eqnarray}\n\\left|B\\right\\rangle ^{(0)} & = & a_{b}^{\\dagger}a_{j}a_{c}^{\\dagger}a_{k}\\left|0\\right\\rangle \\otimes\\left|\\chi_{B}\\right\\rangle ,\n\\label{eq:nonintBiexciton}\n\\end{eqnarray}\nand $E_B^{(0)} = \\varepsilon_b-\\varepsilon_j+\\varepsilon_c-\\varepsilon_k$ where the superscript ``$(0)$'' signifies that a noninteracting formalism is used.\nIn the above, $a^\\dagger_b$ and $a_j$ are electron creation and annihilation operators in quasiparticle states $b$ and $j$, respectively, as defined in Sec.~\\ref{sec:methods}, and $\\left|\\chi_{B}\\right\\rangle $ is the spin part of the biexciton wavefunction. \n\nIntuitively, this approximation is only valid in the limit where the kinetic energy is much larger than the exciton binding energy (\\textit{i.e.}, for very small quasi-0D QDs in the very strong confinement limit, as shown in Fig.~\\ref{fig:AR}), and it quickly breaks down with increasing QD size.\\cite{Philbin2018} Furthermore, this approximation results in computed AR lifetimes that are orders of magnitude too long for quasi-1D nanorods and quasi-2D nanoplatelets (Fig.~\\ref{fig:AR}).\\cite{Philbin2018, Philbin2020b} While this approximation to the initial biexcitonic state is conceptually and computationally simple, it leads to discrepancies between theoretical predictions and experimental measurements on the volume dependence of AR lifetimes in colloidal QDs that persisted for over $20$~years.\n\n\\begin{figure*}[htb]\n\\includegraphics[width=\\textwidth]{figures\/Fig6_Auger.pdf}\n\\caption{\\label{fig:AR}Biexciton Auger recombination (AR) lifetimes, $\\tau_{\\text{AR}}$, for CdSe quantum dots as a function of volume (left). Calculations\\cite{Philbin2018} are compared to experimentally measured lifetimes by Taguchi \\textit{et al.}\\cite{Taguchi2011} (black squares), Htoon \\textit{et al.}\\cite{Htoon2003} (black triangles), and Klimov \\textit{et al.}\\cite{Klimov2000} (black asterisks), demonstrating excellent agreement between the interacting formalism and experiment. AR lifetimes for CdSe nanorods as a function of volume (middle). Calculated lifetimes\\cite{Philbin2018} are shown with those measured by Taguchi \\textit{et al.}\\cite{Taguchi2011} (black squares), Htoon \\textit{et al.}\\cite{Htoon2003} (black triangles), and Zhu \\textit{et al.}\\cite{Zhu2012} (black asterisks). AR lifetimes for $4$~monolayer CdSe nanoplatelets as a function of nanoplatelet area. Calculated lifetimes\\cite{Philbin2020a} compare well with measurements by Philbin \\textit{et al.}\\cite{Philbin2020b} (black squares), She \\textit{et al.}\\cite{She2015} (black triangles), and Li and Lian\\cite{Li2017} (black asterisks). For all systems, the interacting formalism predicts the same volume scaling as experiment while the noninteracting formalism predicts a steeper scaling with NC volume.}\n\\end{figure*}\n\nBeyond this approximation, the initial biexcitonic state can be written as a combination of two excitonic states that includes electron-hole correlations within each exciton\\cite{Refaely-Abramson2017, Philbin2018} but that ignores correlations between excitons, which are typically two (or more) orders of magnitude weaker. The initial biexcitonic state within this formalism, which we previously termed the interacting formalism, is given by\n\\begin{eqnarray}\n\\left|B\\right\\rangle & = & \\sum_{b,j}\\sum_{c,k}c_{b,j}^{n}c_{c,k}^{m}a_{b}^{\\dagger}a_{j}a_{c}^{\\dagger}a_{k}\\left|0\\right\\rangle\n\\otimes\\left|\\chi_{B}\\right\\rangle ,\n\\label{eq:B I}\n\\end{eqnarray}\nwhere the excitonic coefficients $c_{b,j}^{n}$ and $c_{b,j}^{m}$\nare determined by solving the Bethe-Salpeter equation,\\cite{Rohlfing2000} as detailed in Sec.~\\ref{sec:methods}. In this formalism the energy of this biexcitonic state is $E_B = E_n+E_m$, which is simply the sum of the two exciton energies. Thus, a deterministic calculation of the AR lifetime can be performed using~\\cite{Philbin2018}\n\\begin{widetext}\n\\begin{eqnarray}\n\\tau_{\\text{AR}}^{-1} & = & \\frac{2\\pi}{\\hbar Z_{B}}\\sum_{B}e^{-\\beta E_{B}}\\sum_{a,i}\\left|\\sum_{b,c,k}c_{b,i}^{n}c_{c,k}^{m}V_{abck}\\right|^{2}\\delta\\left(E_{B}-\\varepsilon_{a}+\\varepsilon_{i}\\right)\\label{eq:dIntAR}\\\\\n & & +\\frac{2\\pi}{\\hbar Z_{B}}\\sum_{B}e^{-\\beta E_{B}}\\sum_{a,i}\\left|\\sum_{j,c,k}c_{a,j}^{n}c_{c,k}^{m}V_{ijck}\\right|^{2}\\delta\\left(E_{B}-\\varepsilon_{a}+\\varepsilon_{i}\\right).\\nonumber \n\\end{eqnarray}\n\\end{widetext}\nIn Eq.~\\eqref{eq:dIntAR}, the first term on the right hand side indicates the electron channel, in which the electron of the final state is excited, and the second term corresponds to the hole channel, in which the hole of the final state is excited. The final states are still approximated by noninteracting electron-hole pairs, $\\vert S \\rangle = a_a^\\dagger a_i \\vert 0 \\rangle$, for which $E_S = \\varepsilon_{a}-\\varepsilon_{i} $. This representation of the final state is a reasonable approximation given that the final states are high in energy (Fig.~\\ref{fig:AllProcesses}), above the dissociation energy of the exciton. Eq.~\\eqref{eq:dIntAR} was first shown to predict quantitatively accurate AR lifetimes for QDs and nanorods\\cite{Philbin2018} and was then extended and applied to large core-shell QDs~\\cite{Philbin2020a} and nanoplatelets~\\cite{Philbin2020b} using stochastic orbital techniques to reduce the computational cost of the interacting formalism given by Eq.~\\eqref{eq:dIntAR}. Specifically, stochastic orbitals were used to sample the final states \\textit{via} the stochastic resolution of the identity\\cite{Takeshita2017,Dou2019} and also to represent the Coulomb operator.\\cite{Neuhauser2016} The overall computational scaling of the stochastic implementation of the interacting formalism was multiple factors of the system size lower than the deterministic implementation.\\cite{Philbin2020a} \n\nWe have yet to find a system in which Eq.~\\eqref{eq:dIntAR} and the underlying approximation of treating the initial biexcitonic state as a product of two correlated excitonic states (Eq.~\\eqref{eq:B I}) fail to agree with experimental AR lifetimes. However, future work may need to treat the initial state as a fully-correlated biexcitonic state given by\n\\begin{eqnarray}\n\\left|B\\right\\rangle & = & \\sum_{b,c,j,k}c_{b,c,j,k}a_{b}^{\\dagger}a_{j}a_{c}^{\\dagger}a_{k}\\left|0\\right\\rangle \\otimes\\left|\\chi_{B}\\right\\rangle .\\label{eq:fullBiexciton}\n\\end{eqnarray}\nIn particular, there should be a size at which the AR lifetimes become independent of the volume of the QD (perhaps above the biexcitonic radius), where such exciton-exciton correlation becomes important. For example, recent advances in synthesizing single NCs that have multiple, spatially separated excitonic sites\\cite{Cui2019,Koley2021,Philbin2021arxiv} may require the inclusion of all possible quasiparticle-quasiparticle correlations to accurately model the decay of biexcitonic states.\n\n\\begin{figure*}[htb]\n\\includegraphics[width=0.85\\textwidth]{figures\/Fig7_CoreShell.pdf}\n\\caption{\\label{fig:Core-Shell} (a) Projected electron (top) and hole (bottom) carrier densities of the ground excitonic state for CdSe\/CdS (left) and CdSe\/ZnS (right) core-shell quantum dots with a core diameter of 2.2~nm and different shell thicknesses.\\cite{Philbin2020a} (b) The large contribution of surface modes to the reorganization energy in small CdSe NCs can be mitigated by the addition of a passivating shell, lowering the overall reorganization energy. (c) Auger recombination lifetimes, $\\tau_\\text{AR}$, (top) and root-mean-square exciton radii, $r_\\text{e-h} = \\sqrt{\\langle r_\\text{e-h}^2 \\rangle}$, (bottom) of CdSe\/CdS and CdSe\/ZnS core-shell quantum dots as a function of shell monolayers for a CdSe core diameter of 2.2~nm.\\cite{Philbin2020a} (d) The reorganization energies of bare CdSe quantum dots are significantly larger than those of CdSe cores with 3 monolayers of CdS shell. The black triangle corresponds to the experimentally measured reorganization energy of a CdSe\/CdS core-shell particle with a core diameter of 4~nm and 3~monolayers of shell by Talapin \\textit{et al.}\\cite{Weller2008}}\n\\end{figure*}\n\nReturning to the scaling of the AR lifetime with NC size, the universal volume scaling of the AR lifetime with the volume of the QD ($\\tau_{\\text{AR}} \\propto V_{\\text{QD}}$) is shown in Fig.~\\ref{fig:AR}. This volume scaling can be understood from analyzing the volume dependence of the Coulomb coupling and density of final states used to calculate the AR lifetime in Eq.~\\eqref{eq:fermisGoldenRule}. The density of final states increases linearly with the volume of the QD, as it also does for nanorods and nanoplatelets.\\cite{Rabani2010,Baer2012,Philbin2018,Philbin2020b} However, the decreasing Coulomb coupling between the initial biexcitonic state and final high energy excitonic states with increasing system size counteracts the increasing number of final states. We previously reported that these Coulomb couplings decrease with the square of the QD volume in the interacting formalism, such that the overall AR lifetime increases linearly with the volume of the QD, as found experimentally. In AR lifetime calculations that utilize noninteracting biexcitonic states (Eq.~\\eqref{eq:nonintBiexciton}), the Coulomb couplings decrease too fast with the QD volume, which was responsible for the disagreement between theoretical predictions and experimental measurements of the scaling of the AR lifetime with QD volume. The inclusion of electron-hole correlations in the initial biexcitonic state leads to a less steep volume dependence of these Coulomb couplings, as they are related to the electron-hole overlap. Increasing this overlap by properly accounting for the attractive interactions between electrons and holes, as is done in the interacting formalism, leads to increased Coulomb couplings.\n\nThus far, we have been concerned with understanding the decay of initial biexcitonic states \\textit{via} AR. The decay of multiexcitonic states can be modeled by classical master equations (\\textit{i.e.