diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjlqu" "b/data_all_eng_slimpj/shuffled/split2/finalzzjlqu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjlqu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nClifford analysis is, from the very beginning, considered to be a refinement of harmonic analysis\nfor Clifford algebra (or spinor) valued functions. The perfect description of this statement is\nthe Fischer decomposition of spinor-valued polynomials.\n\nLet us first recall the Fischer decomposition of the space\n$\\cP$ of complex-valued polynomials in the Euclidean space $\\bR^m.$ \nDenote by $\\cH_k$ the space of $k$-homogeneous harmonic polynomials in $\\bR^m.$ \nThen, under a~natural action of the orthogonal group $O(m),$ the space \n$\\cP$ has an irreducible (not multiplicity free) decomposition\n\\begin{equation}\\label{Fischer}\n\\cP=\\bigoplus_{k=0}^{\\infty}\\bigoplus_{p=0}^{\\infty}r^{2p}\\cH_k\n\\end{equation}\nwhere $r^2=x_1^2+\\cdots +x_m^2$ for the vector variable $\\underline{x}=(x_1,\\ldots,x_m)\\in\\bR^m.$\n\nFor spinor-valued polynomials,\nthere is a refinement of this decomposition.\nLet $\\bC_m$ be the complex Clifford algebra generated by vectors of the standard basis $(e_1,\\ldots,e_m)$ of $\\bR^m.$\nRecall that the $L$-action of the Pin group $Pin(m)$ on functions $f:\\bR^m\\to\\bC_m$ is defined by\n\\begin{equation}\\label{Laction}\n[L(s)(f)](\\underline{x}) = s\\,f(s^{-1}\\underline{x}s),\\ s\\in Pin(m)\\text{\\ \\ and\\ \\ }\\underline{x}=(x_1,\\ldots,x_m)\\in\\bR^m.\n\\end{equation}\nDenote by $\\bS$ a~basic spinor representation for $Pin(m).$ \nAs is well-known, the spinor space $\\bS$ can be realized inside the Clifford algebra $\\bC_m.$ \nLet us remark that, under the $L$-action, the space $\\cP(\\bS)=\\cP\\otimes\\bS$ of spinor valued polynomials forms a~$Pin(m)$-module.\nDenote by $\\cM_k(\\bS)$ the space of $k$-homogeneous polynomials $P\\in\\cP(\\bS)$ which are (left) monogenic, \ni.e., which satisfy the Dirac equation $\\underline{D}P=0$ where $$\\underline{D}=e_1\\pa_{x_1}+\\cdots+e_m\\pa_{x_m}.$$ \nNow we are ready to state the Fischer decomposition (sometimes called also Almansi decomposition) for this case. Namely, under the $L$-action, the space \n$\\cP(\\bS)$ has an irreducible (not multiplicity free) decomposition \n\\begin{equation}\\label{FischerPin}\n\\cP(\\bS)=\\bigoplus_{k=0}^{\\infty}\\bigoplus_{p=0}^{\\infty}\\underline{x}^p\\cM_k(\\bS)\n\\end{equation}\nwith $\\underline{x}=e_1x_1+\\cdots+e_mx_m.$ See \\cite{BSES,MR,rya}.\nAs $\\cH_k\\otimes\\bS=\\cM_k(\\bS)\\oplus\\underline{x}\\cM_{k-1}(\\bS)$ and $\\underline{x}^2=-r^2,$\nit is easy to see that (\\ref{FischerPin}) is a~real refinement of (\\ref{Fischer}).\n\nThe main aim of the underlying paper is to show that\nthere exists a natural further refinement of the monogenic Fischer decomposition \\eqref{FischerPin}. It is \nquite surprising that such a finer Fischer decomposition was not described earlier.\nIt was the study of special solutions of the Dirac equation which led to such a~refinement.\nBy special solutions we mean just solutions having their values\nin a chosen subspace $V$ of the Clifford algebra $\\bC_m$. There are a lot of possibilities for a choice \nof $V,$ but it is clearly preferable to choose the subspace $V$ having some special properties. \n\nTypical examples\nare solutions of the Dirac equation having values in spinor\nsubspaces of the Clifford algebra. \nThis case is closely related to the \n$L$-action \\eqref{Laction}.\nIndeed, it is well-known that the Clifford algebra $\\bC_m$, considered as a \n$Pin(m)$-module by left multiplication, decomposes into many equivalent spinor submodules. \nMoreover, for every choice of the spinor\nsubmodule, the $Pin(m)$-module of spinor-valued solutions has quite analogous properties.\n \nAnother interesting example of special solutions of the Dirac equation is given by the so-called generalized Moisil-Th\\'eodoresco system (GMT system for short).\nA~lot of interest has recently been paid to GMT systems (see \\cite{DLS} and the references there).\nIn this case, the space $V$ is supposed to be invariant under another (both side) action of the Pin group, namely the so-called $H$-action.\nThe $H$-action on Clifford algebra valued functions $f:\\bR^m\\to\\bC_m$ is given by\n\\begin{equation}\\label{Haction}\n[H(s)(f)](\\underline{x}) = s\\,f(s^{-1}\\underline{x}s)s^{-1},\\ s\\in Pin(m)\\text{\\ \\ and\\ \\ }\\underline{x}\\in\\bR^m.\n\\end{equation}\nIn what follows, we shall use the language of differential forms. \nIndeed, following \\cite{BDS}, we identify naturally the Clifford algebra $\\bC_m$ with the Grassmann algebra $\\La^*(\\bC^m)$ \nand we study the space $\\cP^*=\\cP\\otimes\\La^*(\\bC^m)$ of polynomial differential forms instead of Clifford algebra valued polynomials.\nThen the $H$-action translates into a~natural action of the orthogonal group $O(m)$ on $\\cP^*$ and the Dirac operator $\\underline{D}$ corresponds to the operator $d+d^*.$ Here $d$ and $d^*$ are, respectively, the standard de Rham differential and its adjoint (see \\eqref{dd^*} below). \nAs an $O(m)$-module, the \nGrassmann algebra $\\La^*(\\bC^m)$ has a multiplicity free irreducible decomposition\n$$\\Lambda^*(\\bC^m)=\\bigoplus_{s=0}^m\\Lambda^s(\\bC^m)$$\nwith $\\Lambda^s(\\bC^m)$ being the space of $s$-vectors over $\\bC^m.$ A~GMT system is then defined as the homogeneous system obtained by restricting the operator $d+d^*$ to functions having values in the space\n$$V=\\bigoplus_{s\\in S}\\Lambda^s(\\bC^m)$$\nfor some (suitable) subset $S\\subset\\{0,1,\\ldots,m\\},$ i.e.\n$$(d+d^*)P=0\\mbox{\\ \\ \\ for $V$-valued $P.$}$$\nIn particular, for $V=\\Lambda^s(\\bC^m),$ the corresponding GMT system coincide with\nthe so-called Hodge-de Rham system\n\\begin{equation}\\label{Hodge}\nd P=0,\\, d^* P=0.\n\\end{equation}\n\n\n\nVarious versions of GMT systems of PDE's were studied for a long time (in particular in low dimensions) and they \nwere used in many different applications. \nApplications in numerical analysis and engineering sciences can be found in \\cite{gs1,gs2}.\nIn a~review paper \\cite{spr}, you can find various generalizations of the well-known Hodge-de Rham decomposition of smooth 1-forms, including decompositions for quaternionic and Clifford algebra valued functions and the Almansi (i.e., monogenic Fischer) decomposition. For applications in theory of electromagnetic fields we can refer to \\cite{S_Maxwell}.\n\nThe Fischer decomposition always played a key role in Clifford analysis. \nIn \\cite{lav_Fischer}, the Fischer decomposition for the $H$-action has been recently applied to inframonogenic functions introduced in \\cite{MPS}. Moreover, in \\cite{lav_Fischer}, the obtained results for the space $\\cP^*$ are translated back into the framework of Clifford analysis. \nFor yet another application, we can refer to \\cite{lavSL3}.\n\nRecently, the Fischer decomposition (together with the Cauchy-Kovalevskaya extension) was used systematically for construction of orthogonal bases in the spaces of homogeneous polynomial solutions.\nIn the classical Clifford analysis, it has a quite long history (see \\cite{BGLS} for historical account,\nvarious results can be found in \\cite{Bock2010c,Bock2010a,Bock2009,BG,CacGueMal,CacGueBock,BockCacGue,cac,CM06,CM07,CM08,\nFCM,FM,Gurlebeck1999,NGue2009,lavSL2,step2,Malonek1987,MS,mor09,som,van}).\nAnalogous results in Hermitean Clifford analysis are described in \\cite{ckH, kerH,GTinH, GT2H}.\nFinally, in \\cite{DLS4}, the Fischer decomposition for the $H$-action plays a~key role in constructing orthogonal bases not only for the spaces $H^s_k$ of solutions to the Hodge-de Rham system but even for the spaces of homogeneous solutions of an arbitrary generalized\nMoisil-Th\\'eodoresco system. \n\n\n\nIn this paper, we will establish (using results from \\cite{hom}) a form of the Fischer decomposition appropriate for the $H$-action. The theory of the Howe duality developed in \\cite{how} shows us that we may\nexpect a further refinement of the monogenic Fischer decomposition (\\ref{FischerPin}), see \\cite{DLS3} for details.\nIndeed, this is visible from the form of invariants contained in the polynomial spaces considered.\nFor scalar valued functions, invariant polynomials are generated by powers of $r^2,$\nand the basic equation is the Laplace equation.\nFor spinor-valued polynomials with the left action, we have to look for invariants with\nvalues in the space of endomorphisms of the spinor space (which is, basically, the corresponding\nClifford algebra) and there is just a new invariant $\\underline{x},$ acting as a refinement of $r^2.$\nThe corresponding basic equation is the Dirac equation.\n\nIn the case of the both side action, \nwe deal with the space $\\cP^*$ of $\\La^*(\\bC^m)$-valued\npolynomials. The space of invariants with values in the space of endomorphisms of the Grassmann algebra $\\La^*(\\bC^m)$ is now much richer. It is generated by two elements ${x}$ and ${x}^*$ which correspond to the differential operators $d^*$ and $d$ by the Fischer duality (see \\eqref{xx^*} below). \nConsequently, the corresponding basic system of equations is the Hodge-de Rham system (\\ref{Hodge})\nand the space of invariants consists of polynomials in $x$ and $x^*.$ Actually, due to the fact\nthat ${x}^2=({x}^*)^2=0,$ such invariants are generated by the set \n\\begin{equation}\\label{Omega}\n\\Omega=\\{1, x, x^*, xx^*, x^*x, xx^*x, x^*xx^*, \\ldots\\}.\n\\end{equation}\nMoreover, denote by $\\cP^*_k$ the space of $k$-homogeneous polynomial forms $P\\in\\cP^*$\nand by $H^s_k$ the space of $\\La^s(\\bC^m)$-valued polynomial forms $P\\in\\cP^*_k$ which satisfy the Hodge-de Rham system (\\ref{Hodge}). \nThen, using results from \\cite{hom}, we shall deduce in Section 2 the corresponding Fischer decomposition for the $H$-action.\n\n\n\\begin{thm}\\label{tFischer}\nThe space $\\cP^*=\\cP\\otimes\\La^*(\\bC^m)$ decomposes as follows:\n\\begin{equation}\\label{etFischer}\n\\cP^*=\\cP^*_{(0,0)}\\oplus\\left(\\bigoplus_{s=1}^{m-1}\\bigoplus_{k=0}^{\\infty}\\cP^*_{(s,k)}\\right)\\oplus\\cP^*_{(m,0)}\\text{\\ \\ \\ with\\ \\ \\ }\n\\cP^*_{(s,k)}=\\bigoplus_{w\\in\\Omega}wH^s_k.\n\\end{equation}\nMoreover, in (\\ref{etFischer}), all\n$O(m)$-modules $H^s_k$ are non-trivial, irreducible and mutually inequivalent and all\n$\\cP^*_{(s,k)}$ are corresponding $O(m)$-isotypic components of $\\cP^*.$\n\\end{thm}\n\nNow we show that the Fischer decomposition of the space $\\cP^*$ given in Theorem \\ref{tFischer} is a~refinement of\nthe monogenic Fischer decomposition (\\ref{FischerPin}).\nIndeed, when we identify the Clifford algebra $\\bC_m$ with the Grassmann algebra $\\La^*(\\bC^m)$ on the space $\\cP^*$ we know that \n$$\\underline{D}=d+d^*\\text{\\ \\ \\ and\\ \\ \\ }-\\underline{x}=x+x^*.$$\nConsequently, the space of spherical monogenics of order $k$ is given by $$\\cM_k=\\{P\\in\\cP^*_k;\\ (d+d^*)P=0\\}.$$ \nMoreover, recall that the Laplace operator $\\Delta$ is given by $\\Delta=\\sum_{j=1}^m\\pa^2_{x_j}$ and put \n$$\\Ker_k\\Delta=\\{P\\in\\cP^*_k;\\ \\Delta P=0\\}.$$\n\nAs we mentioned before, under the $L$-action, $\\bC_m\\simeq\\La^*(\\bC^m)$ decomposes into many copies of basic spinor representations $\\bS$ of $Pin(m)$ and so\nthe whole space $\\cM_k$ of spherical monogenics is not irreducible. Indeed,\nunder the $L$-action,\nthe space $\\cM_k$\ndecomposes into many copies of irreducible modules $\\cM_k(\\bS)$.\nIn particular, we have thus that $\\Ker_k\\Delta=\\cM_k\\oplus(x+x^*)\\cM_{k-1}$\nand, by (\\ref{Fischer}), we get easily the following decomposition of the space $\\cP^*$\n\\begin{equation}\\label{FischerPin*}\n\\cP^*=\\bigoplus_{k=0}^{\\infty}\\bigoplus_{p=0}^{\\infty}r^{2p}(\\cM_k\\oplus(x+x^*)\\cM_{k-1}).\n\\end{equation}\nIn an obvious sense, the decomposition (\\ref{FischerPin*}) is equivalent to (\\ref{FischerPin}). \n\nIn Section 3, we shall prove the following theorem which tells us that, under the $H$-action,\nthe spaces $\\cM_k$ and $(x+x^*)\\cM_{k-1}$ decompose again into many irreducible pieces \nbut, in this case, these pieces are not equivalent and \nthey have a~different representation character.\n\n \n\\begin{thm}\\label{tdecompM}\nFor $k\\geq 1,$ the following statements hold:\n\\begin{itemize}\n\\item[(a)] $\\cM_k=\\left(\\bigoplus_{s=0}^m H^s_k\\right)\\oplus\\left(\\bigoplus_{s=1}^{m-1}M_{s,k}\\right)$\n\\item[] where $M_{s,k}=[(k-1+m-s)x^*-(k-1+s)x]H^s_{k-1}.$ \n\\item[(b)]\n$(x+x^*)\\cM_{k-1}=\\left(\\bigoplus_{s=0}^m(x+x^*)H^s_{k-1}\\right)\\oplus\\left(\\bigoplus_{s=1}^{m-1}W^s_k\\right)$\n\\item[] where $W^s_k=[(k-2+m-s)xx^*-(k-2+s)x^*x]H^s_{k-2}.$\n\\end{itemize}\n\\end{thm}\n\nUsing Theorem \\ref{tdecompM}, we get from the decomposition \\eqref{FischerPin*} directly a~finer decomposition of the space $\\cP^*$ which is irreducible with respect to the $H$-action.\n \nThe results stated in Theorems \\ref{tFischer} and \\ref{tdecompM} remain valid also for real valued polynomial forms, that is,\nin the case when the complex Grassmann algebra $\\La^*(\\bC^m)$ is replaced with the real one $\\La^*(\\bR^m).$ \nIndeed, it is sufficient to realize that irreducible $O(m)$-representations $\\La^s(\\bC^m)$ are all of real type, see \\cite[p. 163]{GM}. \n\n \n\n\n \n\\section{A proof of the Fischer decomposition for the $H$-action}\\label{stFischer}\n\nIn this section, we give a~proof of Theorem \\ref{tFischer} stated in Introduction. \nLet $\\cP_k$ stand for the space of $k$-homogeneous (complex-valued) polynomials of $\\cP$ and let\n$\\cP^s_k=\\cP_k\\otimes\\Lambda^s(\\bC^m).$\nThen\nit is easy to see that\n\\begin{equation}\\label{decompP}\n\\cP^*=\\bigoplus_{s=0}^m\\bigoplus_{k=0}^{\\infty}\\cP^s_k.\n\\end{equation}\nLet us remark that a~polynomial form $P$ belongs to $\\cP^s_k$ if and only if\n\\begin{equation}\n\\label{skform}\nP=\\sum_I P_I\\; dx_I\n\\end{equation}\nwhere the sum is taken over all finite strictly increasing sequences $I=\\{i_j\\}_{j=1}^s$ of numbers of the set $\\{1,\\cdots, m\\},$ $P_I\\in\\cP_k$ and $dx_I=dx_{i_1}\\wedge\\cdots\\wedge dx_{i_s}.$\nThe contraction $dx_j\\krat$ is defined as\n$$dx_j\\krat\\; dx_I=\\sum_{k=1}^s(-1)^{k-1}\\delta_{ji_k}dx_{I\\setminus\\{i_k\\}}\\text{ and }\ndx_j\\krat\\; P=\\sum_I P_I\\; dx_j\\krat\\; dx_I$$ for a~polynomial form $P.$ \nThen we have that\n\\begin{equation}\n\\label{dd^*}\nd=\\sum_{j=1}^m\\; \\pa_{x_j} dx_j\\wedge\\text{\\ \\ \\ and\\ \\ \\ }\nd^*=-\\sum_{j=1}^m\\;\\pa_{x_j} dx_j\\krat, \n\\end{equation}\n\\begin{equation}\n\\label{xx^*}\nx=-\\sum_{j=1}^m x_j\\; dx_j\\wedge\\text{\\ \\ \\ and\\ \\ \\ }\nx^*=\\sum_{j=1}^m x_j\\; dx_j\\krat.\n\\end{equation}\nIt is easy to see that $d,$ $d^*,$ $x$ and $x^*$ are $O(m)$-invariant operators on the space $\\cP^*.$\n\nNow we describe explicitly an irreducible decomposition of $O(m)$-modules\n$$\\Ker_k^s\\;\\Delta=\\{P\\in\\cP_k^s;\\ \\Delta P=0\\}.$$\nThe following key result is obtained in \\cite{hom}.\n\n\\begin{lem}{\\label{thforms}}\nGiven $0\\leq s\\leq m$ and $k\\in\\NN_0,$ we have that\n$$\\Ker_k^s\\Delta= H^s_k\\oplus U^s_k\\oplus V^s_k\\oplus W^s_k$$ where\n$H^s_k,$\n$U^s_k,$ $V^s_k$ and $W^s_k$ are irreducible $O(m)$-modules\nwith the following properties:\n\\begin{itemize}\n\\item[(a1)] $H^s_k=\\{P\\in \\cP^s_k;\\ dP=0,\\ d^*P=0\\}$ and $\\Ker_0^s\\Delta= H^s_0=\\cP_0^s.$\n\\item[(a2)] In addition, $H^s_k=\\{0\\}$ for $s\\in\\{0,m\\}$ and $k\\geq 1.$ Otherwise, all $O(m)$ modules $H^s_k$ are non-trivial, irreducible and mutually inequivalent.\n\\item[(b)] $U^s_k=xH^{s-1}_{k-1}\\simeq H^{s-1}_{k-1}$ for $1\\leq s\\leq m$ and $k\\geq 1,$ and $U^s_k=\\{0\\}$ otherwise.\n\\item[(c)] $V^s_k=x^*H^{s+1}_{k-1}\\simeq H^{s+1}_{k-1}$ for $0\\leq s\\leq m-1$ and $k\\geq 1,$ and $V^s_k=\\{0\\}$ otherwise.\n\\item[(d)] $W^s_k=[(k-2+m-s)xx^*-(k-2+s)x^*x]H^s_{k-2}\\simeq H^s_{k-2}$\\\\\nfor $1\\leq s\\leq m-1$ and $k\\geq 2,$ and $W^s_k=\\{0\\}$ otherwise.\n\\end{itemize}\n\\end{lem}\n\nNow we are ready to prove Theorem \\ref{tFischer}. \n\n\\begin{proof}[Proof of Theorem \\ref{tFischer}]\nAs $\\Ker^s_k\\Delta=\\cH_k\\otimes\\La^s(\\bC^m)$ the Fischer decomposition (\\ref{Fischer}) yields\n$$\\cP_k^s=\\bigoplus_{p=0}^{[k\/2]}r^{2p}\\Ker^s_{k-2p}\\Delta,$$\nwhere for $y\\in\\bR,$ $[y]$ denotes the greatest integer not greater than $y.$\nConsequently, by Lemma \\ref{thforms}, we get the decomposition\n\\begin{equation}\\label{decompPsk}\n\\cP_k^s= H^s_k\\oplus\n\\bigoplus_{p=0}^{[k\/2]}r^{2p}U^s_{k-2p}\\oplus\n\\bigoplus_{p=0}^{[k\/2]}r^{2p}V^s_{k-2p}\\oplus\n\\bigoplus_{p=0}^{[k\/2]}r^{2p}Z^s_{k-2p}\n\\end{equation}\nwhere $Z^s_k=r^2 H^s_{k-2}\\oplus W^s_k.$\nSince $r^2=-(xx^*+x^*x)$ Lemma \\ref{thforms} implies that,\nfor $0\\leq s\\leq m$ and $k\\geq 2,$ \n$$Z^s_k=(xx^*)H^s_{k-2}\\oplus (x^*x)H^s_{k-2}.$$\nMoreover, it is easy to see that \n$$r^{2p}U^s_k=(xx^*)^pxH^{s-1}_{k-1},\\ \\ \\\nr^{2p}V^s_k=(x^*x)^px^*H^{s+1}_{k-1}\\text{\\ \\ and}$$\n$$r^{2p}Z^s_k=(xx^*)^{p+1}H^s_{k-2}\\oplus (x^*x)^{p+1}H^s_{k-2}.$$\nNow to complete the proof\nit suffices to use the decompositions (\\ref{decompP}) and (\\ref{decompPsk}).\n\\end{proof}\n\nAt the end of this section we collect the well-known relations we need later on. \nPut, for linear operators $T_1$ and $T_2$ on $\\cP^*,$\n$\\{T_1,T_2\\}=T_1T_2+T_2T_1$ and $[T_1,T_2]=T_1T_2-T_2T_1.$ Then we have that (see e.g. \\cite{hom} or \\cite{BDS})\n\n\\begin{lem}\\label{lrels} Let $E$ be the Euler operator and $\\hat{E}$ be the skew Euler operator, i.e. $$E=\\sum_{j=1}^m x_j\\pa_{x_j}\\text{\\ \\ \\ and\\ \\ \\ }\n\\hat{E}=\\sum_{j=1}^m(dx_j\\;\\wedge)(dx_j\\krat).$$\nThen we have that $EP=kP$ and $\\hat{E}P=sP$ for each $P\\in\\cP^s_k.$\n\nFurthermore, the following relations hold:\n\\begin{equation*}\n\\begin{array}{lll}\n\\{x,x\\}=0, &\\{x^*,x^*\\}=0, &\\{x,x^*\\}=-r^2,\\medskip\\\\{}\n\\{d,d\\}=0, &\\{d^*,d^*\\}=0, &\\{d,d^*\\}=-\\Delta,\\medskip\\\\{}\n\\{x^*,d\\}=E+\\hat{E}, &\\{x,d^*\\}=E-\\hat{E}+m, &\\{x^*,d^*\\}=0=\\{x,d\\}.\n\\end{array}\n\\end{equation*}\n\\end{lem}\n\n\nUsing Lemma \\ref{lrels}, we may give, for example, an explicit description of the projections of the space $\\Ker_k^s\\Delta$ onto the pieces $H^s_k,$ $U^s_k,$ $V^s_k$ and $W^s_k.$ \n\n\\begin{prop}\\label{chforms}\nGiven $0\\leq s\\leq m$ and $k\\in\\NN_0,$ put $c_1=k-2+s$ and $c_2=k-2+m-s.$\nFurthermore, let $\\pi_1,$ $\\pi_2,$ $\\pi_3$ and $\\pi_4$ be the projections of the space $\\Ker_k^s\\Delta$ onto the subspaces $H^s_k,$\n$U^s_k,$ $V^s_k$ and $W^s_k,$ respectively. Then we have that\n\\begin{equation*\n\\pi_4=\\left\\{\n\\begin{array}{ll}\n\\frac{c_2xx^*-c_1x^*x}{c_1c_2(c_1+c_2+2)}\\;dd^* &\\text{for\\ \\ }1\\leq s\\leq m-1\\text{\\ and\\ }k\\geq 2,\\medskip\\\\{}\n0, &\\text{otherwise.}\n\\end{array}\n\\right.\n\\end{equation*}\nMoreover, denoting $\\pi=1-\\pi_4,$ we have that\n\\begin{equation*}\n\\begin{array}{ll\n&\\pi_2 =\\left\\{\n\\begin{array}{ll}\n\\frac{1}{c_2+2}\\;xd^*\\pi &\\text{for\\ \\ }1\\leq s\\leq m\\ \\text{\\ and\\ }k\\geq 1,\\medskip\\\\{}\n0, &\\text{otherwise;}\n\\end{array}\n\\right.\\medskip\\\\{}\n\n&\\pi_3 =\\left\\{\n\\begin{array}{ll}\n\\frac{1}{c_1+2}\\;x^*d\\pi &\\text{for\\ \\ }0\\leq s\\leq m-1\\text{\\ and\\ }k\\geq 1,\\medskip\\\\{}\n0, &\\text{otherwise;}\n\\end{array}\n\\right.\\medskip\\\\{}\n\n&\\pi_1 =1-\\pi_2-\\pi_3-\\pi_4.\n\\end{array}\n\\end{equation*} \n\\end{prop}\n\n\\begin{proof}\nLet $P\\in\\Ker^s_k\\Delta$ be given. Then, by Lemma \\ref{thforms}, there are uniquely determined $P_1\\in H^s_k,$ $P_2\\in H^{s-1}_{k-1},$ $P_3\\in H^{s+1}_{k-1}$ and $P_4\\in H^s_{k-2}$ such that $$P=P_1+xP_2+x^*P_3+(c_2xx^*-c_1x^*x)P_4.$$\nBy Lemma \\ref{lrels},\nit is easy to see that\n$$dd^*P=c_1c_2(c_1+c_2+2)P_4=-d^*dP,$$ which easily implies the formula for $\\pi_4.$\n\nMoreover, $\\pi(P)=P_1+xP_2+x^*P_3.$ By Lemma \\ref{lrels}, we have that \n$$d^*\\pi(P)=d^*xP_2=(c_2+2)P_2\\text{\\ \\ \\ and\\ \\ \\ }d\\pi(P)=dx^*P_3=(c_1+2)P_3,$$\nfrom which the formulae for the projections $\\pi_2$ and $\\pi_3$ may be derived.\n\\end{proof}\n\n\n\n\\section{Decomposition of monogenic polynomial forms}\n \nIn this section, we give a~proof of Theorem \\ref{tdecompM} stated in Introduction.\nTo prove Theorem \\ref{tdecompM} we need some lemmas.\n\n\n\\begin{lem}\\label{lMsk}\nFor $1\\leq s\\leq m-1$ and $k\\geq 1,$\nwe have that\n$$\\left(xH^s_{k-1}\\oplus x^*H^s_{k-1}\\right)\\cap\\cM_k=M_{s,k}.$$\nHere $M_{s,k}=[(k-1+m-s)x^*-(k-1+s)x]H^s_{k-1}.$\n\\end{lem}\n\n\\begin{proof}\nLet $P_1,P_2\\in H^s_{k-1}$ and put $P=xP_1+x^*P_2.$ It suffices to show that $(d+d^*)P=0$\nif and only if $$P_1=-\\frac{k-1+s}{k-1+m-s}P_2.$$\nBy virtue of Lemma \\ref{lrels}, it is easy to see that\n$$(d+d^*)P=(E+m-\\hat E)P_1+(E+\\hat E)P_2=(k-1+m-s)P_1+(k-1+s)P_2,$$\nwhich completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{ldUV}\nFor $1\\leq s\\leq m-1$ and $k\\geq 1,$ we have that\n$$xH^s_{k-1}\\oplus x^*H^s_{k-1}=(x+x^*)H^s_{k-1}\\oplus M_{s,k}. $$\n\\end{lem}\n\n\\begin{proof}\nObvious.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{tdecompM}]\nPut$$\\tilde\\cM_k=\\left(\\bigoplus_{s=0}^m H^s_k\\right)\\oplus\\left(\\bigoplus_{s=1}^{m-1}M_{s,k}\\right).$$\nThen, by Lemma \\ref{lMsk}, it is easy to see that\n$\\tilde\\cM_k\\subset\\cM_k.$\nMoreover, by Lemma \\ref{thforms}, $W^s_k=(x+x^*)M_{s,k-1}.$\nFinally, using Lemma \\ref{ldUV} and Lemma \\ref{thforms},\nwe obtain that\n$$\n\\Ker_k\\Delta=\\tilde\\cM_k\\oplus(x+x^*)\\tilde\\cM_{k-1}\\subset\\cM_k\\oplus(x+x^*)\\cM_{k-1}=\\Ker_k\\Delta,\n$$\nwhich completes the proof.\n\\end{proof}\n\n\n\\subsection*{Acknowledgments}\n\nR. L\\'avi\\v cka and V. Sou\\v cek acknowledge the financial support from the grant GA 201\/08\/0397.\nThis work is also a part of the research plan MSM 0021620839, which is financed by the Ministry of Education of the Czech Republic.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Acknowledgements}\nThis work is supported by PPARC's Particle Physics Peer Review Panel. Grateful thanks must also go to Peter Blythe, Claudia Eberlein and Robert Smith for useful discussion. \n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA basic problem in optimal transportation is the question on whether optimal plans are unique and induced by maps. The crucial result in this direction is the celebrated one of Brenier \\cite{Brenier87} granting that for $\\mu,\\nu\\in\\probt{\\mathbb{R}^d}$ with $\\mu$ absolutely continuous w.r.t. the Lebesgue measure and cost=squared-distance, indeed optimal plans are unique and induced by maps. An important generalization has been given by McCann \\cite{McCann01} for the same problem on Riemannian manifolds: he shows that the unique optimal map can be written as $\\exp(-\\nabla\\varphi)$, where $\\varphi$ is a Kantorovich potential. As a byproduct of McCann's argument, we also know that for $\\mu$-a.e. $x$ the geodesic connecting $x$ to $\\exp(-\\nabla\\varphi(x))$ is unique. We can express both the fact that the unique optimal plan is induced by a map and the uniqueness of geodesics by looking at the optimal transport problem as a dynamical problem, i.e. by minimizing\n\\[\n\\iint_0^1|\\dot\\gamma_t|^2\\,\\d t\\,{\\mbox{\\boldmath$\\pi$}}(\\gamma), \n\\] \nover all measures ${\\mbox{\\boldmath$\\pi$}}$ on $C([0,1],M)$ such that $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\mu$, $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\nu$, ${\\rm{e}}_t:C([0,1],M)\\to M$ being the evaluation map given by ${\\rm{e}}(\\gamma):=\\gamma_t$. Then McCann's result can be read as the uniqueness of the minimizer ${\\mbox{\\boldmath$\\pi$}}$ and the fact that such ${\\mbox{\\boldmath$\\pi$}}$ is induced by some map $T:M\\to C([0,1],M)$ (which in fact takes its values in the space of constant speed minimizing geodesics) in the sense that ${\\mbox{\\boldmath$\\pi$}}=T_\\sharp\\mu$. We refer to \\cite{Villani09} and \\cite{AmbrosioGigli11} for an overview of the subject.\n\n\\medskip\n\nIn the pioneering works of Lott-Villani \\cite{Lott-Villani09} and Sturm \\cite{Sturm06I}, \\cite{Sturm06II}, an abstract notion of lower Ricci curvature bound on metric measure spaces has been given, and since then a great interest has been given by the community to the understanding of the geometric\/analytic properties of these spaces. In \\cite{AmbrosioGigliSavare11-2}, a strengthening of the original ${\\sf CD}(K,\\infty)$ condition as defined by Lott-Sturm-Villani has been proposed: this new condition, called Riemannian Ricci curvature bound and denoted by ${\\sf RCD}(K,\\infty)$, enforces in some weak sense a Riemannian-like behavior of the space. According to the analysis done in \\cite{Erbar-Kuwada-Sturm13}, \\cite{AMS}, \\cite{Gigli13} a natural finite-dimensional analogue of the ${\\sf RCD}(K,\\infty)$ notion can be given by requiring that the space is both ${\\sf RCD}(K,\\infty)$ and satisfies the (reduced) curvature-dimension condition ${\\sf CD}^*(K,N)$ as defined in \\cite{BacherSturm10}.\n\nAim of this short note is to prove the analogue of Brenier-McCann's theorem on ${\\sf RCD}^*(K,N)$ spaces, our result being:\n\\begin{theorem}[Optimal maps]\\label{thm:1}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$ and $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space. Then for every $\\mu,\\nu\\in\\probt X$ with $\\mu\\ll\\mathfrak m$ there exists a unique plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$. Furthermore, this plan is induced by a map and concentrated on a set of non-branching geodesics, i.e. there is a Borel set $\\Gamma\\subset C([0,1],X)$ such that ${\\mbox{\\boldmath$\\pi$}}(\\Gamma)=1$ and for every $t\\in[0,1)$ the map ${\\rm{e}}_t:\\Gamma\\to X$ is injective.\n\\end{theorem}\nHere by ${\\rm{OptGeo}}(\\mu,\\nu)$ we are denoting the set of minimizers of the dynamical version of the optimal transport as discussed above.\nTo some extent, the `hard work' needed for the proof of this result has been already carried out in \\cite{RajalaSturm12} and \\cite{Gigli12a} where it has been proved the following theorem:\n\\begin{theorem}[Optimal maps on ${\\sf RCD}(K,\\infty)$ spaces]\\label{thm:2a}\nLet $K\\in\\mathbb{R}$ and $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}(K,\\infty)$ space. Then for every $\\mu,\\nu\\in\\probt X$ with $\\mu,\\nu\\ll\\mathfrak m$ there exists a unique plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and this plan is induced by a map and concentrated on a set of non-branching geodesics.\n\\end{theorem}\nMore precisely, in \\cite{RajalaSturm12} it has been worked around the delicate issue concerning the non-branching assumption, showing that on ${\\sf RCD}(K,\\infty)$ spaces every optimal geodesic plan between absolutely continuous measures must be concentrated on a set of non-branching geodesics. Then, still in \\cite{RajalaSturm12}, it has been observed how such result coupled with the technique used in \\cite{Gigli12a} to prove existence and uniqueness of optimal maps in the non-branching case yield Theorem \\ref{thm:2a}.\n\n\nHere we start from this results and obtain Theorem \\ref{thm:1} using the enhanced compactness granted by the finite dimensionality together with quite standard ideas in optimal transport theory.\n\nAn interesting fact about Theorem \\ref{thm:1} is that it can be equivalently reformulated in the following way:\n\\begin{theorem}[Exponentiation]\\label{thm:2b}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$, $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space and $\\varphi:X\\to\\mathbb{R}$ a $c$-concave function ($c=\\frac{{\\sf d}^2}{2}$). Then for $\\mathfrak m$-a.e. $x\\in X$ there exists exactly one geodesic $\\gamma$ such that $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$.