}, classical rate equations) that use decay rates of single excitons and biexcitons. These methods have proven to be surprisingly accurate for modeling the decay of a general number of excitons, $N_{\\text{exc}}$, in a NC\\cite{Ben-Shahar2018,Yan2021} as well as Auger heating,\\cite{Guzelturk2021} or the long-lived heating of the NC lattice that occurs due to the sequence of AR events and subsequent hot carrier cooling.\\cite{Achermann2006,Harvey2018} To this end, the rate that $N_{\\text{exc}}$ excitons decays to $(N_{\\text{exc}}-1)$ excitons can be well-approximated by modeling AR as a bimolecular collision between excitons, such that the overall AR rate ($K_{\\text{AR}}$) is given by \n\\begin{eqnarray}\nK_{\\text{AR}} & = & {N_{\\text{exc}} \\choose 2} k_{\\text{AR}}\\,, \\label{eq:totalARRate}\n\\end{eqnarray}\nwhere $k_{\\text{AR}}$ is the inverse of the biexciton AR lifetime ($k_{\\text{AR}}=\\tau_{\\text{AR}}^{-1}$) and $N_{\\text{exc}} \\choose 2$ is the binomial coefficient equal to $N_{\\text{exc}}(N_{\\text{exc}}-1)\/2$.\n\nThe modeling of AR in terms of a bimolecular collision between excitons seems to be consistent with our findings that the interacting formalism, which includes the physics of electrons and holes forming correlated electron-hole pairs (excitons), predicts accurate AR lifetimes. The noninteracting formalism lends itself to modeling the total AR rate as a trimolecular collision between either two quasi-holes and a quasi-electron or two quasi-electrons and a quasi-hole. Given that the noninteracting formalism predicts biexciton AR lifetimes that are far too long, we believe that modeling the total AR decay rate as a trimolecular collision is inappropriate in semiconductor NCs, especially in QDs with radii that are comparable to the exciton Bohr radius of the material and in all nanorods and nanoplatelets.\\cite{Zhu2012} \n\n\n\n\n\\section{Role of interfaces in nanocrystals}\n\\label{sec:interfaces}\nThe decay channels described thus far, such as hot exciton cooling and AR, are dictated by electron-hole correlations, exciton-phonon couplings, and exciton-exciton interactions. In addition to size, dimensionality, and material composition, these interactions can be tuned in NCs by the synthesis of heterostructures, such as core-shell NCs.\\cite{Frederick2010,Jain2016,Sagar2020} The core-shell interface introduces another point of control that enables wave function engineering.\n\nFor example, CdSe\/CdS NCs have a quasi-type II band alignment due to the valence band offset between these bulk materials. The interplay of quantum confinement, band alignment, and electron-hole correlation confines the hole density to the core while the electron density delocalizes into the CdS shell.\\cite{Kong2018} On the other hand, CdSe\/ZnS core-shell systems have a type I band alignment that confines both the electron and hole to the core.\\cite{Zhu2010} These behaviors are well-captured by our atomistic electronic structure framework, as illustrated in Fig.~\\ref{fig:Core-Shell}a. The qualitative differences in wave functions in single-material NCs versus heterostructures have large effects on the magnitudes of both EXPC and AR lifetimes.\n\nAs described in Sec.~\\ref{sec:exph}, phonon modes localized to the surface of NCs have significant contributions to the overall EXPC, especially in NCs that are in the strongly confined regime (Fig.~\\ref{fig:Core-Shell}b). This result suggests that EXPC can be mitigated by treatment of the NC surface, such as through the introduction of a passivating shell. For CdSe\/CdS core-shell QDs, the overall reorganization energy can be almost an order of magnitude smaller than that of bare CdSe cores, as shown in Fig.~\\ref{fig:Core-Shell}d, depending on the core size. This effect is a direct consequence of the quasi-type II band alignment that confines the exciton hole to the CdSe core. As the hole channel is the dominant channel for EXPC,\\cite{Jasrasaria2021} hole localization suppresses coupling of the exciton to surface modes and low-frequency acoustic modes that are delocalized over the NC.\n\nGrowth of a passivating shell on CdSe core NCs also has a profound effect on the AR lifetimes. Fig.~\\ref{fig:Core-Shell}c highlights the dramatic differences between AR lifetimes in CdSe\/CdS versus CdSe\/ZnS systems.\\cite{Philbin2020a} The type I band alignment in CdSe\/ZnS means that the root-mean-square exciton radius, or average electron-hole radial coordinate, is relatively independent of ZnS shell thickness after the growth of one shell monolayer. In the quasi-type II systems, however, the root-mean-square exciton radius grows linearly with the number of CdS shell monolayers as the electron delocalizes over the CdS shell. These results directly affect the AR lifetime, which depends on electron-hole wavefunction overlap \\textit{via} the Coulomb coupling. The CdSe\/ZnS QDs show AR lifetimes that do not change with growth of ZnS shell while those of CdSe\/CdS QDs increase dramatically with growth of CdS shell.\n\nIn addition to core-shell QDs, dimers and superlattices of NCs and NC heterostructures are being developed and studied theoretically.\\cite{Williams2009,Evers2015,Lee2018,Ondry2019,Ondry2021,Notot2022} These materials offer the potential for significant engineering and control of both electronic and phononic properties, enabling the realization of new phenomena.\n\n\n\\section{Outlook}\n\\label{sec:conclusions}\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.5\\textwidth]{figures\/Fig8_Cooling.pdf}\n\\caption{\\label{fig:Cooling} The calculated absorption spectrum (top) and density of excitonic states (bottom) for a wurtzite 3.9~nm CdSe quantum dot. The vertical lines in the top panel indicate the oscillator strength of the transition from the ground state to that excitonic state. The inset depicts the cooling process schematically, indicating that exciton cooling occurs through a cascade of states.}\n\\end{figure}\n\nThus far, we have described our framework for computing the electronic\/vibronic properties of confined semiconductor NCs of experimentally relevant sizes. Our approach includes electron-hole correlations, which are key to accurately describing excited-state phenomena, and exciton-phonon coupling, which are essential for understanding room-temperature optical properties and phonon-mediated exciton dynamics. Calculations using our approach yield very good agreement with experimental measurements of observables, such as fundamental and optical gaps, phonon lifetimes, reorganization energies, and AR lifetimes, for II-VI and III-V materials of a variety of sizes, dimensionalities, and compositions.\n\nThe main short-term goal is to use these tools to address the longstanding controversy surrounding the phonon bottleneck. The original hypothesis~\\cite{Nozik2001} of the phonon bottleneck in NCs is based on a single-particle picture of the \\textit{electronic} states, for which the energy spacing between states near the band edge becomes several hundreds of meV. In this case, because the phonon frequencies in these systems are $\\sim$30~meV and lower, a multiphonon process would be required for phonon-mediated relaxation, which would be extremely slow. In this electron-hole picture, one requires Auger-like, Coulomb-mediated coupling to break the phonon bottleneck. This picture is translated to a relatively high density of \\textit{excitonic } states due to the dense spectrum of holes. Fig.~\\ref{fig:Cooling} illustrates the calculated absorption spectrum of a wurtzite 3.9~nm CdSe quantum dot, which is made up of a few very bright excitonic states that have large oscillator strengths and several dim excitonic states that have small oscillator strengths. While the energy spectrum is relatively sparse near the ground excitonic state, the largest excitonic energy gap in this system is 25~meV, and the energy spacing quickly decreases for states higher in energy.\\cite{Zunger1997} This would result in a cascade of relaxation events to dark\/bright excitons that would be relatively fast due to the small energy spacing, breaking the phonon bottleneck.\n\nFuture work will focus on dynamical processes to describe spectral lineshapes, providing means to further assess and improve the approach, as well as integrating this model with a framework for nonadiabatic dynamics to simulate the exciton cooling process and delineate the timescales and mechanism of cooling as a function of excitation energy, NC dimensionality, and NC size. \n\n\\begin{acknowledgments}\nE.R. acknowledges support from the Department of Energy, Photonics at the Thermodynamic Limits Energy Frontier Research Center, under grant no. DESC0019140. Methods used in this work were provided by the Center for Computational Study of Excited State Phenomena in Energy Materials (C2SEPEM), which is funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, via contract no. DE-AC02-05CH11231, as part of the Computational Materials Sciences Program. Computational resources were provided by the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231. D.J. acknowledges the support of the Computational Science Graduate Fellowship from the U.S. Department of Energy under grant no. DE-SC0019323. J.P.P. acknowledges support from the Harvard University Center for the Environment.