\n\\end{theorem}\nThis result can be naturally interpreted as a definition of what is the exponential map evaluated at `minus the gradient of a $c$-concave function $\\varphi$': for every $x\\in X$ such that the geodesic $\\gamma$ with $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$ is unique, we put $\\exp(-t\\nabla\\varphi):=\\gamma_t$, thus somehow `reversing' the proof of Brenier-McCann theorem. The role of Theorem \\ref{thm:2b} is to ensure that this map is well defined for $\\mathfrak m$-a.e. $x\\in X$.\n\nNotice that to some extent Theorem \\ref{thm:2b} is the best one we can expect about exponentiation on a metric measure space. To see why just consider the case of a smooth complete Riemannian manifold $M$ with boundary. Then given $x\\in M$ and $v\\in T_xM$, the value of $\\exp(v)$ is well defined only if there is $y\\in M$ such that $\\nabla\\frac{{\\sf d}^2(\\cdot,y)}2=-v$ (neglecting smoothness issues), and functions of the kind $\\frac{{\\sf d}^2(\\cdot,y)}2$ are the prototype of $c$-concave functions. \n\nTheorem \\ref{thm:1} has some simple but interesting consequences, the first being:\n\\begin{corollary}\\label{cor:1}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$ and $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space. Then for every $x\\in \\mathop{\\rm supp}\\nolimits(\\mathfrak m)$ the following holds: for $\\mathfrak m$-a.e. $y$ there is only one geodesic connecting $y$ to $x$.\n\\end{corollary} \nThis can be easily seen choosing $\\nu:=\\delta_x$ in Theorem \\ref{thm:1}.\nIn \\cite{R2012a} the conclusion of Corollary \\ref{cor:1} was proven under the assumption that the ${\\sf CD}(K,N)$ condition holds along every geodesic.\nHowever, ${\\sf RCD}^*(K,N)$ a priori only gives the ${\\sf CD}^*(K,N)$ condition along every geodesic between any two measures with bounded densities, see \\cite{Erbar-Kuwada-Sturm13}.\nThus Corollary \\ref{cor:1} is not a direct consequence of \\cite[Theorem 4]{R2012a}.\nA further consequence of this corollary is the following:\n\\begin{corollary}\\label{cor:2}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$ and $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space. Then the space satisfies the ${\\sf MCP}(K,N)$ condition.\n\\end{corollary} \nFrom \\cite{R2012b} we know that every ${\\sf CD}(K,N)$ space satisfies the ${\\sf MCP}(K,N)$ condition in the sense of \\cite{O2007},\nmeaning that between any absolutely continuous measure and a dirac mass there exists a geodesic that satisfies the ${\\sf MCP}(K,N)$ condition.\nIn Corollary \\ref{cor:2} we obtain a more strict version of the ${\\sf MCP}(K,N)$ condition, considered in \\cite{Sturm06II}, with\na global selection of distributions of geodesics between points such that using these geodesics\nthe ${\\sf MCP}(K,N)$ condition always holds. Since by Corollary \\ref{cor:1} the geodesics are essentially unique, in fact\nany choice of geodesics in an ${\\sf RCD}^*(K,N)$ space will work for the ${\\sf MCP}(K,N)$ condition.\n\nThe difficult part in proving Corollary \\ref{cor:2} relies in proving a sort of self-improving property for the ${\\sf CD}^*(K,N)$ condition: this has been the scope of \\cite{Cavalletti-Sturm12}, where such result has been proved under the non-branching assumption. Yet, such additional hypothesis was made only to get the result of Corollary \\ref{cor:1} above. Given that in the ${\\sf RCD}^*(K,N)$ it holds without the a priori non-branching assumption, Corollary \\ref{cor:2} follows.\n\nA final remark which is worth to make, in particular in connection with Sobolev calculus as developed in \\cite{AmbrosioGigliSavare11}, is the following:\n\\begin{corollary}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$, $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space, $\\mu,\\nu\\in\\probt X$ with $\\mu\\ll\\mathfrak m$ and ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ be the unique optimal geodesic plan given by Theorem \\ref{thm:1}. Then $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\ll\\mathfrak m$ for every $t\\in[0,1)$. \n\nFurthermore, if $\\mu,\\nu$ have bounded support (resp. $K=0$) and $\\mu$ and has density bounded above by some constant $C$, then $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\leq C(t)\\mathfrak m$ for any $t\\in[0,1)$ and some constant $C(t)$ depending only on $C,t,K,N$ and the supports of $\\mu,\\nu$ (resp. on $C,t,N$). If $K<0$ and either $\\mu$ or $\\nu$ have unbounded support, then the optimal geodesic plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ can be written as ${\\mbox{\\boldmath$\\pi$}}=\\sum_{n\\in\\mathbb{N}}{\\mbox{\\boldmath$\\pi$}}_n$ with ${\\mbox{\\boldmath$\\pi$}}_n$ non negative Borel measures on ${\\rm{Geo}}(X)$ such that $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\leq C_n(t)\\mathfrak m$ for any $t\\in[0,1)$ and some constants $C_n(t)$ depending only on $C,t,K,N,n$.\n\\end{corollary}\nThe simple proof follows by localizing the ${\\sf CD}^*(K,N)$ condition along the optimal geodesic plan.\n\n\\smallskip\n\\noindent {\\bf Acknowledgement.}\nThis paper was partly written during the program ``Interactions Between Analysis and Geometry'' at the Institute for Pure and Applied Mathematics (IPAM)\nat University of California, Los Angeles. The authors thank the institute for the excellent research environment.\nT.R. also acknowledges the support of the Academy of Finland project no. 137528.\n\n\n\\section{Preliminaries}\n\nWe assume the reader to be familiar with optimal transport and the definition of spaces with Ricci curvature bounded from below in the sense of Lott-Sturm-Villani. Here we just recall some basic notation.\n\nGiven a geodesic, complete and separable metric space $(X,{\\sf d})$, the set $\\probt X$ is the set of Borel probability measures on it with finite second moment. By ${\\rm{Geo}}(X)$ we denote the space of constant speed minimizing geodesics on $X$ endowed with the $\\sup$-distance.\n\nGiven such metric space $(X,{\\sf d})$ and $\\mu,\\nu\\in\\probt X$, a Borel probability measure ${\\mbox{\\boldmath$\\pi$}}$ on ${\\rm{Geo}}(X)$ is called optimal geodesic plan from $\\mu$ to $\\nu$ provided $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\mu$, $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\nu$ and it achieves the minimum of\n\\[\n\\int {\\sf d}^2(\\gamma_0,\\gamma_1)\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma),\n\\]\namong all Borel probability measure ${\\mbox{\\boldmath$\\pi$}}'$ on ${\\rm{Geo}}(X)$ such that $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}'=\\mu$, $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}'=\\nu$. The set of all optimal geodesic plans is denoted by ${\\rm{OptGeo}}(\\mu,\\nu)$. Notice that ${\\rm{OptGeo}}(\\mu,\\nu)$ is never empty under the above assumption.\n\nA function $\\varphi:X\\to\\mathbb{R}\\cup\\{-\\infty\\}$ not identically $-\\infty$ is called $c$-concave provided there is $\\psi:X\\to\\mathbb{R}\\cup\\{-\\infty\\}$ such that\n\\[\n\\varphi(x)=\\inf_{y\\in X}\\frac{{\\sf d}^2(x,y)}{2}-\\psi(y).\n\\]\nGiven a $c$-concave function $\\varphi$, its $c$-transform $\\varphi^c:X\\to\\mathbb{R}\\cup\\{+\\infty\\}$ is defined by\n\\[\n\\varphi^c(y):=\\inf_{x\\in X}\\frac{{\\sf d}^2(x,y)}{2}-\\varphi(x).\n\\] \nIt turns out that for every $c$-concave function $\\varphi$ it holds $\\varphi^{cc}=\\varphi$. The $c$-superdifferential $\\partial^c\\varphi$ of a $c$-concave function $\\varphi$ is the subset of $X^2$ of those couples $(x,y)$ such that\n\\[\n\\varphi(x)+\\varphi^c(y)=\\frac{{\\sf d}^2(x,y)}2,\n\\]\nand for $x\\in X$, the set $\\partial^c\\varphi(x)\\subset X$ is the set of those $y$'s such that $(x,y)\\in\\partial^c\\varphi(y)$.\n\nIt can be proved that a Borel probability measure ${\\mbox{\\boldmath$\\pi$}}$ on ${\\rm{Geo}}(X)$ belongs to ${\\rm{OptGeo}}(({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}},({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}})$ if and only if there is a $c$-concave function $\\varphi$ such that $\\mathop{\\rm supp}\\nolimits({\\rm{e}}_0,{\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\subset\\partial^c\\varphi$. Any such $\\varphi$ is called Kantorovich potential from $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}$ to $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}$. It is then easy to check that for any Kantorovich potential $\\varphi$ from $\\mu$ to $\\nu$, every ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and every $t\\in[0,1]$, the function $t\\varphi$ is a Kantorovich potential from $\\mu$ to $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$.\n\n\n\nNotice that Kantorovich potentials can be chosen to satisfy the following property, slightly stronger than $c$-concavity:\n\\[\n\\varphi(x)=\\inf_{y\\in\\mathop{\\rm supp}\\nolimits(\\nu)}\\frac{{\\sf d}^2(x,y)}{2}-\\varphi^c(y),\n\\]\nwhich shows in particular that if $\\mathop{\\rm supp}\\nolimits(\\nu)$ is bounded, then $\\varphi$ can be chosen to be locally Lipschitz.\n\n\n\nWe turn to the formulation of the ${\\sf CD}^*(K,N)$ condition, coming from \\cite{BacherSturm10}, to which we also refer for a detailed discussion of its relation with the ${\\sf CD}(K,N)$ condition\n (see also \\cite{Cavalletti-Sturm12} and \\cite{Cavalletti12}).\n\nGiven $K \\in \\mathbb{R}$ and $N \\in [1, \\infty)$, we define the distortion coefficient $[0,1]\\times\\mathbb{R}^+\\ni (t,\\theta)\\mapsto \\sigma^{(t)}_{K,N}(\\theta)$ as\n\\[\n\\sigma^{(t)}_{K,N}(\\theta):=\\left\\{\n\\begin{array}{ll}\n+\\infty,&\\qquad\\textrm{ if }K\\theta^2\\geq N\\pi^2,\\\\\n\\frac{\\sin(t\\theta\\sqrt{K\/N})}{\\sin(\\theta\\sqrt{K\/N})}&\\qquad\\textrm{ if }00$ let $G_M\\subset {\\rm{Geo}}(X)$ be defined by \n\\[\nG_M:=\\Big\\{\\gamma\\in {\\rm{Geo}}(X)\\ :\\ \\rho_0(\\gamma_0),\\rho_1(\\gamma_1),{\\sf d}(\\gamma_0,\\bar x),{\\sf d}(\\gamma_1,\\bar x)\\leq M\\Big\\}.\n\\]\nFor $M$ large enough we have ${\\mbox{\\boldmath$\\pi$}}(G_M)>0$, thus the plan ${\\mbox{\\boldmath$\\pi$}}_M:=c_M{\\mbox{\\boldmath$\\pi$}}\\restr{G_M}$ is well defined, $c_M:={\\mbox{\\boldmath$\\pi$}}(G_M)^{-1}$ being the normalizing constant. Put $\\mu_0^M:=({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M$, $\\mu_1^M:=({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M$ and notice that $\\mu_0^M,\\mu_1^M\\ll\\mathfrak m$ and that by construction and since optimality is stable by restriction we get ${\\mbox{\\boldmath$\\pi$}}_M\\in{\\rm{OptGeo}}( \\mu_0^M, \\mu_1^M)$. Hence the uniqueness part of Theorem \\ref{thm:optmap} yields that ${\\mbox{\\boldmath$\\pi$}}_M$ is the only optimal plan from $\\mu_0^M$ to $\\mu_1^M$. Being $(X,{\\sf d},\\mathfrak m)$ a ${\\sf CD}(K,N)$ space it is also a ${\\sf CD}(K,\\infty)$ space and thus fact that ${\\rm Ent}_\\mathfrak m(\\mu_0^M),{\\rm Ent}_\\mathfrak m(\\mu_1^M)<\\infty$ (because both have bounded densities) give ${\\rm Ent}_\\mathfrak m(({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M)<\\infty$ for every $t\\in[0,1]$. In particular, $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M\\ll\\mathfrak m$ for every $t\\in[0,1]$. Since $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M\\uparrow ({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\mu_t$ as $M\\to\\infty$, we deduce $\\mu_t\\ll\\mathfrak m$ for every $t\\in[0,1]$.\n\nWe turn to \\eqref{eq:boundpoint}. Assume for a moment $t=0$, $s=1$ and that the supports of $\\mu_0,\\mu_1$ are bounded and notice that in this case to prove \\eqref{eq:boundpoint} is equivalent to prove that for any Borel set $G\\subset{\\rm{Geo}}(X)$ it holds\n\\begin{equation}\n\\label{eq:intcd}\n\\begin{split}\n-\\int_G\\rho_{r}^{-\\frac1N}(\\gamma_{r})\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma)\\leq& -\\int_G\\rho_0(\\gamma_0)^{-\\frac1N}\\sigma^{(1-t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))+\\rho_1(\\gamma_1)^{-\\frac1N}\\sigma^{(t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma).\n\\end{split}\n\\end{equation}\nFix such Borel set $G\\subset{\\rm{Geo}}(X)$, assume without loss of generality that ${\\mbox{\\boldmath$\\pi$}}(G)>0$ and define ${\\mbox{\\boldmath$\\pi$}}_G:={\\mbox{\\boldmath$\\pi$}}(G)^{-1}{\\mbox{\\boldmath$\\pi$}}\\restr G$. Let $T_t:X\\to{\\rm{Geo}}(X)$ be the maps given by Theorem \\ref{thm:optmap} and notice that the identity ${\\mbox{\\boldmath$\\pi$}}=(T_t)_\\sharp({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$ ensures $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_G={\\mbox{\\boldmath$\\pi$}}(G)^{-1}\\chi_G\\circ T_t({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$. In other words, letting $\\rho_{G,t}\\mathfrak m=({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_G$, a direct consequence of the fact that ${\\mbox{\\boldmath$\\pi$}}$ is concentrated on a set of non-branching geodesics is that we have\n\\begin{equation}\n\\label{eq:simpleresc}\n\\rho_{G,t}(\\gamma_t)={\\mbox{\\boldmath$\\pi$}}(G)^{-1}\\rho_t(\\gamma_t),\\qquad{\\mbox{\\boldmath$\\pi$}}-a.e.\\ \\gamma\\in G.\n\\end{equation}\nIt is clear that ${\\mbox{\\boldmath$\\pi$}}_G$ is optimal from $\\rho_{G,0}\\mathfrak m$ to $\\rho_{G,1}\\mathfrak m$ and by the uniqueness part of Theorem \\ref{thm:optmap} we know that it is the only optimal plan, hence the ${\\sf CD}^*(K,N)$ condition and the fact that $\\rho_{G,0}\\mathfrak m,\\rho_{G,1}\\mathfrak m$ have bounded support (because we assumed $\\mu_0,\\mu_1$ to have bounded support), yield\n\\[\n-\\int\\rho_{r}^{-\\frac1N}(\\gamma_{r})\\,\\d{\\mbox{\\boldmath$\\pi$}}_G(\\gamma)\\leq -\\int\\rho_0(\\gamma_0)^{-\\frac1N}\\sigma^{(1-t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))+\\rho_1(\\gamma_1)^{-\\frac1N}\\sigma^{(t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))\\,\\d{\\mbox{\\boldmath$\\pi$}}_G(\\gamma),\n\\]\nwhich, taking into account \\eqref{eq:simpleresc}, is \\eqref{eq:intcd}.\n\nThe assumption that $\\mu_0,\\mu_1$ have bounded support can be removed with the same truncation argument used at the beginning of the proof. To deal with the case of arbitrary $0\\leq t0$. Then there exists a geodesic $(\\mu_t)$ from $\\mu$ to $\\nu$ such that $\\mu_t\\ll\\mathfrak m$ for every $t\\in[0,1)$.\n\\end{lemma}\n\\begin{proof}\nLet $(\\nu^n)\\subset\\probt X$ be a sequence of absolutely continuous measures weakly converging to $\\nu$ and with uniformly bounded supports and ${\\mbox{\\boldmath$\\pi$}}^n\\in{\\rm{OptGeo}}(\\mu,\\nu^n)$ the unique optimal plan given by Theorem \\ref{thm:optmap}. Then the bound \\eqref{eq:boundpoint} gives, after dropping the term involving $\\rho_1$, the inequality\n\\begin{equation}\n\\label{eq:sar}\n\\rho_t(\\gamma_t)\\leq \\rho_0(\\gamma_0)(\\sigma_{K,N}^{(1-t)}({\\sf d}(\\gamma_0,\\gamma_1)))^{-N},\\qquad{\\mbox{\\boldmath$\\pi$}}^n-a.e.\\ \\gamma .\n\\end{equation}\nBy the definition of the distortion coefficients $\\sigma_{K,N}^{(1-t)}(\\theta)$ we see that for some function $f:[0,1)\\to\\mathbb{R}^+$ depending on $K$, $N$ and ${\\rm diam}(\\mathop{\\rm supp}\\nolimits(\\mu)\\cup(\\cup_n\\mathop{\\rm supp}\\nolimits(\\nu^n)))$, it holds $(\\sigma_{K,N}^{(1-t)}({\\sf d}(\\gamma_0,\\gamma_1)))^{-N}\\leq f(t)$ and thus \\eqref{eq:sar} and the bound $\\mu\\leq C\\mathfrak m$ give\n\\[\n\\mu^n_t:=({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}^n\\leq Cf(t)\\mathfrak m,\\qquad\\forall t\\in[0,1).\n\\]\nThis bound is independent on $n\\in\\mathbb{N}$, hence with a simple compactness argument based on the fact that $(X,{\\sf d},\\mathfrak m)$ is proper we get the conclusion by letting $n\\to\\infty$.\n\\end{proof}\n\nWe shall also use the following lemma, whose proof was given in \\cite{FigalliGigli11} (see also \\cite{Gigli11}) for the case of Riemannian manifolds; yet, the argument is only metric and can be repeated without any change. We report it just for completeness.\n\\begin{lemma}\\label{le:lip}\nLet $(X,{\\sf d})$ be a proper geodesic space, $\\varphi$ a $c$-concave function and $\\Omega\\subset X$ the interior of $\\{\\varphi>-\\infty\\}$. Then $\\varphi$ is locally bounded and locally Lipschitz on $\\Omega$ and for every compact set $K\\subset\\Omega$ the set $\\cup_{x\\in K}\\partial^c\\varphi(x)$ is bounded and not empty.\n\\end{lemma}\n\\begin{proof} \nBeing $c$-concave, $\\varphi$ is the infimum of a family of continuous functions, hence upper-semicontinuous and thus locally bounded from above. We prove that it is locally bounded from below by contradiction. Thus, recall that $(X,{\\sf d})$ is proper, assume that there exists a sequence $(x_n)\\subset \\Omega$ converging to some $x_\\infty\\in\\Omega$ such that $\\varphi(x_n)\\to-\\infty$ as $n\\to\\infty$. For every $n\\in\\mathbb{N}$ let $y_n\\in X$ be such that\n\\begin{equation}\n\\label{eq:yn}\n\\varphi(x_n)\\geq \\frac{{\\sf d}^2(x_n,y_n)}{2}-\\varphi^c(y_n)-1,\\qquad\\forall n\\in\\mathbb{N},\n\\end{equation}\nand notice that this bound and the fact that $\\varphi(x_n)\\to-\\infty$ yield that $\\varphi^c(y_n)\\to+\\infty$ as $n\\to\\infty$. Thus from \n\\[\n\\mathbb{R}\\ni \\varphi(x_\\infty)\\leq \\frac{{\\sf d}^2(x_\\infty,y_n)}{2}-\\varphi^c(y_n),\\qquad\\forall n\\in\\mathbb{N},\n\\]\nwe deduce that $\\frac{{\\sf d}^2(x_\\infty,y_n)}{2}\\to+\\infty$ as well as $n\\to\\infty$ and therefore also that\n\\[\n\\lim_{n\\to\\infty}\\frac{{\\sf d}^2(x_n,y_n)}{2}\\to+\\infty.\n\\]\nIn particular, without loss of generality we can assume ${\\sf d}(x_n,y_n)\\geq 1$ for every $n\\in\\mathbb{N}$. Now let $\\gamma^n:[0,{\\sf d}(x_n,y_n)]\\to X$ be a geodesic from $x_n$ to $y_n$ parametrized by arc-length. We claim that \n\\begin{equation}\n\\label{eq:claimcn}\n\\sup_{B_1(\\gamma^n_1)}\\varphi\\to-\\infty,\\qquad \\textrm{as } n\\to\\infty.\n\\end{equation}\nIndeed, for $x\\in B_1(\\gamma^n_1) $ we have\n\\[\n\\begin{split}\n\\varphi(x)&\\leq \\frac{{\\sf d}^2(x,y_n)}2-\\varphi^c(y_n)\\leq \\frac{({\\sf d}(x,\\gamma^n_1)+{\\sf d}(\\gamma^n_1,y_n))^2}2-\\varphi^c(y_n)\\\\\n&\\leq \\frac{{\\sf d}^2(x_n,y_n)}2-\\varphi^c(y_n)\\leq\\varphi(x_n)+1,\n\\end{split}\n\\]\nhaving used \\eqref{eq:yn} in the last inequality. Given that the $x_n$'s were chosen so that $\\varphi(x_n)\\to-\\infty$ as $n\\to\\infty$, our claim \\eqref{eq:claimcn} is proved. \n\nUp to pass to a subsequence, we can assume that $(\\gamma^n_1)$ converges to some $z\\in X$. From \\eqref{eq:claimcn} it easily follows that in the internal part of $B_1(z)$ the function $\\varphi$ is identically $-\\infty$. Given that ${\\sf d}(x,z)=1$, this fact contradicts the assumption that $x\\in\\Omega$. Hence $\\varphi$ is locally bounded.\n\nNow let $\\bar x\\in\\Omega$ and $r>0$ be such that $B_{2r}(x)\\subset\\Omega$. Pick $x\\in B_r(\\bar x)$ and let $(y_n)$ be such that $\\varphi(x)=\\lim_{n}\\frac{{\\sf d}^2(x,y_n)}{2}-\\varphi^c(y_n)$. We claim that there exists a constant $C$ depending only on $\\bar x, r$ and $\\varphi$ such that $(y_n) \\subset B_C(\\bar x)$. In proving this we may assume that ${\\sf d}(x,y_n)>r$ for all $n$. \nPick unit speed geodesics $\\gamma^n:[0,{\\sf d}(x,y_n)]\\to X$ from $x$ to $y_n$ and notice that\n\\[\n\\begin{split}\n\\varlimsup_{n\\to\\infty}\\varphi(x)-\\varphi(\\gamma^n_r)\\geq \\varlimsup_{n\\to\\infty}\\frac{{\\sf d}^2(x,y_n)}{2}-\\frac{{\\sf d}^2(\\gamma^n_r,y_n)}{2}=\\varlimsup_{n\\to\\infty}r{\\sf d}(x,y_n)-\\frac{r^2}{2}.\n\\end{split}\n\\]\nBy construction we have $x,\\gamma^n_r\\in B_{2r}(\\bar x)\\subset \\Omega$ thus by what we previously proved we know that the leftmost side of the above inequality is bounded by some constant depending only on $\\bar x, r$ and $\\varphi$. Hence the sequence $(y_n)$ is bounded and we directly get that any limit point belongs to $\\partial^c\\varphi(x)$, which therefore is non-empty. The very same argument also shows that $C:=\\cup_{x\\in B_r(\\bar x)}\\partial^c\\varphi(x)$ is bounded. In particular we get \n\\[\n\\varphi(x)=\\min_{y\\in C}\\frac{{\\sf d}^2(x,y)}{2}-\\varphi^c(y),\\qquad\\forall x\\in B_r(\\bar x),\n\\]\nand since for $y\\in C$ the functions $x\\mapsto \\frac{{\\sf d}^2(x,y)}{2}-\\varphi^c(y)$ are uniformly Lipschitz, we deduce the local Lipschitz continuity of $\\varphi$ as well.\n\\end{proof}\n\\begin{theorem}[Exponentiation and optimal maps] Let $K\\in \\mathbb{R}$, $N\\in[1,\\infty)$, $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space, $\\varphi$ a $c$-concave function and $\\Omega\\subset X$ the interior of $\\{\\varphi>-\\infty\\}$. Then for $\\mathfrak m$-a.e. $x\\in\\Omega$ there exists a unique geodesic $\\gamma$ with $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$.\n\nIn particular, for every $\\mu,\\nu\\in\\probt X$ with $\\mu\\ll\\mathfrak m$ there exists a unique optimal geodesic plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and this plan is induced by a map and concentrated on a set of non-branching geodesics.\n\\end{theorem}\n\\begin{proof} Existence trivially follows from the fact that $\\partial^c\\varphi(x)$ is non-empty for every $x\\in\\Omega$ and the fact that $(X,{\\sf d})$ is geodesic. For uniqueness we argue by contradiction. For $x\\in\\Omega$ let $G(x)\\subset{\\rm{Geo}}(X)$ be the set of $\\gamma$'s such that $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$ and assume that there is a compact set $K_1\\subset \\Omega$ such that $\\mathfrak m(K_1)>0$ and $\\#G(x)\\geq 2$ for every $x\\in K_1$. By Lemma \\ref{le:lip} we know that for some $L>0$ we have ${\\sf d}(\\gamma_0,\\gamma_1)\\leq L$ for any $x\\in K_1$ and $\\gamma\\in G(x)$ so that the geodesics in $\\cup_{x\\in K_1}G(x)$ are equi-Lipschitz.\n\n\nFor some $a>0$ the compact set $K_2\\subset K_1$ of $x$'s such that ${\\rm diam}G(x)\\geq a$ is such that $\\mathfrak m(K_2)>0$. Pick such $a$ and $K_2$. For $t\\in[0,1]$ put $G_t(x):=\\{\\gamma_t:\\gamma\\in G(x)\\}\\subset X$ and consider the set $\\mathcal K\\subset K_2\\times [0,1]$ of $(x,t)$'s such that ${\\rm diam}G_t(x)\\geq\\frac a2$. It is easy to check that $\\mathcal K$ is closed and the continuity of geodesics grants that for any $x\\in K_2$ the set of $t$'s such that $(x,t)\\in \\mathcal K$ has positive $\\mathcal L^1$-measure. By Fubini's theorem, there is $t_0\\in[0,1]$ such that the compact set $K_3\\subset K_2$ of $x$'s such that ${\\rm diam}G_{t_0}(x)\\geq\\frac a2$ has positive $\\mathfrak m$-measure. Notice that necessarily $t_0>0$. With a Borel selection argument we can find a Borel map $T:K_3\\to X$ such that $T(x)\\in G_{t_0}(x)$ for every $x\\in K_3$. Let $x_0\\in X$ be such that $T_\\sharp(\\mathfrak m\\restr {K_3})(B_{\\frac a6}(x_0))>0$ and put $A:=T^{-1}(B_{\\frac a6}(x_0))$, so that $\\mathfrak m(A)>0$. By construction, the map $A\\ni x\\mapsto G_{t_0}(x)\\setminus B_{\\frac{a}3}(x_0)$ is Borel and has non-empty values, thus again with a Borel selection argument we can find Borel map $S:A\\to X$ such that $S(x)\\in G_{t_0}(x)\\setminus B_{\\frac{a}3}(x_0) $ for every $x\\in A$.\n\nLet $\\mu:=\\mathfrak m(A)^{-1}\\mathfrak m\\restr A$, $\\nu_1:=T_\\sharp\\mu$ and $\\nu_2:=S_\\sharp\\mu$. By construction $\\nu_1$ and $\\nu_2$ have disjoint support, and in particular $\\nu_1\\neq \\nu_2$. Furthermore, the function $t_0\\varphi$ is a Kantorovich potential both from $\\mu$ to $\\nu_1$ and from $\\mu$ to $\\nu_2$. Apply Lemma \\ref{le:ac} to both $(\\mu,\\nu_1)$ and $(\\mu,\\nu_2)$ to find geodesics $(\\mu^i_t)$, $i=1,2$, from $\\mu$ to $\\nu_1,\\nu_2$ respectively such that $\\mu^i_t\\ll\\mathfrak m$ for every $t\\in[0,1)$, $i=1,2$. By construction, for $t$ sufficiently close to 1 we have $\\mu^1_t\\neq \\mu^2_t$. Fix such $t$, let ${\\mbox{\\boldmath$\\pi$}}^i\\in{\\rm{OptGeo}}(\\mu,\\mu^i_t)$, $i=1,2$ and notice that ${\\mbox{\\boldmath$\\pi$}}^1\\neq {\\mbox{\\boldmath$\\pi$}}^2$ and that $\\mathop{\\rm supp}\\nolimits(({\\rm{e}}_0,{\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}^i)\\subset\\partial^c(tt_0\\varphi)$, $i=1,2$. \n\nThus for the plan ${\\mbox{\\boldmath$\\pi$}}:=\\frac12({\\mbox{\\boldmath$\\pi$}}^1+{\\mbox{\\boldmath$\\pi$}}^2)$ it also holds $\\mathop{\\rm supp}\\nolimits(({\\rm{e}}_0,{\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}})\\subset \\partial^c(tt_0\\varphi)$ and thus is optimal. Moreover it satisfies $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}},({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\ll\\mathfrak m$ and, by construction, is not induced by a map. This contradicts Theorem \\ref{thm:optmap}, concluding the proof of the first part of the statement.\n\nFor the second part, notice that if the optimal geodesic plan is not unique or not induced by a map, there must be ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ which is not induced by a map. With a restriction argument we can then assume that $\\mu:=({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}$, and $\\nu:=({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}$ have bounded support, with $\\mu\\ll\\mathfrak m$. But in this case there is a locally Lipschitz Kantorovich potential from $\\mu$ to $\\nu$ and the first part of the statement gives the conclusion.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nGiven a sequence of independent and identically distributed standard complex Gaussian random variables $(\\xi_n)$, consider the Gaussian power series \n $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ defined in the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C}: |z| < 1\\}$ and the zero set of $f(z)$, that is, \n$$\\mathscr{Z} = \\{z \\in \\mathbb{D}: f(z) = 0\\}.$$\nThe zero set $\\mathscr{Z}$ constitutes a point process on $\\mathbb{D}$. The joint intensity $p$ of the process $\\mathscr{Z}$ is defined as\n $$p(z_1,z_2, \\ldots, z_n) = \\lim_{\\epsilon\\to 0} \\frac{\\mathbb{P}_{\\epsilon}(z_1, z_2, \\ldots, z_n)}{\\pi^n \\epsilon^{2n}}$$ where \n$\\mathbb{P}_\\epsilon(z_1, z_2, \\ldots, z_n)$ is the probability that simultaneously for all $1\\leq i \\leq n$, the function $f(z)$ has a zero in the disc of centre $z_i$ and radius $\\epsilon>0$. \nRecently, Peres and Vir\\'ag \\cite{Peres_Virag} obtained the striking fact that the point process $\\mathscr{Z}$ is a determinantal process. In fact, they proved that, for all $z_1, z_2, \\ldots, z_n$ in $\\mathbb{D}$, \n $$p(z_1, z_2, \\ldots, z_n) = \\det\\left(\\frac{1}{\\pi(1 - z_k \\overline{z_j})^2}\\right)_{k,j=1}^n.$$\nThat is, \n $$p(z_1, z_2, \\ldots, z_n) = \\det\\left(K(z_k, z_j)\\right)_{1\\leq k,j\\leq n}$$ where $K(z, w) = \\pi^{-1}(1 - z \\overline{w})^{-2}$ is the classical Bergman kernel in $\\mathbb{D}$. (A thorough discussion on determinantal point processes can be found in the book by Hough et al. \\cite{Hough}.)\nThe study of the zeros of the general random series \n $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ where the restrictions of independence and identical distribution are relaxed upon the random variables $(\\xi_n)$ is an important open problem. This paper considers this question in the following context: {\\it Determine sequences of dependent Gaussian random variables $(\\xi_n)$ such that the zero set of the random series $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ is a determinantal point process as in the case of i.i.d random variables.} We recall that a matrix $(a_{k,j})$ is called a Toeplitz matrix if $a_{k,j}$ depends only on the difference $k-j$, that is, for all $k,j$ and for any integer $\\ell$ such that $a_{k+\\ell, j+\\ell}$ is defined, $a_{k+\\ell, j+\\ell} = a_{k,j}.$ \nWe consider a complex infinite Toeplitz matrix $G$ that is hermitian and positive definite and we assume that $G$ admits a classical inverse in the sense that there exists a hermitian positive definite matrix $G^{-1}$ such that $G G^{-1} = G^{-1} G = I$. Since $G$ is positive definite and Toeplitz, then there exits a positive definite function $\\gamma$ on the integers such that $G_{k,j} = \\gamma(k-j) = {\\overline{\\gamma(j-k)}}$ for all $k,j$. (We shall assume without loss of generality that $G_{k,k} = \\gamma(0) = 1$ for all $k$.)\nThen by the classical Bochner theorem, one can associate to $G$ a (unique) probability measure $\\mu$ on the unit circle $\\mathbb{T}$ such that \n\\begin{eqnarray*}\n\\gamma(n) = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} d\\mu(\\theta),\\,\\, \\mbox{ for all } n \\in \\mathbb{Z}.