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe two envelope paradox is well known, and a number of approaches exist to explain it \\cite{wiki}. Our approach is based on probability theory. \n\nWe formally define the problem, and prove that switching does not improve expected value. We then resolve the paradox by explaining where the \"naive\" calculation goes wrong. Finally we discuss how having some prior information changes the problem. \n\n\\section{Expected value and optimal strategy}\n\nLet Y be the base amount (i.e. the envelopes contain Y and 2Y), it is a random variable with distribution \\(F_1\\), on a probability space \\(\\Omega\\). \n\nLet X(Y) and X'(Y) represent the amounts in the chosen envelope and the other envelope respectively. To represent the choice of envelope, we have the distribution \\(F_2\\) on a probability space with two events \\( \\mathcal{E} = \\{\\mathcal{E}_1, \\mathcal{E}_2\\} \\). X is a function \\( X (Y(\\omega), \\epsilon ) \\) where \\( ( \\omega, \\epsilon) \\in \\Omega \\times \\mathcal{E} \\). When \\( \\epsilon = \\mathcal{E}_1 \\), we happen to choose the smaller envelope, so X = Y and X' = 2Y. And when \\( \\epsilon = \\mathcal{E}_2 \\), X = 2Y and X' = Y. The choices are equally likely i.e. \\( \\mathbb{P}(\\mathcal{E}_1) = \\int_{\\mathcal{E}_1} dF_2 = 1\/2\\) and \\(\\mathbb{P}(\\bar{\\mathcal{E}_2}) = \\int_{\\bar{\\mathcal{E}_2}} dF_2 = 1\/2 \\). \n\nLet V(Y) be the final value the player gets after deciding to switch or not. It has distribution \\(F_3\\) on two events \\( \\{ \\mathcal{S}, \\bar{\\mathcal{S}} \\} \\) representing respectively switching (V = X'), or not switching (V = X). For example if the strategy is to always switch, \\( F_3 \\) is deterministic with \\( \\mathbb{P}(\\mathcal{S}) = \\int_{\\mathcal{S}} dF_3 = 1, \\mathbb{P}(\\bar{\\mathcal{S}}) = \\int_{\\bar{\\mathcal{S}}} dF_3 = 0 \\). More generally, \\(F_3\\) is allowed to depend on the observed value of X, as the player can decide to switch after seeing the value in the chosen envelope. But not on \\(F_1 \\) which the player has no information about.\n\nOur key result is that the payoff is the same for all possible switching strategies. \n\n\\begin{theorem}\n\\(E[V] = \\frac{3}{2} E[Y]\\), for all possible switching strategies \\(F_3 \\). \n\\end{theorem}\n\n\\begin{proof}\nThe expected value of V over all probability measures \\( F_1 \\), \\( F_2 \\) and \\( F_3 \\) is: \n\\begin{equation}\n\\label{ev}\nE[V] = \\int_{\\Omega} E[V|Y] dF_1\n\\end{equation}\nFor a given Y, we have the choice of switching or not switching i.e.\n\\begin{align*}\nE[V|Y] = \\int V(Y) dF_3 = \\int_{\\bar{\\mathcal{S}}} X(Y)dF_3 + \\int_{\\mathcal{S}} X'(Y) dF_3\n\\end{align*}\nand expanding the terms to show the choice of envelope \\(\\mathcal{E}_1\\) and \\(\\mathcal{E}_2\\):\n\\begin{align*}\nE[V|Y] & = \\int_{\\bar{\\mathcal{S}}} \\left( \\int_{\\mathcal{E}_1} Y dF_2 + \\int_{\\mathcal{E}_2} 2Y dF_2 \\right) dF_3 + \\int_{\\mathcal{S}} \\left( \\int_{\\mathcal{E}_1} 2Y dF_2 + \\int_{\\mathcal{E}_2} Y dF_2 \\right) dF_3 \\\\\n & = \\int_{\\mathcal{E}_1} \\left( \\int_{\\bar{\\mathcal{S}}} Y dF_3 + \\int_{\\mathcal{S}} 2Y dF_3 \\right) dF_2 + \\int_{\\mathcal{E}_2} \\left( \\int_{\\bar{\\mathcal{S}}} 2Y dF_3 + \\int_{\\mathcal{S}} Y dF_3 \\right) dF_2 \n\\end{align*}\nNow, since we don't know which envelope was chosen (i.e. whether we are in \\(\\mathcal{E}_1 \\) or \\( \\mathcal{E}_2 \\)) when we make the choice to switch or not (being in \\( \\mathcal{S}\\) or \\( \\bar{\\mathcal{S}} \\)), \\( F_3 \\) is independent of \\(F_2\\), so we can take out the integrands:\n\\begin{equation}\n\\label{indep}\nE[V|Y] = \\left( \\int_{\\bar{\\mathcal{S}}} Y dF_3 + \\int_{\\mathcal{S}} 2Y dF_3 \\right) \\int_{\\mathcal{E}_1} dF_2 + \n\\left( \\int_{\\bar{\\mathcal{S}}} 2Y dF_3 + \\int_{\\mathcal{S}} Y dF_3 \\right) \\int_{\\mathcal{E}_2} dF_2\n\\end{equation}\nThus, plugging in the actual values for \\( F_2 \\)\n\\begin{align*}\nE[V|Y] & = \\left( \\int_{\\bar{\\mathcal{S}}} Y dF_3 + \\int_{\\mathcal{S}} 2Y dF_3 \\right) \\frac{1}{2} + \\left( \\int_{\\bar{\\mathcal{S}}} 2Y dF_3 + \\int_{\\mathcal{S}} Y dF_3 \\right) \\frac{1}{2} \\\\\n & = \\frac{1}{2} \\left( \\int_{\\bar{\\mathcal{S}}}\\left( Y+2Y \\right) dF_3 + \\int_{\\mathcal{S}} \\left( 2Y+Y \\right) dF_3 \\right) \\\\\n & = \\frac{3}{2} Y \\left( \\int_{\\bar{\\mathcal{S}}} dF_3 + \\int_{\\mathcal{S}} dF_3 \\right) = \\frac{3}{2} Y \n\\end{align*}\nNow plugging this back into \\eqref{ev}, the overall expectation of V is \n\\begin{equation}\n\\label{th1}\nE[V] = \\frac{3}{2} \\int_{\\Omega} Y dF_1 = \\frac{3}{2} E[Y] \n\\end{equation}\n\\end{proof}\n\nThus the expected value is always the same, regardless of the switching strategy \\(F_3\\), including never switching, and seeing X or not seeing X makes no difference.\nNow, we can see there is no paradox. \n\n\\section{Resolving the paradox}\n\n\\begin{corollary} The expected value from switching to the second envelope is the same as the expected value of keeping the first, i.e. E[V] = E[X].\n\\end{corollary}\n\\begin{proof}\nBy definition of X, \n\\begin{align*}\nE[X|Y] = \\int_{\\mathcal{E}_1} Y dF_2 + \\int_{\\mathcal{E}_2} 2Y dF_2 = \\frac{1}{2} Y + \\frac{1}{2} 2Y\n\\end{align*}\nsince \\( \\mathbb{P}(\\mathcal{E}_1) = \\mathbb{P}(\\mathcal{E}_2) = 1\/2 \\).\nTaking expectation over Y, we get\n\\begin{equation}\n\\label{ex}\nE[X] = \\frac{3}{2} E[Y] \n\\end{equation}\ntherefore from \\eqref{th1}, \\(E[X] = E[V] \\).\n\\end{proof}\n\n\nThe \"paradox\" is that it may naively seem like \\(E[V] > E[X]\\). To see how it arises and why it is incorrect, let's restate the E[V] calculation in terms of X: \n\\begin{equation}\n\\label{evx}\nE[V|X] = \\int_{\\bar{\\mathcal{S}}} X dF_3 + \\int_{\\mathcal{S}} \\left( \\int_{\\mathcal{E}_1} 2X dF_2 + \\int_{\\mathcal{E}_2} \\frac{1}{2}X dF_2 \\right) dF_3 \\\\\n\\end{equation}\nNaively treating X as a constant, since \\( \\mathbb{P}(\\mathcal{E}_1) = \\mathbb{P}(\\mathcal{E}_2) = 1\/2 \\) it seems like \n\\begin{equation}\n\\label{wrong}\nE[V|X] \\stackrel{?}{=} \\mathbb{P}(\\bar{\\mathcal{S}}) X + \\mathbb{P}(\\mathcal{S}) \\frac{5}{4} X \n\\end{equation}\nwhich implies that \\(E[V|X] > X\\), i.e. any non-zero switching probability is a strict improvement. In fact always switching i.e. \\( \\mathbb{P}(\\mathcal{S}) = 1 \\) is optimal and gives a 25\\% gain over never switching. \n\nThe root of the apparent paradox is that \\eqref{wrong} is incorrect, because in \\eqref{evx}, \\(X \\) is not actually a constant in \\( \\mathcal{E} \\). This is counter-intuitive because we can compute \\( E[V|X] \\) after seeing the actual value of X, so X seems like it should be constant. But here we are evaluating on \\( \\mathcal{E} \\), the choice of envelope, which of course affects the value of X. More precisely \\(X dF_2 \\) is \\(X(Y, \\epsilon) dF_2(\\epsilon)\\), i.e. it is not the same X in the two \\(dF_2\\) integrals, since \\( \\epsilon \\) is the variable being integrated on. \nThe event \\( \\epsilon \\) cannot by definition be treated as a constant when evaluating it's probabilities. \n\nBut \\( Y \\) \\emph{is} a constant in \\( \\mathcal{E} \\), since the event \\( \\omega \\in \\Omega \\) is determined. So we can use the fact that \\( X(Y, \\mathcal{E}_1 ) = Y \\), and \\( X(Y, \\mathcal{E}_2) = 2Y\\). Also, as in \\eqref{indep}, the choice \\( F_3 \\) is independent of \\( F_2 \\). Thus we get: \n\\begin{align*}\nE[V|X] & = \\mathbb{P}(\\bar{\\mathcal{S}}) X + \\mathbb{P}(\\mathcal{S}) \\left( \\int_{\\mathcal{E}_1} 2Y dF_2 + \\int_{\\mathcal{E}_2} \n\\frac{1}{2} 2Y dF_2 \\right) \\\\\n & = \\mathbb{P}(\\bar{\\mathcal{S}}) X + \\mathbb{P}(\\mathcal{S}) \\left( \\frac{1}{2} 2Y + \\frac{1}{2}\\frac{1}{2} 2Y \\right) \\\\\n& = \\mathbb{P}(\\bar{\\mathcal{S}}) X + \\mathbb{P}(\\mathcal{S}) \\frac{3}{2} Y\n\\end{align*}\nNow for the given X, we can take expectations over all values of Y, \n\\begin{align*}\nE[V|X] = E[E[V|X, Y]] & = \\mathbb{P}(\\bar{\\mathcal{S}}) X + \\mathbb{P}(\\mathcal{S}) \\frac{3}{2} E[Y]\n\\end{align*}\nAnd, using \\eqref{ex}, we see that: \\( E[V] = \\frac{3}{2} E[Y] \\), i.e. there's no paradox.\n\nA simple example to illustrate: Suppose we open the chosen envelope and see X =\\$100. Contrary to the naive estimate, we are not actually in a state where the other envelope has an equal chance of containing \\$50 or \\$200. Rather, we are in state where a hidden Y has already been chosen and we are looking at Y or 2Y, with equal chance. To see this more clearly, imagine after the envelopes are filled, they are cloned into many instances of the game in parallel (not repeated!), and X is an average of the observed value. By ergodicity, the expected value in the one-shot game is the same as the average value in the parallel games. Since we expect to observe an average of X = 3Y\/2, by seeing X = \\$100, we \"learn\" that Y = 2X\/3 = \\$66.66... and the average value of switching or not switching remains \\$100.\n\n\\section{Open vs closed envelope} \nConsider the variation of the problem based on whether the player gets to see the value \\(X \\) or not before making the decision to switch. In all of the above, knowing the actual value of of \\(X \\) does not change the optimal strategy. Thus in both the open and closed versions, the answer remains that switching makes no difference. Indeed we don't know anything about \\(F_1 \\) so knowing one value doesn't help us decide if \\( X \\) is \\(Y \\) or \\(2Y\\). \n\n\\section{Prior information on distribution} \nIf we (the player) have some prior knowledge of \\(F_1\\), then, given X, we may know if it's more likely to be Y or 2Y, i.e. if we are in \\(\\mathcal{E}_1 \\) or \\( \\mathcal{E}_2 \\), which we can use in deciding to switch or not, i.e. \\(F_3\\) can be a function of \\(\\epsilon \\). Therefore the step \\eqref{indep} where we factor out \\(dF_2 \\) from \\(dF_3 \\) is no longer valid and it is no longer true that all strategies have the same \\(E[V] \\). In fact switching does sometimes lead to gains, and knowing the value of \\(X \\) before deciding makes a difference too.\n\nFor example, if we know the average E[Y], then we would use the strategy: switch if and only if \\( X < \\frac{3}{2} E[Y] \\). Another example is, if we know \\( Y_{max} \\) the largest possible value of Y, then when \\( X > Y_{max} \\), naturally we should never switch because we know for sure that we are in \\(\\mathcal{E}_2 \\). Similarly, if we know \\( Y_{min} \\) the smallest possible value of Y, then when \\( X < 2Y_{min} \\), we should always switch because we know for sure that we are in \\(\\mathcal{E}_1 \\). \n\nMore broadly, if we know \\( F_1 \\), after seeing \\( X \\), we can estimate \\( \\mathbb{P}(\\mathcal{E}_1 | X) \\) and \\( \\mathbb{P}(\\mathcal{E}_2 | X) \\), and choose a mixed strategy \\(F_3\\) whereby the probability of not switching, \\( \\mathbb{P}(\\bar{\\mathcal{S}}) \\), is higher when \\( \\mathbb{P}(\\mathcal{E}_2|X) \\) is higher, and \\( \\mathbb{P}(\\mathcal{S}) \\) is higher if \\( \\mathbb{P}(\\mathcal{E}_1 |X) \\) is higher. \n\nIn general, the switching strategy can be optimized to take advantage of any prior information about \\(F_1\\). A few interesting cases are covered in \\cite{mcdon}.