\n\\end{eqnarray*}\nWe shall assume throughout that the probability measure $\\mu$ satisfies the following condition: \n{\\bf Condition (C):} The measure $\\mu$ is absolutely continuous and its density $\\varphi$ is strictly positive almost everywhere on $\\mathbb{T}$ with respect to the Lebesgue measure on $\\mathbb{T}$. \n\nWe now consider a discrete-time complex Gaussian process $(\\xi_n)_{n\\in \\mathbb{N}}$ with zero mean, covariance matrix $G^{-1}$ and zero pseudo-covariance matrix, that is, for all $n,m\\in \\mathbb{N}$,\n $$\\mathbb{E}(\\xi_n) = 0, \\,\\,\\,\\mathbb{E}(\\xi_n \\overline{\\xi_m}) = \\left(G^{-1}\\right)_{n,m} \n\\mbox{ and } \\mathbb{E}(\\xi_n \\xi_m) = 0.$$\nThe existence of such process $(\\xi_n)$ is classical (see for example Miller \\cite{Miller} and references therein.) In the case where $G$ (and hence $G^{-1})$ is a real matrix one can simply take a real Gaussian process $(\\zeta_n)$ of covariance matrix $G^{-1}$ and write $\\xi_n = (\\zeta_n + i \\zeta_n')\/\\sqrt{2}$ where $(\\zeta_n')$ is an independent copy of $(\\zeta_n)$.\nWe shall consider the Gaussian analytic function \n $$f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1},\\,\\, z\\in \\mathbb{D}.$$\nOur main finding is that the zero set of the Gaussian analytic function $f(z)$\nis a determinantal point process governed by the Bergman kernel just as it is for the case of i.i.d random variables. \nThe main result of this paper is the following.\n \\begin{theorem}\\label{mainth} \n Let $G$ be an infinite, invertible, hermitian and positive definite Toeplitz matrix such that the associated probability measure $\\mu$ satisfies condition $(C)$ and the inverse $G^{-1}$ is such that\n\\begin{eqnarray} \\label{sdse34rfs}\n\\sup_{n,m} |(G^{-1})_{n,m}| < \\infty.\n\\end{eqnarray}\nIf $(\\xi_n)_{n \\in \\mathbb{Z}}$ is a centred complex Gaussian process with covariance matrix $G^{-1}$ and zero pseudo-covariance matrix, then the zero set of the Gaussian analytic function\n$$f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1},\\,\\, z\\in \\mathbb{D}$$\n is a determinantal point process governed by the Bergman kernel. That is the joint intensity $p$ of the zeros of $f(z)$ is given by\n $$p(z_1, z_2, \\ldots, z_n) = \\det\\left(\\frac{1}{\\pi(1 - z_k \\overline{z_j})^2}\\right)_{1\\leq k,j\\leq n},\\,\\,\\, z_1, z_2, \\ldots, z_n \\in \\mathbb{D}.$$ \n\\end{theorem}\nIf $(\\zeta_n)$ is a centred Gaussian process with covariance matrix $G$ (that is a Toeplitz matrix), the zero set of $f(z) = \\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ does not necessarily have the same distribution as $\\sum_{n=1}^\\infty \\chi_n z^{n-1}$ for i.i.d Gaussian variables $(\\chi_n)$. (This is discussed in \\cite{Mukeru_al}. An example is also given in section 6.1.) However if we take instead of $(\\zeta_n)$ a sequence $(\\xi_n)$ with covariance matrix $G^{-1}$ then the corresponding zero set has the same distribution as for i.i.d. Gaussian variables. \n This implies that in terms of the corresponding zero sets sequences of Gaussian variables whose covariance matrix is the inverse of a Toeplitz matrix are more closer to the sequence of i.i.d Gaussian variables than sequences of variables whose covariance matrix is a Toeplitz matrix. This looks awkward but one should remember that the distribution of a Gaussian vector depends more directly on the inverse of its covariance matrix rather than the covariance matrix itself. \nLemma \\ref{lemmaone} in section \\ref{sdsefer} gives an interesting property of sequences of Gaussian variables whose covariance is an inverse of a Toeplitz matrix. It looks like such sequences are of some independent interest that requires further investigation. \n\nThe question of dependent random variables can be raised in connection with other determinantal point processes, for instance the point processes obtained by Krishnapur \\cite{Krishnapur}. It is also relevant in the context of \nPfaffian processes studied by Matsumoto and Shirai \\cite{Matsumoto}. \n The rest of the paper is organised as follows. Section 2 contains some basic well-known facts about the zeros of the $f(z)$ and the Szeg\\\"o kernel. Section 3 contains a connection between the covariance kernel of $f(z)$ and a Hardy space defined by the spectral measure of $(\\xi_n)$. In section 4 we obtain some properties of the sequence $(\\xi_n)$. In section 5 we provide an important connection between the classical Mobius transformation and the covariance kernel of $f(z)$ which is key to the proof of the main result. The last section contains some examples that illustrate the main result. \n\n\n\n\n\n\n\\section{Szeg\\\"o kernel and Hardy spaces} \n \nThe starting point in the study of the zeros of any zero-mean Gaussian analytic function $f$ in a planar domain is the following general expression for its joint intensity function (Peres and Vir\\'ag \\cite{Peres_Virag}): \n \\begin{eqnarray} \\label{eqnads23001}\n p(z_1, z_2, \\ldots, z_n) = \\frac{\\mathbb{E}\\left(|f'(z_1) f'(z_2) \\ldots f'(z_n)|^2\\,|f(z_1) = f(z_2) = \\ldots=f(z_n) = 0\\right)}{\\pi^n \\det(A)}\n \\end{eqnarray}\n or equivalently \n \\begin{eqnarray} \\label{eqnads231}\n p(z_1, z_2, \\ldots, z_n) = \\frac{\\mbox{perm}(C - B A^{-1} B^*)}{\\pi^n \\det(A)}\n \\end{eqnarray}\n where $A, B$ and $C$ are the $n \\times n$ matrices\n $$A = (\\mathbb{E}(f(z_k) \\overline{f(z_j)})), B = (\\mathbb{E}(f'(z_k) \\overline{f(z_j)})) \\mbox{ and } C = (\\mathbb{E}(f'(z_k) \\overline{f'(z_j)}))$$ \nand $\\mbox{perm}$ denotes the permanent of a matrix. For the classical Gaussian power series $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ with independent and identically distributed (i.i.d) random variables $(\\xi_n)$,\n $$ \\mathbb{E}(f(z) \\overline{f(w)})) = \\sum_{n=0}^\\infty (z \\overline{w})^{n-1} = \\frac{1}{1 - z \\overline{w}}, \\,\\, z, w \\in \\mathbb{D}.$$\nThat is, $$ \\mathbb{E}(f(z) \\overline{f(w)})) = \\mathbb{K}(z, w)$$ where $\\mathbb{K}$ is the classical Szeg\\\"o kernel. This means that the covariance kernel of $f(z)$ is the Szeg\\\"o kernel.\nThe classical Hardy space\n $\\mathbb{H}^2(\\mathbb{D})$ is the class of holomorphic functions $f$ in the unit disc $\\mathbb{D}$ for which \n $$\\sup_{0 \\leq r < 1} \\int_{\\mathbb{T}} \\left|f(r e^{2\\pi i \\theta})\\right|^2 d\\theta = \\lim_{r \\to 1} \\int_{\\mathbb{T}} \\left|f(r e^{2\\pi i \\theta})\\right|^2 d\\theta < \\infty$$ where \n $\\mathbb{T}$ is the unit circle $\\mathbb{R}\/\\mathbb{Z}$. Equivalently $\\mathbb{H}^2(\\mathbb{D})$ is the class of holomorphic functions $f(z) = \\sum_{n=0}^\\infty a_n z^n$, $z \\in \\mathbb{D}$, $a_n \\in \\mathbb{C}$ such that $\\sum_{n=0}^\\infty |a_n|^2 < \\infty.$ \nIt is a Hilbert space with the inner product: \n \\begin{eqnarray} \\label{ew23edfr1e1}\n\\langle f, g\\rangle = \\sum_{n=0}^\\infty a_n \\overline{b_n},\\,\\,\\,\\, f = \\sum_{n=0}^\\infty a_n z^n, \\,\\,g = \\sum_{n=0}^\\infty b_n z^n.\n\\end{eqnarray}\nAny function $f(z) = \\sum_{n=0}^\\infty a_n z^n$ in $\\mathbb{H}^2(\\mathbb{D})$ is such that its radial limit\n$$\\tilde{f}(\\theta) = \\lim_{r\\to 1} f(r e^{2\\pi i \\theta}) = f(e^{2\\pi i \\theta})= \\sum_{n=0}^\\infty a_n e^{2\\pi i n \\theta}$$\nexists almost everywhere in $\\mathbb{T}$ and $\\tilde{f} \\in L^2(\\mathbb{T})$. Moreover\n$$\\langle f, g\\rangle = \\int_{\\mathbb{T}} \\tilde{f}(\\theta) \\overline{\\tilde{g}(\\theta)} d\\theta = \\int_{\\mathbb{T}} f(e^{2\\pi i \\theta}) \\overline{g(e^{2\\pi i \\theta})} d\\theta$$ and \n $$\\|f\\|^2_{H^2(\\mathbb{D})} = \\|\\tilde{f}\\|_{L^2(\\mathbb{T})} = \\sum_{n=0}^\\infty |a_n|^2.$$\n(See Katznelson \\cite[p 98]{Katznelson}.) \n\nIt is well-known that $\\mathbb{H}^2(\\mathbb{D})$ is a reproducing kernel Hilbert space whose kernel is the Szeg\\\"o kernel. This means that for each $y \\in \\mathbb{D}$ and for each $f \\in \\mathbb{H}^2(\\mathbb{D})$, the function\n$\\mathbb{K}(.,y): \\mathbb{D} \\to \\mathbb{C}$ defined by $\\mathbb{K}(.,y)(x) = \\mathbb{K}(x, y)$ is such that\n $$f(y) = \\langle f, \\mathbb{K}(.,y) \\rangle.$$\n(We refer to the book by Paulsen \\cite{Paulsen} for a background on reproducing kernel Hilbert spaces.)\nThe main argument of Peres and Vir\\'ag is to make use of these connections between the Hardy space $\\mathbb{H}^2(\\mathbb{D})$ and the Szeg\\\"o kernel.\n\n\n\n\n\\section{Inverse Toeplitz matrices and weighted Hardy spaces} \nIn the general case where the covariance matrix of the variables $(\\xi_n)$ is the matrix $G^{-1}$ (where $G$ is an infinite hermitian and positive definite Toeplitz matrix), the covariance kernel of the function $f(z)$ is given by\n \\begin{eqnarray} \\label{sdds34rewdss}\n\\mathbb{K}_G(z,w) = \\mathbb{E}\\left(f(z) \\overline{f(w)}\\right) = \\sum_{k,j=1}^\\infty \\left(G^{-1}\\right)_{k,j} z^{k-1} \\left(\\overline{w}\\right)^{j-1}, \\,\\, z, w \\in \\mathbb{D}.\n\\end{eqnarray}\nFor the convergence of the series in (\\ref{sdds34rewdss}) it is enough to assume that \n $\\sup_{k,j} |(G^{-1})_{k,j}| < \\infty$\n(that is condition (\\ref{sdse34rfs}) in Theorem \\ref{mainth}.)\nOne can write $$ \\mathbb{K}_G(z,w) = Z^{T} G^{-1} \\overline{W}$$\nwhere $Z^T = (1, z, z^2, \\ldots)$ and $W = (1, w, w^2, \\ldots)^T$. \nClearly in the particular case where $G$ is the identity matrix $\\mathbb{K}_G$ is the Szeg\\\"o kernel . An important tool in the proof of the main result is the fact that there is a reproducing kernel Hilbert space whose kernel is the covariance kernel $\\mathbb{K}_G$. Assume that the Toeplitz matrix $G$ is given by $G_{k,j} = \\gamma(k-j) = {\\overline{\\gamma(j-k)}}$ for a function $\\gamma$ defined on the integers. We shall assume without loss of generality that $G_{k,k} = \\gamma(0) = 1$ for all $k$.\nSince $G$ is a Toeplitz matrix and it is positive definite then by the classical Bochner theorem, there exists a probability measure $\\mu$ on the unit circle $\\mathbb{T}$ such that \n\\begin{eqnarray} \\label{sdwdsdserfds}\n\\gamma(n) = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} d\\mu(\\theta),\\,\\, \\mbox{ for all } n \\in \\mathbb{Z}.\n\\end{eqnarray}\nWe have assumed throughout that $\\mu$ is absolutely continuous. Its density $\\varphi$ is called the spectral density function of the matrix $G$ and is such that \n\\begin{eqnarray} \\label{dsdsedererss}\n\\gamma(n) = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\varphi(\\theta) d\\theta,\\,\\, \\mbox{ for all } n \\in \\mathbb{Z}\n\\end{eqnarray}\nwhich (under some conditions) implies in return that \n\\begin{eqnarray} \\label{dsdswedwew334}\n\\varphi(\\theta) = \\sum_{n \\in \\mathbb{Z}} \\gamma(n) e^{-2\\pi i n \\theta},\\,\\, \\theta \\in \\mathbb{T}. \n\\end{eqnarray}\nConsider the sub-space $H^2_G(\\mathbb{D})$ of the Hardy space $H^2(\\mathbb{D})$ of functions $f(z) = \\sum_{n=0}^\\infty a_n z^n$, $z \\in \\mathbb{D}$, $a_n \\in \\mathbb{C}$ such that \n $$\\|f(e^{2\\pi i \\theta})\\|^2_{L^2(\\mu)} = \\int_{\\mathbb{T}} |f(e^{2\\pi i \\theta})|^2 d\\mu(\\theta) < \\infty$$ and set\n $$\\|f\\|_{H^2_G(\\mathbb{D})} = \\|f(e^{2\\pi i \\theta})\\|_{L^2(\\mu)}.$$\nClearly if $$\\|f\\|^2_{H^2_G(\\mathbb{D})} = \\int_{\\mathbb{T}} |f(e^{2\\pi i \\theta})|^2 \\varphi(\\theta) d\\theta = 0,$$ then the fact that $\\varphi > 0$ almost everywhere in $\\mathbb{T}$ yields\n$$\\|f\\|^2_{H^2(\\mathbb{D})} = \\int_{\\mathbb{T}} |f(e^{2\\pi i \\theta})|^2 d\\theta = 0.$$ \nThis implies that $f = 0$ everywhere in $\\mathbb{D}$. Moreover the norm $\\|.\\|_{H^2_G(\\mathbb{D})}$ is complete. \n Indeed, let $(f_k)$ be a sequence of functions in $H^2_G(\\mathbb{D})$ such that \n$$\\lim_{k,j\\to \\infty} \\|f_k - f_j\\|^2_{H^2_G(\\mathbb{D})} = \\lim_{k,j\\to \\infty} \\|f_k(e^{2\\pi i \\theta}) - f_j(e^{2\\pi i \\theta})|^2 _{L^2(\\mu)}= 0.$$ Since $L^2(\\mu)$ is complete, then the sequence $(f_k(e^{2\\pi i \\theta}))$ has a limit $\\ell: \\mathbb{T} \\to \\mathbb{C}$ in $L^2(\\mu)$. That is,\n \\begin{eqnarray*} \n\\lim_{k\\to \\infty} \\int_{\\mathbb{T}} |f_k(e^{2\\pi i \\theta}) - \\ell(\\theta)|^2 \\varphi(\\theta) d\\theta = 0.\n\\end{eqnarray*}\nAgain since $\\varphi >0$ almost everywhere, then \n\\begin{eqnarray} \\label{kaziba1}\n\\lim_{k\\to \\infty} \\int_{\\mathbb{T}} |f_k(e^{2\\pi i \\theta}) - \\ell(\\theta)|^2 d\\theta = 0.\n\\end{eqnarray}\n This means that $\\ell$ is also the limit of the sequence $(f_k(e^{2\\pi i \\theta}))$ in \n$L^2(\\mathbb{T})$. It follows that $\\ell \\in L^2(\\mathbb{T})$. We can consider the Fourier series of $\\ell$ and write\n $\\ell(\\theta) = \\sum_{n = -\\infty}^{\\infty} \\hat \\ell(n) e^{2\\pi i n \\theta} $ (this series converges in $L^2(\\mathbb{T})$) and $$\\sum_{n = -\\infty}^{\\infty} |\\hat \\ell(n)|^2 = \\|\\ell\\|^2_{L^2(\\mathbb{T})} < \\infty.$$ \nSet\n $$f_{n,k}(z) = \\sum_{n=0}^\\infty a_{n,k} z^n.$$ \n Then (\\ref{kaziba1}) implies\n $$\\lim_{k \\to \\infty} \\sum_{n \\geq 0} |a_{n,k} - \\hat \\ell(n)|^2 + \\sum_{n < 0} |\\hat \\ell(n)|^2 = 0.$$\nHence $ \\hat \\ell(n) = 0$ for all $n < 0$. Then consider the function \n $g(z) = \\sum_{n = 0}^\\infty \\hat \\ell(n) z^n$. It is now clear that $(f_k)$ converges to $g$ both in $H^2(\\mathbb{D})$ and $H^2_G(\\mathbb{D})$. \n\nWe define the inner product on $H^2_G(\\mathbb{D})$ by:\n \\begin{eqnarray} \\label{ew23edfr1eswe}\n\\langle f, g\\rangle = \\int_{\\mathbb{T}} f(e^{2\\pi i \\theta}) \\overline{g(e^{2\\pi i \\theta})} d\\mu(\\theta).\n\\end{eqnarray}\nClearly $H^2_G(\\mathbb{D})$ is a Hilbert space. We want to show that $\\mathbb{H}^2_G(\\mathbb{D})$ is in fact a reproducing kernel Hilbert space whose kernel is $\\mathbb{K}_G$. \n \\begin{theorem} \\label{sdsd34edsds}\nThe space $\\mathbb{H}^2_G(\\mathbb{D})$ is a reproducing kernel Hilbert space whose kernel is \n $\\mathbb{K}_G$ given by\n $$ \\mathbb{K}_G(z,w) = Z^{T} G^{-1} \\overline{W},\\,\\,z, w \\in \\mathbb{D}$$\n with $Z = (z^n)_{n \\in \\mathbb{N}}$ and $W = (w^n)_{n \\in \\mathbb{N}}$.\n\\end{theorem}\n\n\\begin{proof}\nWe shall first prove that the monomials $z^n$ are in the reproducing kernel Hilbert space associate to $\\mathbb{K}_G$. That is, \n $$z^n = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\,\\,\\overline{\\mathbb{K}_G(e^{2\\pi i \\theta}, z)} d\\mu(\\theta).$$\nSince for all $w, y \\in \\mathbb{D}$,\n $$ \\mathbb{K}_G(w, y) = \\sum_{k, j = 1}^\\infty (w)^{k-1} (\\overline{y})^{j-1} (G^{-1})_{k,j} = \\overline{\\mathbb{K}_G(y, w)},$$ then \n\n\\begin{eqnarray*}\n\\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\,\\,\\overline{\\mathbb{K}_G(e^{2\\pi i \\theta}, z)} d\\mu(\\theta) & = & \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\,\\,\\mathbb{K}_G(z, e^{2\\pi i \\theta}) d\\mu(\\theta) \\\\\n & = & \\sum_{j=1}^{\\infty} z^{j-1} \\sum_{k=1}^\\infty (G^{-1})_{k,j} \\int_{\\mathbb{T}} e^{2\\pi i (n - k+1) \\theta} d\\mu(\\theta) \\\\\n& = & \\sum_{j=1}^{\\infty} z^{j-1} \\sum_{k=1}^\\infty (G^{-1})_{k,j} \\,\\gamma(n-k+1) \\\\\n& = & \\sum_{j=1}^{\\infty} z^{j-1} \\sum_{k=1}^\\infty (G^{-1})_{k,j}\\, G_{n+1,k} \\\\\n& = & z^n.\\\\\n\\end{eqnarray*}\nTo complete of the proof it suffices to determine an orthonormal basis $\\{P_k(z): k=1,2,\\ldots\\}$ of $H^2_G(\\mathbb{D})$ and prove that it is the case that\n $$\\sum_{k=1}^\\infty P_k(z) \\overline{P_k(w)} = \\mathbb{K}_G(z, w),\\,\\,\\mbox{ for all } z, w \\in \\mathbb{D}.$$\nFirst, it is clear that in the Hilbert space $H^2_G(\\mathbb{D})$, \nfor all $k,j \\in \\mathbb{N}$, \n \\begin{eqnarray*}\n \\langle z^k, z^j \\rangle & = & \\int_{\\mathbb{T}} e^{2\\pi i(k-j) \\theta} d\\mu(\\theta) \\\\\n & = & \\gamma(k-j).\n \\end{eqnarray*}\nNext we shall take $(P_k)$ to be the orthonormal basis deduced from \nthe sequence of polynomials $(1, z, z^2, \\ldots, z^k, \\ldots)$ by the classical Gram--Schmidt process. \nConsider an infinite lower-triangular matrix $A = (a_{k,j})$ (that is $a_{k,j} = 0$ for $j>k$) such that \n for each $k$,\n\\begin{eqnarray} \\label{sdewewe3reew}\nz^{k-1} = a_{k,1} P_1(z) + a_{k,2} P_2(z) + \\ldots+ a_{k,k} P_k(z).\n\\end{eqnarray}\nThen since $\\{P_k: k=1,2,\\ldots\\}$ is orthonormal, then for $j\\leq k$, \n\\begin{eqnarray*}\n\\langle z^{k-1}, z^{j-1} \\rangle = a_{k,1} \\overline{a_{j,1}} + a_{k,2} \\overline{a_{j,2}} + \\ldots+ a_{k,j} \\overline{a_{j,j}}.\n\\end{eqnarray*}\nMoreover, using \n$ \\langle z^{k-1}, z^{j-1} \\rangle = \\gamma(k-j)$, it follows that\n $$a_{k,1} \\overline{a_{j,1}} + a_{k,2} \\overline{a_{j,2}} + \\ldots+ a_{k,j} \\overline{a_{j,j}} = \\gamma(k-j) = G_{k,j}.$$\nThis yields\n $A A^* = G$ where $A^*$ is the conjugate transpose of $A$.\nIt follows from (\\ref{sdewewe3reew}) that \n \\begin{eqnarray*}\n \\begin{pmatrix}\n P_1(z)\\\\\n P_2(z)\\\\\n P_3(z)\\\\\n \\vdots\\\\\n \n P_k(z)\\\\\n \\vdots\n \\end{pmatrix}\n = A^{-1} \n \\begin{pmatrix}\n 1\\\\\n z\\\\\n z^2\\\\\n \\vdots\\\\\n z^{k-1} \\\\\n\\vdots \n \\end{pmatrix} = A^{-1} Z\n \\end{eqnarray*}\n for a lower-triangular matrix $A$ such that $A A^* = G$. It is clear that since $A$ is a lower triangular matrix, then $A^{-1}$ is also a lower triangular matrix and moreover $P_k(z)$ is fully determined by the first $k$ rows of $A$. \nNow clearly,\n \\begin{eqnarray*}\n \\sum_{k=1}^\\infty P_k(z) \\overline{P_k(w)} & = & \\lim_{n \\to \\infty} \\sum_{k=1}^n P_k(z) \\overline{P_k(w)} \\\\\n& = &\\lim_{n \\to \\infty} \\left(\\left(A_n\\right)^{-1} Z_n\\right)^T \\overline{\\left(A_n\\right)^{-1} W_n} \\\\\n& = & \\lim_{n\\to \\infty} Z_n^T (G_n)^{-1} \\overline{W_n} \\\\\n& = & Z^T G^{-1} \\overline{W}\n \\end{eqnarray*}\nwhere $A_n$ (resp. $G_n$) is the block of $A$ (resp. $G$) consisting of the first $n$ rows and columns of $A$ (resp. $G$) and\n$Z_n = (1, z, z^2, \\ldots, z^{n-1})$ and $W_n = (1, w, w^2, \\ldots, w^{n-1})$.\nIt follows that \n $$ \\sum_{k=1}^\\infty P_k(z) \\overline{P_k(w)} = \\mathbb{K}_G(z, w)$$ which concludes the proof. \n\n\\end{proof}\n\n\\begin{corollary}\nThe Gaussian analytic function $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ (where $(\\xi_n)$ has covariance matrix $G^{-1}$) has the same distribution with the function\n $g(z) = \\sum_{n=1}^\\infty \\chi_k P_n(z)$ where $(\\chi_n)$ is a sequence of standard i.i.d complex Gaussian random variables and $(P_n(z))$ are the polynomials defined by the matrix $G$ as in the proof of Theorem \\ref{sdsd34edsds}. \n\\end{corollary}\nIt is so because the two random functions have the same covariance kernel:\n $$\\mathbb{E}(f(z) \\overline{f(w)}) = \\mathbb{E}(g(z) \\overline{g(w)}) = \\sum_{k=1}^\\infty P_n(z) \\overline{P_n(w)} = Z^T G^{-1} \\overline{W}.$$\n\n\n\n\n\n \\section{Some properties of the sequence $(\\xi_n)$ of covariance $G^{-1}$} \\label{sdsefer}\nHere we obtain important properties of the sequence $(\\xi_n)$ of covariance matrix $G^{-1}$ that will be useful for the proof of main result. \n\n\\begin{lemma}\\label{lemmaone}\nAssume that $(\\xi_n)_{n\\in \\mathbb{N}}$ is a centred complex Gaussian process with zero \\\\ pseudo-covariance and covariance matrix $G^{-1}$ where $G$ is an infinite hermitian positive definite Toeplitz matrix. Then for each $n \\geq 2$, the conditional joint distribution of the sequence $(\\xi_n, \\xi_{n+1}, \\xi_{n+2}, \\ldots)$ under the condition $\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0$ is equal to the unconditional joint distribution of $(\\xi_{1}, \\xi_{2}, \\xi_{3}, \\ldots)$. \nThat is,\n $$((\\xi_n, \\xi_{n+1}, \\xi_{n+3}, \\ldots)|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0) \\stackrel{d}{=} (\\xi_{1}, \\xi_{2}, \\xi_3, \\ldots)$$\n \\end{lemma}\n\n\\begin{proof}.\nSet $$S_1 = (\\xi_1, \\xi_2, \\ldots), \\,\\, S_n = (\\xi_{n}, \\xi_{n+1}, \\ldots),\\,\\,\\mbox{ for all } n\\geq 1.$$\nThen it is well-known that the covariance matrix of \n$$(S_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$$ is the Schur complement of $\\mbox{Cov}(\\xi_1, \\xi_2, \\ldots, \\xi_{n-1})$ in the overall covariance matrix $\\mbox{Cov}(S_1) = G^{-1}.$ It is obtained by taking the matrix $G^{-1}$, take its inverse, that is, $G$, delete the rows and columns corresponding to the random variables $\\xi_1, \\xi_2, \\ldots, \\xi_{n-1}$ and take the inverse of the resulting matrix. Now deleting the first $n-1$ rows and columns of the infinite Toeplitz matrix $G$ yields the very same matrix $G$. It follows that \nthe covariance matrix of $(S_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$ is just $G^{-1}$. This implies that \n $(S_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$ has the same distribution as $(\\xi_1, \\xi_2, \\ldots)$. \n \\hfill \\qed\n\\end{proof}\nAn immediate consequence of this lemma is: \n\\begin{corollary} \\label{corol1}\nFor any sequence $(\\alpha_k)$ of complex numbers and for any integer $n \\geq 1$,\n $$\\left.\\left(\\sum_{k=1}^\\infty \\alpha_{k} \\xi_{n+k-1}\\right)\\right|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$$ has the same distribution with $$\\sum_{k=1}^\\infty \\alpha_k \\xi_{k}$$ provided the involved series converge almost surely. \nIn particular,\n $$(\\xi_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{k} = 0) \\stackrel{d}{=} \\xi_{n-k} \\,\\, \\mbox{ for all } 1 \\leq k < n.$$\n \\end{corollary}\n\n \n\n\n\\section{Mobius transformation and the kernel $\\mathbb{K}_G$}\n\\subsection{Mobius transformation}\nFor $w \\in \\mathbb{D}$, consider as in Peres and Vir\\'ag \\cite{Peres_Virag}, the Mobius transformation of the unit disc $\\mathbb{D}$:\n $$T_w(z) = \\frac{z-w}{1-z \\overline{w}},\\,\\, z\\in \\mathbb{D}.$$\nIn the case where $G$ is the identity matrix (or equivalently the random variables $(\\xi_k)$ are independent and identically distributed), Peres and Vir\\'ag \\cite{Peres_Virag} proved the following lemma:\n\\begin{lemma}[Peres and Vir\\'ag] \\label{lemmaPV}\nAssume that $(\\xi_k)$ is the sequence of standard i.i.d complex Gaussian random variables. Then for any $w$ fixed in $\\mathbb{D}$, under the condition $f(w) = 0$, the random function $f(z)$ has the same distribution with \n $$T_{w}(z) f(z) = \\left(\\frac{z-w}{1- z \\overline{w}}\\right) f(z),\\,\\,z \\in \\mathbb{D},$$ that is, \n$$(f(z)| f(w) = 0) \\stackrel{d}{=} T_{w}(z) f(z)$$ where\n $\\stackrel{d}{=}$ denotes equality in distribution. \n In general for $w_1, w_2, \\ldots, w_n$ fixed in $\\mathbb{D}$, \n \\begin{eqnarray} \n(f(z)| f(w_1)=0, f(w_2)=0, \\ldots, f(w_n) = 0) \\stackrel{d}{=} T_{w_1}(z) T_{w_2}(z) \\ldots T_{w_n}(z) f(z).\n\\end{eqnarray} \n\\end{lemma}\nA closer look at Peres and Vir\\'ag's proofs reveals that their main result (that is Theorem \\ref{mainth} in the case where $G$ is the identity matrix) is a consequence of Lemma \\ref{lemmaPV}. This implies that if we prove that Lemma \\ref{lemmaPV} holds true in the general case of a Toeplitz matrix $G$, then the same argument as in Peres and Vir\\'ag \\cite{Peres_Virag} will complete the proof of Theorem \\ref{mainth}.\nIn other words, in order to prove our main result, it is sufficient to prove that the following lemma holds. \n\n\\begin{lemma} \\label{lemmaSM}\nLet $G$ be an invertible infinite hermitian Toeplitz matrix such that its associated measure $\\mu$ is absolutely continuous with density $\\varphi >0$ almost everywhere on $\\mathbb{T}$. Assume that $(\\xi_k)_{k\\in \\mathbb{N}}$ is a centred Gaussian process with covariance matrix $G^{-1}$ and zero pseudo-covariance. Let\n $$f(z) = \\sum_{k=1}^\\infty \\xi_k z^{k-1},\\,\\,\\, z\\in \\mathbb{D}.$$\nThen for any $w$ fixed in $\\mathbb{D}$, \n$$(f(z)| f(w) = 0) \\stackrel{d}{=} T_{w}(z) f(z).$$ Moreover, \n for $w_1, w_2, \\ldots, w_n$ fixed in $\\mathbb{D}$, \n \\begin{eqnarray} \n(f(z)| f(w_1)=0, f(w_2)=0, \\ldots, f(w_n) = 0) \\stackrel{d}{=} T_{w_1}(z) T_{w_2}(z) \\ldots T_{w_n}(z) f(z). \n\\end{eqnarray} \n\\end{lemma}\nIt is now an easy matter to prove that Lemma \\ref{lemmaSM} yields Theorem \\ref{mainth} based on Peres and Vir\\'ag arguments. \n\\subsection{Proof of Theorem \\ref{mainth}.}\nFor all fixed $z_1, z_2, \\ldots, z_n$, $w_1, w_2, \\ldots, w_n$ in $\\mathbb{D}$,\n the conditional joint distribution of $$(f(z_1), f(z_2), \\ldots, f(z_n)|f(w_1) = f(w_2) = \\ldots = f(w_n)=0)$$ is equal to the non-conditional joint distribution of $$\\left(T_{w_1}(z_1) f(z_1), T_{w_2}(z_2) f(z_2), \\ldots, T_{w_n}(z_n) f(z_n)\\right).$$ Taking the derivatives, it follows as in \\cite[corollary 13]{Peres_Virag} \nthat the conditional joint distribution of \n $$(f{'}(z_1), f{'}(z_2), \\ldots, f'(z_n)|f(z_1) = f(z_2) = \\ldots = f(z_n) = 0)$$is the same as the unconditional joint distribution of \n $$(\\Upsilon'(z_1) f(z_1), \\Upsilon'(z_2) f(z_2), \\ldots, \\Upsilon'(z_n) f(z_n))$$\nwhere \n $$\\Upsilon(z) = T_{z_1}(z) T_{z_2}(z) \\ldots T_{z_n}(z).$$ \nThis follows from the fact that\n $$T_{z}'(z) = \\frac{1}{1- |z|^2}\\,\\,\\mbox{ and } \\,\\,T_z(z) = 0,\\,\\,\\,\\,z\\in \\mathbb{D}. $$\nAt this stage, we make use of relation (\\ref{eqnads23001}) to obtain\n \\begin{eqnarray*} \n p_0(z_1, z_2, \\ldots, z_n) & = & \\frac{\\mathbb{E}\\left(|f'(z_1) f'(z_2) \\ldots f'(z_n)|^2\\,|f(z_1) = f(z_2) = \\ldots=f(z_n) = 0\\right)}{\\pi^n \\det(A)} \\\\\n & = & \\frac{\\mathbb{E}\\left(|\\Upsilon'(z_1) f(z_1) \\Upsilon'(z_2) f(z_2) \\ldots \\Upsilon'(z_n) f(z_n)|^2\\right)}{\\pi^n \\det(A)} \\\\\n & = & \\frac{\\mathbb{E}\\left(|f(z_1) f(z_2) \\ldots f(z_n)|^2\\right) \\prod_{k=1}^n |\\Upsilon'(z_k)|^2 }{\\pi^n \\det(A)} \\\\\n \\end{eqnarray*}\nUsing the classical Cauchy determinant formula, Peres and Vir\\'ag \\cite{Peres_Virag} showed that \n $$ \\prod_{k=1}^n |\\Upsilon'(z_k)| = \\det(A_0)$$ \nwhere \n $$A_0 = \\left(\\frac{1}{1- z_k \\overline{z_j}}\\right)_{k,j=1}^n.$$\nMoreover since it is well-known that if $X_1, X_2, \\ldots, X_n$ are random variables with joint Gaussian distribution with mean 0 and covariance matrix $\\Sigma$, \n $$\\mathbb{E}\\left(|X_1 X_2 \\ldots X_n|^2\\right) = \\mbox{perm}(\\Sigma),$$ it follows that, \n \\begin{eqnarray} \\label{dsdefesdws}\n p_0(z_1, z_2, \\ldots, z_n) = \\frac{\\mbox{perm}(A)(\\det(A_0))^2 }{\\pi^n \\det(A)}.\n \\end{eqnarray}\nNow elementary operations on the matrix $A$ yields\n \\begin{eqnarray*}\n\\mbox{perm}(A) & = & \\mbox{perm}(A_0) \\prod_{k=1}^n \\left(\\frac{1}{|1-z_k|^2}\\right)\\\\\n\\det(A) & = & \\det(A_0) \\prod_{k=1}^n \\left(\\frac{1}{|1-z_k|^2}\\right).\n\\end{eqnarray*}\nHence (\\ref{dsdefesdws}) yields\n \\begin{eqnarray*} \n p_0(z_1, z_2, \\ldots, z_n) = \\frac{\\mbox{perm}(A_0)\\det(A_0) }{\\pi^n}\n \\end{eqnarray*}\nand it is proven in Peres and Vir\\'ag \\cite[rel. (27)]{Peres_Virag} that \n $$\\mbox{perm}(A_0)\\det(A_0) = \\det\\left(\\frac{1}{(1 - z_k \\overline{z_j})^2}\\right)_{k,j=1}^n.$$\nThis concludes the proof. \\hfill \\qed\n\n\\subsection{Proof of Lemma \\ref{lemmaSM}.} Peres and Virag's proof is based on the invariance property of the Szeg\\\"o kernel with respect to Mobius transformations that are conformal mappings. This property does not hold for the general kernel $\\mathbb{K}_G$. Our proof is more general. \n (a) In the case where $w = 0$, it is an immediate consequence of Corollary \\ref{corol1}. Indeed, \n\\begin{eqnarray*}\n(f(z)| f(w) = 0) & = & (f(z)|f(0) = 0) = (f(z)|\\xi_1=0) = \\left(\\sum_{k=2}^\\infty \\xi_k z^{k-1}|\\xi_1 = 0\\right)\\\\\n& = & z \\left(\\sum_{k=2}^\\infty \\xi_k z^{k-2}|\\xi_1 = 0\\right) = z \\left(\\sum_{k=1}^\\infty \\xi_{k+1} z^{k-1}|\\xi_1 = 0\\right)\\\\\n&\\stackrel{d}{=} & z \\sum_{k=1}^\\infty \\xi_{k} z^{k-1} \n\\end{eqnarray*}\nwhere the equality in distribution follows from Corollary \\ref{corol1}.\\\\\n(b) For a general $w \\in \\mathbb{D}$, set \n $$F(z) = (f(z)|f(w) = 0), \\,z \\in \\mathbb{D}.$$ \nClearly, the covariance kernel of the random function $F(z)$ is given by\n \\begin{eqnarray*}\n\\mathbb{E}(F(z) \\overline{F(y)}) & = & \\mathbb{E}\\left(f(z) \\overline{f(y)}\\right) - \\frac{\\mathbb{E}\\left(f(z) \\overline{f(w)}\\right) \\mathbb{E}\\left(f(w) \\overline{f(y)}\\right)}{\\mathbb{E}\\left(f(w) \\overline{f(w)}\\right)}\\\\\n& = & \\mathbb{K}_G(z, y)-\\frac{\\mathbb{K}_G(z, w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)}\n\\end{eqnarray*}\nwhere $z, y \\in \\mathbb{D}$. This is clearly a kernel function and we shall denote it by $\\mathscr{K}_1$, that is:\n $$\\mathscr{K}_1(z, y) = \\mathbb{K}_G(z, y)-\\frac{\\mathbb{K}_G(z, w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)},\\,\\,z, y\\in \\mathbb{D}.$$\nWe also denote by $\\mathscr{K}_2$ the covariance kernel of the random function $T_w(z) f(z)$, that is,\n\\begin{eqnarray*}\n\\mathscr{K}_2(z, y) & = & \\mathbb{E}\\left(T_w(z) f(z) \\overline{T_w(y) f(y)}\\right)\\\\\n & = & T_w(z) \\mathbb{K}_G(z, y) \\overline{T_w(y)}.\n\\end{eqnarray*}\nTo show that the (Gaussian) random functions $F(z)$ and $T_w(z) f(z)$ have the same distribution, it is sufficient to show that their covariance kernels $\\mathscr{K}_1$ and $\\mathscr{K}_2$ are identical. So we shall prove the following important identity: For all $w, z, y \\in \\mathbb{D}$,\n $$\\mathbb{K}_G(z, y)-\\frac{\\mathbb{K}_G(z, w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)} = T_w(z) \\mathbb{K}_G(z, y) \\overline{T_w(y)}.$$ \nWe shall prove that $\\mathscr{K}_1$ and $\\mathscr{K}_2$ are both reproducing kernels of a certain Hilbert space, namely the subspace $\\mathscr{H}$ of $H^2_G(\\mathbb{D})$ of the functions that vanish at the point $w$. That is, \n $$\\mathscr{H} = \\{g \\in H^2_G(\\mathbb{D}): g(w) = 0\\}.$$ \n(To see that $\\mathscr{H}$ is closed in $H^2_G(\\mathbb{D})$, assume that $g_k = \\sum_{n=0}^\\infty a_{n,k} z^n$ converges to $g = \\sum_{n=0}^\\infty b_n z^n$. Then as discussed earlier, this convergence also holds with respect to the $H^2(\\mathbb{D})$-norm and hence $\\lim_{k \\to \\infty} \\sum_{n=0}^\\infty |a_{n,k} - b_n|^2 = 0$. Then \nfor each $z \\in \\mathbb{D}$ (by the Cauchy-Schwarz inequality)\n $$\\lim_{k \\to \\infty} |g_{k}(z) - g(z)|^2 = \\lim_{k \\to \\infty} |\\sum_{n=0}^\\infty (a_{n,k} - b_n) z^k|^2 \\leq \\lim_{k \\to \\infty} \\sum_{n=0}^\\infty |a_{n,k} - b_n|^2 \\sum_{n=0}^\\infty |z|^{2n} = 0.$$ In particular since $g_{k}(w) = 0$ for all $n$, then $g(w) = 0$ and hence $g \\in \\mathscr{H}$.)\n \nTo prove the claim that $\\mathscr{K}_1$ and $\\mathscr{K}_2$ are reproducing kernels $\\mathscr{H}$, we shall prove that for any function $g \\in \\mathscr{H}$ and any $y \\in \\mathbb{D}$,\n\\begin{eqnarray} \\label{sdsde34rfds}\ng(y) = \\langle g, \\mathscr{K}_1(.,y)\\rangle = \\langle g, \\mathscr{K}_2(.,y)\\rangle\n\\end{eqnarray}\n where\n$\\mathscr{K}_1(.,y)$ is the function defined by\n $$\\mathscr{K}_1(.,y): \\mathbb{D} \\to \\mathbb{C},\\, z \\mapsto \\mathscr{K}_1(z,y)$$ and similarly for $\\mathscr{K}_2.$\nClearly,\n \\begin{eqnarray*}\n\\langle g, \\mathscr{K}_1(.,y)\\rangle & = & \\left\\langle g, \\mathbb{K}_G(., y)-\\frac{\\mathbb{K}_G(., w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)}\\right \\rangle \\\\\n& = & \\left\\langle g, \\mathbb{K}_G(., y)\\right\\rangle - \\left(\\frac{\\overline{\\mathbb{K}_G(w, y)}}{\\overline{\\mathbb{K}_G(w, w)}}\\right) \\left\\langle g, \\mathbb{K}_G(., w)\\right \\rangle\\\\\n& = & g(y) - \\left(\\frac{\\overline{\\mathbb{K}_G(w, y)}}{\\mathbb{K}_G(w, w)}\\right)\\, g(w)\\\\\n& = & g(y) \n \\end{eqnarray*} \nsince $g(w) = 0$.\\\\\nFor the kernel $\\mathscr{K}_2$, we shall make use of the explicit inner product in $H^2_G(\\mathbb{D})$ in relation (\\ref{ew23edfr1eswe}), and show that\n\\begin{eqnarray}\\label{sdsdesafa1}\ng(y) = \\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\overline{\\mathscr{K}_2(e^{2 \\pi i \\theta},y)} d\\mu(\\theta).\n\\end{eqnarray}\nClearly, by definition of the kernel $\\mathscr{K}_2$, \n\\begin{eqnarray} \\label{ewdsdaswe45}\n\\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\overline{\\mathscr{K}_2(e^{2 \\pi i \\theta},y)} d\\mu(\\theta)\n= T_w(y) \\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\overline{T_w(e^{2 \\pi i \\theta})} \\, \\overline{\\mathbb{K}_G(e^{2 \\pi i \\theta},y)} d\\mu(\\theta).\n\\end{eqnarray}\nNow note that \n $$\\overline{T_w(e^{2 \\pi i \\theta})} = \\frac{e^{-2 \\pi i \\theta} -\\overline{w}}{1 - w e^{-2 \\pi i \\theta}} = \\frac{1 -\\overline{w} e^{2 \\pi i \\theta} }{e^{2 \\pi i \\theta} - w}.$$\n Therefore\n \\begin{eqnarray*}\n \\langle g, \\mathscr{K}_2(.,y)\\rangle & = & T_w(y) \\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\left(\\frac{1 -\\overline{w}\\ e^{2 \\pi i \\theta} }{e^{2 \\pi i \\theta} - w}\\right) \\, \\overline{\\mathbb{K}_G(e^{2 \\pi i \\theta},y)} d\\mu(\\theta)\\\\\n & = & T_w(y) \\left \\langle g\\ p, \\mathbb{K}_G(., y)\\right\\rangle\n \\end{eqnarray*} \nwhere $p$ is the function defined by \n $$p(z) = \\frac{1 -\\overline{w} z }{z - w} = \\frac{1}{T_w(z)},\\,\\,\\,z \\in \\mathbb{D}.$$ \n \nSince $g \\in H^2_G(\\mathbb{D})$, then the product $g p$ is also in $H^2_G(\\mathbb{D})$. Indeed,\n $$\\int_{\\mathbb{T}} |g(e^{2\\pi i \\theta})|^2 |p(e^{2\\pi i \\theta})|^2 d\\mu(\\theta) \\leq C \\int_{\\mathbb{T}} |g(e^{2\\pi i \\theta})|^2 d\\mu(\\theta) < \\infty$$ where\n $$C = \\sup_{\\theta \\in \\mathbb{T}} \\left|p(e^{2 \\pi i \\theta})\\right|^2 = \\sup_{\\theta \\in \\mathbb{T}} \\left|\\frac{1 -\\overline{w} e^{2\\pi i \\theta }}{e^{2\\pi i \\theta} - w}\\right|^2 < \\infty$$\n(because $w \\in \\mathbb{D}$ and hence $e^{2\\pi i \\theta} - w \\ne 0$). \nIt follows that \n $$\\left \\langle g p, \\mathbb{K}_G(., y)\\right\\rangle = g(y) p(y) = g(y) \\left(T_w(y)\\right)^{-1}.$$\n Hence\n $$ \\langle g, \\mathscr{K}_2(.,y)\\rangle = T_w(y) g(y) \\left(T_w(y)\\right)^{-1} = g(y).$$ \nThis yields (\\ref{sdsde34rfds}) and concludes the proof of \n $$F(z) = (f(z)| f(w) = 0) \\stackrel{d}{=} T_w(z) f(z).$$ \nNow the general case that\n$$(f(z)| f(w_1)=0, f(w_2)=0, \\ldots, f(w_n) = 0) \\stackrel{d}{=} T_{w_1}(z) T_{w_2}(z) \\ldots T_{w_n}(z) f(z)$$\n follows immediately by an induction argument. This concludes the proof of Lemma \\ref{lemmaSM}. \\hfill \\qed \n \n\n\n\\section{Illustrating examples}\n\n\n\n\\subsection{Explicit inverse of tridiagonal Toeplitz matrices}\\label{exampl1}\nGiven a real number $q$ such that $|q|< 1\/2$, consider the Toeplitz matrix\n $$G = \\left(\\gamma(k-j)\\right)_{k,j=1}^\\infty$$ where \n \\begin{eqnarray*}\n \\gamma(k) = \\left \\{ \\begin{array}{cc}\n 1 & \\mbox{ if } k=0 \\\\\n q & \\mbox{ if } |k| = 1\\\\\n 0 & \\mbox{ otherwise. }\n \\end{array}\n \\right. \n \\end{eqnarray*}\nThe spectral density function $\\varphi$ of $G$ (i.e. the density of the corresponding measure $\\mu$) is given by\n$$\\varphi(\\theta) = 1 + q e^{2 \\pi i \\theta} + q e^{-2 \\pi i \\theta} = 1 + 2 q \\cos(2\\pi \\theta),\\,\\, \\theta \\in \\mathbb{T}.$$\nLet $G_n$ be the submatrix of $G$ formed by its first $n$ rows and first $n$ columns. Then the inverse $G_n^{-1}$ of $G_n$ is the symmetric matrix given by (see da Fonseca and Petronilho \\cite{da Fonseca}): \n $$\\left(G_n^{-1}\\right)_{k, j} = (-1)^{k+j} \\frac{q^{j-k}}{|q|^{j-k+1}} \\frac{U_{k-1}(\\alpha) U_{n-j}(\\alpha)}{U_n(\\alpha)},\\,\\,\\,1\\leq k \\leq j\\leq n$$\nwhere $$\\alpha = \\frac{1}{2 |q|}$$ and $(U_k)$ is the sequence of Chebyshev polynomials of second kind given by:\n \\begin{eqnarray*}\n U_0 & = & 1\\\\\n U_1(x) & = & 2 x\\\\\n U_{k+1} (x) & = & 2 x U_k(x) - U_{k-1}(x),\\,\\,k = 1,2,\\ldots\n \\end{eqnarray*}\nExplicitly, for $|x| > 1$, \n $$U_k(x) = \\frac{\\left(x + \\sqrt{x^2-1}\\right)^{k+1} - \\left(x - \\sqrt{x^2-1}\\right)^{k+1}}{2 \\sqrt{x^2-1}}.$$\nTaking the limit of $G_n^{-1}$ as $n \\to \\infty$, it can be easily checked that the infinite matrix $G$ is indeed invertible and its inverse is the infinite symmetric matrix given for $k \\leq j$ by\n \\begin{eqnarray*}\n\\left(G^{-1}\\right)_{k,j} & = & \\lim_{n\\to \\infty} \\left(G_n^{-1}\\right)_{k,j}\\\\\n& = & \\frac{(-2 q)^{j - k} \\left(1 + \\sqrt{1 - 4 |q|^2}\\right)^{-j} \\left(\\left(1 + \\sqrt{1 - 4 |q|^2}\\right)^k - \\left(1 - \\sqrt{1 - 4 |q|^2}\\right)^k\\right)}{\\sqrt{1 - 4|q|^2}}. \n \\end{eqnarray*}\nFor example, if $q = -\\frac{1}{3}$, the inverse of the infinite matrix $G$ is given by \n $$\\left(G^{-1}\\right)_{k, j} = \\left(\\frac{3}{\\sqrt{5}}\\right) \\left(\\frac{3 - \\sqrt{5}}{2}\\right)^j \\left(\\left(\\frac{3 + \\sqrt{5}}{2}\\right)^\n k-\\left(\\frac{3 - \\sqrt{5}}{2}\\right)^{k}\\right),\\,\\,\\mbox{ for } j \\geq k.$$\nNow some involved (but elementary) calculations yield that the kernel function $\\mathbb{K}_G$ defined by the matrix $G$ is given by\n \\begin{eqnarray}\\label{sdwwe34r}\n\\mathbb{K}_G(z, w) = Z^T G^{-1} \\overline{W} = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}}\n\\end{eqnarray}\n where $\\psi$ is the function defined in the unit disc $\\mathbb{D}$ \nby\n $$\\psi(z) =\\left(\\frac{2}{|q|}\\right)^{1\/2} \\left(\\frac{1}{a + b z}\\right)$$ with\n \\begin{eqnarray*}\n a & = & \\sqrt{|q|^{-1} + \\sqrt{q^{-2} - 4}} \\\\\n b & = & (2\/a)\\sign(q).\n \\end{eqnarray*}\nFor example for $q = -1\/3$, \n$$\\psi(z) = \\frac{5^{1\/4}(3 + \\sqrt{5}) \\left(\\frac{3}{2(5 + 3\\sqrt{5})}\\right)^{1\/2}}{z - \\frac{3+ \\sqrt{5}}{2}}.$$\nNote that relation (\\ref{sdwwe34r}) means that\n $$\\mathbb{K}_G(z, w) = \\psi(z) \\mathbb{K}(z, w) \\overline{\\psi(w)}$$ where $\\mathbb{K}$ is the classical Szeg\\\"o kernel (which is the covariance kernel associated to $\\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ for i.i.d $(\\zeta_n)$). \nThen for a sequence $(\\xi_n)$ of Gaussian random variables with covariance matrix $G^{-1}$, this implies that the random functions $\\sum_{n=1}^\\infty \\xi_n z^{n-1}$ and $\\psi(z) \\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ have the same distribution, that is,\n $$\\sum_{n=1}^\\infty \\xi_n z^{n-1} \\stackrel{d}{=} \\psi(z) \\sum_{n=1}^\\infty \\zeta_k z^{k-1}$$\n (for i.i.d $(\\zeta_n)$). Since clearly $\\psi(z) \\ne 0$ everywhere in $\\mathbb{D}$, it follows that the zeros of $\\sum_{n=1}^\\infty \\xi_n z^{n-1}$ have the same distribution as the zeros of $\\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ and therefore they constitute a determinantal point process as predicted by Theorem 1. \n\nIn particular the intensity of the zeros of $f(z)$ is also given \n$$p(z) = \\frac{1}{\\pi(1 - |z|^2)^2},\\,\\,z \\in \\mathbb{D}$$ as it is the case for i.i.d variables.\nTo emphasize that we need to consider the $G^{-1}$ as the covariance matrix instead of $G$, consider a sequence of Gaussian variables $(\\tau_n)$ with covariance matrix $G$ and the function \n$g(z) = \\sum_{n=1}^\\infty \\tau_n z^n$. Using relation (\\ref{eqnads231}), it is easy to derive that the intensity of the corresponding zero set is\n $$p(z) = \\frac{1}{\\pi(1 - |z|^2)^2} \\left(1 - \\frac{q^2(1 - |z|^2)^2}{(1 + q z + q \\overline{z})^2}\\right),\\,\\,\\,\\, z \\in \\mathbb{D}.$$\nHence clearly the zeros of $g(z)$ do not have the same distribution as the zeros of $f(z) = \\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ for i.i.d $(\\zeta_n)$ since the corresponding intensity is $p(z) = \\pi^{-1}(1 - |z|^2)^{-2}$. \n\nFinally since the function $\\varphi(\\theta) = 1 + 2 q \\cos(2 \\pi \\theta)$ is bounded on $\\mathbb{T}$, then the set \n$H^2_G(\\mathbb{D})$ is equal to $H^2(\\mathbb{D})$ but with a different norm: \n $$\\|g\\|^2_{H^2_G(\\mathbb{D})} = \\int_{\\mathbb{T}} |g(\\theta)|^2 \\varphi(\\theta) d\\theta = \\int_{\\mathbb{T}} |g(\\theta)|^2 (1 + 2 q \\cos(2 \\pi \\theta)) d\\theta.$$\nTherefore with this norm, $H^2(\\mathbb{D})$ is the reproducing kernel Hilbert space given by the kernel\n $$K_G(z, w) = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}} $$ where the function $\\psi(z)$ is given above. \n\n\n\n\\subsection{Inverse of the Kac-Murdock-Szeg\\\"o matrix}\nThe same property is also observed for the classical Kac-Murdock-Szeg\\\"o matrix. It is the Toeplitz matrix $G$ defined for a complex number $q$ by \n $$ (G)_{k,j} = q^{|k-j|}.$$ \nFor $q$ real with $|q| < 1$, the spectral density function $\\varphi$ of $G$ is given by\n $$\\varphi(\\theta) = \\sum_{n=-\\infty}^\\infty q^{|n|} e^{2\\pi i n \\theta} = \\frac{1 - q^2}{1 - 2q \\cos(2 \\pi \\theta) + q^2}. $$\n(Here $\\varphi(\\theta)$ can be seen as the classical Poisson kernel in the unit disc.)\n Under these conditions ($q$ real and $|q|< 1$), $G$ is symmetrical and invertible and its inverse is well-known and given by:\n \\begin{eqnarray*}\n\\left(G^{-1}\\right)_{k,j} = \\left\\{ \\begin{array} {ccc}\n (1-q^2)^{-1} & \\mbox{ if } k=j=1\\\\\n (1+q^2)(1-q^2)^{-1} & \\mbox{ if } k=j\\geq 2\\\\\n -q (1-q^2)^{-1} & \\mbox{ if } |k-j| = 1\\\\\n 0 & \\mbox{ otherwise.} \n \\end{array}\n \\right.\n \\end{eqnarray*}\n(More details on the spectral properties of Kac-Murdock-Szeg\\\"o matrix for a complex parameter are given in Fikioris \\cite{Fikioris}.)\n Then clearly the corresponding kernel is \n$$\\mathbb{K}_G(z, w) = \\frac{(1- q z)(1 - q \\overline{w})}{(1-q^2)(1 - z \\overline{w})} = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}} $$ \n where $$\\psi(z) = \\frac{1- q z}{\\sqrt{1-q^2}}.$$ \nIn general for $q \\in \\mathbb{C}$ with $|q| < 1$, let\n \\begin{eqnarray*}\nG_{k,j} = \\left\\{ \\begin{array} {ccc}\n q^{|k-j|} \\mbox{ if } k \\geq j \\\\\n (\\overline{q})^{|k-j|} \\mbox{ otherwise. }\n \\end{array}\n \\right.\n \\end{eqnarray*} \nThen $G$ is hermitian and invertible and its inverse is given by\n \\begin{eqnarray*}\n\\left(G^{-1}\\right)_{k,j} = \\left\\{ \\begin{array} {ccc}\n (1-|q|^2)^{-1} & \\mbox{ if } k=j=1\\\\\n (1+|q|^2)(1-|q|^2)^{-1} & \\mbox{ if } k=j\\geq 2\\\\\n -q (1-|q|^2)^{-1} & \\mbox{ if } k-j = 1\\\\\n -\\overline{q}(1-|q|^2)^{-1} & \\mbox{ if } k - j = -1\\\\\n 0 & \\mbox{ otherwise.} \n \\end{array}\n \\right.\n \\end{eqnarray*}\nThis yields \n $$\\mathbb{K}_G(z, w) = Z G^{-1} \\overline{W} = \\frac{(1- q z)(1 - \\overline{q} \\overline{w})}{(1-|q|^2)(1 - z \\overline{w})} = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}} $$ \n where $$\\psi(z) = \\frac{1- q z}{\\sqrt{1-|q|^2}}.$$ \n\\subsection{Inverse fractional Gaussian noise} \nGiven $0 < h < 1$, the classical complex fractional Gaussian noise of Hurst index $h$ is a sequence $\\{\\Delta_n\\}_{n=1}^\\infty$ of centred Gaussian random variables such that $\\mathbb{E}(\\Delta_n \\Delta_m) = 0$ and with covariance structure\n \\begin{eqnarray} \\label{ew32wswaws2121}\n\\gamma(k) := \\mathbb{E}(\\Delta_n \\overline{\\Delta_{n+k}}) = {\\scriptstyle\\frac{1}{2}}|k+1|^{2h} + {\\scriptstyle\\frac{1}{2}}|k-1|^{2h} -|k|^{2h},\\,\\,\\,\\, k, n\\in \\mathbb{N}.\n\\end{eqnarray}\nThe covariance matrix of $\\{\\Delta_n\\}_{n=0}^\\infty$ is the Toeplitz matrix $G$ given by\n $$G = \\left(\\gamma(k-j)\\right)_{k,j=1}^\\infty.$$ \n(The particular case $h = 1\/2$ corresponds i.i.d random variables.)\nIt is well-known that the matrix $G$ is invertible (see for example \\cite{Dambrogi-ola}.) Unfortunately an explicit inverse of $G$ is not known. Consider its inverse matrix $G^{-1}$. A sequence of Gaussian random variables with covariance matrix $G^{-1}$ shall be called the inverse fractional Gaussian noise of index $h$. \nSome properties of the zeros of the random polynomial $\\sum_{k=0}^n \\Delta_k x^k$ and the power series $\\sum_{k=0}^{\\infty} \\Delta_n x^n$ where $(\\Delta_n)$ is the fractional Gaussian noise are given in \\cite{Mukeru_2018} and \\cite{Mukeru_al}. Here we are interested in the function $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ where $(\\xi_n)$ is the inverse fractional Gaussian noise. \nIt is well-known (using an argument by Sinai \\cite[Theorem 2.1]{Sinai}) that the matrix $G$ admits a spectral density function $\\varphi_h$ given by\n \\begin{eqnarray} \\label{dsdfesw23}\n\\varphi_h(\\theta) = C(h)|e^{2 \\pi i \\theta}-1|^2\\left(\\sum_{n=-\\infty}^\\infty \\frac{1}{|\\theta +n|^{2h+1}}\\right),\\, \\theta \\in \\mathbb{T}, \\theta \\ne 0\n\\end{eqnarray}\n where $C(h)$ is a normalising constant given by\n $$C(h) = -\\frac{\\zeta(-2h)}{2 \\zeta(1+2h)}$$ where $\\zeta(.)$ is the Riemann zeta function. \nClearly, \n\\begin{eqnarray} \\label{dsdfesw2312s}\n\\varphi_h(\\theta) &=& 4 C(h) \\left(\\sin^2 \\pi \\theta\\right) \\sum_{n=0}^\\infty \\left(\\frac{1}{(n+\\theta)^{2h+1}} + \\frac{1}{(n+ 1-\\theta)^{2h+1}}\\right) \\nonumber\\\\\n & = &4 C(h) \\left(\\sin^2 \\pi \\theta\\right) (\\zeta(2h+1, \\theta) + \\zeta(2h+1, 1-\\theta))\n\\end{eqnarray}\n where $\\zeta(.,.)$ is the classical Hurwitz zeta function. \n\nIt is not difficult to see that the function $\\varphi_h$ is continuous on $(0, 1)$ and satisfies \n \\begin{eqnarray*} \\label{sddsdwdwwsas}\n\\varphi_h(t) = O(t^{1-2h} (1-t)^{1-2h}),\\,\\, \\mbox{ for } t \\mbox{ near } 0 \\mbox{ or } 1.\n\\end{eqnarray*}\nThis implies that both functions $\\varphi(t)$ and $1\/\\varphi(t)$ are integrable on the unit circle. The inverse matrix $G^{-1}$ is therefore such that \n $$\\left(G^{-1}\\right)_{k,j} = \\int_{\\mathbb{T}} \\frac{e^{-2\\pi i (k-j) t}}{\\varphi_h(t)} dt,\\,\\,\\mbox{ for } j+ k \\to \\infty.$$\n(See D'Ambrogi-Ola \\cite{Dambrogi-ola}.)\nThis yields that $G^{-1}$ is asymptotically a Toeplitz matrix in the sense that for each $k,j$ fixed,\n $$\\lim_{n\\to \\infty} \\left(G^{-1}\\right)_{k+n,j+n} = \\int_{\\mathbb{T}} \\frac{e^{-2\\pi i (k-j) t}}{\\varphi_h(t)} dt = \\widehat{\\left(1\/\\varphi_h\\right)} (k-j).$$\nThis implies in particular that $\\left(G^{-1}\\right)_{k,j} \\to 0$ for $k+j \\to \\infty$ and hence \n$\\sup_{k,j}|(G^{-1})_{k,j}| < \\infty$ which guarantees that \nfor each $z, w \\in \\mathbb{D}$ the series $Z^{T} G^{-1} \\overline{W}$ converges (for $Z = (1, z, z^2, \\ldots)$ and $W = (1, w, w^2, \\ldots))$. \nThe corresponding space $H^2_G(\\mathbb{D})$ is the class of functions $ g \\in H^2(\\mathbb{D})$ such that\n $$\\int_{\\mathbb{T}} \\left|g(e^{2 \\pi i \\theta})\\right|^2 \\varphi_h(\\theta) d\\theta < \\infty.$$ \nFor $h$ varying in $(0,1)$, this yields a family of sub-spaces of the Hardy space $H^2(\\mathbb{D})$. \n The exact entries of the matrix $G^{-1}$ are not known and therefore we do not have an explicit representation of the kernel $\\mathbb{K}_G(z, w)$ as in the first two examples. However Theorem \\ref{mainth} yields that if $(\\xi_n)_{n\\in \\mathbb{N}}$ is a zero-mean complex Gaussian sequence of covariance matrix $G^{-1}$ and zero pseudo-covariance, then the zeros of $f(z) = \\sum_{n =1}^\\infty \\xi_n z^{n-1}$ constitute a determinantal point process. \n \nAs in the general case one can compute from the sequence of polynomials $1, z, z^2, \\ldots$ a sequence of orthonormal polynomials $\\{P_n(z): n=1,2,\\ldots\\}$ and deduce that if $(\\chi_n)$ is a sequence of i.i.d standard Gaussian random variables then the zeros of $f(z) = \\sum_{n=1}^\\infty \\chi_n P_n(z)$ constitute a determinantal point process. \n \nIn the limit case where $h = 0$, the fractional Gaussian noise with index $h$ is such that the covariance matrix $G$ is given by \n $$G_{k,k} = 1, G_{k,k+1} = G_{k+1, k} = -1\/2 \\mbox{ and } G_{k,j} = 0 \\mbox{ for } |k-j| \\geq 2,$$ and it is not invertible. However it still determines a determinantal point process. \nThe spectral density function of $G$ is \n $$\\varphi_0(\\theta) = 1 - \\cos(2\\pi \\theta),\\,\\, 0 \\in \\mathbb{T}.$$\n From the sequence of polynomials $(1, z, z^2, z^3, \\ldots)$, we derive the orthonormal sequence: \n \\begin{eqnarray*}\n P_n(z) = \\left(\\frac{2}{n(n+1)}\\right)^{1\/2}(1 + 2 z + 3 z^2 + \\ldots+ n z^{n-1}),\\,\\,\\,n = 1,2,\\ldots\n \\end{eqnarray*}\nIn this case, the kernel of $\\mathbb{H}^2_G(\\mathbb{D})$ is explicitly given by:\n\\begin{eqnarray*} \\label{Saf_Kernel}\n\\mathbb{K}_0(z, w) = \\sum_{n=1}^\\infty P_n(z) \\overline{P_n(w)} = \\frac{2}{(1-z)(1-\\overline{w})(1- z \\overline{w})},\\,\\,\\,z, w \\in \\mathbb{D}.\n \\end{eqnarray*}\n This is exactly the limit case of the kernel given in Example \\ref{exampl1} when the parameter $q$ approaches $ -1\/2$. \nThen for a sequence $(\\chi_n)$ of i.i.d standard Gaussian variables, the zeros of \n \\begin{eqnarray*}\n f(z) = \\sum_{n=1}^\\infty \\chi_n P_n(z),\\,\\,\\,\\, z \\in \\mathbb{D}\n \\end{eqnarray*}\nconstitute a determinantal point process. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\n\\noindent Spontaneous symmetry breaking is usually treated as a\nphenomenon that arises from properties of an asymmetric quantum mechanical vacuum\nstate. In particular, the non-invariance of the vacuum\nstate with respect to a symmetry is said to lead to spontaneous\nsymmetry breakdown. While this picture is clearly valid and useful, it\nis not generally appreciated that in relativistic theories of quantum mechanics, \nit is strictly a matter of convention which\narises from the (usually implicit) choice of quantization\nsurface~\\cite{Dirac:1949cp}. Indeed, the standard viewpoint ---the\ninstant form--- arises from choosing to view dynamics in Minkowski\nspace as the evolution of families of parallel spaces at various\ninstants of time. An alternate view of dynamics is to consider the\nevolution of families of parallel spaces tangent to the light cone;\ni.e.~null planes~\\cite{Dirac:1949cp,Weinberg:1966jm,Susskind:1967rg,Bardakci:1969dv,Kogut:1969xa,Leutwyler:1977vy}. In this\nviewpoint ---the front form--- the momentum operator has a spectrum\nconfined to the open positive half-line and therefore the vacuum of\nthe interacting theory may be regarded as the structureless Fock-space\nvacuum, which is an invariant with respect to all internal symmetries,\nand spontaneous symmetry breaking must be attributed to properties of\nthe dynamical Poincar\\'e generators. Therefore in the front form,\nspontaneous chiral symmetry breaking is a property of operators rather\nthan of a complicated vacuum state. Naturally one expects that\nphysics is independent of the choice of quantization surface. However,\nfor theories like QCD where the detailed dynamics are largely\nintractable, one may suppose that the two forms of dynamics lead to\ndistinct insights into the behavior of the theory at strong coupling.\nOur goal in this paper is to argue that this is indeed the case.\n\nThe fundamental point which we wish to emphasize in this paper is\nthat, in contrast to the instant form, where spontaneous symmetry\nbreaking lives entirely in the non-trivial vacuum, in the front form,\nsymmetry breaking is expressed entirely through the fact that the\nHamiltonians, or dynamical Poincar\\'e generators, do not commute with\nthe internal symmetry charges. The resulting commutation relations\namong space-time generators and internal symmetry generators in QCD\nimply powerful constraints on the spectrum and spin of the hadronic\nworld~\\cite{Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn,Weinberg:1994tu}.\nThere have been many studies of spontaneous chiral symmetry breaking\non null\nplanes~\\cite{Jersak:1969zg,Leutwyler:1969av,Weinberg:1969hw,Weinberg:1969db,Feinberg:1973qb,Eichten:1973ip,Casher:1973vh,Casher:1974xd,Carlitz:1974sg,Sazdjian:1974gk,Wilson:1994fk,Wilson:1994gn,Susskind:1994wr,Kim:1994rm,Burkardt:1995eb,Burkardt:1996pa,Burkardt:1998dd,Yamawaki:1998cy,Itakura:2001yt,Burkardt:2002yf,Wu:2003vn,Lenz:2004tw,Dalley:2004re,Brodsky:2008xm,Brodsky:2008xu,Brodsky:2009zd,Ji:2009jc,Strikman:2010pu,Chang:2011mu,Alberg:2012wr,Brodsky:2010xf,Schweitzer:2012hh,Brodsky:2012ku}.\nIn many cases the emphasis has been on learning detailed information\nabout the dynamical mechanism of chiral symmetry breaking in QCD and\nin models. Here our approach is much less ambitious; we assume that\nchiral symmetry is broken spontaneously by complicated and not-well\nunderstood dynamics, and we then determine the constraints that follow\nfrom this assumption. In particular, we are interested primarily in\nformulating the model-independent consequences of chiral symmetry\nbreaking on null-planes. A fundamental assumption we make is that\nphysics must be independent of the choice of quantization\nsurface. Nowhere in this study do we find anything resembling a\ncontradiction of this basic assumption. Indeed, this assumption of\nwhat one might call ``form invariance'' leads to various constraints\nwhich reveal a great deal about the nature and consistency of chiral\nsymmetry breaking on null-planes. On general grounds, the null-plane\nchiral symmetry charges annihilate the vacuum. Therefore, in order\nthat spontaneous chiral symmetry breaking take place, the chiral\nsymmetry axial-vector current on the null-plane cannot be\nconserved~\\cite{Jersak:1969zg,Leutwyler:1969av}. This property leads\nto a simple proof of Goldstone's theorem on a null-plane, which is\ncompletely decoupled from any assumptions about the formation of\nsymmetry-breaking condensates. A second consistency condition is that\nthe part of the QCD vacuum energy that is dependent on the quark\nmasses should be invariant with respect to the choice of\ncoordinates. This condition recovers the Gell-Mann-Oakes-Renner\nrelation~\\cite{GellMann:1968rz} in the null-plane description, and\nleads to a general prescription for relating all instant-form\nchiral-symmetry breaking condensates to the vacuum expectation values\nof chiral singlet null-plane QCD operators.\n\nIt is difficult to find a general solution of the null-plane operator\nalgebra~\\cite{Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn,Weinberg:1994tu}. However,\nthere is a non-trivial limit in which a solution can be found. One\nexpects that, in general, the chiral symmetry breaking part of the\nnull-plane energy has an energy scale comparable to $\\Lambda_{QCD}$\nand therefore is not parametrically small. However, assuming that this\nis small (which is the case parametrically for baryon operators at\nlarge-$N_c$), while the chiral symmetry breaking part of the spin\nHamiltonians is of natural size, allows a non-trivial solution of the\noperator algebra which closes to the Lie brackets of $SU(2N)$, thus\nrecovering the basic group theoretical structure of the constituent\nquark model. This result, originally found by\nWeinberg~\\cite{Weinberg:1994tu} working with current-algebra sum rules\nin special Lorentz frames, is shown in this context to be a general\nconsequence of the null-plane QCD Lie algebraic constraints which are\nvalid in any Lorentz frame.\n\nThe paper is organized as follows. In section~\\ref{sec:poincare}, the\nnull-plane coordinates and conventions are introduced, and the\nfront-form Poincar\\'e algebra is obtained. The null-plane Hamiltonians\nand the Lie brackets that they satisfy are identified, and the\nmomentum eigenstates are constructed. In\nsection~\\ref{sec:chiralityintro} the null-plane internal symmetry\ncharges are introduced, and the commutators that mix Poincar\\'e and\nchiral generators are obtained. Using these commutators, a general\nproof of Goldstone's theorem is given, and a polology analysis is\ngiven which elucidates the structure of the axial-vector current on\nthe null-plane. The special case of QCD with $N$ flavors of light\nquarks is considered in section~\\ref{sec:qcd}. The QCD Lagrangian is\nexpressed in the null-plane coordinates, and the chiral symmetry\nbreaking Hamiltonians and the constraints that they satisfy are\nderived. The issue of condensates in the null-plane formulation is\naddressed in detail; the Gell-Mann-Oakes-Renner formula is recovered\nin the front-form and a general method for relating instant-form\ncondensates to front-form condensates is presented.