\n\\section*{Acknowledgement}\nI would like to thank Jacob Eliosoff for pointing me to this problem and for helpful discussions. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Why quantum gravity?}\nThe subject of quantum gravity emerged as part of the unification\nprogram that led to electromagnetism and the electroweak model.\nWe'd like to unify all forces of Nature. Forces other than gravity\nare certainly of a quantum nature. Thus we cannot hope to have a\nfully unified theory before quantizing gravity.\n\nTo come clean about it right from the start, we should stress that\nthere is no compelling experimental reason for quantizing gravity.\nFor all we know, gravity could stand alone with respect to all\nother forces, and simply be exactly classical in all regimes.\nThere is no evidence at all that the gravitational field ever\nbecomes quantum~\\footnote{Exercise for the student: discuss the\ngravitational field of a photon undergoing the double slit\nexperiment. Could you collapse the wave function by measuring the\ngravitational field? Lay out all possibilities in the form of\nthought-experiments. You may find it interesting to make contact\nwith the old problem of how classical and quantum systems\ninteract. Then repeat the exercise with a double slit experiment\nwhere gravitons are used instead of photons.}. Yet this hasn't\ndeterred a large number of physicists from devoting lifetimes to\nthis pursuit.\n\nAssaults on the problem currently follow two main trends: string\/M\ntheory~\\cite{pol,stringref} and loop quantum\ngravity~\\cite{rovelli,carlip}. Both have merits and deficiencies,\ncommented extensively elsewhere. As a poor third we mention\nRegge-calculus (and lattice techniques), non-commutative geometry,\nand several other methods none of which has fared better or worse\nthan the two main strands.\n\nThis course is {\\it not} about those theories. Rather it's about\nthe question: {\\it Where might experiment fit into these\ntheoretical efforts of quantizing gravity?} A middle ground has\nrecently emerged -- phenomenological quantum gravity. The\nrequirements are simple: a phenomenological formalism must provide\na believable approximation limit for more sophisticated\napproaches; it must also make clear contact with experimental\nanomalies that don't fit into our current understanding of the\nworld. The following argument illustrates what we mean by this.\n\nWhen physicists find themselves at a loss they often turn to dimensional\nanalysis. Following this simplistic philosophy we estimate the scales where\nquantum gravity effects may become relevant by\nbuilding quantities with dimensions\nof energy, length and time from $\\hbar$ (the quantum), $c$ (relativity)\nand $G$ (gravity). These are called the Planck energy $E_P$, the Planck\nlength $l_P$ and the Planck time $t_P$. For instance\n$E_p=\\sqrt{\\hbar c^5\/G}\\approx 1.2\\times 10^{19}GeV\\approx 2.2\\times 10^{-5}g$.\nQuantum gravitational effects are expected to kick in for energies above\n$E_P$ or lengths and durations smaller than $l_p$ and $t_p$.\nBeware: dimensional analysis can be too naive.\n\n\nWe expect quantization of gravity to take the form of a theory in\nwhich space and time are discretized. General relativity is a\ntheory of curved space-time. Thus, quantum gravity should quantize\nnot only curvature but actual durations and lengths. Here\nquantization means to replace a continuum by a discrete structure.\nThis may be done, as a first approximation, just as Planck did in\nthe first quantum theory, when he simply proposed that the energy\nof a harmonic oscillator would be a multiple of a fixed energy,\n$\\hbar \\omega$. The same could be true for space and time: space\nis granular, time has an atomic structure. This is, however, an\napproximation. As we know, what actually happens in quantum\nmechanics is that observables are replaced by operators with a\ndiscrete spectrum, whose eigenvalues represent the possible\noutcomes of a measurement. Similarly, in loop quantum gravity,\narea and volume become operators with a discrete spectrum and the\ngeometry becomes a quantum state (a spin foam, more specifically).\nStill, we may use the ``Planck-style'' of quantization as a basis\nfor a phenomenological model.\n\n\nThe point of this simplified quantization approach is that now we\nhave a springboard for contact with experiment. The argument is\nbased on a paradox similar in flavor to the Zeno paradox in\nancient Greek philosophy (which incidentally concerned the absurd\napparent conflict between discrete and continuum). Whatever\nquantum gravity may turn out to be, it is expected to agree with\nspecial relativity when the gravitational field is weak or absent,\nand also with all experiments probing the nature of space-time on\nscales much larger than $l_P$ (or energy scales smaller than\n$E_P$). The granules of space-time should be invisible unless we\nexamine these scales with a powerful ``microscope''.\n\nThis immediately gives rise to a simple question: {\\it In whose\nreference frame is $l_P$ the threshold for new phenomena?} For\nsuppose that there is a physical length scale which measures the\nsize of spatial structures in quantum space-times, such as the\ndiscrete area and volume predicted by loop quantum gravity. Then\nif this scale is $l_P$ in one inertial reference frame, special\nrelativity suggests it will be different in another observer's\nframe -- a straightforward implication of Lorentz-Fitzgerald\ncontraction, easily derived from the Lorentz transformations. In\nother words the border between classical and quantum gravity is\nnot invariant or well defined. Similar arguments can be made with\nenergy and time.\n\nThere are two obvious answers to the problem. On the one hand,\nLorentz transformations may be correct on all scales, such that\nthe Planck length is sensitive to Lorentz contraction. In this\ncase, quantum gravity picks up a preferred frame in which the\nPlanck length is the border between classical and quantum gravity.\nOn the other hand, it could be that quantum gravity is the ether\nwind, and that all other effects baffle the Michelson-Morley\nexperiment.\n\nA distinct possibility is for quantum gravity to respect the\nprinciple of relativity, but require a revision of the Lorentz\ntransformations at extreme scales. Such transformations should\nleave the Planck length invariant. A toy model for this is to let\nthe speed of light be length dependent, and go to infinity for\n$l_p$. A more sophisticated version is non-linear relativity, also\ncalled doubly special relativity (DSR), developed in the next\nsection. This is the prototype of the argument leading to\nphenomenological quantum gravity.\n\nThe strength of this last approach is that a relation to\nobservations is quickly obtained in this way. Indeed, DSR explains\nultra high energy cosmic ray anomalies. This illustrates what is\nmeant by phenomenological quantum gravity. The theoretical problem\nis too hard. Perhaps it needs a bit of fresh air, called\nexperiment. A simplified formalism could then be set up with the\nflavors of attempts at a full solution. That is, such a formalism\ncan act as a target for low-energy approximations to the the full\nsolution, and can also make immediate contact with experiment. A\nbridge between theory and experiment has been set up.\n\n\n\n\n\\section{Nonlinear Relativity}\n\n\n\n\"Doubly Special Relativity\" (DSR) ~\\cite{amelstat,gli,leejoao,leejoao1} is a\nsemiclassical theory, formulated in flat space-time, yet\nsignificant at extremely high energy scales. It is based on a\nnon-linear extension of the laws of Special Relativity\n~\\cite{leejoao}. Just as Einstein's relativity theory is\n\"special\" because it holds invariant the speed of light (c) as a\nfundamental relativistic scale, non-linear relativity is \"doubly\nspecial\" because it fixes, in addition, a relativistic {\\it\nenergy} scale. We stress the word \"relativistic\" in order to\nhighlight what type of fundamental scale we are dealing with. For\ninstance, $\\hbar$ is not a relativistic scale since it does not\naffect the transformations between inertial observers like $c$\ndoes at very high velocities. The new fixed energy scale will play\na role, independent of $c$, in relativistic transformations at\nvery high energies.\n\nPart of the justification for the introduction of an invariant\nenergy scale into Special Relativity can be found in the lineage\nof Einstein's theory. Galileo's original expression for the energy\nof a fundamental particle, $E=p^2\/2m$, is linearly invariant under\nthe transformations he defined circa 1600 A.D.. These Galilean\nTransformations describe the relativity of inertial motion. They\nstate that position and time coordinates measured in a \"primed\"\nframe moving at velocity $v$ in the $x$-direction with respect to\na lab frame at rest are expressed as $x'=x-vt$ and $t'=t$,\nrespectively. A length $\\ell$, a time interval $t$ and the\nvelocity of a particle moving at speed\n$v_1$ in the moving frame would then be written \\begin{eqnarray} v_1'&=&v_1-v \\\\\n\\ell '&=&\\ell \\\\ t'&=&t .\\end{eqnarray}\n\nRoughly 200 years later Maxwell formulated his equations\ndescribing electricity and magnetism and introduced the notion of\na constant value for the speed of light. In order for this\ninvariance to hold whilst satisfying Galilean transformations, the\nnotion of a ``preferred observer'' had to be introduced. The\npreference was made manifest by introducing a uniform cosmic\nbackground called the ``ether''. In other words, the relativity of\ninertial observers was lost.