\nSection~\\ref{sec:consequences} explores the consequences of the QCD\nnull-plane operator algebra. In particular, a simple solution of the\noperator algebra is given which contains the spin-flavor symmetries of\nthe constituent quark model. In section~\\ref{sec:conc} we summarize\nour findings and conclude. \\vskip0.3in\n\\noindent {\\it Nota bene}: \nWe have made use of the many general\nreviews of null-plane (or light-front)\nquantization~\\cite{Hornbostel:1990ya,Perry:1994kp,Zhang:1994ti,Burkardt:1995ct,Harindranath:1996hq,Brodsky:1997de,Perry:1997uv,Miller:1997cr,Venugopalan:1998zd,Heinzl:2000ht,Miller:2000kv,Diehl:2003ny,Belitsky:2005qn}, as well as reviews that focus\nprimarily on chiral symmetry related\nissues~\\cite{Mustaki:1994mf,Yamawaki:1998cy,Itakura:2001yt}.\nIn order to provide a self-contained description of the subject\nof chiral-symmetry breaking on a null-plane, there is a significant \namount of review material in this paper. \n\n\n\n\\section{Space-time symmetry in the front form}\n\\label{sec:poincare}\n\n\\subsection{A null plane defined}\n\n\\noindent In the front-form of relativistic Hamiltonian dynamics, one\nchooses the initial state of the system to be on a light-like plane,\nor null-plane, which is a hypersurface of points $x$ in Minkowski\nspace such that $x\\cdot n =\\tau$ (see fig.~\\ref{fig:nullplane}). Here\n$n$ is a light-like vector which will be chosen below, and $\\tau$ is a\nconstant which plays the role of time. We will refer to a null-plane\nas $\\Sigma_n^\\tau$. The subgroup of the Poincar\\'e group that maps\n$\\Sigma_n^\\tau$ to itself is called the stability group of the\nnull-plane and determines the kinematics within the null-plane. The\nremaining three Poincar\\'e generators map $\\Sigma_n^\\tau$ to a new\nsurface, $\\Sigma_n^{\\tau'}$, and therefore describe the evolution of the\nsystem in time. The front-form is special in that it has seven\nkinematical generators, the largest stability group of all of the\nforms of dynamics~\\cite{Dirac:1949cp}. It stands to reason that in\ncomplicated problems in relativistic quantum mechanics one would\nprefer a formulation which has the fewest number of Hamiltonians to\ndetermine.\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[scale=0.54]{nullplane2.pdf}\n \\caption{A null plane is a surface tangent to the light cone. The\n null-plane Hamiltonians map the initial light-like surface onto\n some other surface and therefore describe the dynamical\n evolution of the system. The energy $P^-$ translates the system\n in the null-plane time coordinate $x^+$, whereas the spin\n Hamiltonians $F_r$ rotate the initial surface about the surface\n of the light cone.}\n \\label{fig:nullplane}\n\\end{figure}\n\\subsection{Choice of coordinates}\n\n\\noindent Consider the light-like vectors $n^\\mu$ and $n^{\\ast\\mu}$\nwhich satisfy $n^2 = n^{\\ast 2} = 0$ and $n \\cdot n^\\ast = 1$. Here we will\nchoose these vectors such that \n\\begin{equation}\nn^\\mu \\equiv \\ft{1}{\\sqrt{2}} (1, 0, 0, -1) \\quad , \\quad\nn^{\\ast \\mu} \\equiv \\ft{1}{\\sqrt{2}} (1, 0, 0, 1)\n\\, .\n\\label{nullvectors}\n\\end{equation}\nWe will take the initial surface to be the null-plane\n$\\Sigma_n^0$. A coordinate system adapted to null-planes is then given by\n\\begin{eqnarray}\nx^+ \\equiv x \\cdot n = \\ft{1}{\\sqrt{2}} (x^0 + x^3)\n\\, , \\qquad\nx^- \\equiv x \\cdot n^\\ast = \\ft{1}{\\sqrt{2}} (x^0 - x^3)\n\\label{eq:LCcoordinates}\n\\end{eqnarray} which we take as the time variable and ``longitudinal'' position, respectively~\\footnote{This\n is known as the Kogut-Soper convention~\\cite{Kogut:1969xa}. Our\n metric and other notational conventions can be found in\n Appendix~\\ref{npconventions} and in Ref.~\\cite{Brodsky:1997de}.}. The remaining coordinates, ${\\bf\n x}_\\perp=( {x}^1, {x}^2)$ provide the ``transverse''\nposition. Denoting the null-plane contravariant coordinate\nfour-vector by ${\\tilde x}^\\mu=(x^+,{x}^1,{x}^2,x^-)=(x^+,{\\bf x}_\\perp,x^-)$, then\none can write\n\\begin{eqnarray}\n{\\tilde x}^\\mu\\ = \\ {\\cal C}^\\mu_\\nu\\ x^\\nu \\ .\n\\label{eq:ETtoLC}\n\\end{eqnarray} The matrix ${\\cal C}^\\mu_\\nu$, given explicitly in\nAppendix~\\ref{npconventions}, allows one to transform all Lorentz\ntensors from instant-form to front-form coordinates. In particular,\nthe null-plane metric tensor is given by\n\\begin{eqnarray}\n{\\tilde g}_{\\mu\\nu}\\ =\\ ({\\cal C}^{-1})^\\alpha_\\mu\\ g_{\\alpha\\beta}\\ ({\\cal C}^{-1})^\\beta_\\nu \\ .\n\\label{eq:LCmetric1}\n\\end{eqnarray}\n\nThe energy, canonical to the null-plane time variable $x^+$ is\n$p^-=p_+$ , and the momentum canonical to the longitudinal position\nvariable $x^-$ is $p^+=p_-$. Therefore, the on-mass-shell condition\nfor a relativistic particle of mass $m$ yields the null-plane\ndispersion relation:\n\\begin{eqnarray}\np^-\\ =\\ \\frac{{\\bf p}^2_\\perp\\ +\\ m^2}{2p^+} \\ .\n\\label{eq:LCdisp}\n\\end{eqnarray} This dispersion relation reveals several interesting generic\nfeatures of the null-plane formulation. Firstly, the dispersion\nrelation resembles the non-relativistic dispersion relation of a\nparticle of mass $p^+$ in a constant potential. Secondly, we see that\nthe positivity and finiteness of the null-plane energy of a free\nmassive particle requires $p^+>0$. Only massless particles with\nstrictly vanishing momentum can have $p^+=0$. This implies that pair\nproduction is subtle, and the vacuum state is in some sense simple,\nwith the exception of contributions that are strictly from $p^+=0$\nmodes~\\cite{Hornbostel:1990ya,Perry:1994kp,Zhang:1994ti,Burkardt:1995ct,Harindranath:1996hq,Brodsky:1997de,Perry:1997uv,Venugopalan:1998zd,Heinzl:2000ht,\n Diehl:2003ny,Belitsky:2005qn}.\n\n\n\\subsection{The null-plane Poincar\\'e generators}\n\n\\noindent In this section we will review the\nLie brackets of the Lorentz generators in the front form~\\footnote{Here\n we follow closely the development of\n Refs.~\\cite{Bardakci:1969dv,Kogut:1969xa,Leutwyler:1977vy}. See also Ref.~\\cite{Ballesteros:1995mi}.}. The Poincar\\'e algebra\nin our convention is:\n\\begin{eqnarray}\n& & \\lbrack\\, P^\\mu \\, ,\\, P^\\nu\\, \\rbrack\\ =\\ 0 \\qquad , \\qquad \\lbrack\\, { M}_{\\mu\\nu} \\, ,\\, P_\\rho\\, \\rbrack\\ \n=\\ i\\, (\\, g_{\\nu\\rho} P_\\mu\\, -\\, g_{\\mu\\rho} P_\\nu \\, ) \n\\nonumber \\\\\n& &\\lbrack\\, { M}_{\\mu\\nu} \\, ,\\, { M}_{\\rho\\sigma}\\, \\rbrack\\ = \n\\ i\\, (\\, g_{\\mu\\sigma} { M}_{\\nu\\rho}\\, + \\, g_{\\nu\\rho} { M}_{\\mu\\sigma}\\, - \\, g_{\\mu\\rho} { M}_{\\nu\\sigma}\\, -\n\\, g_{\\nu\\sigma} { M}_{\\mu\\rho}\\, ) \\ ,\n\\label{eq:ETpoincare}\n\\end{eqnarray} where ${ M}_{ij}=\\epsilon_{ijk}{ J}_k$ and ${\n M}_{i0}={ K}_i$ with ${ J}_i$ and ${ K}_i$ the generators\nof rotations and boosts, respectively. Using ${\\cal C}^\\mu_\\nu$ we\ncan transform from the instant-form to the front-form giving ${\\tilde\n { P}}^\\mu=({ P}^+,{ P}^1,{ P}^2,{ P}^-)$, \n${\\tilde { M}}_{r+}=-{\\tilde { M}}_{+r}={F}_r$, \n${\\tilde { M}}_{r-}=-{\\tilde { M}}_{-r}={E}_r$, \n${\\tilde { M}}_{rs}=\\epsilon_{rs}{J}_3$, and\n${\\tilde { M}}_{+-}=-{\\tilde { M}}_{-+}={ K}_3$, where we have\ndefined\n\\begin{eqnarray}\n{ P}^+ &=& \\textstyle{1\\over\\sqrt{2}}\\, (\\, { P}^0\\, +\\, { P}^3\\, )\\qquad , \\qquad { P}^-\\ =\\ \\textstyle{1\\over\\sqrt{2}}\\, (\\, { P}^0\\, -\\, { P}^3\\, )\\ ;\n\\nonumber \\\\\n{ E}_r &=& \\textstyle{1\\over\\sqrt{2}}\\, (\\, { K}_r\\, + \\, \\epsilon_{rs}{ J}_s\\, )\\quad , \\quad \n\\quad{ F}_r\\ =\\ \\textstyle{1\\over\\sqrt{2}}\\, (\\, { K}_r\\, - \\, \\epsilon_{rs}{ J}_s\\, )\\ .\n\\label{eq:LCpoingens}\n\\end{eqnarray} Here ${ P}_+={ P}^-$ is the null-plane energy while ${ P}_- = { P}^+$ is the longitudinal momentum.\n(Note that the indices $r,s,t,\\ldots$ are transverse indices that range over $1,2$.\nSee Appendix~\\ref{npconventions}.)\n\nIt is straightforward to show that ${ P}^+$, ${ P}_r$, ${ K}_3$, ${ E}_r$,\nand ${ J}_3$ are kinematical generators that leave the null plane $x^+=0$\nintact. These seven generators form the stability group of the null plane. \nIt is useful to classify the subgroups of the Poincar\\'e algebra by\nconsidering the transformation properties of the generators with respect\nto longitudinal boosts, which serve to rescale the generators. Writing\n\\begin{eqnarray}\n& & \\lbrack\\, { K}_3 \\, ,\\, A\\, \\rbrack\\ =\\ -i \\gamma\\, A\n\\label{eq:goodness}\n\\end{eqnarray}\nwhere $A$ is a generator, one finds ${ E}_r$ and ${ P}^+$ have $\\gamma=1$,\n${ J}_3$, ${ K}_3$ and ${ P}_r$ have $\\gamma=0$, and ${ P}^-$ and ${ F}_r$ have $\\gamma=-1$.\nThe Poincar\\'e generators have subgroups $G_\\gamma$ labeled by $\\gamma$, and\nthere exist two seven-parameter subgroups $S_\\pm$ with a semi-direct product\nstructure $S_\\pm=G_0\\times G_\\pm$. Therefore the stability group coincides\nwith the subgroup $S_+$.\nThe non-vanishing commutation relations among these generators are:\n\\begin{eqnarray}\n& & \\lbrack\\, { K}_3 \\, ,\\, { E}_r\\, \\rbrack\\ =\\ -i { E}_r \\quad , \\quad \\lbrack\\, { K}_3 \\, ,\\, { P}^+\\, \\rbrack\\ =\\ -i { P}^+ \\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { J}_3 \\, ,\\, { E}_r\\, \\rbrack\\ =\\ i\\epsilon_{rs} { E}_s \\quad , \\quad \n\\lbrack\\, { J}_3 \\, ,\\, { P}_r\\, \\rbrack\\ =\\ i\\epsilon_{rs} { P}_s \\ ;\n\\nonumber \\\\\n& & \\qquad \\qquad \\quad\n\\lbrack\\, { E}_r \\, ,\\,{ P}_s \\, \\rbrack\\ =\\ -i\\delta_{rs} { P}^+ \\ .\n\\label{eq:LCSGcomms}\n\\end{eqnarray}\n\nBy contrast, ${ P}^-$ and ${ F}_r$ are the Hamiltonians which consist\nof the subgroup $G_{-1}$; they are\nthe dynamical generators which move physical states away from the\n$x^+=0$ surface (see fig.~\\ref{fig:nullplane}). The non-vanishing commutators among the stability\ngroup generators and the Hamiltonians are:\n\\begin{eqnarray}\n& & \\lbrack\\, { K}_3 \\, ,\\, { P}^-\\, \\rbrack\\ =\\ i { P}^- \\quad , \\quad \\lbrack\\, { E}_r \\, ,\\, { P}^-\\, \\rbrack\\ =\\ -i { P}_r \\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { K}_3 \\, ,\\, { F}_r\\, \\rbrack\\ =\\ i { F}_r \\quad , \\quad \\lbrack\\, { J}_3 \\, ,\\, { F}_r\\, \\rbrack\\ =\\ i\\epsilon_{rs} { F}_s \\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { P}_r \\, ,\\, { F}_s\\, \\rbrack\\ =\\ i\\delta_{rs}{ P}^- \\quad , \\quad \\lbrack\\, { P}^+ \\, ,\\, { F}_r\\, \\rbrack\\ =\\ i{ P}_r\\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { E}_r \\, ,\\, { F}_s\\, \\rbrack\\ =\\ -i\\left(\\,\\delta_{rs}\\,{ K}_3\\,+\\,\\epsilon_{rs} { J}_3\\,\\right)\\ .\n\\label{eq:LCHSGcomms}\n\\end{eqnarray} \nThis algebraic structure is isomorphic to the Galilean group of\ntwo-dimensional quantum mechanics where one identifies\n$\\lbrace\\,P^-,E_r,P_r,J_3,P^+\\,\\rbrace$ with the Hamiltonian, Galilean\nboosts, momentum, angular momentum, and mass, respectively. This\nisomorphism is responsible for the similarities between the front form\nand nonrelativistic quantum mechanics that we noted in the dispersion\nrelation, and was originally noted in the context of the\ninfinite momentum frame of instant-form\ndynamics~\\cite{Weinberg:1966jm,Susskind:1967rg} which has a similar\ndispersion relation.\n\n\\subsection{Null-plane momentum states and reduced Hamiltonians}\n\n\\noindent As momentum is a kinematical observable, it is convenient to work with momentum eigenstates,\nsuch that\n\\begin{eqnarray}\nP_r\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle &=& p_r|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ ;\\\\\nP^+\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle &=& p^+|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:momonmom}\n\\end{eqnarray}\nThe action of the boosts on momentum states follows directly from the commutation relations in eq.~\\ref{eq:LCSGcomms}\nand is given by\n\\begin{eqnarray}\ne^{-iv_r E_r}e^{-i\\omega K_3}|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, \\rangle = |\\, e^\\omega\\,p^+\\, ,\\, {\\bf p}_\\perp+p^+{\\bf v}_\\perp\\,\\rangle \\ .\n\\label{eq:boostsdefined}\n\\end{eqnarray}\nOne can then define the unitary boost operator\n\\begin{eqnarray}\n{\\cal U}(p^+,p_r)=e^{-i\\beta_r E_r}e^{-i\\beta_3 K_3} \\ ,\n\\label{eq:boostop}\n\\end{eqnarray}\nwith $\\beta_r\\equiv p_r\/p^+$ and $\\beta_3\\equiv\\log(\\sqrt{2}p^+\/M)$ \nwhich boosts the state at rest to one with arbitrary momentum:\n\\begin{eqnarray}\n{\\cal U}(p^+,p_r)|\\, M\/\\sqrt{2}\\, ,\\, {\\bf 0}\\, \\rangle = |\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:boostfromrest}\n\\end{eqnarray}\nThe action of the boosts on the momentum states is then easily found to be\n\\begin{eqnarray}\nE_r\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle &=& ip^+\\frac{d}{dp_r}|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ ;\\\\\nK_3\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle &=& ip^+\\frac{d}{dp^+}|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:boostonmom}\n\\end{eqnarray}\n\nUnitarity of the boost operators fixes the normalization of the momentum states up to a constant. We assume the covariant\nnormalization:\n\\begin{eqnarray}\n\\langle\\, p^{+{\\prime}}\\, ,\\, {\\bf p}_\\perp^{\\,\\prime}\\, | \\, p^+\\, ,\\, {\\bf p}_\\perp\\, \\rangle\\ =\\ \n(2\\pi)^3\\,2\\,p^+\\,\n\\delta(\\,p^{+{\\prime}}\\,-\\,p^+\\,)\\,\\delta^2(\\,{\\bf p}_\\perp^{\\,\\prime}\\,-\\,{\\bf p}_\\perp \\,) \\ ,\n\\label{eq:HBnormalization}\n\\end{eqnarray}\nand the corresponding completeness relation\n\\begin{eqnarray}\n{\\bf 1}\\, =\\, \n\\int\\,\\frac{dp^+d^2{\\bf p}_\\perp}{(2\\pi)^3 2p^+}\\,\n|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, \\rangle\\, \\langle\\, p^+\\, ,\\, {\\bf p}_\\perp\\, | \\ .\n\\label{eq:HBcomplete}\n\\end{eqnarray}\n\n\n\nWe can now find angular momentum operators, ${\\cal J}_r$ and ${\\cal J}_3$, that are\nvalid in any frame by boosting from an arbitrary momentum state to a state at rest,\nacting with the angular momentum generators $J_r=\\epsilon_{rs}(F_s-E_s)\/\\sqrt{2}$ and $J_3$, and then boosting back\nto the arbitrary momentum state. That is,\n\\begin{eqnarray}\n{\\cal J}_i|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ =\\ {\\cal U}(p^+,p_r)\\;J_i\\;{\\cal U}^{-1}(p^+,p_r)\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:generalAM}\n\\end{eqnarray}\nUsing this procedure one finds angular momentum operators that are valid in any frame:\n\\begin{eqnarray}\n{\\cal J}_3 &=& { J}_3\\ +\\ \\epsilon_{rs}\\,{ E}_r{ P}_s\\,\\left(1\/{ P}^+\\right) \\ ; \\label{eq:npJ3generalA} \\\\\n{\\cal J}_r &=& \\epsilon_{rs}\\,\\big\\lbrack\\,{ P}^+{ F}_s\\ -\\ { P}^-{ E}_s\\ +\\ \\epsilon_{st}{ P}_t{\\cal J}_3\\ +\\ { P}_s{ K}_3\\,\\big\\rbrack\\,\\left(1\/M\\right) \\ .\n\\label{eq:npJ3generalB}\n\\end{eqnarray}\nInverting eq.~\\ref{eq:npJ3generalB} one then finds the following expressions for the Hamiltonians:\n\\begin{eqnarray}\n{ P}^- &=& \\left(1\/2{ P}^+\\right)\\,\\big\\lbrack\\, { P}_1^2\\ +\\ { P}_2^2\\ +\\ M^2 \\,\\big\\rbrack \\ ; \\nonumber \\\\\n{ F_1}\\ &=& \\left(1\/{ P}^+\\right)\\,\\big\\lbrack\\,-{ P}_1{ K}_3\\ +\\ { P}^-{ E}_1\\ -\\ { P}_2{\\cal J}_3\\ -\\ M{\\cal J}_2\\,\\big\\rbrack \n\\ ; \\nonumber \\\\\n{ F}_2\\ &=& \\left(1\/{ P}^+\\right)\\,\\big\\lbrack\\,-{ P}_2{ K}_3\\ +\\ { P}^-{ E}_2\\ +\\ { P}_1{\\cal J}_3\\ +\\ M{\\cal J}_1\\, \\big\\rbrack \\ .\n\\label{eq:npHams}\n\\end{eqnarray} \nA striking feature of the null-plane formulation is that the\nfundamental dynamical objects are the {\\it products} $M^2$ and $M{\\cal\n J}_{r}$, rather than the generators themselves. Following\nRef.~\\cite{Leutwyler:1977vy}, we will refer to these objects as\nreduced Hamiltonians. The reduced Hamiltonians, together with \n${\\cal J}_3$, commute with all kinematical generators and\nsatisfy the algebra of $U(2)$. This is conveniently demonstrated\nby making use of the Pauli-Lubanski vector\n\\begin{eqnarray}\n{ W}^\\mu = \\ft12 \\varepsilon^{\\mu\\nu\\rho\\sigma} { P}_\\nu { M}_{\\rho\\sigma} \\ ,\n\\end{eqnarray}\nwhich satisfies $W^\\mu P_\\mu=0$ and the non-trivial commutation relations:\n\\begin{eqnarray}\n\\lbrack\\, { M}_{\\mu\\nu} \\, ,\\, W_\\rho\\, \\rbrack & =& i\\, (\\, g_{\\nu\\rho} W_\\mu\\, -\\, g_{\\mu\\rho} W_\\nu \\, ) \\ ;\\\\\n\\lbrack\\, W^\\mu \\, ,\\,W^\\nu\\, \\rbrack &=& -i\\varepsilon^{\\mu\\nu\\rho\\sigma} W_\\rho P_\\sigma \\ .\n\\label{eq:ETpoincareW}\n\\end{eqnarray}\nOne then finds general, compact expressions for the angular momentum operators:\n\\begin{eqnarray}\n{\\cal J}_3 \\ = \\ { W}^+\/{ P}^+ \\quad , \\quad\nM{\\cal J}_r \\ = \\ { W}_r\\ -\\ { P}_r\\;{ W}^+\/{ P}^+ \\ .\n\\end{eqnarray}\nBy considering the commutation relations among $W_\\mu$, $P^\\mu$ and $M_{\\mu\\nu}$\none confirms that\n\\begin{eqnarray}\n[\\, {\\cal J}_3 \\, ,\\, M{\\cal J}_r \\, ] \\ =\\ i\\,\\epsilon_{rs} M{\\cal J}_s\n\\ \\ \\ & , &\\ \\ \\ \n[\\, {\\cal J}_3 \\, ,\\, M^2 \\, ]\\, = \\, 0 \\ ; \\nonumber \\\\\n{} [\\, M{\\cal J}_r \\, ,\\, M{\\cal J}_s \\, ] \\ =\\ i\\,\\epsilon_{rs} M^2 {\\cal J}_3 \n\\ \\ \\ & ,& \\ \\ \\\n[\\, M^2 \\, ,\\, M{\\cal J}_r \\, ]\\, = \\, 0 \\ .\n\\label{eq:dynalgarbB}\n\\end{eqnarray} \nHence, the reduced Hamiltonians together with the stability group\ngenerator ${\\cal J}_3$ satisfy the algebra of $U(2)$, and the\nproblem of finding a Lorentz invariant description of a relativistic\nquantum mechanical system is thus equivalent to finding a representation\nof the three reduced Hamiltonians which satisfy this algebra~\\footnote{Since \nthe mass operator, $M=\\sqrt{p_\\mu p^\\mu}$, commutes with the spin operators, this\nalgebra can clearly be expressed in the canonical form: \n$[\\, {\\cal J}_i \\, ,\\, {\\cal J}_j \\, ]\\, = \\, i\\epsilon_{ijk}J_k$ and $[\\, M \\, ,\\, {\\cal J}_i \\, ]\\, = \\, 0$.}. \nSince the essence of Lorentz invariance resides in these Lie brackets, and they\nrequire knowledge of the reduced Hamiltonians, in theories with complicated\ndynamics like QCD, the formulation of the theory at weak coupling ---where QCD is\ndefined as a continuum quantum field theory--- will\nlack manifest Lorentz invariance, which is tied up with the detailed\ndynamics of the theory, and is as complicated to achieve as finding\nthe spectrum of the theory.\n\nWe can write a general momentum eigenstate as:\n\\begin{eqnarray}\n|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle = |\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle\\otimes | \\lambda \\, ,\\, n\\, \\rangle \\ .\n\\label{eq:HBdefined}\n\\end{eqnarray}\nHere $n$ are additional variables that may be needed to specify the\nstate of a system at rest, and $\\lambda$ is helicity, the eigenvalue of ${\\cal J}_3$:\n\\begin{eqnarray}\n{\\cal J}_3\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ =\\ \\lambda\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ ,\n\\label{eq:J3eigen}\n\\end{eqnarray} \nand therefore, using eq.~\\ref{eq:npJ3generalA}, we have\n\\begin{eqnarray}\nJ_3\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \n\\ =\\ \\left(\\,\\lambda+i\\epsilon_{rs}\\,p_r\\frac{d}{dp_s}\\,\\right)\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ ,\n\\label{eq:J3atrest}\n\\end{eqnarray} \nwhich completes the catalog of the action of the stability group generators on the momentum states.\nIt is useful to write\n\\begin{eqnarray}\n|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ =\\ {\\cal U}(p^+,p_r)|\\, M\/\\sqrt{2}\\, ,\\, {\\bf 0}\\, ; \\lambda\\, ,\\,n\\,\\rangle\n\\ \\equiv \\ a_n^\\dagger\\left(p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\right)\\;|\\, 0\\,\\rangle \\ ,\n\\label{eq:vacuumdefined}\n\\end{eqnarray}\nwhere $a_n^\\dagger$ is an operator that creates the momentum state when acting on the null-plane vacuum, $|\\, 0\\,\\rangle$.\nWhat is special about the null-plane description is that the\nkinematical generators (with the exception of ${\\cal J}_3$) act on\nstates in a manner independent of the inner variables $n$. And the\nreduced Hamiltonians act exclusively on the inner variables in a\nmanner independent of the momentum. Therefore, one may view the\nPoincar\\'e algebra by the direct sum of ${\\cal K}$ and ${\\cal D}$, where\n${\\cal K}=\\lbrace\\,E_r,P_r,K_3,P^+\\,\\rbrace$ contains all stability\ngroup generators with the exception of ${\\cal J}_3$ which is grouped\nwith the reduced Hamiltonians, ${\\cal D}=\\lbrace\\,{\\cal J}_3,M{\\cal\n J}_r,M^2\\rbrace$~\\cite{Leutwyler:1977vz}. \n\nThe structure of the Poincar\\'e algebra in the front-form is well\nsuited to the study of systems with complicated dynamics like QCD, as\nthe dynamical generators are directly related to the most important\nobservable quantities, namely the energy and the angular momentum of\nthe system, while momenta and boosts are purely kinematical and\ntherefore are easy to implement~\\footnote{By contrast, in the instant form\n of dynamics, the energy and the boosts are dynamical. As boosts are\n not among the observables, one refers only to the one Hamiltonian\n corresponding to energy.}. The reduced Hamiltonians will have a\nfundamental role to play in the description of chiral symmetry\nbreaking on null planes.\n\n\n\n\\section{Chiral symmetry in the front form}\n\\label{sec:chiralityintro}\n\n\\subsection{Null plane charges and the chiral algebra}\n\n\\noindent Consider a Lagrangian field theory that has an\n$SU(N)_R\\otimes SU(N)_L$ chiral symmetry. Let us assume that this\nsystem has a null-plane Lagrangian formulation which allows one, by\nthe standard Noether procedure, to obtain the currents ${\\tilde\n J}_\\alpha^\\mu(x)$ and ${\\tilde J}_{5\\alpha}^\\mu(x)$, which are\nrelated to the symmetry currents via ${\\tilde J}^\\mu_{L\\alpha} =\n({\\tilde J}^\\mu_\\alpha - {\\tilde J}^\\mu_{5\\alpha})\/2$ and ${\\tilde\n J}^\\mu_{R\\alpha} = ({\\tilde J}^\\mu_\\alpha + {\\tilde\n J}^\\mu_{5\\alpha})\/2$. We will further assume that the Lagrangian\ncontains an operator that explicitly breaks the chiral symmetry in the\npattern $SU(N)_R\\otimes SU(N)_L\\rightarrow SU(N)_F$ and is governed by\nthe parameter $\\epsilon_\\chi$ such that as $\\epsilon_\\chi\\rightarrow\n0$, the symmetry is restored at the classical level. The general\nrelation between currents and their associated charges is given by\n\\begin{eqnarray}\n{Q}(n\\cdot x)\\, =\\, \\int\\, d^4y\\, \\delta(n\\cdot (x\\, -\\, y\\,)\\, )\\, n\\cdot {J}(y) \\ ,\n\\label{eq:Qgendef}\n\\end{eqnarray} \nwhere the vector $n_\\mu$ selects the initial quantization\nsurface, which we take to be the null plane $\\Sigma_n^\\tau$. Therefore, the null-plane chiral symmetry charges are \n\\begin{eqnarray}\n{\\tilde Q}_\\alpha \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{\\alpha}(x^-, {\\vec x}_\\perp) \\ ; \\label{eq:npchargesgenA}\\\\\n{\\tilde Q}^5_\\alpha(x^+) \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ,\n\\label{eq:npchargesgenB}\n\\end{eqnarray}\nwhere the axial charges have been given explicit null-plane time dependence as they are not conserved\ndue to the explicit breaking operator in the Lagrangian.\nThese charges satisfy the $SU(N)_R\\otimes SU(N)_L$ chiral algebra,\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha\\, ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\, i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma\n\\ & , &\\ \n[\\, {\\tilde Q}_5^\\alpha(x^+)\\, ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\, i\\,f^{\\alpha\\beta\\gamma}\\, {\\tilde Q}_5^\\gamma(x^+) \\ ;\n\\label{eq:LCalga}\n\\end{eqnarray}\n\\begin{eqnarray}\n[\\, {\\tilde Q}_5^\\alpha(x^+) \\, ,\\, {\\tilde Q}_5^\\beta(x^+)\\, ] \\, =\\, i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma \\ .\n\\label{eq:LCalgb}\n\\end{eqnarray}\nWe further assert that both types of chiral charges annihilate the vacuum. That is,\n\\begin{eqnarray}\n{\\tilde Q}^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, {\\tilde Q}_5^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, 0 \\ .\n\\label{eq:vacuumgen}\n\\end{eqnarray} \nThis is the statement that the front-form vacuum is invariant with\nrespect to the full $SU(N)_R\\otimes SU(N)_L$ symmetry. In particular,\nthis implies that there can be no vacuum condensates that break\n$SU(N)_R\\otimes SU(N)_L$ on a null-plane. This may seem to be an odd\nassumption, since the chiral charge is directly related to the\naxial-vector current through eq.~\\ref{eq:npchargesgenB}, and in\ngeneral one would expect that this current has a Goldstone boson pole\ncontribution, in turn implying that the chiral charges acting on the\nvacuum state excite massless Goldstone bosons. Below we will confirm the assertion,\neq.~\\ref{eq:vacuumgen}, by using standard current-algebra polology to\nshow that indeed the Goldstone boson pole contribution to the null-plane\naxial-vector current is absent.\n\n\\subsection{Symmetries of the reduced Hamiltonians}\n\n\\noindent Mixed commutators among the Poincar\\'e generators and internal symmetry generators\ncan be expressed generally as~\\cite{Feinberg:1973qb}:\n\\begin{eqnarray}\n\\lbrack\\, {Q}_\\alpha(n\\cdot x) \\, ,\\, P^\\mu\\rbrack \\, &=&\\, -i\\;n^\\mu \n\\int\\, d^4y\\, \\delta( n\\cdot (x\\, -\\, y\\,)\\, )\\, \\partial_\\nu J^\\nu_\\alpha(y) \\ ; \\\\\n\\lbrack\\, {Q}_\\alpha(n\\cdot x) \\, ,\\, M^{\\mu\\nu}\\rbrack \\,&=&\\,\ni\\int\\, d^4y\\, \\delta( n\\cdot (x\\, -\\, y\\,)\\, )\\,\\left(n^\\mu y^\\nu \\ -\\ n^\\nu y^\\mu \\right) \\partial_\\kappa J^\\kappa_\\alpha(y) \\ .\n\\end{eqnarray}\nFrom these expressions one then obtains the mixed commutator between the Pauli-Lubanski vector\nand the internal symmetry charges:\n\\begin{eqnarray}\n\\hskip-1.3em\\lbrack{Q}_\\alpha(n\\cdot x) , W_\\nu\\rbrack = \\ft i2 \\varepsilon_{\\nu\\delta\\rho\\sigma}\n\\int d^4y \\delta( n\\cdot (x - y))\n\\bigg\\lbrack M^{\\delta\\rho}n^\\sigma - \\left(n^\\delta y^\\rho -n^\\rho y^\\delta \\right)P^\\sigma\n\\bigg\\rbrack \\partial_\\kappa J^\\kappa_\\alpha(y).\n\\end{eqnarray}\nUsing these expressions, one finds the commutation relations between null-plane chiral charges and the\nreduced Hamiltonians:\n\\begin{eqnarray}\n\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack \\, &=&\\, -2i\\,P^+\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ; \\label{eq:mixedchiralc}\\\\\n\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M{\\cal J}_r\\rbrack \\, &=&\\, i\\,\\epsilon_{rs}\\,P^+\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\Gamma}_s\\,\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ,\n\\label{eq:mixedchirald}\n\\end{eqnarray}\nwhere $\\Gamma_s\\equiv E_s - P^+ x_s $. Here and in what follows, we are assuming that $SU(N)_F$ is \nunbroken and therefore $\\partial_\\mu {\\tilde J}^\\mu_{\\alpha}=0$ and\nthe reduced Hamiltonians commute with the $SU(N)_F$ charges:\n\\begin{eqnarray}\n\\lbrack\\, {\\tilde Q}_\\alpha\\, ,\\, M^2\\rbrack &=& \\lbrack\\, {\\tilde Q}_\\alpha\\, ,\\, M{\\cal J}_r\\rbrack \\ = \\ 0\\ .\n\\label{eq:mixedvector}\n\\end{eqnarray}\n\n\\subsection{Goldstone's theorem on a null plane}\n\n\\noindent In the instant form, a symmetry has three possible fates in\nthe quantum theory: the symmetry remains exact and the current is\nconserved, the symmetry is spontaneously broken and again the current\nis conserved, or the symmetry is anomalous and the current is not\nconserved. The front form realizes a fourth possibility: the symmetry\nis spontaneously broken and the associated current in not conserved.\nThis fourth possibility is necessary in the front form because the\nvacuum is invariant with respect to all internal symmetries. In general, we\ncan write\n\\begin{eqnarray}\n\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ = \\ \\epsilon_{\\chi}\\ {\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ,\n\\label{eq:genpcac}\n\\end{eqnarray}\nwhere $\\epsilon_{\\chi}$ is the parameter that gauges the amount of chiral symmetry breaking that is present in the Lagrangian.\nUsing the short hand,\n\\begin{eqnarray}\n|\\,h\\,\\rangle\\equiv \\,|\\, p^+\\, ,\\, {\\vec p}_\\perp\\,;\\, \\lambda\\,,\\, h\\,\\rangle \\ ,\n\\label{eq:HME2}\n\\end{eqnarray} for the momentum eigenstates, we take the matrix element of\neq.