\n\nWhile Michaelson and Morley worked to prove the constancy of the\nspeed of light proposed by Maxwell, Einstein's work in the early\n1900s demolished the concept of the ether by unifying the\npreviously disjoint notions of time and space in new laws\ndescribing the relativity of inertial motion. His discovery of the\nequivalence of mass and energy led him to a revised definition of\nparticle energy, the quadratic \"dispersion relation\"\n$E^2=p^2c^2+m^2c^4$. This was no longer, however, invariant under\nGalilean transformations. In order to reestablish observer\nindependent laws, new relativistic transformations, the Lorentz\ntransformations, had to be formulated. With respect to a frame at\nrest, a particle will have coordinates in a frame moving at\nvelocity $v$ in the $x$-direction given by, \\begin{eqnarray}\nt'&=&\\gamma (t-{vx\\over c^2}) \\\\ x'&=&\\gamma (x-vt) \\\\ y'&=&y \\\\\nz'&=&z , \\end{eqnarray} where $\\gamma=1\/\\sqrt{1-v^2\/c^2}$. With respect to\nenergy and momenta coordinates these transformations become, \\begin{eqnarray}\nE'&=&\\gamma(E-vp_x)\n\\\\ p_x'&=&\\gamma(p_x-{Ev\\over c^2}) \\\\ p_y'&=&p_y \\\\ p_z'&=&p_z .\n\\end{eqnarray} Galileo's linear velocity addition laws, invariant lengths\nand invariant time intervals were accordingly transformed into\nnonlinear velocity addition laws, length contraction and time\ndilation formulae given by, \\begin{eqnarray} v_1'&=&{v_1 +v \\over 1+v_1v} \\\\\n\\ell '&=&{\\ell\\over\\gamma}\n\\\\ \\Delta t'&=&\\gamma\\Delta t .\\end{eqnarray} With these new laws governing\ninertial motion, Einstein enforced the equality of all observers.\n\nIn this vein, in order to preserve the relativity of inertial\nobservers, DSR theories deform the Lorentz transformations such\nthat the modified dispersion relation,\n$E^2=\\textbf{p}^2c^2+m^2c^4+f(E,\\textbf{p}^2;E_0)$ ~\\cite{\nleejoao}, is non-linearly invariant under their action. $E_0$ is\nthe invariant energy scale. The $f$-function in this expression\nrepresents a generalized way of introducing the energy-dependence\ninto the Lorentz transformations, as we will see below.\n\nHistory aside, there are two significant motivational areas for\nintroducing a fixed energy scale into Einstein's relativistic\ntransformations, one motivated by issues in quantum gravity\ntheories (e.g. \\cite{leejoao}) and one by anomalous observations\nof the cosmos (e.g. \\cite{am}). Included in the first is the idea\nof a fundamental energy (or length) scale, such as would arise in\na theory of quantum gravity. This scale would take on a certain\nvalue in one frame of reference, but when boosted to another frame\nof reference it would assume another value according to the\nLorentz transformations. In fact, there would exist such a frame\nin which the given energy scale would appear to surpass the\nlimiting energy value predicted by quantum gravity. It is thus\nparadoxical for a \"fundamental\" scale to be relativistic, and so,\nin principle, we want to modify the transformations such that they\nhold this fundamental scale fixed while preserving observer\nindependence~\\cite{amelstat,gli,leejoao}. This modification would\nmanifest itself only as we approach extremely high energy scales.\nOtherwise, by the correspondence principle, if we set this energy\nscale to infinity (or length scale to zero), which is equivalent\nto not having a limiting scale at all, we recover Special\nRelativity and its Lorentz transformations. This is reflective of\nthe $c\\rightarrow\\infty$ limit taking Lorentz transformations to\nGalilean transformations, and the $\\hbar\\rightarrow 0$ limit\ntaking quantum mechanics to classical mechanics.\n\nAnother theoretical motivation within the realm of quantum\ngravity, is the fact that theories of Non-Commutative Geometries,\nStringy Space-Time Foams, and Loop Quantum Gravity all predict\nmodified dispersion relations ~\\cite{leejoao}. So, the hope would\nbe to come to a deformed DSR dispersion relation that is in accord\nwith one of these theoretical predictions. But, naturally,\nobservational support is needed. As we will show below, deformed\ndispersion relations, manifest at high energy scales, can be used\nto clarify anomalous observations of high energy particle\ninteractions. In particular, an initial goal for DSR theories was\nto explain the Ultra High Energy Cosmic Rays (UHECRs) that are\nobserved to collide with the Earth's atmosphere, but which are\ntheoretically predicted not to exist due to threshold interactions\nof cosmic rays with the Cosmic Microwave Background (CMB) as they\ntravel through space ~\\cite{review,crexp,cosmicray,leejoao1} (see\nalso~\\cite{amel,amel1,liouv}).\n\nThroughout these notes we will choose the Planck Energy,\n$E_p\\approx \\sqrt{\\hbar c^5\/G}=10^{19}GeV$, as our fundamental energy\nscale, where $\\hbar$ is Planck's constant, $c$ is the speed of\nlight, and $G$ is Newton's gravitational constant. This energy is\nthe fundamental scale for many quantum gravity theories and is\nchosen as an explicit example here for simplicity. We do not\nignore the fact that it may {\\it not} be precisely this value that\nshould come into play in the DSR equations. The exact value must\nbe predicted uniquely and consistently by both quantum gravity and\nobservation.\n\n\n\\subsection{The Lorentz group}\n\nBefore launching into the construction of DSR theories we will\nreview some basics of the structure of the Lorentz Group, which is\nan example of the general class of Lie groups, pivotal to\nunderstanding the relativistic transformations to hand. The\n\"generators\" of the Lorentz group actions are the infinitesimal\ntransformations \\begin{equation} L_{ab}=p_a{\\partial\\over\\partial p^b}-\np_b{\\partial\\over\\partial p^a},\\end{equation} where $p$ denotes the\nenergy-momentum 4-vector such that the lettered indexes run over\nthe values $(0,1,2,3)$ and Latin indexes, $i,j,etc.$ run over the\nspatial indexes $(1,2,3)$. The metric signature we will use\nthroughout the notes is $(-,+,+,+)$. The $L_{ij}$ are rotations\nabout the $j$-axis, and the $L_{0j}$ are boosts in the\n$j$-direction.\n\nThe commutation relations of the Lorentz generators form the {\\it\nalgebra} of the Lorentz group. The generators, in general, do not\ncommute, which allows for the algebra to be {\\it closed}. This\nmeans that the action of any two generators applied in sequence\nand then replied in reverse sequence will result in an action that\nis an element of the original group. Defining the boosts and\nrotations to be $B_j\\equiv L_{0j}$ and $R_j\\equiv L_{ij}$,\nrespectively, the Lorentz group algebra can be written,\n\\begin{eqnarray} \\left[B_i,B_j\\right]&=&-R_k \\\\ \\left[R_i,R_j\\right]&=&iR_k \\\\\n\\left[B_i,R_j\\right]&=&iB_k \\\\ \\left[B_i,R_i\\right]&=&0 .\\end{eqnarray}\n\nExponentiating these generators gives us the full finite Lorentz\ntransformations, $p_a'=e^{\\theta L_{ab}}p_b$. Consider, for\ninstance, the component $L_{01}=p_0{\\partial\\over\\partial p^1}-\np_1{\\partial\\over\\partial p^0}$. Exponentiating it and letting it\nact on the energy-momenta coordinates will give the finite Lorentz\ntransformation shown above. We do this by tailor expanding the\nexponential function. For the case of energy, $p_0$, the\nexpansion reads $p_0'=(1+\\theta\nL_{01}+{{\\theta}^2\\over{2!}}L_{01}^2\n+{{\\theta}^3\\over{3!}}L_{01}^3+...)p_0$. It can be shown that the\naction of $L_{01}$ on energy and momentum is given by\n$L_{01}p_1=p_0$ and $L_{01}p_0=p_1$, respectively. Thus, \\begin{eqnarray}\np_0'&=&p_0+\\theta\np_1+{{\\theta}^2\\over{2!}}p_0+{{\\theta}^3\\over{3!}}p_1+...\n\\\\ &=&p_0cosh\\theta+p_1sinh\\theta. \\end{eqnarray} Similarly,\n$p_1'=p_1cosh\\theta+p_0sinh\\theta$. These expressions tell us\nthat Lorentz transformations are rotations in hyperbolic space. We\ncan express the boost factor as\n$\\gamma={1\\over\\sqrt{1-v^2\/c^2}}=cosh\\theta$, where the velocity\nis given by $v=tanh\\theta$.\n\nA final group property that we will need in the construction of\nour DSR theory is a {\\it realization}. A realization of the\nLorentz group is a set of generators that has the same algebra as\nthe original group. It is comprised of the generators\n$K_i=U^{-1}B_iU$ and $J_i=U^{-1}R_iU$, for any function $U$ such\nthat the Lorentz algebra relations hold. For instance, we show\n\\begin{eqnarray}\n[K_i,K_j]&=&U^{-1}B_iUU^{-1}B_jU-[i\\leftrightarrow j] \\\\\n&=&U^{-1}[B_i,B_j]U \\\\ &=&-iU^{-1}R_kU \\\\ &=&-iJ_k .\\end{eqnarray} Let us\nnow look explicitly at what happens to the Lorentz algebra in DSR\ntheory.\n\n\n\\subsection{Implications of a Non-Quadratic Invariant}\n\nThe DSR theory we will be discussing keeps the value of the\nfundamental Planck energy scale, $E_p$, constant. In so doing, it\npreserves the Lorentz algebra, such that the DSR generators\nconstitute a realization of the Lorentz Group ~\\cite{leejoao}.\nRecall that the deformed dispersion relation introduced above,\n$E^2=\\textbf{p}^2c^2+m^2c^4+f(E,\\textbf{p}^2;E_p)$, incorporates\nan extra term representing a new dependence on a fundamental\nenergy, $E_p$. This function, $f$, is a map from energy-momentum\nspace into itself. We can thus write this as a function acting on\nenergy-momentum coordinates. In particular, we can write the\nmodified boosts of DSR as \\begin{equation} K_i=U^{-1}(E_p)B_iU(E_p),\\end{equation} where\n$B_i$ are the special relativistic linear boost generators. The\n$E_p$-dependence now appears in the map, $U$, and must be\nnon-linear in order to keep $E_p$ invariant ~\\cite{leejoao1} (see\nalso \\cite{fock,man,step}). The rotation transformations in DSR\nremain unchanged under this action, such that the deformed boost\ngenerators are all that is necessary to satisfy the aims that DSR\nlooks to achieve ~\\cite{leejoao}. We do not wish to modify the\nrotations anyways since rotational invariance has been proved\naccurate to extreme precision. The transformed boosts would then\nhold invariant the nonlinear expression,\n$m^2=\\eta^{ab}U_a(p)U_b(p)$.\n\n To clearly illustrate the effects that DSR has on Special\nRelativity, we will introduce the $E_p$ energy scale into the\n$U$-map via a particular example ~\\cite{leejoao}. Consider the\ndeformed boost generator given by \\begin{equation} K_i=L_{0i}+{p_i\\over E_p}D,\n\\end{equation} where $D=p_a{\\partial\\over\\partial p^a}$ is the {\\it\ndilatation} generator. It can be shown that this simple form\nleaves $E_p$ invariant as well as the Lorentz algebra unchanged.\nThe corresponding $U$-map, the modification to the original\nLorentz boost generator, can thus be written \\begin{equation} U=e^{{E\\over\nE_p}D}, \\end{equation} where $E=p_0$ always. Expanding this in a tailor\nseries, we can find its action on the energy-momenta coordinates.\nThe exact expression is \\begin{equation} U(p_a)={p_a\\over 1-{E\\over E_p}},\\end{equation}\nwhere we have a new factor in the denominator that, by the\ncorrespondence principle, reduces to unity if we let\n$E_p\\longrightarrow\\infty$. If, on the other hand, $E=E_p$, the\nexpression blows up, creating the invariant behaviour of $E_p$ we\ndesire.\n\nExponentiation of the $K_i$ generator gives us the finite form for\nthe transformed boosts. However, we need not perform this\ncalculation. Instead, we can derive their form in a\nmuch simpler manner by applying our $U$-map to the regular special\nrelativistic boost equation ~\\cite{leejoao}. That is, \\begin{eqnarray}\nU(E')=\\gamma(U(E)-vU(\\textbf{p})) \\\\\nU(p')=\\gamma(U(\\textbf{p})-vU(E)), \\end{eqnarray} where we have set $c=1$\nfor simplicity. By tailor expansion it can be proved that indeed\n$e^{\\theta U^{-1}B^iU}=U^{-1}e^{\\theta B^i}U$. To make this\nexplicit, $U$ linearizes the physical momentum, which is then\nboosted by $e^{\\theta B^i}$ and then converted back to a physical\nmomentum via $U^{-1}$: \\begin{equation} p_{physical}'\n\\stackrel{U^{-1}}{\\longleftarrow}p_{linear}' \\stackrel{e^{\\theta\nB^i}}{\\longleftarrow}p_{linear}\n\\stackrel{U}{\\longleftarrow}p_{physical} . \\end{equation}\n\nUsing the particular form for the $U$-map shown\nabove~\\cite{leejoao}, we can solve for $E'$ on the left hand side.\nThis is equivalent to applying the $U^{-1}$ map, thus giving us\nour deformed transformations. These now hold invariant the\nmass-squared expression\n\\begin{equation}\nm^2={\\eta^{ab}p_ap_b\\over (1-{E\\over\nE_p})^2}\n\\end{equation}\nFor a boost in the $x$-direction, the transformations\nassume the form~\\cite{leejoao}, \\begin{eqnarray} E'&=&{\\gamma(E-vp_x)\\over\n1+(\\gamma-1){E\\over E_p}-\\gamma {vp_x\\over E_p}}\\nonumber\\\\\np'{_x}&=&{\\gamma(p_x-vE)\\over\n1+(\\gamma-1){E\\over E_p}-\\gamma {vp_x\\over E_p} }\\nonumber\\\\\np'{_y}&=&{p_y\\over 1+(\\gamma-1){E\\over E_p}-\\gamma {vp_x\\over E_p}\n}\\nonumber\\\\\np'{_z}&=&{p_z\\over 1+(\\gamma-1){E\\over E_p}-\\gamma {vp_x\\over E_p}\n} .\\end{eqnarray} The numerators of these expressions are the familiar\nspecial relativistic ones, whereas the denominators fully\nrepresent the deformation. Note that in letting\n$E_p\\rightarrow\\infty$, we recover the special relativistic boosts\nin accordance with the correspondence principle.\n\nA simple analysis of these equations lends immediate insight into\nhow they are different from regular special relativistic boosts.\nMost importantly, plugging $(E_p,0,0,0)$ in for $(E,p_x,p_y,p_z)$\ngives $(E_p,-vE_p,0,0)$, demonstrating that $E_p$ is indeed an\ninvariant quantity. The regular boosts would have given us extra\n$\\gamma$-factors in front of the $E$ and $p_x$ values, which are\nnow cancelled by the new factor in the denominator of the deformed\nboosts. Because of this limit, $E0$) is precisely opposite to that needed to support\nobservational data. That is, the equations predict a value of\n$\\alpha$ that {\\it decreases} in time, whereas observations\nsupport an {\\it increasing} alpha ~\\cite{bsm}. The\ntheory can fit the data, however, for two very different physical\nscenarios. Firstly, if there exists a type of dark matter such\nthat the $B^2$ contribution to the electromagnetic lagrangian\ndominates the $E^2$ contribution, then $\\alpha$ will increase in\ntime within observational bounds. Some cosmological defect\ntheories, specifically those of superconducting strings, satisfy\nthis requirement, but it is more or less unappealing since most\nmatter types do not have this property.\n\nIt is, on the other hand, possible to explain the observational\ndata supporting an $\\alpha$ value that was lower in the past if\n$\\omega<0$~\\cite{bsm}. A negative scalar coupling\nimplies a negative energy density for $\\psi$, which in turn tells\nus that $\\psi$ is a \"ghost\" field. \"Ghost fields\" coupled to\nmatter have often been deemed undesirable since they will\nconsistently dump positive energy into matter. However, it is the\nspecific forms of the equations of motion that will dictate just\nhow this interchange is mediated such that it may, in fact, not be\nproblematic. We will ignore the \"runaway\" pathological behavior of\nghosts at the quantum level on the grounds that a scalar field\n$\\phi$ is non-renormalizable and should not be quantized, just as\nin classical general relativity.\n\nIt has been shown ~\\cite{bkj} that the simplest varying constant\ntheories, such as the Brans-Dicke and Bekenstein models described\nabove, predict a non-singular and cyclic universe given a scalar\ncoupling $\\omega<0$. In particular, the Bekenstein model\nequations with negative $\\omega$ are the same as the Brans-Dicke\nequations in the Einstein frame. In both cases, a regularly\noscillating universe results if the scalar field is uncoupled to\nmatter. On the other hand, a cyclic universe with \"bounces\" of\never-increasing amplitudes is produced if the field {\\it is}\ncoupled to matter near the bounce, such that positive energy is\ntransferred into the matter field from the negative energy field.\nThe universe, with this additional positive energy density will be\nhotter and will thus have to grow to a larger size in order to\nreach the critical temperature for a turnaround. Note that these\nmodels occur strictly within the radiation-dominated epoch,\nmeaning that the scalar fields couple to radiation. Assuming that\nwe are presently in the first matter-dominated era, we can\ncalculate the number of bounces needed to get to the value of the\nconstants that are measured today ~\\cite{bkj}. In this manner, it\nis possible to justify the extreme values of the parameters simply\ngiven the large age of our universe.\n\nWe have so far reviewed a large number of motivations for varying\n\"constant\" scenarios. Among them are included quantum gravity\ntheories with dilaton scalar fields resembling the scalar fields\nthat describe varying constants, theories such as DSR which\npredict deformed dispersion relations, and observations pointing\nto the variability of the fine structure constant. In all cases,\nscalar fields play the pivotal role of describing how these\nfundamental parameters of nature vary throughout space-time. We\nknow that scalar fields appear in other roles in physics, such as\nthe cosmological constant that drives inflation, so it\nis more than relevant to ask what the crucial role of scalar\nfields in the universe is in general. Is their primary role to\nvary the fundamental constants of nature? Are they there simply\nto source inflation? Or, are they merely an unobservable\nbyproduct of higher dimensional theories such as string theories?\nPerhaps these theories are intimately connected, but to even begin\nto address these questions, one must broaden the scope of study.\nFor instance, an extension of the Bekenstein model to the\nelectroweak scenario been done ~\\cite{varsm} in order to see if\nthere exists any evidence of a scalar field affecting the standard\nmodel of particle physics. With respect to cyclic universes, there\nare other non-singular models ~\\cite{rob,varun} sourced other than\nby negative energy scalar fields, which could help shed light on\nthe necessity and\/or importance of scalar fields in bouncing\nmodels.\n\n\\subsection{A Variable Speed of Light}\nEven after the proposal of special relativity in 1905 many varying\nspeed of light theories were considered, most notably by Einstein\nhimself~\\cite{einsvsl}. VSL was then rediscovered and forgotten on\nseveral occasions. For instance, in the 1930s, VSL was used as an\nalternate explanation for the cosmological redshift\n\\cite{stew,buc,wald} (these theories conflict with fine structure\nobservations). None of these efforts relate to recent VSL\ntheories, which are firmly entrenched in the successes (and\nremaining failures) of the hot big bang theory of the universe. In\nthis sense the first ``modern'' VSL theory was J. W. Moffat's\nground breaking paper \\cite{moffat93}, where spontaneous symmetry\nbreaking of Lorentz symmetry leads to VSL and an elegant solution\nto the horizon problem.\n\nSince then there has been a growing literature on the subject,\nwith several groups working on different aspects of VSL (for a\ncomprehensive review see~\\cite{vslreview}).\nHere we examine the simplest and most conservative implementations\ncurrently being considered. The first resembles Brans-Dicke theory\nin that two metrics are considered, one for gravity,\nanother for matter (this is comparable to the metrics used in the\nEinstein and Jordan frame in Brans-Dicke theory).\nThe second implementation is a generalization of Bekenstein's\ntheory in which all coupling strengths become dynamical.\n\n\n\\subsubsection {Bimetric VSL theories}\\label{bivsl}\nThis approach was initially proposed by J. W. Moffat and M. A. Clayton\n\\cite{mofclay}, and by I. Drummond \\cite{drum}. It does not sacrifice\nthe principle of relativity and special care is taken with the damage\ncaused to the second principle of special relativity.\nIn these theories the speeds of the various\nmassless species may be different, but special relativity is still\n(linearly) realized within each sector. Typically the speed of the graviton\nis taken to be different from that of massless matter particles.