~\\ref{eq:mixedchiralc} between momentum eigenstates, which gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle\\, &=&\\, -2i\\,p^+\\,\n\\epsilon_{\\chi}\\, \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, | {\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+) |\\, h\\, \\rangle \\ ; \n\\label{eq:mixedchiral1}\n\\end{eqnarray} If the right hand side of this equation vanishes for all $h$ and\n$h'$, then there can be no chiral symmetry breaking of any kind. \nTherefore, in order that the chiral symmetry be spontaneously broken, the\nchiral current cannot be conserved and we have the following\nconstraint~\\cite{Kim:1994rm,Yamawaki:1998cy} in the limit $\\epsilon_\\chi\\rightarrow 0$:\n\\begin{eqnarray}\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, |{\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+)|\\, h\\, \\rangle &\\longrightarrow & \\frac{1}{\\epsilon_\\chi}\\ +\\ \\ldots \\ ,\\label{eq:pizeromodea}\n\\end{eqnarray}\nwhere the dots represent other possible terms that are non-singular in the limit $\\epsilon_\\chi\\rightarrow 0$.\nNow we will show that this condition implies the existence of $N^2-1$ Goldstone bosons~\\footnote{Note that if we took eq.~\\ref{eq:pizeromodea}\nas a constraint on the operator ${\\tilde P}_\\alpha$ rather than on its matrix elements, then this constraint would be viewed\nas a constraint on the zero-modes of the operator~\\cite{Kim:1994rm,Yamawaki:1998cy}. Here we work entirely with matrix elements.}. \nWe will assume that ${\\tilde P}_\\alpha$ is an interpolating operator for Lorentz-scalar fields $\\phi_\\alpha^i$, and \ntherefore we can write\n\\begin{eqnarray}\n{\\tilde P}_\\alpha(x) \\ =\\ \\sum_i {\\cal Z}_i\\,\\phi_\\alpha^i(x) \\ \n\\label{eq:zfactors}\n\\end{eqnarray}\nwhere the ${\\cal Z}_i$ are overlap factors. Using the reduction\nformula we relate the matrix elements of field operators between\nphysical states to transition amplitudes. Of course here it is understood\nthat there is no selection rule which would forbid these transitions.\nThe S-matrix element for the transition $h(p)\\rightarrow h'(p')+\\phi^i_\\alpha(q)$ can be defined by\n\\begin{eqnarray}\n\\langle \\, h'\\,;\\, \\phi_\\alpha^i(q)\\, |\nS |\\, h\\, \\rangle &\\equiv& i(2\\pi)^4\\,\\delta^4(\\, p\\,-\\, p'-q)\\,\n{\\cal M}^i_\\alpha (\\,p',\\,\\lambda',\\, h'\\,;\\,p,\\,\\lambda,\\, h\\,) \\nonumber \\\\\n&=& i\\int d^4x\\,e^{-iq\\cdot x}\\,\\left(-q^2+M_{\\phi^i}^2\\right)\\,\\langle\\, h'\\, |\\, \\phi_\\alpha^i(x)\\,|\\, h\\, \\rangle\n\\label{eq:HME3}\n\\end{eqnarray}\nwhere ${\\cal M}^i_\\alpha$ is the Feynman amplitude and in the second line we have used\nthe reduction formula. It then follows that\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, \\phi_\\alpha^i(x)\\,|\\, h\\, \\rangle & =& \n-e^{iq\\cdot x}\\,{1\\over{q^2-M_{\\phi^i}^2}}\\, {\\cal M}^i_\\alpha(q) \\ .\n\\label{eq:reductiongen}\n\\end{eqnarray}\nUsing this formula together with eq.~\\ref{eq:zfactors} in eq.~\\ref{eq:mixedchiral1} then gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle &=& 2i\\,p^+\\,\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\sum_i \\frac{\\epsilon_\\chi\\, {\\cal Z}_i}\n{2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2}-M_{\\phi^i}^2}\\,{\\cal M}^i_\\alpha(q) \\nonumber \\\\ \n&=&{} -2i\\,p^+\\, (2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\sum_i \\frac{\\epsilon_\\chi\\, {\\cal Z}_i}\n{M_{\\phi^i}^2}\\,{\\cal M}^i_\\alpha(q^-) \\ ,\n\\label{eq:mixedchiralproof}\n\\end{eqnarray} \nwhere in the second line we have used the momentum delta\nfunctions. In order that the right hand side not vanish in the\nsymmetry limit, there must be at least one field $\\phi_\\alpha^i$ whose\nmass-squared vanishes proportionally to $\\epsilon_\\chi$ as\n$\\epsilon_\\chi\\rightarrow 0$. We will denote this field as\n$\\pi^\\alpha\\equiv \\phi_\\alpha^1$ with \n\\begin{eqnarray} M_\\pi^2 \\ =\\ c_p\\,\n\\epsilon_\\chi \\ ,\n\\label{eq:pionmass}\n\\end{eqnarray} \nwhere $c_p$ is a constant of proportionality. There are therefore\n$N^2-1$ massless fields $\\pi_\\alpha$ in the symmetry limit, which we\nidentify as the Goldstone bosons. It is noteworthy that this proof\nrelies entirely on physical matrix elements; i.e. there is no need to\nassume the existence of a vacuum condensate that breaks the chiral\nsymmetry. Of course, in instant-form QCD, we know that the\nproportionality constant in eq.~\\ref{eq:pionmass} contains the quark\ncondensate. This issue will be resolved below in\nsection~\\ref{sec:qcd}. While we have carried out this proof in the\ncase of $SU(N)_R\\otimes SU(N)_L$ broken to $SU(N)_V$, it is clearly\neasily generalized to other systems.\n\nWe can now write ${\\tilde P}_\\alpha \\ =\\ {\\cal Z}\\,\\pi_\\alpha\\ +\\ \\ldots$\nwhere the dots represent non-Goldstone boson fields, and \n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \n\\langle\\, h'\\, |\\, {\\bar{\\cal Z}}\\,M_\\pi^2\\,\\pi_\\alpha(x)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:pcacgen}\n\\end{eqnarray}\nwhere ${\\bar{\\cal Z}}\\equiv{\\cal Z}\/c_p$. Here, as in the usual current algebra\nmanipulations, we have assumed that only the Goldstone bosons couple to the\naxial-vector current, and it is now a standard exercise to determine the overlap factor. First, define\nthe Goldstone-boson decay constant, $F_\\pi$, via\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, \\pi_\\beta\\, \\rangle & \\equiv& -i\\,p^\\mu\\;F_\\pi\\,\\delta_{\\alpha\\beta}\\;e^{ip\\cdot x} \\ ,\n\\label{eq:piondecaydef}\n\\end{eqnarray}\nwhere $|\\, \\pi_\\beta\\, \\rangle \\equiv |\\, p^+\\, ,\\, {\\vec p}_\\perp\\,;\\, 0,,\\, \\pi_\\beta\\, \\rangle$.\nTaking the divergence of the current and raising eq.~\\ref{eq:pcacgen} to an operator relation yields\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,{\\bar{\\cal Z}}\\,M_\\pi^2\\,\\pi_\\alpha(x) \\,|\\, \\pi_\\beta\\, \\rangle & =& F_\\pi\\,M_\\pi^2\\,\\delta_{\\alpha\\beta}\\;e^{ip\\cdot x} \\ .\n\\label{eq:piondecaydef2}\n\\end{eqnarray}\nThe normalization of the Goldstone-boson field,\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,\\pi_\\alpha(x)\\,|\\, \\pi_\\beta\\, \\rangle &= & \\delta_{\\alpha\\beta}\\;e^{ip\\cdot x} \\ ,\n\\label{eq:pionfieldnorm}\n\\end{eqnarray}\nthen gives $\\bar{\\cal Z}=F_\\pi$ and we recover the standard operator relation\n\\begin{eqnarray}\n\\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x) & =& F_\\pi\\,M_\\pi^2\\,\\pi_\\alpha(x) \\ .\n\\label{eq:pcacgen2}\n\\end{eqnarray}\nWe can now express the mixed Lie bracket, eq.~\\ref{eq:mixedchiralproof}, as \n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle \n&=& -2i\\,p^+\\, (2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}F_\\pi\\,{\\cal M}_\\alpha(q^-) \\ ,\n\\label{eq:mixedfinal}\n\\end{eqnarray}\nwhere here ${\\cal M}_\\alpha(q^-)$ is the Feynman amplitude for the transition $h(p)\\rightarrow h'(p')+\\pi_\\alpha(q)$.\nWe see that while the chiral current is not conserved, its divergence is proportional to an S-matrix element.\nNoting that \n\\begin{eqnarray}\n\\hskip-1.3em\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle \n&=& 2p^+q^-\\langle\\, h'\\, |{\\tilde Q}^5_\\alpha(x^+) |\\, h\\, \\rangle \\ =\\ -2p^+\\langle\\, h'\\, |i\\,\\frac{d}{dx^+}{\\tilde Q}^5_\\alpha(x^+) |\\, h\\, \\rangle \\ ,\n\\label{eq:mixeddef}\n\\end{eqnarray}\nand from the definition of the chiral charge, eq.~\\ref{eq:npchargesgenB},\n\\begin{eqnarray}\n\\langle\\, h'\\, |{\\tilde Q}^5_\\alpha(x^+) |\\, h\\, \\rangle &=& \n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\langle\\, h'\\, |{\\tilde J}^+_{5\\alpha} (0) |\\, h\\, \\rangle \\ ,\n\\label{eq:chiralchargecurrent}\n\\end{eqnarray}\none finds, using eq.~\\ref{eq:mixedfinal},\n\\begin{eqnarray}\n{\\cal M}_\\alpha(q^-)\n& =& \n\\frac{i\\,q^-}{F_\\pi}\\, \\langle\\, h'\\, |\\, {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:matcurr}\n\\end{eqnarray}\nor, in Lorentz-invariant form,\n\\begin{eqnarray}\n{\\cal M}_\\alpha(q) & =& \n\\frac{i\\ q_\\mu}{F_\\pi}\\, \\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:HME8a}\n\\end{eqnarray}\nwhich is the standard current-algebra result. In order to confirm\nsome of these properties in a better-known fashion, and to address the\nassumption we have made that the chiral charges annihilate the vacuum,\nwe will now consider current algebra polology on the\nnull-plane.\n\n\\subsection{Polology and the chiral invariant vacuum}\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[scale=0.25]{polology.pdf}\n \\caption{Above shows the standard instant-form polology; the matrix element of the chiral current\nhas a Goldstone-boson pole piece, and a non-pole piece. These two contributions cancel in the symmetry limit\nensuring a conserved chiral current. Below shows the standard front-form polology; the Goldstone-boson pole contribution\nis absent and therefore the current is not conserved but rather has a divergence which is proportional to the matrix element\nfor the emission or absorption of a Goldstone boson.}\n \\label{fig:polo}\n\\end{figure}\n\n\\noindent Our starting point is the matrix element between hadronic states $h$\nand $h'$ of the axial-vector current, which can be written in a general\nway as~\\cite{Weinberg:1995mt,Weinberg:1996kr}\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\, =\\, \n\\frac{i F_\\pi\\, q^\\mu}{q^2-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \n\\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N}\n\\label{eq:HME1}\n\\end{eqnarray}\nwhere as before $q=p-p^\\prime$. Using translational invariance, we have\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle \\, =\\, \ne^{iq\\cdot x}\\,\\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ .\n\\label{eq:HME4}\n\\end{eqnarray}\nIt follows that \n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \ni\\, q_\\mu\\,\\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle \\nonumber \\\\\n& =& \ne^{iq\\cdot x}\\, \\Bigl\\lbrack\\,\n\\frac{-F_\\pi\\, q^2}{q^2-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N}\\, \\Bigr\\rbrack \\ ,\n\\label{eq:HME5}\n\\end{eqnarray}\nand using\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \n\\langle\\, h'\\, |\\, F_\\pi\\,M_\\pi^2\\,\\pi_\\alpha(x)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:HME6}\n\\end{eqnarray}\nand the reduction formula, eq.~\\ref{eq:reductiongen}, reproduces eq.~\\ref{eq:HME8a}.\nNote that in null-plane coordinates eq.~\\ref{eq:HME1} gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\, =\\, \n\\frac{i F_\\pi\\, q^+}{2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2}-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \n\\langle\\, h'\\, |\\, {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\ .\n\\label{eq:LCC2}\n\\end{eqnarray}\nWe therefore have\n\\begin{eqnarray}\n\\lim_{{q+,{\\vec q}_\\perp}\\rightarrow 0}\\langle\\, h'\\, |\\, {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ =\\ \n\\langle\\, h'\\, |\\, {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\ .\n\\label{eq:LCC3}\n\\end{eqnarray}\nBy comparing with eq.~\\ref{eq:chiralchargecurrent}, it is clear that\nthe null-plane chiral charges, by construction, do not excite the\nGoldstone boson states. The property, eq.~\\ref{eq:vacuumgen}, of\nvacuum annihilation which we assumed above, is therefore a general\nproperty of the null-plane chiral charges.\n\nAgain consider the space-integrated current divergence in the front-form, but now using eq.~\\ref{eq:HME5}. One finds\n\\begin{eqnarray}\n&&\\int\\, d x^-\\, d^2 {\\bf x_\\perp}\\,\\langle\\, h'\\, |\\, \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle \\ = \\\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\,\\langle\\, h'\\, |\\, \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\nonumber \\\\\n&&\\qquad = \n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\,\n\\Bigl\\lbrack\\,\n\\frac{-F_\\pi\\, \\left( 2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2} \\right)}{2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2}-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\,\n\\Bigl\\rbrack \\nonumber \\\\\n&&\\qquad =\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\, i\\ q_\\mu\\, \\langle\\, h'\\, |\\, {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\nonumber \\\\\n&&\\qquad =\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\, F_\\pi\\,{\\cal M}_\\alpha(q^-) \\ ,\n\\label{eq:HME9a}\n\\end{eqnarray}\nwhere in the third line the momentum delta functions have been used,\nand in the last line we have used eq.~\\ref{eq:HME8a} and eq.~\\ref{eq:LCC3}. Now using\neq.~\\ref{eq:mixedchiralc}, we see that we have recovered\neq.~\\ref{eq:mixedfinal}. In this derivation we see explicitly that the\nGoldstone-boson pole does not contribute to the divergence of the\naxial-current. It is for this reason that the current cannot be\nconserved. For purposes of comparison, recall that in the instant\nform, one has\n\\begin{eqnarray}\n\\int\\, d^3 x\\,\\langle\\, h'\\, |\\, \\partial_\\mu\\,{J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \n(2\\pi)^3\\,\\delta^3(\\,{\\vec q}\\,)\n\\,\\langle\\, h'\\, |\\, \\partial_\\mu\\,{J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ ; \\nonumber \\\\\n& =& \n(2\\pi)^3\\,\\delta^3(\\,{\\vec q}\\,)\\,\n\\Bigl\\lbrack\\,\n\\frac{-F_\\pi\\, q_0^2}{q_0^2-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\, {J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\,\n\\Bigl\\rbrack \\ ; \\nonumber \\\\\n& \\mapright{M_\\pi\\rightarrow 0}& \n(2\\pi)^3\\,\\delta^3(\\,{\\vec q}\\,)\\,\n\\Bigl\\lbrack\\,\n-F_\\pi \\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\, {J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\,\n\\Bigr\\rbrack \\ ; \\nonumber \\\\\n& =& 0 \\ ,\n\\label{eq:HME9}\n\\end{eqnarray}\nwhere in the last line, eq.~\\ref{eq:HME8a} has once again been used.\nHere there is a cancellation between the pole and non-pole parts of the matrix element\nwhich ensure that the integrated current divergence vanishes in the chiral limit.\nThis analysis, which is expressed pictorially in fig.\\ref{fig:polo}, suggests that the front-form and instant-form axial-vector currents\nare related, at the operator level, through\n\\begin{eqnarray}\n{\\tilde J}^\\mu_{5\\alpha}\\ =\\ {J}^\\mu_{5\\alpha}\\ -\\ ({J}^\\mu_{5\\alpha})_{\\it GB\\,pole}\n\\label{eq:HME11}\n\\end{eqnarray} \nwhere the second term on the right is the purely Goldstone-boson pole part of the axial-vector current.\nWe will see that this peculiar realization of chiral symmetry does indeed emerge in QCD.\n\nIt is useful to define new objects which give a matrix-element representation of the \ninternal-symmetry charges~\\cite{Weinberg:1969hw,Weinberg:1969db}:\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde Q}^5_\\alpha(x^+) \\,|\\, h\\, \\rangle \\, =\\, \n(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\n\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\,\n\\delta_{\\lambda'\\lambda} \\ ;\n\\label{eq:HME40}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde Q}_\\alpha \\,|\\, h\\, \\rangle \\, =\\, \n(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\n\\lbrack\\, T_\\alpha \\,\\rbrack_{h}\\, \\delta_{h h'}\\,\n\\delta_{\\lambda'\\lambda} \\ . \n\\label{eq:HME41}\n\\end{eqnarray}\nThese definitions are particularly useful as they allow the preservation of the Lie-algebraic structure of the operator algebra in the\ncase where correlation functions are given purely by single-particle states. The matrix element for Goldstone boson emission and absorption is:\n\\begin{eqnarray}\n{\\cal M}_\\alpha (\\,p',\\,\\lambda',\\, h'\\,;\\,p,\\,\\lambda,\\, h\\,) \\ =\\ \\frac{i}{F_\\pi}\\,\n(\\,M^2_h\\ -\\ M^2_{h'}\\,)\\, \\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\,\n\\delta_{\\lambda'\\lambda} \\ . \n\\label{eq:HME42}\n\\end{eqnarray}\nAs one might expect, in the limit that chiral symmetry is restored through a second-order phase transition,\nthe matrix $\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}$ becomes a true symmetry generator~\\cite{Weinberg:1995fe}. \nIn this limit, one also expects that the states $h'$ and $h$ become degenerate. In order that the matrix element of\neq.~\\ref{eq:HME42} not vanish in this limit, ${F_\\pi}$ must approach zero in the symmetry limit in precisely the same way~\\cite{Weinberg:1995fe}. \nThe role of ${F_\\pi}$ as an order parameter of chiral symmetry breaking is then apparent in eq.~\\ref{eq:mixedfinal},\nas the mixed-Lie bracket vanishes as ${F_\\pi}\\rightarrow 0$. Therefore, ${F_\\pi}$ is an order parameter of chiral \nsymmetry breaking on the null-plane.\n\n\n\\subsection{Broken chiral symmetry and spin}\n\\label{susec:spin}\n\n\\noindent Using the results of the previous two sections one finds\n\\begin{eqnarray}\n&&\\hspace{-.4in}\\langle\\, h'\\,,\\,\\lambda' |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\,,\\,\\lambda \\rangle = \n\\delta_{\\lambda',\\lambda}(2\\pi)^3\\,2\\,p^+\\,\\delta(q^+)\\delta^2({\\vec q}_\\perp)\n\\left(M_h^2-M_{h'}^2\\right)\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\n\\label{eq:mixedchiral1out}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&&\\hspace{-.92in}\\langle\\, h'\\,,\\,\\lambda' |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M{\\cal J}_\\pm\\rbrack |\\, h\\,,\\,\\lambda \\rangle\\ =\\ \n\\delta_{\\lambda',\\lambda\\pm 1}\\;(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\nonumber\\\\\n&&\\times\\;\\bigg\\lbrack M_h\\,c_\\pm(h,\\lambda)\\,\\lbrack\\, X_\\alpha(\\lambda\\pm 1) \\,\\rbrack_{h' h}\\,-\\,\nM_h'\\,c_\\mp(h',\\lambda')\\,\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\bigg\\rbrack,\n\\label{eq:mixedchiral2out}\n\\end{eqnarray}\nwhere ${\\cal J}_\\pm\\equiv{\\cal J}_1\\pm i{\\cal J}_2$ and $c_\\pm(h,\\lambda)\\equiv\\sqrt{j_h(j_h+1)-\\lambda(\\lambda\\pm1)}$.\nEq.~\\ref{eq:mixedchiral2out} has been obtained by a direct evaluation of the left-hand side using the usual\nangular momentum ladder relations and eq.~\\ref{eq:HME40}.\nWritten in this form, it is clear that in the presence of spontaneous symmetry breaking, the mixed Lie brackets between\nthe reduced Hamiltonians and the chiral charge are directly related to Goldstone-boson transition amplitudes and are\nnon-vanishing in the symmetry limit. The spin reduced Hamiltonians imply constraints on Goldstone-boson transitions that \nchange the helicity by one unit.\n\nAn important consequence of eqs.~\\ref{eq:mixedchiral1out} and ~\\ref{eq:mixedchiral2out} which will \nprove useful below is that chiral symmetry breaking remains relevant even when there are no mass \nsplittings. If we take $M_h\\;=\\;M_{h'}$, then chiral symmetry breaking arises solely through the\ntransverse spin operator, ${\\cal J}_r$, which is dynamical on the null-plane. That is,\n\\begin{eqnarray}\n&&\\hspace{-.92in}\\langle\\, h'\\,,\\,\\lambda' |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, {\\cal J}_\\pm\\rbrack |\\, h\\,,\\,\\lambda \\rangle\\ =\\ \n\\delta_{\\lambda',\\lambda\\pm 1}\\;(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\nonumber\\\\\n&&\\times\\;\\bigg\\lbrack \\,c_\\pm(h,\\lambda)\\,\\lbrack\\, X_\\alpha(\\lambda\\pm 1) \\,\\rbrack_{h' h}\\,-\\,\n\\,c_\\mp(h',\\lambda')\\,\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\bigg\\rbrack \\ .\n\\label{eq:mixedchiral2outb}\n\\end{eqnarray}\nIn this case, Goldstone's theorem must be obtained from the relation\n\\begin{eqnarray}\n\\hspace{-.32in}\\langle\\, h'\\, |\\;M\\;\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, {\\cal J}_r\\rbrack |\\, h\\, \\rangle\\, &=&\\, -i\\,\\epsilon_{rs}\\,p^+\\,\n\\epsilon_{\\chi}\\, \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, |{x}_s\\, {\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+) |\\, h\\, \\rangle ,\n\\label{eq:mixedchiral2}\n\\end{eqnarray}\nand its corresponding constraint\n\\begin{eqnarray}\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, |{x}_s\\,{\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+)|\\, h\\, \\rangle &\\longrightarrow & \\frac{1}{\\epsilon_\\chi}\\ +\\ \\ldots \n\\label{eq:pizeromodeb}\n\\end{eqnarray}\nin the symmetry limit, ${\\epsilon_\\chi}\\rightarrow 0$.\nFollowing the same steps as for the mass-squared reduced Hamiltonian, we have\n\\begin{eqnarray}\n\\hskip-1.8em\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, {\\cal J}_r\\rbrack |\\, h\\, \\rangle\n= \n-\\epsilon_{rs}\\,\\frac{p^+}{M_h}\\, (2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\sum_i \\frac{\\epsilon_\\chi\\, {\\cal Z}_i}\n{M_{\\phi^i}^2}\\left(\\frac{\\partial}{\\partial q_s}\\;{\\cal M}^i_\\alpha(q)\\right)\n\\label{eq:mixedchiralproof2}\n\\end{eqnarray} which again leads, via the same logic presented above, to\nGoldstone's theorem. Therefore, even if $M^2$ commutes with the chiral\ncharges, the chiral symmetry breaking contained in the spin\nHamiltonians implies the presence of massless states. Evaluating\neq.~\\ref{eq:mixedchiralproof2} in the rest frame, where\n$p^+\\rightarrow M_h\/\\sqrt{2}$ and ${\\cal J}_r\\rightarrow{J_r}$, and using eq.~\\ref{eq:HME8a}, gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde J}^+_{5\\alpha}\\, ,\\, {J}^r\\rbrack |\\, h\\, \\rangle\n&=& \ni\\frac{1}{\\sqrt{2}}\\epsilon^{rs}\\langle\\, h'\\, |\\; {\\tilde J}^s_{5\\alpha}\\; |\\, h\\, \\rangle \\ ,\n\\label{eq:wetheorem}\n\\end{eqnarray}\nwhich is simply the statement that the axial current transform as a vector operator.\n\n\\subsection{General operator algebra and the chiral basis}\n\n\\noindent A physical system with an $SU(N)_R\\otimes SU(N)_L$ chiral\nsymmetry broken to the vector subgroup $SU(N)_F$ may be expressed as a\ndynamical Hamiltonian system which evolves with null-plane time,\nwhose reduced Hamiltonians satisfy the $U(2)$ algebra of eq.~\\ref{eq:dynalgarbB},\nand in addition have non-vanishing Lie brackets with the non-conserved\nchiral charges. In operator form the reduced Hamiltonians satisfy:\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\beta_5(x^+)\\, ,\\, M^2\\, ] \\ \\neq \\ 0 \\quad ; \\quad\n[\\, {\\tilde Q}^\\beta_5(x^+)\\, ,\\, M {\\cal J}_\\pm\\, ]\\ \\neq \\ 0 \\ , \n\\label{eq:Qcomm2agen}\n\\end{eqnarray}\nwhich express the spontaneous breaking of the chiral symmetry. \nEq.~\\ref{eq:Qcomm2agen} has the general operator solution\n\\begin{eqnarray}\nM^2\\, &=&\\, M_{\\bf 1}^2\\, +\\, \\sum_{\\cal R}\\, M_{\\cal R}^2 \\ \\ ; \\nonumber \\\\\nM {\\cal J}_\\pm\\, &=&(M {\\cal J}_\\pm)\\, _{\\bf 1}\\, +\\, \\sum_{\\cal R}\\, (M {\\cal J}_\\pm)\\, _{\\cal R}\n\\label{eq:MESONScommgen}\n\\end{eqnarray} \nwhere ${\\bf 1}$ denotes the singlet $SU(N)_R\\otimes SU(N)_L$\nrepresentation, $({\\bf 1},{\\bf 1})$, and ${\\cal R}=({\\bf {\\cal\n R}_{R}},{\\bf {\\cal R}_{L}})$ is a non-trivial representation. Note\nthat all three symmetry-breaking reduced Hamiltonians must transform\nin the same way. This follows directly from eqs.~\\ref{eq:mixedchiralc}\nand~\\ref{eq:mixedchirald}. \n\nIt is useful to give a heuristic description of the consequences of\nthis algebraic structure. Consider an interpolating field operator,\n$a_h^\\dagger$ which creates a momentum state $h$ out of the vacuum;\nthat is,\n\\begin{eqnarray}\na_h^\\dagger\\, |\\, 0\\,\\rangle\\, =\\, |\\, h\\,\\rangle\\, \\ .\n\\label{eq:hdef}\n\\end{eqnarray}\nHere and below for simplicity we will suppress the flavor indices.\nBecause the null-plane chiral charges annihilate the vacuum, ${\\tilde Q}_5\\, |\\, 0\\,\\rangle\\, =\\, 0 $,\none has\n\\begin{eqnarray}\n{\\tilde Q}_5 \\, |\\, h\\,\\rangle\\, &=&\\, \n\\lbrack\\, {\\tilde Q}_5 \\, ,\\, a_h^\\dagger\\, \\rbrack \\, |\\, 0\\,\\rangle\\ . \n\\label{eq:htransf}\n\\end{eqnarray}\nNow we will assume that the interpolating field operator $a_h^\\dagger$ has definite chiral transformation properties with respect \nto the chiral charge in the sense that \n\\begin{eqnarray}\n\\lbrack\\, {\\tilde Q}_5 \\, ,\\, a_h^\\dagger\\, \\rbrack \\, =\\, C'\\, a_{h'}^\\dagger\\, +\\, C''\\, a_{h''}^\\dagger\\, +\\, \\ldots \\ ,\n\\label{eq:htransfprops}\n\\end{eqnarray}\nwhere $C',C'',\\ldots$ are group-theoretic factors.\nThis is simply the statement that the field operators $\\lbrace a_h, a_{h'},a_{h''},\\ldots\\rbrace$ are in a non-trivial $SU(N)_R\\otimes SU(N)_L$ representation. It\nthen follows from eq.~\\ref{eq:htransf} that\n\\begin{eqnarray}\n{\\tilde Q}_5 \\, |\\, h\\,\\rangle\\, &=&\\, C'\\,|\\, h'\\,\\rangle\\, + C''\\,|\\, h''\\,\\rangle\\, +\\,\\ldots \\ ,\n\\label{eq:htransf2}\n\\end{eqnarray} \nand therefore the states $\\lbrace h, {h'},{h''},\\ldots\\rbrace$ are also in an $SU(N)_R\\otimes SU(N)_L$ representation~\\footnote{\nNote that the instant-form interpolating operators also fill out \n$SU(N)_R\\otimes SU(N)_L$ representations. However, the instant-form charges do not annihilate the vacuum,\n i.e. $Q_5|\\,0\\, \\rangle \\equiv |\\, {\\bf\\omega}\\, \\rangle$, it follows\n that ${Q}_5\\, |\\, h\\,\\rangle\\, =\\, \\lbrack\\, {Q}_5\\, ,\\,\n a_h^\\dagger\\, \\rbrack \\, |\\, 0\\,\\rangle\\, +\\, |\\, h\\,;\\, {\\bf\\omega}\n \\,\\rangle$. Therefore ${Q}^5 \\, |\\, h\\,\\rangle\\, =\\, C'\\,|\\,\n h'\\,\\rangle\\, + C''\\,|\\, h''\\,\\rangle\\, +\\,\\ldots\\,+ \\, |\\, h\\,;\\,\n {\\bf\\omega} \\,\\rangle$ and the utility of chiral symmetry as a\n classification symmetry is lost.}.\n\nOne then has, for instance, \n\\begin{eqnarray}\n\\langle\\,h'\\,|\\,{\\tilde Q}_5\\, |\\, h\\,\\rangle & =& C' \\ ; \\\\\n\\langle\\,h''\\,|\\,{\\tilde Q}_5\\, |\\, h\\,\\rangle &=& C'' \\ ,\n\\label{eq:piontrans}\n\\end{eqnarray} \nwhich are, via eq.~\\ref{eq:HME40}, Goldstone-boson transition matrix elements.\nIf $\\lbrace h, h',h'',\\ldots\\rbrace$ are in an irreducible\nrepresentation, then the $C$'s are completely determined by the\nsymmetry (i.e. are Clebsch-Gordon coefficients), while if the\nrepresentation is reducible, then the $C$'s will depend on the mixing\nangles which mix the irreducible representations. Therefore through\nthe study of Goldstone-boson transitions one learns about the chiral\nrepresentations filled out by the physical states~\\footnote{Here it should be\nstressed that the chiral multiplet structure of the states is useful\nonly when the null-plane chiral charges mediate transitions\nbetween single-particle states~\\cite{Weinberg:1969hw,Weinberg:1969db}. Multi-particle\nstates obscure the algebraic consequences of null-plane chiral symmetry.}.\nTo learn more about the chiral representations, one considers the\nmixed Lie brackets, eqs.~\\ref{eq:mixedchiral1out} and \\ref{eq:mixedchiral2out}.\nKnowledge of the transformation properties of\nthe chiral-symmetry breaking reduced Hamiltonians gives\ninformation about how the hadron masses and spins are related, and therefore in\nhow the irreducible representations mix with each other when the\nsymmetry is broken. \n\nA natural null-plane basis can be written as\n\\begin{eqnarray}\n|\\, k^+\\, ,\\, {\\vec k}_\\perp\\, ;\\, \\lambda\\,, h\\, ,\\, ({\\bf {\\cal R}_{R}},{\\bf {\\cal R}_{L}})\\, \\rangle \\ .\n\\label{eq:HBpluschiraldefined}\n\\end{eqnarray} \nWhile the mass eigenstates are eigenstates of helicity, they clearly\nare not eigenstates of $SU(N)_R\\otimes SU(N)_L$ when the symmetry is\nspontaneously broken. Nevertheless, the chiral basis is useful when\nthe state $h$ can only appear in a finite number of chiral\nrepresentations, even though $h$ may be in an infinite-dimensional\nreducible chiral representation, as is the case generally in QCD at large-${N_c}$~\\cite{'tHooft:1973jz,Weinberg:1990xn}. In the\nchiral basis, the reduced Hamiltonian matrix $M^2$ is then of finite\nrank, even though there can be submatrices of infinite rank (and\ntherefore the Fock expansion in the number of constituents is\ninfinite). Ultimately, the utility of the chiral basis is\ndetermined by comparison with experiment~\\cite{Adler:1965ka,Weisberger:1965hp,Gilman:1967qs,Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn,Weinberg:1994tu,Beane:1999hp,Beane:2002ud,Beane:2002td}.