\nThis is implemented by introducing two metrics (or tetrads in the\nformalism of \\cite{drum,drum1}), one for gravity and one for\nmatter. The model was further studied by \\cite{clay1}\n(scalar-tensor model), \\cite{clay} (vector model), and\n\\cite{covvsl,bass,bass1}.\n\nWe now sketch the scalar-tensor model. It uses a scalar field\n$\\phi$ that is minimally coupled to a gravitational field\ndescribed by the metric $g_{\\mu\\nu}$. However the matter couples\nto a different metric, given by\n\\begin{equation}\n\\hat{g}_{\\mu\\nu}=g_{\\mu\\nu}+B\\partial_\\mu\\phi\\partial_\\nu\\phi.\n\\end{equation}\nThus there is a space-time, or graviton metric $g_{\\mu\\nu}$, and a\nmatter metric $\\hat{g}_{\\mu\\nu}$. The total action is\n\\begin{equation}\nS=S_g+S_{\\phi}+\\hat{S}_{\\rm M},\n\\end{equation}\nwhere the gravitational action is as usual\n\\begin{equation}\nS_g=\\frac{1}{16\\pi }\\int dx^4{\\sqrt {-g}} (R(g)-2\\Lambda),\n\\end{equation}\n(notice that the cosmological constant $\\Lambda$ could also,\nnon-equivalently, appear as part of the matter action). The scalar\nfield action is\n\\begin{equation}\nS_\\phi=\\frac{1}{16\\pi }\\int dx^4{\\sqrt {-g}}\\,\n\\Bigl[\\frac{1}{2}g^{\\mu\\nu}\\partial_\\mu\\phi\\partial_\\nu\\phi-V(\\phi)\\Bigr],\n\\end{equation}\nleading to the the stress-energy tensor,\n\\begin{equation}\nT^{\\mu\\nu}_\\phi= \\frac{1}{16\\pi } \\Bigl[\ng^{\\mu\\alpha}g^{\\nu\\beta}\\partial_\\alpha\\phi\\partial_\\beta\\phi\n-\\frac{1}{2}g^{\\mu\\nu}g^{\\alpha\\beta}\\partial_\\alpha\\phi\\partial_\\beta\\phi\n+g^{\\mu\\nu}V(\\phi) \\Bigr].\n\\end{equation}\nThe matter action is then written as usual, but using the metric\n${\\hat g}_{\\mu\\nu}$. Variation with respect to $g_{\\mu\\nu}$ leads\nto the gravitational field equations,\n\\begin{equation}\nG^{\\mu\\nu}=\\Lambda g^{\\mu\\nu}\n +8\\pi T^{\\mu\\nu}_\\phi\n +8\\pi \\frac{\\sqrt{-\\hat{g}}}{\\sqrt {-g}}\\hat{T}^{\\mu\\nu}.\n\\end{equation}\nIn this theory the speed of light is not preset, but becomes a\ndynamical variable predicted by a special wave equation\n\\begin{equation}\\label{dynabivsl}\n\\bar{g}^{\\mu\\nu}\\hat{\\nabla}_\\mu\\hat{\\nabla}_\\nu\\phi+KV^\\prime\n[\\phi]=0\\,\n\\end{equation}\nwhere the biscalar metric $\\bar g$ is defined in \\cite{mofclay}.\n\nThis model not only predicts a varying speed of light (if the\nspeed of the graviton is assumed to be constant), but also allows\nsolutions with a de Sitter phase that provides sufficient\ninflation to solve the horizon and flatness problems. This is\nachieved without the addition of a potential for the scalar field.\nThe model has also been used as an alternative explanation for the\ndark matter \\cite{drum1} and dark\nenergy\\cite{bass,bass1}.\n\n\n\\subsubsection{``Lorentz invariant'' VSL theories}\\label{livsl}\nIt is also possible to preserve the essence of Lorentz invariance\nin its totality and still have a space-time (as opposed to energy\ndependent) varying $c$. One possibility is that Lorentz invariance\nis spontaneously broken, as proposed by J. W. Moffat~\\cite{moffat93,moff2}\n(see also \\cite{jacobson}). Here the\nfull theory is endowed with exact local Lorentz symmetry; however\nthe vacuum fails to exhibit this symmetry. For example an $O(3,1)$\nscalar field $\\phi^a$ (with $a=0,1,2,3$) could acquire a time-like\nvacuum expectation value (VEV), providing a preferred frame and\nspontaneously breaking local Lorentz invariance to $O(3)$\n(rotational invariance). Such a VEV would act as the preferred\nvector $u^a=\\phi^a_0$; however the\nfull theory would still be locally Lorentz invariant. Typically in\nthis scenario the speed of light undergoes a first or second\norder phase transition to a value more than 30 orders of magnitude\nsmaller, corresponding to the presently measured speed of light.\nInterestingly, before the phase transition the entropy of the\nuniverse is reduced by many orders of magnitude, but afterwards\nthe radiation density and entropy of the universe vastly increase.\nThus the entropy increase follows the arrow of time determined by\nthe spontaneously broken direction of the timelike VEV $\\phi^a_0$.\nThis solves the enigma of the arrow of time and the second law of\nthermodynamics.\n\nAnother example is the covariant and locally Lorentz invariant\ntheory proposed in \\cite{covvsl}. In that work definitions were\nproposed for covariance and local Lorentz invariance that remain\napplicable when the speed of light $c$ is allowed to vary. They\nhave the merit of retaining only those aspects of the usual\ndefinitions which are invariant under unit transformations, and\nwhich can therefore legitimately represent the outcome of an\nexperiment. In the\nsimplest case a scalar field is then defined $\\psi=\\log( c\/ c_0)$,\nand minimal coupling to matter requires that \\begin{equation}\\label{alphan}\n\\alpha_i\\propto g_i\\propto \\hbar c\\propto c^{q} \\end{equation} with $q$ a\nparameter of the theory. The action may be taken to be\n\\begin{equation} S= \\int d^4x \\sqrt{-g}(\ne^{a\\psi}( R-2\\Lambda +{\\cal L}_{\\psi}) +{ 16\\pi }e^{b\\psi}{\\cal L}_m )\n\\end{equation}\nand the simplest dynamics for $\\psi$ derives from:\n\\begin{equation} {\\cal L}_{\\psi}=-\\kappa(\\psi)\n\\nabla_\\mu\\psi\\nabla^\\mu\\psi\n\\end{equation}\nwhere $\\kappa(\\psi)$ is a dimensionless coupling function. For\n$a=4$, $b=0$, this theory is nothing but a unit transformation\napplied to Brans-Dicke theory. More generally, it's only when\n$b+q=0$ that these theories are scalar-tensor theories in\ndisguise. In all other cases it has been shown that a unit\ntransformation may always be found such that $c$ is a constant but\nthen the dynamics of the theory becomes much more complicated.\nThus we should label these theories varying speed of light theories.\n\n\nIn these theories the cosmological constant $\\Lambda$ may depend\non $c$, and so act as a potential driving $\\psi$. Since the vacuum\nenergy usually scales like $c^4$ we may take $\\Lambda\\propto\n(c\/c_0)^n=e^{n\\psi}$ with $n$ an integer. In this case, if we set\n$a=b=0$ the dynamical equation for $\\psi$ is :\n\\begin{equation}\\label{dynacovvsl} \\nabla_\\mu\\nabla^\\mu \\psi ={32\\pi \\over c^4\\kappa}{\\cal\nL}_m +{1\\over \\kappa} n\\Lambda \\end{equation} Thus it is possible\nthat the presence of Lambda drives changes in the speed\nof light, a matter examined (in another context) in \\cite{vslsn}.\n\n\nParticle production and second quantization for this model has\nbeen discussed in \\cite{covvsl}. Black hole solutions were also\nextensively studied \\cite{vslbh}. Predictions for the classical\ntests of relativity (gravitational light deflection, gravitational\nredshift, radar echo delay, and the precession of the perihelion\nof Mercury) were also shown to differ distinctly from their\nEinstein counterparts, while still evading experimental\nconstraints~\\cite{vslbh}. Other interesting results were the\ndiscovery of Fock-Lorentz space-time\\cite{man,step} as the\n``free'' solution, and fast-tracks (tubes where the speed of light\nis much higher) as solutions driven by cosmic\nstrings~\\cite{covvsl}.\n\nBeautiful as these two theories may be, their application to\ncosmology is somewhat cumbersome.\n\n\\subsubsection{The simplistic cosmological motivation}\nLike inflation~\\cite{infl}, modern VSL theories were motivated by\nthe ``cosmological problems'' -- the flatness, entropy,\nhomogeneity, isotropy and cosmological constant problems of Big\nBang cosmology (see~\\cite{am,basker}). The definition\nof a cosmological arrow of time was also a strong consideration.\n\n\nBut at its most simplistic, VSL was inspired by the horizon problem.\nAs we go back into our past the present comoving horizon breaks down\ninto more and more comoving causally connected regions. These\ndisconnected early days of the Universe prevent a physical\nexplanation for the large scale features we observe -- the ``horizon\nproblem''. It does\nnot take much to see that a larger speed of light in the early\nuniverse could open up the horizons~\\cite{moffat93,am} (see\nFigs.~\\ref{fig1} and \\ref{fig2}). More mathematically,\nthe comoving horizon is given by\n$r_h=c\/\\dot a$, so that a solution to the horizon problem requires\nthat in our past $r_h$ must have decreased in order to causally\nconnect the large region we can see nowadays. Thus\n\\begin{equation}\n{\\ddot a\\over \\dot a}-{\\dot c\\over c}>0\n\\end{equation}\nthat is, either we have accelerated expansion (inflation), or a\ndecreasing speed of light, or a combination of both. This argument\nis far from general: a contraction period ($\\dot a<0$, as in the\nbouncing universe), or a static start for the universe ($\\dot\na=0$) are examples of exceptions to this rule.\n\nHowever the horizon problem is just a warm up for the other\nproblems.\nMore recently structure formation has been the leading driving\nforce in these scenarios. This is still very much unaccomplished,\nin spite of recent efforts and consequently is left as an exercise\nfor the student.\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=hor1.eps,width=6 cm,angle=-90}}\n\\caption{A conformal diagram (in which light travels at $45^\\circ$).\nThis diagram reveals that the sky is a cone in 4-dimensional\nspace-time. When we look far away we look into the past; there is\nan horizon because we can only look as far away as the Universe is\nold. The fact that the horizon is very small in the very early\nUniverse, means that we can now see regions in our sky outside\neach others' horizon. This is the horizon problem of standard Big\nBang cosmology.} \\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=hor2.eps,width=6 cm,angle=-90}}\n\\caption{Diagram showing the horizon structure in a model in which\nat time $t_c$ the speed of light changed from $c^-$ to $c^+\\ll\nc^-$. Light travels at $45^\\circ$ after $t_c$ but it travels at a\nmuch smaller angle to the spatial axis before $t_c$. Hence it is\npossible for the horizon at $t_c$ to be much larger than the\nportion of the Universe at $t_c$ intersecting our past light cone.