\n\n\\section{QCD in the front form}\n\\label{sec:qcd}\n\n\\subsection{Basic instant-form conventions}\n\n\\noindent In this section, we will review the relevant symmetry\nproperties of the instant-form QCD Lagrangian for purposes of\nestablishing conventions which will clarify the null-plane\ndescription. Consider the QCD Lagrangian with $N$ flavors of light\nquarks and $N_c$ colors:\n\\begin{equation}\n{\\cal L}_{\\scriptstyle\\rm QCD} (x)\n=\n \\bar\\psi (x)\\Big\\lbrack\\ft{i}2\\left(\n\\stackrel{{}_\\rightarrow}{D}_\\mu - \\stackrel{{}_\\leftarrow}{D}_\\mu\n\\right)\n\\gamma^\\mu - \\mathbb{M} \\Big\\rbrack\\psi (x)\n-\n\\ft14 F_{\\mu\\nu}^a (x) F^{\\mu\\nu}_a (x)\n\\,\n\\label{QCDET}\n\\end{equation}\nwhere $\\mathbb{M}$ is the quark mass matrix, for now taken as a diagonal $N\\times N$ matrix, and\nthe covariant derivatives are \n\\begin{equation}\n\\stackrel{{}_\\rightarrow}{D}_\\mu\n\\, = \\,\n\\stackrel{{}_\\rightarrow}{\\partial}_\\mu\n- \\, i g \\, t^a A_\\mu^a (x)\n\\, , \\qquad\n\\stackrel{{}_\\leftarrow}{D}_\\mu\n\\, = \\,\n\\stackrel{{}_\\leftarrow}{\\partial}_\\mu\n+ \\, i g \\, t^a A_\\mu^a (x)\n\\, ,\n\\end{equation}\nwhere $g$ is the strong coupling constant, and indices $a,b,\\ldots$ are taken as adjoint indices of the $SU(3)$-color gauge group.\nThe Lagrangian is invariant with respect to baryon number and \nsinglet axial transformations\n\\begin{equation}\n\\psi\\rightarrow e^{-i\\theta} \\psi \\ \\ \\ ,\\ \\ \\ \\psi\\rightarrow e^{-i\\theta \\gamma_5} \\psi \\ ,\n\\end{equation}\nwith associated currents\n\\begin{equation}\n{J}^\\mu\\ =\\ \\bar\\psi \\gamma^\\mu\\psi \\ \\ \\ , \\ \\ \\ {J}^\\mu_{5}\\ =\\ \\bar\\psi \\gamma^\\mu\\gamma_5 \\psi \\ ,\n\\end{equation}\nand with divergences\n\\begin{equation}\n\\partial_\\mu {J}^\\mu\\ =\\ 0 \\ \\ \\ ,\\ \\ \\ \n\\partial_\\mu {J}^\\mu_{5}\\ =\\ 2i \\bar\\psi\\, \\mathbb{M}\\, \\gamma_5 \\psi \\ -\\ N \\frac{g^2}{16 \\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\;\ntr\\left( F_{\\mu\\nu}F_{\\rho\\sigma}\\right) \\ ,\n\\end{equation}\nwhere the singlet axial symmetry is of course anomalous. In addition, the Lagrangian is invariant with respect to \nthe symmetry transformations\n\\begin{equation}\n\\psi\\rightarrow e^{-i\\theta_\\alpha T_\\alpha} \\psi \\ \\ \\ , \\ \\ \\ \\psi\\rightarrow e^{-i\\theta_\\alpha T_\\alpha \\gamma_5} \\psi \\ ,\n\\end{equation}\nwhere the $T_\\alpha$ are $SU(N)$ generators (see appendix).\nBy the standard Noether procedure one defines the associated currents,\n\\begin{equation}\n{J}^\\mu_\\alpha\\ =\\ \\bar\\psi \\gamma^\\mu T_\\alpha \\psi \\ \\ \\ , \\ \\ \\ \n{J}^\\mu_{5\\alpha}\\ =\\ \\bar\\psi \\gamma^\\mu\\gamma_5 T_\\alpha \\psi \\ ,\n\\end{equation}\nrespectively, with divergences\n\\begin{equation}\n\\partial_\\mu {J}^\\mu_\\alpha\\ =\\ -i \\bar\\psi\\, [\\, \\mathbb{M}\\, ,\\, T_\\alpha \\, ]\\, \\psi \\ \\ \\ , \\ \\ \\ \n\\partial_\\mu {J}^\\mu_{5\\alpha}\\ =\\ i \\bar\\psi\\, \\{ \\, \\mathbb{M}\\, ,\\, T_\\alpha \\, \\}\\, \\gamma_5 \\psi \\ .\n\\end{equation}\nTherefore, with $N$ degenerate flavors the QCD Lagrangian is $SU(N)_F$ invariant and\nin the chiral limit where $\\mathbb{M}$ vanishes, there is an\n $SU(N)_R\\otimes SU(N)_L$ chiral symmetry generated by the currents\n${J}^\\mu_{L\\alpha} = ({J}^\\mu_\\alpha - {J}^\\mu_{5\\alpha})\/2$ and ${J}^\\mu_{R\\alpha} = ({J}^\\mu_\\alpha + {J}^\\mu_{5\\alpha})\/2$.\n\nThe energy-momentum tensor may be written as\n\\begin{equation}\n{T}^{\\mu\\nu}\\ =\\ - g^{\\mu\\nu} {\\cal L}_{\\rm\\scriptstyle QCD} \\ -\\ \nF^{\\mu\\rho}_a \\, F^{\\nu}_{a \\, \\rho}\n\\ +\\ \n\\frac{i}{2}\n\\bar\\psi\n\\stackrel{{}_\\leftrightarrow}{D}{\\!}^\\mu \\gamma^\\nu\n\\psi\n\\, .\n\\label{EMtensorQCD}\n\\end{equation}\nFrom the energy-momentum tensor we can form the Hamiltonian,\n\\begin{eqnarray}\nP^0 \\ =\\ \\int\\, d^3 \\bit{x}\\, {T}^{00}\\ .\n\\end{eqnarray}\nHere we will assume that chiral symmetry is spontaneously broken through the formation of the condensate\n\\begin{eqnarray}\n\\mathbb{M}\\,\\langle\\, \\Omega \\, |\\,\\bar\\psi\\psi \\,|\\, \\Omega\\, \\rangle\\ =\\ \n\\mathbb{M}\\,\\langle\\, \\Omega \\, |\\,\\frac{\\partial {\\mit T}^{00}}{\\partial \\mathbb{M}}\\,|\\, \\Omega\\, \\rangle\\ =\\\n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0}{\\partial \\mathbb{M}}\\ \\neq\\ 0 \\ ,\n\\end{eqnarray}\nwhere we have used the Feynman-Hellmann theorem, $|\\, \\Omega\\, \\rangle$ represents the (complicated) instant-form QCD vacuum state,\nand ${\\cal E}_0$ is the QCD vacuum energy. It is straightforward to show that the condensate transforms as \nthe $(\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ representation of $SU(N)_R\\otimes SU(N)_L$. \nWe can compute the vacuum energy in the low-energy effective field theory; i.e. chiral perturbation theory ($\\chi$PT)~\\cite{Weinberg:1978kz,Gasser:1983yg},\nas well. And therefore, \n\\begin{eqnarray} \n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0}{\\partial \\mathbb{M}}\\ =\\\n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0^{\\chi{\\rm PT}}}{\\partial\n \\mathbb{M}} \\ , \n\\end{eqnarray} \nwhere ${\\cal E}_0^{\\chi{\\rm PT}}$ is the $\\chi$PT vacuum energy. In\nthe non-linear realization of the chiral group\nthe Goldstone boson field may be written as\n$U(x)=\\exp{\\left(i\\pi_\\alpha(x) T_\\alpha\/F_\\pi\\right)}$, and the\nleading quark mass contribution to the $\\chi$PT Lagrangian is\n\\begin{eqnarray}\n{\\cal L}^{\\chi{\\rm PT}}_{\\rm\\scriptstyle QCD} \\ =\\ v\\;tr\\left( U \\mathbb{M}^\\dagger + U^\\dagger \\mathbb{M}\\right) \\ +\\ \\ldots \\ ,\n\\end{eqnarray}\nwith $v=M_\\pi^2\\,F_\\pi^2\/\\mathbb{M}$ and with $M_\\pi$ the Goldstone boson mass. One then obtains the Gell-Mann-Oakes-Renner formula~\\cite{GellMann:1968rz}.\n\\begin{eqnarray}\n-{\\mathbb{M}\\,\\langle\\, \\Omega \\, |\\,\\bar\\psi\\psi \\,|\\, \\Omega\\, \\rangle}\\ =\\ \\textstyle{1\\over 2} \\,N\\,M_\\pi^2\\,F_\\pi^2 \\ \\ +\\ \\dots \\ .\n\\label{eq:ifgmor}\n\\end{eqnarray}\nIt will be a principle goal in what follows to determine what takes the place of this relation in null-plane QCD.\n\n\\subsection{Null plane representation}\n\n\\noindent The QCD Lagrangian in the null-plane coordinates is obtained by generalizing the results\ngiven in Appendices~\\ref{FFdecompose} and \\ref{FGdecompose} to the interacting case~\\footnote{We follow the notation\nand conventions of Ref.~\\cite{Belitsky:2005qn}.}. (Note that we work in light-cone gauge, $A^+=0$, throughout.) The \nQCD equations of constraint for the non-dynamical degrees of freedom are\n\\begin{equation}\n\\psi_- \\ =\\ \\frac{1}{2 i\\! \\stackrel{{}_\\rightarrow}{\\partial^+}}\\left( -i \\bit{\\gamma}^r\\!\\stackrel{{}_\\rightarrow}{D^r}\\! \\ +\\ \\mathbb{M} \\right) \\gamma^+\\psi_+\n\\quad , \\quad\n\\psi^\\dagger_- \\ =\\ \\psi^\\dagger_+ \\gamma^- \\left( i \\bit{\\gamma}^r\\!\\stackrel{{}_\\leftarrow}{D^r}\\! \\ -\\ \\mathbb{M} \\right) \\frac{1}{2 i\\! \\stackrel{{}_\\leftarrow}{\\partial^+}}\n\\end{equation}\nfor the redundant quark degrees of freedom, and\n\\begin{eqnarray}\n\\partial^+ A^-_a \\ =\\ \\frac{1}{\\partial^+}\nD^r_{ab} \\partial^+ \\bit{A}^r_{b} \\ -\\ g \\frac{1}{\\partial^+} \\bar\\psi_+ \\gamma^+ t^a \\psi_+ \\ ,\n\\label{eq:freegaugeEOMsol1withI}\n\\end{eqnarray}\nfor the redundant gauge degrees of freedom.\nThe null-plane QCD Lagrangian can then be expressed in terms of the dynamical degrees of freedom as\n\\begin{eqnarray}\n{\\tilde {\\cal L}}_{\\scriptstyle\\rm QCD} & = &\ni \\bar\\psi_+ \\gamma^+ \\partial^- \\psi_+\n-\n\\ft{i}{2}\n\\bar\\psi_+ \\bit{\\gamma}^r \\gamma^+ \\bit{\\gamma}^s\nD^r \\frac{1}{\\partial^+} D^s \\psi_+\n\\nonumber\\\\\n&+&\\!\\!\\!\n\\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\mathbb{M}^2 \\frac{1}{\\partial^+} \\psi_+\n+\n\\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\mathbb{M}\\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)\\frac{1}{\\partial^+}\\psi_+\n-\n\\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\mathbb{M}\\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\psi_+\\right)\n\\nonumber\\\\\n&-&\\!\\!\\!\n\\ft{1}{4} \\bit{F}_{a}^{rs} \\bit{F}_{a}^{rs}\n\\ +\\ \n\\left( \\partial^+ \\bit{A}^r_{a} \\right)\n\\left( \\partial^- \\bit{A}^r_{a} \\right)\n-\n\\ft12\n\\left(\n\\frac{1}{\\partial^+}\nD^r_{ab} \\partial^+ \\bit{A}^r_{b}\n-\ng \\frac{1}{\\partial^+} \\bar\\psi_+ \\gamma^+ t^a \\psi_+\n\\right)^2\n\\, .\n\\end{eqnarray}\nThe price to pay for working with the physical degrees of freedom in\nthe null-plane coordinates is a loss of manifest Lorentz covariance, as well as\nthe appearance of operators which that appear to be non-local in the longitudinal\ncoordinate. As in the instant-form, one should view this Lagrangian\nas providing a perturbative definition of QCD at large momentum\ntransfers, where the longitudinal zero modes play no role.\nNotice that in null-plane QCD there are two kinds of operators that\ndepend on the quark-mass matrix~\\footnote{To minimize clutter, it will prove convenient to \ndefine the operator\n\\begin{eqnarray}\n\\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ \\equiv \\\n\\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)\\frac{1}{\\partial^+}\\psi_+\n-\n\\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\psi_+\\right) \\ .\n\\end{eqnarray}\n}. One is a kinetic term, quadratic in the quark masses, and the other\nis a spin-flip quark-gluon interaction that is linear in the quark masses.\n\nNaturally we expect that null-plane QCD has the same\nsymmetries as instant-form QCD. Consider the $U(1)_R\\otimes U(1)_L$ transformations,\n\\begin{equation}\n\\psi_+\\rightarrow e^{-i\\theta} \\psi_+ \\ \\ \\ , \\ \\ \\ \\psi_+\\rightarrow e^{-i\\theta \\gamma_5} \\psi_+ \\ .\n\\label{chiraltransu1}\n\\end{equation}\nWhile baryon number is unaltered in moving to the null-plane\ncoordinates, this is clearly not the same chiral transformation that\nwe had in the instant form, as that transformation acts on the\nnon-dynamical degrees of freedom, $\\psi_-$, in a distinct manner and\nis therefore complicated on the null-plane. That the chiral symmetry transformations\nare different in the two forms of dynamics is essential for what follows. We will return below to\nthe relation between the chiral symmetries in the instant-form and the\nfront form, as this will be important in understanding the role of\ncondensates on the null-plane. The $U(1)_A$ current and its divergence\nare~\\cite{Wu:2003vn}\n\\begin{equation}\n{\\tilde J}^\\mu_{5}\\ =\\ {J}^\\mu_{5} \\ -\\ i \\bar\\psi \\gamma^\\mu \\gamma^+\\gamma_5\\, \\mathbb{M}\\, \\frac{1}{\\partial^+} \\psi_+\\ ;\n\\label{eq:HME11qcdu1}\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_{5}\\ =\\ \n\\bar\\psi_+ \\gamma^+\\gamma_5 \\, \\mathbb{M}\\, \\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \n\\ -\\ N \\frac{g^2}{16 \\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\;\ntr\\left( F_{\\mu\\nu}F_{\\rho\\sigma}\\right) \\ .\n\\end{equation}\nConsider the $SU(N)_R\\otimes SU(N)_L$ transformations,\n\\begin{equation}\n\\psi_+\\rightarrow e^{-i\\theta_\\alpha T_\\alpha} \\psi_+ \\ \\ \\ , \\ \\ \\ \\psi_+\\rightarrow e^{-i\\theta_\\alpha T_\\alpha \\gamma_5} \\psi_+ \\ .\n\\label{chiraltrans}\n\\end{equation}\nThe currents associated with eq.~\\ref{chiraltrans} are\n\\begin{equation}\n{\\tilde J}^\\mu_\\alpha\\ =\\ {J}^\\mu_\\alpha \\ -\\ \\ft{i}{2} \\bar\\psi \\gamma^\\mu \\gamma^+ \\, [\\, \\mathbb{M}\\, ,\\, T_\\alpha \\, ]\\, \\frac{1}{\\partial^+} \\psi_+\\ ;\n\\end{equation}\n\\begin{equation}\n{\\tilde J}^\\mu_{5\\alpha}\\ =\\ {J}^\\mu_{5\\alpha} \\ -\\ \\ft{i}{2} \\bar\\psi \\gamma^\\mu \\gamma^+\\gamma_5 \\, \\{\\, \\mathbb{M}\\, ,\\, T_\\alpha \\, \\}\\, \\frac{1}{\\partial^+} \\psi_+ \\ ,\n\\end{equation}\nwith divergences\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_\\alpha\\ =\\ \\ft{1}{2} \\bar\\psi \\gamma^+ \\, [\\, \\mathbb{M}^2\\, ,\\, T_\\alpha \\, ]\\, \\frac{1}{\\partial^+} \\psi_+ \\ ;\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}\\ =\\ \\ft{1}{2} \\bar\\psi \\gamma^+ \\gamma_5 \\, [\\, \\mathbb{M}^2\\, ,\\, T_\\alpha \\, ]\\, \\frac{1}{\\partial^+} \\psi_+\n+\\ft{1}{2}\n\\bar\\psi_+ \\gamma^+\\gamma_5 \\{ \\, \\mathbb{M}\\, ,\\, T_\\alpha \\, \\}\\, \\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ .\n\\end{equation}\nFor $N$ degenerate flavors, the quark mass matrix is proportional to the identity, the vector current is conserved, and the axial current\nand the divergence of the axial current are \n\\begin{equation}\n{\\tilde J}^\\mu_{5\\alpha}\\ =\\ {J}^\\mu_{5\\alpha} \\ -\\ i \\bar\\psi \\gamma^\\mu \\gamma^+\\gamma_5\\, T_\\alpha \\, \\mathbb{M}\\, \\frac{1}{\\partial^+} \\psi_+\\ ;\n\\label{eq:HME11qcd}\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}\\ =\\ \n\\bar\\psi_+ \\gamma^+\\gamma_5 \\, T_\\alpha \\, \\mathbb{M}\\, \\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ .\n\\label{eq:chidivqcd}\n\\end{equation}\nHere note in particular that the null-plane axial-vector current in null-plane QCD evidently takes the form, eq.~\\ref{eq:HME11}, expected\non general grounds. \n\n\\subsection{Null-plane charges}\n\n\\noindent The null-plane singlet axial charge is defined as\n\\begin{eqnarray}\n{\\tilde Q}^5 \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{5} \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\bar\\psi_+\\gamma^+ \\gamma_5 \\, \\psi_+ \\ ,\n\\label{eq:QCDu1a}\n\\end{eqnarray}\nwhere we have used eq.~\\ref{eq:HME11qcdu1}. Using the momentum-space representation of $\\psi_+$, given in eq.~\\ref{eq:freediracmomspace}, one finds\n\\begin{eqnarray}\n{\\tilde Q}^5 &=& \\sum_{\\lambda = \\uparrow\\downarrow}2\\lambda\n\\int \\frac{d k^+ d^2 \\bit{k}_\\perp}{2 k^+ (2 \\pi)^3}\n \\left\\{ b^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) b_\\lambda (k^+ , \\bit{k}_\\perp) +\nd^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) d_\\lambda (k^+ , \\bit{k}_\\perp) \\right\\}.\n\\end{eqnarray}\nComparison with eq.~\\ref{eq:freehelquark} one sees that the singlet axial charge coincides (up to a factor of two) with the free-fermion helicity operator.\nThis of course explains why the quark mass term in the free-fermion theory is a chiral invariant; on the null-plane, chiral symmetry breaking\nin the free-fermion theory implies breaking of rotational invariance in the transverse plane.\n\nSimilarly, the null-plane non-singlet vector and chiral charges are, respectively, \n\\begin{eqnarray}\n{\\tilde Q}_\\alpha \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{\\alpha} \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\bar\\psi_+\\gamma^+ \\, T_\\alpha \\psi_+ \\ ; \\\\\n{\\tilde Q}^5_\\alpha \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{5\\alpha} \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\bar\\psi_+\\gamma^+ \\gamma_5 \\, T_\\alpha \\psi_+ \\ ,\n\\label{eq:QCDcc}\n\\end{eqnarray}\nand using the momentum-space representation of $\\psi_+$, given in eq.~\\ref{eq:freediracmomspace}, one finds\n\\begin{eqnarray}\n\\hskip-1.9em{\\tilde Q}_\\alpha &=& \\sum_{\\lambda = \\uparrow\\downarrow}\n\\int \\frac{d k^+ d^2 \\bit{k}_\\perp}{2 k^+ (2 \\pi)^3}\n \\left\\{ b^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T_\\alpha b_\\lambda (k^+ , \\bit{k}_\\perp) -\nd^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T^T_\\alpha d_\\lambda (k^+ , \\bit{k}_\\perp) \\right\\} \\ ; \\\\\n\\hskip-1.9em{\\tilde Q}^5_\\alpha &=& \\sum_{\\lambda = \\uparrow\\downarrow}2\\lambda\n\\int \\frac{d k^+ d^2 \\bit{k}_\\perp}{2 k^+ (2 \\pi)^3}\n \\left\\{ b^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T_\\alpha b_\\lambda (k^+ , \\bit{k}_\\perp) +\nd^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T^T_\\alpha d_\\lambda (k^+ , \\bit{k}_\\perp) \\right\\}.\n\\end{eqnarray}\nOne readily checks that the null-plane chiral algebra, eqs.~\\ref{eq:LCalga} and \\ref{eq:LCalgb}, is satisfied by these charges. \nAs these charge are written as sums of number operators that count the number\nof quarks and anti-quarks, both chiral charges annihilate the vacuum, and we have\n\\begin{eqnarray}\n{\\tilde Q}^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, {\\tilde Q}_5^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, 0 \\ ,\n\\label{eq:vacuumann2}\n\\end{eqnarray}\nas expected on the general grounds presented above. One then has\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha\\, ,\\, \\psi_+ \\, ]\\, =\\, -T^\\alpha\\,\\psi_+ \\ \\ ; \\ \\ \n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, \\psi_+ \\, ]\\, =\\, -\\gamma_5\\,T^\\alpha\\,\\psi_+ \\ .\n\\end{eqnarray}\nBreaking down the fields into left- and right-handed components, \n\\begin{eqnarray}\n\\psi_{+R} \\ =\\ \\ft{1}{2} (1+\\gamma_5) \\psi_+ \\qquad , \\qquad \n\\psi_{+L} \\ =\\ \\ft{1}{2} (1-\\gamma_5) \\psi_+ \n\\end{eqnarray}\nand, using the results of Appendix~\\ref{FFdecompose}, one verifies the fermion transformation properties with respect to $SU(N)_R\\otimes SU(N)_L$:\n\\begin{eqnarray}\n\\psi_{+R}\\ =\\ \\psi_{+\\uparrow}\\ &\\in& \\ ({\\bf 1},{\\bf N}) \\qquad , \\qquad \\psi_{+R}^\\dagger \\ =\\ \\psi_{+\\downarrow}^\\dagger\\ \\in \\ ({\\bf 1},\\bar{\\bf N})\\ ; \\\\\n\\psi_{+L}\\ =\\ \\psi_{+\\downarrow}\\ &\\in& \\ ({\\bf N},{\\bf 1}) \\qquad , \\qquad \\psi_{+L}^\\dagger \\ =\\ \\psi_{+\\uparrow}^\\dagger\\ \\ \\in \\ (\\bar{\\bf N},{\\bf 1})\\ ,\n\\label{eq:chiralassign}\n\\end{eqnarray}\nand the helicity eigen-equations of the quarks\n\\begin{eqnarray}\n\\Sigma_{12}\\;\\psi_{+\\uparrow}& =& {\\textstyle\\frac{1}{2}}\\psi_{+\\uparrow} \\ ;\\\\\n\\Sigma_{12}\\;\\psi_{+\\downarrow}& =& -{\\textstyle\\frac{1}{2}}\\psi_{+\\uparrow} \\ ,\n\\end{eqnarray}\nwhere the helicity operator, $\\Sigma_{12}$, is defined in Appendix~\\ref{FFdecompose}.\n\n\\subsection{Chiral symmetry breaking Hamiltonians}\n\n\\noindent Using the results of the previous section, it is\nstraightforward to find the transformation properties of the\nsymmetry-breaking parts of the reduced Hamiltonians. Define the\noperators~\\footnote{From here forward we will use the definition:\n\\begin{equation}\n\\frac{1}{\\partial_{\\mathbb{M}}^+} \\equiv \\mathbb{M}\\,\\frac{1}{\\partial^+} \\ .\\nonumber\n\\end{equation}\n}:\n\\begin{equation}\n{\\tilde D}_{5\\alpha} \\ \\equiv \\ \\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}\\ =\\ \n\\bar\\psi_+ \\gamma^+\\gamma_5 \\, T_\\alpha \\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ;\n\\label{eq:axcurr2}\n\\end{equation}\n\\begin{equation}\n{\\tilde D}_{5} \\ \\equiv \\ \\bar\\psi_+ \\gamma^+\\gamma_5 \\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ;\n\\end{equation}\n\\begin{equation}\n{\\tilde D}_{\\alpha} \\ \\equiv \\ \\bar\\psi_+ \\gamma^+\\, T_\\alpha \\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ;\n\\end{equation}\n\\begin{equation}\n{\\tilde D} \\ \\equiv \\ \\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ .\n\\end{equation}\nIt is then a textbook exercise to find\n\\begin{eqnarray}\n&&\\hfill[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}^\\beta_5 \\, ] \\, =\\, \\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\;{\\tilde D} \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}^\\gamma \\ ; \n\\label{eq:algsola}\\\\\n&&\\hfill[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}\\, ] \\, =\\, 2\\;{\\tilde D}_5^\\alpha \\ ; \n\\label{eq:algsolb}\\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}^\\beta \\, ]\\, =\\, \\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\;{\\tilde D}_5 \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}_5^\\gamma \\ ; \n\\label{eq:algsolc}\\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}_5\\, ]\\, =\\, 2\\;{\\tilde D}^\\alpha \\ .\n\\label{eq:algsold}\n\\end{eqnarray}\nIt follows that the $2N^2$ operators $({\\tilde D}_{5\\alpha},{\\tilde D}_{5},{\\tilde D}_{\\alpha},{\\tilde D})$ fill out the\n$(\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ representation of $SU(N)_R\\otimes SU(N)_L$. \n\nThe null-plane Hamiltonian $P^-$ is:\n\\begin{eqnarray}\nP^- \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {T}^{-+}\\ ,\n\\end{eqnarray}\nand therefore the chiral-symmetry breaking part of this Hamiltonian is given by:\n\\begin{eqnarray}\nP_{({\\bf N},{\\bf N})}^- \\ \\equiv \\ -\\ft{i}{2}\\,\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde D} \\ .\n\\end{eqnarray}\nOne readily checks that this is consistent with eqs.~\\ref{eq:mixedchiralc} and \\ref{eq:axcurr2}.\n\nOne then finds the symmetry breaking parts of the reduced QCD Hamiltonians:\n\\begin{eqnarray}\nM^2_{({\\bf N},{\\bf N})} & = & -i P^+ \\,\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ; \n\\label{eq:symmbreakhamsa}\\\\\n\\left(M{\\cal J}_r\\right)_{({\\bf N},{\\bf N})} & = & i{\\textstyle{\\frac{1}{2}}}\\epsilon_{rs}\\,P^+\n{\\int}\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\Gamma}_s\\, \\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ,\n\\label{eq:symmbreakhamsb}\n\\end{eqnarray}\nwhere, in addition, we have used eqs.~\\ref{eq:mixedchirald} and \\ref{eq:chidivqcd} to obtain the reduced Hamiltonian for spin.\nAll chiral symmetry breaking in null-plane QCD is contained in these two operators. \n\nUsing eqs.~\\ref{eq:mixedchiralc}, \\ref{eq:mixedchirald} and \\ref{eq:algsola} one finds\n\\begin{eqnarray}\n{}[{\\tilde Q}^\\beta_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M^2\\, ]] &=& -2iP^+\\;\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\;\\left(\\;\n\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,{\\tilde D} \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}^\\gamma\\; \\right) \\ ; \\label{eq:fundcona}\\\\\n{}[{\\tilde Q}^\\beta_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M{\\cal J}_r\\, ]] &=& i\\epsilon_{rs}P^+\\;\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\Gamma_s\\;\\left(\\;\n\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,{\\tilde D} \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}^\\gamma\\; \\right) \\ .\n\\label{eq:fundconb}\n\\end{eqnarray}\nActing on these equations with $\\delta^{\\alpha\\beta}$ and $d^{\\alpha\\beta\\gamma}$, and using the identities in Appendix~\\ref{suN} gives\n\\begin{eqnarray}\n&&-2iP^+ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde D} \\ =\\ \\frac{N}{N^2-1}\\, [{\\tilde Q}^\\alpha_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M^2\\, ]]\\ ; \\\\\n&&-2iP^+ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde D}_\\gamma \\ =\\ d_{\\alpha\\beta\\gamma}\\frac{N}{N^2-4}\\, [{\\tilde Q}^\\beta_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M^2\\, ]]\\ .\n\\end{eqnarray}\nTherefore, eq.~\\ref{eq:fundcona} can be written as\n\\begin{eqnarray}\n\\hskip-1.4em[{\\tilde Q}^\\beta_5 , [ {\\tilde Q}^\\alpha_5 , M^2]]=\\frac{1}{N^2-1}\\delta^{\\alpha\\beta}[{\\tilde Q}^\\gamma_5 , [ {\\tilde Q}^\\gamma_5 , M^2 ]]\n+ \\frac{N}{N^2-4}d^{\\alpha\\beta\\gamma}d^{\\mu\\nu\\gamma}[ {\\tilde Q}^\\mu_5 , [ {\\tilde Q}^\\nu_5 , M^2 ]]\\ ,\n\\end{eqnarray} \nand eq.~\\ref{eq:fundconb} takes the same form but with $M^2$ replaced by $M {\\cal J}_\\pm$. Defining the projection operator\n\\begin{eqnarray} {\\cal P}^{\\alpha\\beta ; \\mu\\nu}\\ \\equiv\\\n\\delta^{\\alpha\\nu}\\delta^{\\beta\\mu}\\ -\\\n\\frac{1}{N^2-1}\\;\\delta^{\\alpha\\beta}\\delta^{\\mu\\nu} \\ -\\\n\\frac{N}{N^2-4}\\;d^{\\alpha\\beta\\gamma}d^{\\mu\\nu\\gamma} \\ , \n\\end{eqnarray} \nwe can express the constraints on the reduced Hamiltonians in compact notation as:\n\\begin{eqnarray}\n{\\cal P}^{\\alpha\\beta ;\\mu\\nu}\\;[{\\tilde Q}^\\mu_5\\, ,\\, [\\, {\\tilde Q}^\\nu_5\\, ,\\, M^2]]\\ =\\ {\\cal P}^{\\alpha\\beta ;\\mu\\nu}\\;[{\\tilde Q}^\\mu_5\\, ,\\, [\\, {\\tilde Q}^\\nu_5\\, ,\\, M {\\cal J}_\\pm]]\\ =\\ 0 \\ .\n\\label{eq:fundmix}\n\\end{eqnarray} \nThese are quite possibly the most important equations in null-plane\nQCD, as they are the mathematical expression of the specific way in\nwhich the internal symmetries and Poincar\\'e symmetries intersect.\nThese equations were obtained originally in\nRefs.~\\cite{Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn} by\nconsidering the most general form of Goldstone-boson-hadron scattering\namplitudes in specially-designed Lorentz frames, and using input from Regge-pole\ntheory expectations of their high-energy behavior. Note that the\nprojection operator, ${\\cal P}^{\\alpha\\beta ; \\mu\\nu}$, has four\nadjoint indices and is, as shown in Ref.~\\cite{Weinberg:1969hw}\nrelated to the interactions of Goldstone bosons (in the t-channel of\nGoldstone-boson-hadron scattering), which are in the adjoint of\n$SU(N)_F$ and whose scattering amplitudes therefore transform as the\nproduct of two adjoints. In the case of two flavors, where ${\\bf\n 3}\\otimes{\\bf 3}={\\bf 1}\\oplus{\\bf 3}\\oplus{\\bf 5}$, it projects out\nthe ${\\bf 5}$-dimensional representation ($I=2$) and in the case of\nthree flavors, where ${\\bf 8}\\otimes{\\bf 8}={\\bf 1}\\oplus{\\bf\n 8}\\oplus{\\bf 8}\\oplus{\\bf 10}\\oplus\\bar{\\bf 10}\\oplus{\\bf 27}$, it\nprojects out the ${\\bf 10}$, $\\bar{\\bf 10}$, and ${\\bf\n 27}$-dimensional representations. As shown above, these are the\nrepresentations that cannot be formed from a single quark bilinear;\ni.e. they are not contained in $(\\bar{\\bf N},{\\bf N})\\oplus({\\bf\n N},\\bar{\\bf N})$, as is clear from direct inspection of\neqs.~\\ref{eq:fundcona} and \\ref{eq:fundconb}.\n\n\n\\subsection{Gell-Mann-Oakes-Renner relation recovered}\n\n\\noindent We are now in a position to address the fate of instant-form QCD chiral-symmetry breaking condensates\nin null-plane QCD. Again using the Feynman-Hellmann theorem we find\n\\begin{eqnarray} \n\\mathbb{M}\\,\\langle\\, 0 \\, |\\,\\frac{\\partial T^{-+}}{\\partial \\mathbb{M}}\\,|\\, 0\\, \\rangle &=& \\mathbb{M}\\,\\frac{\\partial {\\tilde {\\cal E}}_0}{\\partial\n \\mathbb{M}}\\ =\\ \n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0}{\\partial \\mathbb{M}}\\ =\\\n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0^{\\chi{\\rm PT}}}{\\partial\n \\mathbb{M}} \\ , \n\\end{eqnarray} \nwhere $|\\, 0\\, \\rangle$ represents the null-plane QCD vacuum\nstate, and ${\\tilde {\\cal E}}_0$ is the null-plane QCD vacuum energy. In this equation we have also expressed\nthat physics is independent of the choice of coordinates. Therefore calculation\nof the leading quark-mass contribution to the vacuum energy must be independent of the quantization surface,\nand should be the same whether one works with the fundamental degrees of freedom, or with the Goldstone bosons\nin the infrared. One then has\n\\begin{eqnarray} \n\\mathbb{M}\\,\\frac{\\partial {\\tilde {\\cal E}}_0}{\\partial \\mathbb{M}}\\ =\\ -\\mathbb{M}\\,\\langle\\, 0\\, |\\,i\\;\\bar\\psi_+ \\gamma^+\n\\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\,|\\, 0\\, \\rangle +\n\\langle\\, 0\\, |\\, \\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \n\\,|\\, 0\\, \\rangle \\ .\n\\end{eqnarray} \nThe second term must vanish as the chiral charges annihilate the vacuum and therefore there can be no\nchiral-symmetry breaking condensates. Operationally one sees this directly by \ntaking the vacuum expectation value of eq.~\\ref{eq:algsola} which gives\n\\begin{eqnarray} \n\\langle\\, 0\\, |\\, \\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \n\\,|\\, 0\\, \\rangle \\ =\\ 0.\n\\end{eqnarray} \nWe are then left with the null-plane expression of the Gell-Mann-Oakes-Renner relation:\n\\begin{eqnarray} {\\mathbb{M}\\,\\langle\\, 0\\, |\\,i\\;\\bar\\psi_+ \\gamma^+\n \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\,|\\, 0\\, \\rangle}\\ =\\ \\textstyle{1\\over 2} \\,\nN\\,M_\\pi^2\\,F_\\pi^2 \\ \\ +\\ \\dots \\ . \n\\label{eq:npgoar}\n\\end{eqnarray} Hence, a chiral-symmetry breaking condensate in the instant-form\nformulation of QCD has been replaced by a chiral-symmetry conserving\ncondensate in the null-plane formulation. Note that while the operator\nnaively vanishes in the chiral limit, the matrix element is infrared\nsingular and therefore it need not, and indeed cannot, vanish in the\nchiral limit~\\footnote{This expression of the Gell-Mann-Oakes-Renner\n formula was found previously in Ref.