\nAll regions in our past have then always been in causal contact.\nThis is the VSL solution of the horizon problem.} \\label{fig2}\n\\end{figure}\n\n\\subsubsection{The experimental front of VSL}\n\nAs explained in~\\cite{vslreview} varying $e$ and $c$ theories in\ngeneral predict the same $\\alpha(z)$ profiles.\nTo distinguish between varying $e$ and varying $c$ theories\none must look elsewhere. The status of the equivalence principle\nin these theories turns out to be a good solution. The varying $e$\ntheories \\cite{bsm,bek2,olive} violate the weak equivalence\nprinciple, whereas VSL theories do not\\cite{mofwep,mbswep}. The\nE\\\"otv\\\"os parameter\n\\begin{equation}\n\\eta \\equiv {\\frac{2|a_1-a_2|}{a_1+a_2}}\n\\end{equation}\nis of the order $10^{-13}$ in varying $e$ theories, just an order\nof magnitude below existing experimental bounds.\n\nStill, it would be good to find a widely independent confirmation\nof the Webb results.\nA very promising area is cold atom clocks~\\cite{sortais}. The\nsearch for the perfect unit of time has led to the the quest for\nvery stable oscillatory systems, leading to a gain, every ten years, of\nabout one order of magnitude in timing accuracy.\nCold atom clocks may be used as laboratory ``table-top''\nprobes for varying $\\alpha$, with a current sensitivity\nof about $10^{-15}$ per year. For all varying alpha theories\nit is found that at present:\n\\begin{equation}\n{\\dot\\alpha\\over\\alpha}\\approx 2.98\\times 10^{-16} h \\quad {\\rm year}^{-1}\n\\end{equation}\nwith $H_0=100 h$~Km~sec$^{-1}$~Mpc$^{-1}$, $\\Omega_\\Lambda=0.71$\nand $\\Omega_m=0.29$. For $h=0.7$ this\ngives a fractional variation in alpha of about $2\\times 10^{-16}$ per year,\nwhich should soon be within the reach of technology.\nSuch an observation would be\nan incredible vindication of the Webb results. On the other\nhand this effect would become a further annoyance for those\nconcerned with the practicalities of defining\nthe unit of time.\n\n\nSpatial variations of $\\alpha $ are likely to be significant\n\\cite{mbswep} in any varying alpha theory. For any causal theory\nrelative variations in\n$\\alpha $ near a star are proportional to the local gravitational\npotential. The exact relation between the change in $\\alpha $ with\nredshift and in space (near massive objects) is model dependent.\nFor instance, we have\n\\begin{equation}\n{\\frac{\\delta \\alpha }\\alpha }=-{\\frac{\\zeta _s}\\omega }{\\frac{M_s}{\\pi r}}%\n\\approx 2\\times 10^{-4}{\\frac{\\zeta _s}{\\zeta _m}}{\\frac{M_s}{\\pi\nr}} \\label{alphar}\n\\end{equation}\nfor a typical varying $e$ theory, but\n\\begin{equation}\n{\\frac{\\delta \\alpha }\\alpha }=-{\\frac{bq}\\omega }{\\frac{M_s}{4\\pi r}}%\n\\approx 2\\times 10^{-4}{\\frac{M_s}{\\pi r}}\\;, \\label{alphar1}\n\\end{equation}\nfor a VSL theory. Here $M_s$ is\nthe mass of the compact object, $r$ is its radius, and $\\zeta$\nis the ratio between $E^2-B^2$ and $E^2+B^2$. When $\\zeta _m$ (for\nthe dark matter) and $\\zeta _s$ (for, say, a star) have different\nsigns, for a cosmologically {\\it increasing}\n$\\alpha $, varying $e$ theories predict that $%\n\\alpha $ should {\\it decrease} on approach to a massive object.\nAnd indeed one must have $\\zeta _m<0$ in order to fit the Webb\nresults. In VSL, on the contrary, $\\alpha $ {\\it increases}\nnear compact objects (with decreasing $c$ if $q<0$, and increasing $c$ if $%\nq>0$). In VSL theories, near a black hole $\\alpha $ could become\nmuch larger than 1, so that electromagnetism would become\nnon-perturbative with dramatic consequences for particle physics near\nblack holes. In varying-$e$ theories precisely the opposite\nhappens: electromagnetism switches off.\n\n\nThese effects are in principle observable using similar\nspectroscopic techniques to those of Webb, but applied to lines\nformed on the surface of very massive objects near us (in the\nsense of $z\\ll 1$). For that, we\nneed an object with a radius sufficiently close to its\nSchwarzchild radius, such as an AGN, a pulsar or a\nwhite dwarf, for the effect to be non-negligible. Furthermore we\nneed the ``chemistry'' of such an object to be sufficiently\nsimple, so that line blending does not become problematic.\n\n\n\n\\section{A last exercise for the student: MOND}\nIt is sadly the case that we can't finish this review with\nresounding conclusions. Rather we will peter out in the realm of\nuncertainties -- problems which remain unsolved and that may have\nsomething to say about phenomenological quantum gravity. For this\npurpose we have selected the problem of dark matter in the\nUniverse.\n\n\nGalactic rotation curves have long puzzled cosmologists.\nNewtonian theory predicts that they should fall out like\n$v_r\\propto 1\/r^{1\/2}$. This follows from the simple\ncalculation:\n\\begin{equation}\nF=ma \\rightarrow {Mm\\over r^2}=m{v^2\\over r}\\rightarrow v^2={M\\over r}\n\\end{equation}\nInstead we observe them flattening out: $v\\rightarrow v_\\infty$.\nA simple solution is that besides the visible matter there is\na halo of dark matter which dominates gravity on the outskirts\nof the galaxy. This halo has the property $M_{DM} = A r$,\ni.e. it must have a density profile $\\rho_{DM} = B\/ r^2$.\nThus\n\\begin{equation}\n{M_{DM}m\\over r^2}=m{v^2\\over r}\\rightarrow v^2=A\n\\end{equation}\nHistorically this the first hint of dark matter\nin the Universe. It is important to stress that there are now\nmany other reasons to invoke dark matter\\footnote{It's not obvious that\nthe required matter is always the same.}.\n\n\nThis is all very well; however three difficulties are quickly encountered:\n\\begin{itemize}\n\\item{1. }\nThe halos don't appear to be stable when left to evolve according\nto their own gravity. Rather they collapse into a central cusp.\nThis is the drama of every N-body simulation performed so far.\nLack of resolution and physical content is usually blamed.\n\\item{2. } The onset of the terminal velocity seems to be triggered\nnot by a length or mass scale but by an acceleration. This\nhas been measured to be $a_0\\approx 10^{-10}ms^{-2}$.\n\\item{3. } An empirical law has been established called\nthe Tully-Fisher relation\nestablishing that $v_\\infty^4$ is proportional to the luminosity\n(which presumably is proportional to the visible mass). This\nis the equivalent of Kepler's third law.\n\\end{itemize}\nIt is hard to see how dark matter, even if creating a stable\nhalo, could explain the Tully-Fisher relation. There would have\nto be a finely tuned correlation between constant $B$ (appearing in the\ndensity profile for the halo) and the mass in visible matter.\nLikewise the emergence of $a_0$ in the dark matter scenario is\nhard to understand.\nStill, it is possible that future N-body simulations may solve\nthese problems.\n\nDisconcertingly there is a very simple alternative solution, called\nMOND (MOdified Newtonian Dynamics). Perhaps galactic rotation\ncurves are simply telling us that gravity has departed from\nNewton's equations (and that there is no\ndark matter). Changing Newton's gravitational law, however, won't do because\nthis would trigger novel behaviour at a given length scale\nrather than at an acceleration scale. Instead MOND\nposits that {\\it the response law}, $F=ma$, must be modified.\nThe usual law is only valid at high accelerations; for accelerations\nsmaller than $a_0$ we have instead that\n\\begin{equation}\nF=m{a^2\\over a_0}\\, .\n\\end{equation}\nStraightforward application of the MOND prescription leads to\n\\begin{equation}\n {Mm\\over r^2}=m{v^4\\over a_0 r^2}\\rightarrow v^4=M a_0\n\\end{equation}\nThus the Tully-Fisher relation is trivially explained as well as the fact\nthat novel behaviour is triggered by an acceleration.\n\nMOND is an excellent phenomenological description of galactic\nrotation curves. However it makes no sense whatsoever. Applied\ncrudely it violates energy and momentum conservation in ways that\nwould readily conflict with observations. It also has\nno relativistic generalization; we need such a relativistic\ntheory in order, for consistency, to do away with\ndark matter in all regimes. We quote the exemples of\ngravitational lenses, the cosmological expansion, and structure\nformation. Again, we're left in the\nrealm of wishful thinking: for dark matter with regards to\ncomputer power, here with\nrespect to brain power and essential theoretical developments.\n\nWhy might this discussion be relevant to quantum gravity?\nMost obviously because MOND leads to a scary possibility:\n{\\it in trying to quantize gravity we may have chosen the wrong classical\ntheory}. No wonder we're stuck. Our failures could simply\nsignal that we don't yet have the correct classical theory of gravity.\nThis is a speculation, but one with dramatic, far-reaching consequences.\nAs the title of this section shows, whatever one makes of it,\nthis is very much exercise for the student~\\footnote{Warning:\nthis is an exercise\nfor a very keen\ngraduate student hoping to start a high risk, high return project.}.\nSpecifically, here's the problem:\n\\begin{itemize}\n\\item{1.} Find a relativistic version of MOND consistent with\nenergy and momentum conservation and capable of explaining\ngravitational lenses, and all the successes of the Big Bang\nwithout dark matter. \\item{2.} Quantize this theory.\n\\end {itemize}\nThe following points may be interesting hints (but then again,\nthey may also be red herrings):\n\\begin{itemize}\n\\item The Pioneer puzzle, related to the anomalous acceleration\nsuffered by these satellites in their courses outside the Solar system,\nis associated with acceleration $a_P=8.74\\pm 1.33\\times 10^{-10}m s^{-2}$.\n\\item The observed cosmic acceleration is of the same order\nas $a_0$.\n\\item In contradiction with Mach's principle there appears\nto be absolute frames in the Universe for acceleration\n(but not speed). Why?\n\\end {itemize}\nWe'll leave a single reference on this topic:~\\cite{bekmond}.\n\n\n\n\n\n\n\n\\bibliographystyle{aipproc} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}