~\\cite{Wu:2003vn} using the\n methods that will be described below.}. It would be very\ninteresting to define the relevant operator non-perturbatively and\ncalculate this condensate directly, perhaps using transverse lattice\ngauge theory\nmethods~\\cite{Bardeen:1979xx,Burkardt:1998ws,Dalley:1998bj,Dalley:2001gj,Burkardt:2001mf,Burkardt:2001dy,Burkardt:2001jg,Dalley:2002nj,Bratt:2004wq}. Note\nthat {\\it a priori} knowledge of the singlet condensate in\neq.~\\ref{eq:npgoar} is not very different to {\\it a priori} knowledge of the symmetry-breaking quark condensate in\neq.~\\ref{eq:ifgmor}. In both cases, it is necessary to keep the quark masses finite and only\nat the very end take the chiral limit~\\cite{Banks:1979yr}.\n\n\n\\subsection{Condensates on a null-plane}\n\n\\noindent We will now derive the Gell-Mann-Oakes-Renner relation in a different way\nwhich will suggest a general prescription for expressing all\ninstant-form condensates with null-plane condensates.\nWhile the left- and right-handed components of $\\psi_+$ transform irreducibly with respect\nto the null-plane chiral charges, the transformation properties of $\\psi$ are complicated\nby the presence of the non-dynamical component $\\psi_-$. Indeed one finds\n\\begin{eqnarray} [\\, {\\tilde Q}^\\alpha_5\\, ,\\, \\psi \\, ]\\, =\\,\n-\\gamma_5\\,T^\\alpha\\,\\psi \\ -\\\ni\\;\\gamma_5\\;\\gamma^+\\;T^\\alpha\\;\\frac{1}{\\partial^+_\\mathbb{M}} \\psi \\ , \n\\label{eq:psitrans}\n\\end{eqnarray} \nfrom which it follows that \n\\begin{eqnarray}\n\\psi_{R}\\, ,\\, \\psi_{L}\\ \\in \\ ({\\bf 1},{\\bf N})\\oplus ({\\bf N},{\\bf 1}) \\quad , \\quad\n\\psi_{R}^\\dagger\\, ,\\,\\psi_{L}^\\dagger \\ \\in \\ ({\\bf 1},\\bar{\\bf\n N})\\oplus(\\bar{\\bf N},{\\bf 1}) \\ . \\end{eqnarray} \nSince the left- and right-handed components of the quark field\ntransform reducibly with respect to the chiral group, generally\nproducts of bilinear operators of the form $\\bar\\psi\\Gamma\\psi$ will have\ncomplicated reducible chiral transformation properties. However, QCD\noperators built out of these bilinears will always have a component that\ntransforms as a chiral singlet. We will now see, for the simplest\nexample, that this is essential to the consistency of the null-plane\nformulation. Consider the transformation properties of the following\nset of bilinears:\n\\begin{eqnarray}\n{D}^\\alpha_{5} \\ &\\equiv& \\ \\bar\\psi\\,\\gamma_5\\,T^\\alpha\\,\\psi\\ \\ , \\ \\\n{D}_{5} \\ \\equiv \\ \\bar\\psi\\,\\gamma_5\\,\\psi\\ ; \\\\\n{D}^{\\alpha} \\ &\\equiv& \\ \\bar\\psi\\,T^\\alpha\\,\\psi\\ \\ \\ , \\ \\ \\ \\\n{D} \\ \\equiv \\ \\bar\\psi\\,\\psi\\ .\n\\end{eqnarray}\nIs is again simple to check that these operators fill out the\n$(\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ representation of $SU(N)_R\\otimes SU(N)_L$\nwith respect to the instant-form chiral charges $Q_5^\\alpha$.\nNow consider the transformation properties of these operators \nwith respect to the null-plane chiral charges. One finds\n\\begin{eqnarray} \n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D}^\\beta_{5} \\, ]\\,\n=\\, -\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,\\left(\\,D + \\\n i\\;\n \\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\right) \\label{eq:GORf} \\nonumber \\\\\n&&\\quad\\qquad\\qquad\\qquad-\\,d^{\\alpha\\beta\\gamma}\\left(\\,D^\\gamma \\ +\\\n i\\;\\bar\\psi_+ \\gamma^+ T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}}\n \\psi_+\\right) \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D} \\, ]\\,\n=\\, -2\\,D^\\alpha_5 \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D}^\\beta \\, ]\\,\n=\\, -\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,D_5 \\ -\\ d^{\\alpha\\beta\\gamma}\\,D_5^\\gamma\\nonumber\\\\\n&&\\quad\\qquad\\qquad\\qquad\n +\\,f^{\\alpha\\beta\\gamma}\\,\\bar\\psi_+ \\gamma^+ \\gamma_5 T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}}\n \\psi_+\\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D}_5 \\, ]\\,\n=\\, -2\\,D^\\alpha\\ -\\ 2i \\bar\\psi_+ \\gamma^+ T^\\alpha \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ .\n\\end{eqnarray} \nTo close the algebra we must add, in addition, the commutation relations:\n\\begin{eqnarray} \n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\, ]\\ =\\ 0 \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,\\bar\\psi_+ \\gamma^+ T^\\beta \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\, ]\\ =\\ \ni\\,f^{\\alpha\\beta\\gamma} \\, \\bar\\psi_+ \\gamma^+ \\gamma_5 T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,\\bar\\psi_+ \\gamma^+ \\gamma_5 T^\\beta \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\, ]\\ =\\ \ni\\,f^{\\alpha\\beta\\gamma} \\, \\bar\\psi_+ \\gamma^+ T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ .\n\\end{eqnarray}\nHence the full set of operators transform as the reducible $4N^2$-dimensional\n$({\\bf 1},{\\bf 1})\\oplus({\\bf 1},{\\bf {\\cal A}})\\oplus({\\bf {\\cal A}},{\\bf 1})\\oplus (\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ \nrepresentation of $SU(N)_R\\otimes SU(N)_L$, where here ${\\bf {\\cal A}}$ denotes the $SU(N)$ adjoint\nrepresentation. In particular one see that\n\\begin{eqnarray}\n\\bar\\psi\\psi \\in \\ (\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})\\oplus({\\bf 1},{\\bf 1})\\oplus\\ldots \\ ,\n\\end{eqnarray} \nand therefore transforms reducibly. This is verified by direct calculation which gives~\\footnote{Note that the second term, which is breaks chiral symmetry\nand is independent of the interaction does not appear in the free fermion Lagrangian as it is cancelled by a piece coming from the other kinetic term, as must be the\ncase in order that the Lagrangian commute with the helicity operator. }\n\\begin{eqnarray}\n\\hskip-1.3em\\mathbb{M}\\,\\bar\\psi\\psi \\ =\\ -i\\,\\mathbb{M}\\,\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ +\\ \n\\ft{i}{2}\\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+\\ .\n\\label{eq:psibarpsiexpl}\n\\end{eqnarray} \nTaking the vacuum expectation value of eq.~\\ref{eq:psibarpsiexpl} (or eq.~\\ref{eq:GORf}) gives the general\nsolution~\\cite{Wu:2003vn}\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,\\bar\\psi\\psi\\,|\\, 0\\, \\rangle\\ = \\ -\\langle\\, 0\\, |\\,i\\;\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\,|\\, 0\\, \\rangle \\ .\n\\label{eq:WuZhang}\n\\end{eqnarray} \nTherefore only the singlet part of $\\bar\\psi\\psi$ can acquire a vacuum\nexpectation value on the null plane, as must be the case since\n$SU(N)_R\\otimes SU(N)_L$ is a symmetry of the null-plane vacuum\nstate. This argument readily generalizes to any chiral symmetry\nbreaking Lorentz scalar operator, ${\\mathcal O}$, that one can build\nout of products of quark bilinears in instant-form QCD. One can write\n\\begin{eqnarray}\n{\\mathcal O} \\ =\\ \\sum_{\\bf{\\mathcal R}}\\,{\\mathcal O}_{\\bf{\\mathcal R}} \\ =\\ \\sum_{\\tilde{\\bf{\\mathcal R}}}\\,{\\cal O}_{\\tilde{\\bf{\\mathcal R}}}\\ + \\ {\\mathcal O}_{\\tilde{\\bf 1}} \n\\label{eq:gencondensate}\n\\end{eqnarray} \nwhere $\\bf{\\mathcal R}$ is a non-trivial chiral representation with respect to the instant-form chiral charges, $Q_{5\\alpha}$,\nand $\\tilde{\\bf{\\mathcal R}}$ (${\\tilde{\\bf 1}}$) is a non-trivial (the singlet) representation with respect to the \nfront-form chiral charges, ${\\tilde Q}_{5\\alpha}$. Unless protected by another symmetry, ${\\mathcal O}$ has a non-vanishing \nvacuum expectation value, which can be expressed as\n\\begin{eqnarray}\n\\langle\\, \\Omega \\, |\\,{\\mathcal O} \\,|\\, \\Omega\\, \\rangle\\ =\\ \n\\langle\\, \\Omega \\, |\\,\\sum_{\\bf{\\mathcal R}}\\,{\\mathcal O}_{\\bf{\\mathcal R}}\\,|\\, \\Omega\\, \\rangle\\ =\\ \n\\langle\\, 0 \\, |\\,{\\mathcal O}_{\\tilde{\\bf 1}} \\,|\\, 0\\, \\rangle \\ \\neq \\ 0 \\ .\n\\label{eq:gencond}\n\\end{eqnarray}\nNote that the final equality expresses an equivalence between a matrix element evaluated\nin the instant form and one in the front form. This equality ensures that physics is\nunmodified in moving between the two forms of dynamics.\nTherefore all instant-form chiral symmetry breaking QCD condensates map to chiral\nsymmetry conserving condensates in the front-form. The presence of the singlet\npart of the operator can always be traced to the reducible chiral transformation\nproperty of $\\psi$ given in eq.~\\ref{eq:psitrans}. For the case at hand, with \n${\\mathcal O}=\\bar\\psi\\psi$, we have\n\\begin{eqnarray}\n\\langle\\, \\Omega\\, |\\,\\bar\\psi\\psi\\,|\\, \\Omega\\, \\rangle \\ = \\ \\langle\\, 0\\, |\\,\\bar\\psi\\psi\\,|\\, 0\\, \\rangle \\ ,\n\\label{eq:psibarpsicond}\n\\end{eqnarray} \nwhich together with eq.~\\ref{eq:WuZhang}, provides the desired link between the instant-form and front-form expressions of\nthe Gell-Mann-Oakes-Renner relation.\n\nThe general relation, eq.~\\ref{eq:gencond} is important for the\nconsistency of null-plane QCD, as it demonstrates that, as expected,\nthe QCD vacuum energy is unaltered in moving from the instant-form to\nthe front-form description, and these relations must, of course, exist\nin order that the operator product expansion be independent of the\nchoice of quantization surface. We see that a symmetry-breaking\ncondensate can form in the instant-form coordinates with an asymmetric\nvacuum which is equal to a corresponding symmetry-preserving\ncondensate in the null-plane description with a symmetric vacuum. The\ncondensate relation eq.~\\ref{eq:psibarpsicond} is one of an infinite\nnumber of relations which translates condensates which break chiral\nsymmetry in the instant form to null-plane condensates which transform\nas chiral singlets.\n\n\\section{Consequences of the operator algebra}\n\\label{sec:consequences}\n\n\\subsection{Summary of the null-plane QCD description}\n\n\\noindent Before considering the consequences of the null-plane QCD\noperator algebra, we will summarize the picture of chiral symmetry\nbreaking that we have so far established. While the null-plane QCD\nvacuum state is chirally invariant, chiral symmetry is spontaneously\nbroken by the three reduced Hamiltonians that have contributions,\n$M^2_{({\\bf N},{\\bf N})}$ and $\\left(M{\\cal J}_r\\right)_{({\\bf N},{\\bf\n N})}$, which transform as $(\\bar{\\bf N},{\\bf N})\\oplus({\\bf\n N},\\bar{\\bf N})$ with respect to $SU(N)_R\\otimes SU(N)_L$. The\nthree reduced Hamiltonians satisfy the constraints,\neq.~\\ref{eq:fundmix}. In addition to these signatures of chiral\nsymmetry breaking, the three reduced Hamiltonians, together with the\ngenerator of rotations on the transverse plane together generate the\n$U(2)$ dynamical sub-group of the null-plane Poincar\\'e algebra,\neq.~\\ref{eq:dynalgarbB}. And finally, the null-plane vector and\nchiral charges satisfy the $SU(N)_R\\otimes SU(N)_L$ algebra,\neqs.~\\ref{eq:LCalga} and \\ref{eq:LCalgb}. The entire set of\nLie-brackets provide all of the constraints that exist among the\ngenerators of the internal and space-time symmetries in null-plane\nQCD. The consequences of chiral symmetry breaking for the spectrum\nand spin of QCD are contained in the symmetry-breaking parts of the\nreduced Hamiltonians.\n\n\\subsection{Recovery of spin-flavor symmetries}\n\n\\noindent In searching for solutions of the algebraic system that\nmixes the chiral charges and the reduced Hamiltonians, one may worry\nabout the existence of no-go theorems that forbid non-trivial algebras\nthat mix space-time and internal symmetries. In the null-plane\nformulation the no-go theorems are avoided because it is only the\ndynamical part, ${\\cal D}$, of the null-plane Poincar\\'e algebra that\nmixes with the internal symmetry generators~\\cite{Leutwyler:1977vz}.\nUnfortunately, a direct general solution of the null-plane QCD\noperator algebra in the general case appears difficult. However, there\nis a limiting case in which the algebra yields an important\nnon-trivial solution. Here we will treat the QCD operator algebra as\nan abstract operator algebra and consider the limit in which the\nchiral-symmetry breaking part of the reduced Hamiltonian $M^2$ can be\ntreated as a perturbation. However, one should keep in mind that\nmatrix elements of the operator relations between hadronic states\nmust eventually be taken in order to extract observables.\nWe first define\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, M\\, ] \\, \\equiv \\, \\epsilon^\\alpha \\ ,\n\\end{eqnarray}\nand neglect terms of ${\\cal O}(\\epsilon)$. This implies that all\nchiral symmetry breaking occurs in the spin Hamiltonians.\nThis limit is non-trivial, as we have shown above in \nsection~\\ref{susec:spin} that the spin Hamiltonians alone imply the presence\nof Goldstone bosons. In this limit, the QCD operator algebra reduces to\n\\begin{eqnarray}\n[\\, {\\cal J}_i \\, ,\\, {\\cal J}_j \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\, {\\cal J}_k\n\\end{eqnarray}\nwhich generates $SU(2)$ spin, and the $SU(N)_R\\otimes SU(N)_L$ algebra,\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha\\, ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\, i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma \\ \\ , \\ \\ \n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\, i\\,f^{\\alpha\\beta\\gamma}\\, {\\tilde Q}^\\gamma_5 \\ \\ , \\ \\\n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde Q}^\\beta_5\\, ] \\, = \\, i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma \\ .\n\\label{eq:QCDoa}\n\\end{eqnarray}\nThe remaining non-trivial mixed commutator is for the spin Hamiltonian:\n\\begin{eqnarray} \n{\\cal P}^{\\alpha\\beta ;\\mu\\nu}\\;[{\\tilde Q}^\\mu_5\\, ,\\, [\\, {\\tilde\n Q}^\\nu_5\\, ,\\, {\\cal J}_\\pm]]\\ =\\ 0 \\ . \n\\label{eq:OPQcomm2b}\n\\end{eqnarray} \nNow this simplified algebra can be put into a more familiar form. Consider\nan operator $G_{\\alpha i}$ which transforms in the adjoint of $SU(N)$ and as a rotational\nvector in the sense that\n\\begin{eqnarray}\n&&[\\, {\\cal J}_i \\, ,\\, {G}_{\\alpha j} \\, ] \\ = \\ i\\,\\epsilon_{ijk}\\, {G}_{\\alpha k} \\ ; \\\\\n&&[\\, {\\tilde Q}_\\alpha \\, ,\\, {G}_{\\beta i} \\, ] \\ = \\ i\\,f_{\\alpha\\beta\\gamma}\\, {G}_{\\gamma i} \\ .\n\\end{eqnarray}\nIn general, the commutator of $G^{\\alpha i}$ with itself may be expressed as\n\\begin{eqnarray}\n[\\, G_{\\alpha i} \\, ,\\, G_{\\beta j} \\, ] \\, =\\, \ni\\,f_{\\alpha\\beta\\gamma}\\, {\\cal A}_{ij,\\gamma}\\, +\\, i\\,\\epsilon_{ijk}\\, {\\cal B}_{\\alpha\\beta,k} \\ ,\n\\label{eq:su4commgen}\n\\end{eqnarray}\nwhere ${\\cal A}_{ij,\\gamma}={\\cal A}_{ji,\\gamma}$ and ${\\cal\n B}_{\\alpha\\beta,k}={\\cal B}_{\\beta\\alpha,k}$. Now we identify\n$G^{\\alpha 3}\\ \\equiv \\ {\\tilde Q}^\\alpha_5$. From\neq.~\\ref{eq:QCDoa} it then follows that ${\\cal\n A}_{33,\\alpha}={\\tilde Q}_\\alpha$. Rotational invariance then\nimplies ${\\cal A}_{ij,\\alpha}=\\delta_{ij}{\\tilde Q}_\\alpha$. By\nconsidering Jacobi identities of ${\\cal J}_i$ and ${\\tilde Q}_\\alpha$\nwith the commutator in eq.~\\ref{eq:su4commgen} one finds, respectively,\n\\begin{eqnarray}\n&&[\\, {\\tilde Q}_\\gamma \\, ,\\, {\\cal B}_{\\alpha\\beta,i} \\, ] \\, =\\, i\\,f_{\\gamma\\beta\\mu}\\, {\\cal B}_{\\alpha\\mu,i} \\, +\\, i\\,f_{\\gamma\\alpha\\mu}\\, {\\cal B}_{\\beta\\mu,i}\\ ; \\\\\n&&[\\, {\\cal J}_i \\, ,\\, {\\cal B}_{\\alpha\\beta,j} \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\, {\\cal B}_{\\alpha\\beta,k}\\ ,\n\\end{eqnarray}\nwhich simply indicate that ${\\cal B}_{\\alpha\\beta,i}$ transforms as a rank-two $SU(N)$ tensor and a\nrotational vector.\n\nTo obtain ${\\cal B}_{\\alpha\\beta,i}$ we use eq.~\\ref{eq:OPQcomm2b} to find:\n\\begin{eqnarray}\n{\\cal P}_{\\alpha\\beta ;\\mu\\nu}\\;[\\, G_{\\alpha 3}\\, ,\\, G_{\\beta 1}\\, \\pm\\, i\\, G_{\\beta 2} \\, ] \\ = \\ 0 \\ , \n\\label{eq:su4commgen3}\n\\end{eqnarray} \nfrom which it follows that ${\\cal B}_{\\alpha\\beta,2}$ and ${\\cal B}_{\\alpha\\beta,1}$ have a piece proportional\nto $\\delta_{\\alpha\\beta}$ and a piece proportional to $d_{\\alpha\\beta\\gamma}$. \nRotational invariance then determines that ${\\cal B}_{\\alpha\\beta,i}$ is a linear combination of\n$\\delta_{\\alpha\\beta}{\\cal J}_i$ and $d_{\\alpha\\beta\\gamma}G_{\\gamma i}$. The coefficients of these\nterms are determined by considering the Jacobi identity of $G_{\\alpha i}$ with the commutator \nin eq.~\\ref{eq:su4commgen}, together with the relation among $SU(N)$ structure constants\ngiven in Appendix~\\ref{suN}. Finally, one obtains\n\\begin{eqnarray}\n[\\, G_{\\alpha i} \\, ,\\, G_{\\beta j} \\, ] \\, =\\, \ni\\,\\delta_{ij}\\,f_{\\alpha\\beta\\gamma}\\, {\\tilde Q}_\\gamma\\, +\\, \\frac{2}{N}\\,i\\,\\delta_{\\alpha\\beta}\\,\\epsilon_{ijk}\\,{\\cal J}_k \\, + \\, i\\epsilon_{ijk}\\,d_{\\alpha\\beta\\gamma}\\, {G}_{\\gamma k} \\ ,\n\\label{eq:su2N}\n\\end{eqnarray}\nwhich together with \n\\begin{eqnarray} &&[\\, {\\tilde Q}_\\alpha\\, ,\\, G_{\\beta i}\\, ]\\, =\\,\ni\\,f_{\\alpha\\beta\\gamma} \\, G_{\\gamma i} \\ \\ \\ , \\ \\ \\\n[\\, {\\cal J}_i \\, ,\\, G_{\\alpha j} \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\, G_{\\alpha k} \\ ; \\\\\n&&[\\, {\\tilde Q}_\\alpha\\, ,\\, {\\tilde Q}_\\beta\\, ]\\, =\\,\ni\\,f_{\\alpha\\beta\\gamma} \\, {\\tilde Q}_\\gamma \\ \\ \\ , \\ \\ \\ [\\, {\\cal\n J}_i \\, ,\\, {\\cal J}_j \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\, {\\cal J}_k\n\\end{eqnarray} \nclose the algebra of the symmetry group $SU(2N)$. To find the consequences\nof this algebra for observable quantities like the mass-squared matrix and\nthe matrix elements for Goldstone boson emission and absorption, one takes matrix elements of this\nalgebra between hadron states $h'$ and $h$, and neglecting transitions from single-particle to multi-particle\nstates in the completeness sums over intermediate states, one recovers the same algebra but with the replacements\n${\\tilde Q}_\\alpha\\rightarrow \\lbrack\\, T_\\alpha \\,\\rbrack_{h' h}\\,$ and ${\\tilde Q}_{5\\alpha}\\rightarrow \\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\,$,\nand corresponding replacements for $G_{\\beta i}$ and ${\\cal J}_k$.\nThis result, originally found by Weinberg~\\cite{Weinberg:1994tu}, is here shown to\nbe a general consequence of the null-plane QCD operator algebra, valid\nin any Lorentz frame.\n\nIt is important to emphasize that the $SU(2N)$ symmetry found here is\nonly operative in the full interacting field theory. It is therefore\nunrelated to the $SU(2N)$ invariance of the QCD Lagrangian in the\nlimit of no interaction. Indeed we have show above in\nsection~\\ref{susec:spin} that eq.~\\ref{eq:OPQcomm2b}, the main\ningredient in the derivation of $SU(2N)$, in itself implies the\nexistence of Goldstone bosons. In addition, in a special case, this\nsymmetry does emerge in a well-defined limit of QCD. As\n$\\langle\\,h'\\,|\\epsilon^\\alpha|\\, h\\,\\rangle\\sim M_h-M_{h'}$, and\nbaryons within a given large-$N_c$ multiplet have mass splittings that\nscale as $1\/N_c$~\\cite{Witten:1979kh}, the large-$N_c$ QCD scaling\nrules suggest that for baryons $\\epsilon^\\alpha\\sim 1\/N_c$. Of\ncourse, as the matrix element of chiral charges between baryon states\nscales as $N_c$, the $SU(2N)$ symmetry reduces to the contracted\n$SU(2N)$~\\cite{Weinberg:1994tu,Beane:1998xp} for baryons in the\nlarge-$N_c$ limit, as one expects on general\ngrounds~\\cite{Gervais:1983wq,Dashen:1993as,Dashen:1993jt}.\n\nIt is instructive to consider a simple example. Consider the case\n$N=3$. Using the chiral transformation properties of the quarks,\neq.~\\ref{eq:chiralassign}, one sees that a $\\lambda=3\/2$ baryonic\noperator $\\psi_{+\\uparrow}\\psi_{+\\uparrow}\\psi_{+\\uparrow}$ transforms\nas $({\\bf 1},{\\bf 1})$, $({\\bf 1},{\\bf 8})$, or $({\\bf 1},{\\bf 10})$\nwith respect to $SU(3)_R\\otimes SU(3)_L$. Therefore, if the baryon is\na decuplet of $SU(3)_F$ with its $\\lambda=3\/2$ part in the $({\\bf 1},{\\bf 10})$, then one easily checks that its $\\lambda=1\/2$\npart must transform as $({\\bf 3},{\\bf 6})$ or $({\\bf 6},{\\bf 3})$.\nHowever, the different helicity states are unrelated by chiral\nsymmetry in itself. It is the mixed Lie-bracket,\neq.~\\ref{eq:OPQcomm2b}, the expression of broken chiral symmetry in\nthe spin Hamiltonian, that relates the helicities. Indeed taking the\n$\\lambda=1\/2$ decuplet to transform as $({\\bf 3},{\\bf 6})$ together\nwith an octet spin-$1\/2$ field and their negative-helicity partners in\n$({\\bf 10},{\\bf 1})\\oplus({\\bf 6},{\\bf 3})$ together fill out the\n${\\bf 56}$-dimensional representation of $SU(6)$ as is familiar from\nthe quark model. The difference here is that this symmetry arises from\nQCD symmetries and their pattern of breaking, and, in particular, has\nnothing to do with the non-relativistic limit. Hence we see that\nstarting from the formal null-plane QCD operator algebra, the simple\nassumption that the part of the null-plane reduced Hamiltonian, $M^2$,\nthat breaks chiral symmetry is small implies all of the usual\nconsequences of the non-relativistic quark model, without the need of\nany further assumption like the existence of constituent quark degrees\nof freedom~\\cite{Weinberg:1994tu}.\n\n\\section{Conclusion}\n\\label{sec:conc}\n\n\\noindent Usually one views the spontaneous breaking of a symmetry as\nthe non-invariance of the vacuum state with respect to the\nsymmetry. However, in relativistic theories of quantum mechanics, this\npicture is purely a matter of convention. We have seen that the\nfront-form vacuum is a singlet with respect to all symmetries and yet\nspontaneous symmetry breaking can occur via non-conserved currents\nwhose divergences are directly proportional to S-matrix elements for\nthe emission and absorption of Goldstone bosons. One may view the\nnull-plane description as a change of coordinates which moves\ndynamical information out of the vacuum state and into the interaction\noperators of the theory. The primary advantage of working with the\nnull-plane description is that broken chiral symmetry constraints\nbecome manifest in the sense that there are non-trivial Lie brackets\nbetween the Poincar\\'e generators and the broken symmetry\ngenerators. In the instant-form, the chiral constraints that appear\nnaturally in the front-form are present, but require one to work in\nspecial Lorentz frames and to make assumptions about the asymptotic\nbehavior of Goldstone-boson scattering amplitudes.\n\\vskip0.2in\n\nHere we will restate the main conclusions of this paper:\n\n\\vskip0.2in\n\\noindent$\\bullet$ In the front-form, spontaneous chiral symmetry\nbreaking is contained entirely in the three null-plane reduced\nHamiltonians, which encode the mass spectrum and spin content of a\ngiven theory. This must be the case as the null-plane chiral charges\nannihilate the vacuum state, and therefore chiral symmetry breaking\ncannot be attributed to the formation of chiral-symmetry breaking\ncondensates. In null-plane QCD, all chiral symmetry breaking\narises from the symmetry breaking parts of the reduced Hamiltonians,\ngiven explicitly in eqs.~\\ref{eq:symmbreakhamsa} and \\ref{eq:symmbreakhamsb}.\n\n\\vskip0.2in\n\\noindent$\\bullet$ Goldstone's theorem on the null-plane follows\ndirectly from the Lie-brackets between the null-plane Hamiltonians and\nthe chiral charges. A consistent null-plane description of spontaneous\nsymmetry breaking requires that a small explicit symmetry-breaking\noperator be included and that this explicit symmetry breaking be taken\nto zero only at the level of matrix elements of operators. The\ndivergence of the axial-vector current is proportional to the explicit\nsymmetry breaking. Therefore, as the current cannot be conserved in\nthe symmetry limit, the existence of massless states arises as a\nconsequence of the need to cancel the explicit breaking parameter that\nappears in its divergence.\n\n\\vskip0.2in\n\\noindent$\\bullet$ The Gell-Mann-Oakes-Renner relation is recovered in\nnull-plane QCD and a general prescription exists for translating all\nchiral-symmetry breaking condensates in instant-form QCD to\nchiral-singlet condensates in null-plane QCD. It is therefore\nsimplistic to say that the vacuum is trivial in the front-form, since\nthere are necessarily symmetry-preserving condensates which arise from\nmodes with strictly zero longitudinal momentum. In particular,\nin contrast with claims in the literature~\\cite{Brodsky:2008xm,Brodsky:2008xu,Brodsky:2009zd,Brodsky:2010xf},\nwe expect that the QCD vacuum energy is unaltered in moving from\nthe instant-form to the front-form descriptions of QCD, as\nis essential for the consistency of null-plane QCD.\n\n\\vskip0.2in\n\\noindent$\\bullet$ A simple solution of the null-plane operator\nalgebra recovers the spin-flavor symmetry of the constituent quark\nmodel. This result was obtained originally in\nRef.~\\cite{Weinberg:1994tu}, which obtained the algebra of charges and\nHamiltonians by working with sum rules obtained in special Lorentz\nframes, and using input from Regge-pole theory expectations of the\nasymptotic behavior of scattering amplitudes involving Goldstone\nbosons. The results of the present work may be viewed as an attempt to clarify this\noriginal work by formulating it in a Lorentz frame-independent manner\nwhich follows directly from null-plane QCD.\n\n\\vskip0.2in\n\nIn the null-plane formulation of QCD, the loss of manifest Lorentz\ninvariance and locality are, operationally, a result of integrating\nout non-dynamical degrees of freedom. Physically, it is clear that the\nloss of Lorentz invariance is tied to the fact that the essence of\nLorentz invariance lies in the Poincar\\'e Lie brackets that must be\nsatisfied by the spin generators, and, of course, on the null-plane\nspin is dynamical and therefore requires the solution of the theory to\nproperly implement. By contrast, the non-locality of the theory would\nappear to be related to the fact that the null-plane chiral symmetry\nconstraints on observables are properly formulated as sum rules which\nspan many energy scales, and therefore do not exhibit the separation\nof scales that allows a useful description in terms of local\nLagrangian effective field theory. Indeed, it appears that, in some\nsense, scattering amplitudes are the fundamental objects in the\nnull-plane formulation. This is particularly clear from the\nLie-brackets that mix the Poincar\\'e and chiral symmetry generators,\nwhich are given by the S-matrix elements for Goldstone boson emission\nand absorption. From a theoretical standpoint, the most interesting\nconsequences of the results obtained in this paper are apparent only\nin the large-$N_c$ limit, which will be treated separately.\n\n\\vskip0.3in\n\n\\noindent \nI thank Ulf-G.~Mei\\ss ner for valuable comments on the manuscript, and\nT.~Becher, G.~Colangelo, H.~Leutwyler, F.~Niedermayer, and U.~Wenger\nfor useful discussions. I am particularly grateful to the Institute\nfor Theoretical Physics at the University of Bern for providing a\nstimulating work environment during academic year 2010\/2011. The\nAlbert Einstein Center for Fundamental Physics is supported by the\n\"Innovations- und Kooperationsprojekt C-13\" of the \"Schweizerische\nUniversit\\\"atskonferenz SUK\/CRUS\". I gratefully acknowledge the\nhospitality of HISKP-theorie and the support of the Mercator programme\nof the Deutsche Forschungsgemeinschaft during academic year\n2012\/2013. This work was supported in part by NSF CAREER Grant\nPHY-0645570 and continuing grant PHY1206498.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}