Content-Type: multipart/mixed; boundary="-------------0607141006387" This is a multi-part message in MIME format. ---------------0607141006387 Content-Type: text/plain; name="06-203.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-203.keywords" Constructive Quantum Field Theory, Indefinite Metric Gauge theories, Krein selfadjointness, Chern-Simons, Maxwell-Chern-Simons fields, Canonical Quantization ---------------0607141006387 Content-Type: application/x-tex; name="CSKrein-1.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="CSKrein-1.tex" \documentclass[12pt]{article} %\usepackage{amssymb} %\usepackage[left=1.5in,top=1in,right=1in,bottom=1in]{geometry} \newcommand{\mathbb}[1]{\ensuremath{#1}} \usepackage{latexsym} \renewcommand{\d}{\ensuremath{\partial}} \renewcommand{\L}{\ensuremath{\mathcal{L}}} \newcommand{\HR}{\ensuremath{\mathcal{H}_{R}}} \newcommand{\HE}{\ensuremath{\mathcal{H}_{E}}} \newcommand{\EP}{\ensuremath{\mathcal{E}_{+}^{0}}} \newcommand{\FP}{\ensuremath{\mathcal{F}_{+}}} \newcommand{\HS}[1]{\ensuremath{\mathcal{H}_{#1}}} \newcommand{\sgn}[1]{\ensuremath{\mathrm{sgn}\left( #1 \right)}} \newcommand{\supp}{\ensuremath{\mathrm{supp}}} \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\C}{\ensuremath{\mathbb{C}}} \newcommand{\hmr}[2]{\ensuremath{(#1,#2)_{R}}} \newcommand{\hms}[3]{\ensuremath{(#1,#2)_{#3}}} \newcommand{\kmr}[2]{\ensuremath{\{#1,#2\}_{R}}} \newcommand{\hme}[2]{\ensuremath{(#1,#2)_{E}}} \newcommand{\kme}[2]{\ensuremath{\{#1,#2\}_{E}}} \newcommand{\dual}[2]{\ensuremath{\langle #1,#2\rangle}} \newcommand{\wa}{\ensuremath{\hat{w}_{1}}} \newcommand{\wb}{\ensuremath{\hat{w}_{2}}} \newcommand{\wc}{\ensuremath{\hat{w}_{3}}} \newcommand{\va}{\ensuremath{\hat{v}_{1}}} \newcommand{\vb}{\ensuremath{\hat{v}_{2}}} \newcommand{\vc}{\ensuremath{\hat{v}_{3}}} \newcommand{\ea}{\ensuremath{\hat{e}_{\parallel}}} \newcommand{\eb}{\ensuremath{\hat{e}_{c}}} \newcommand{\ec}{\ensuremath{\hat{e}_{\perp,c}}} \newcommand{\lap}{\ensuremath{\triangle}} \newcommand{\proof}{\underline{Proof:}\newline} \newcommand{\qed}{Q.E.D.\newline} \newcommand{\dbar}[1]{\ensuremath{\overline{\overline{#1}}}} \newcommand{\halo}[1]{\ensuremath{\stackrel{\circ}{#1}}} \newcommand{\svec}[1]{\ensuremath{#1}} \renewcommand{\thesection}{\Roman{section}} \newtheorem{axiom}{Axiom} \renewcommand{\theaxiom}{\Roman{axiom}} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lemma}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \begin{document} \title{Canonical Quantization of Lattice Higgs-Maxwell-Chern-Simons Fields:\\Krein Selfadjointness\thanks{The following article has been submitted to the Journal of Mathematical Physics. After it is published, it will be found at \texttt{http://link.aip.org/link/JMAPAQ }}} \author{Daniel A. Bowman\\School of Natural Sciences and Mathematics\\ Ferrum College, Ferrum, VA 24088 \and John L. Challifour\\Departments of Mathematics and Physics\\Indiana University, Bloomington, IN 47405}\maketitle \begin{abstract}It is shown how techniques from constructive quantum field theory may be applied to indefinite metric gauge theories in Hilbert space for the case of a Higgs-Maxwell-Chern-Simons theory on a lattice. The Hamiltonian operator is shown to be Krein essentially selfadjoint by means of unbounded but Krein unitary transformations relating the Hamiltonian to an essentially maximal accretive operator. \end{abstract} \section{Introduction\label{s1}}The mathematical study of Higgs gauge theories has been focussed mainly on gauge invariant quantities undertaken using euclidean functional integrals~\cite{BFSI,FroMarch,Sei1982}. However in the theoretical physics literature, it is mostly the Higgs and gauge fields that are used directly ~\cite{DJT,Dunn1995,Chen:1996, Dunn1998}. In this direction we consider a (2+1) dimensional spacetime Maxwell-Chern-Simons field minimally coupled to a charged scalar field in the indefinite metric framework proposed by Wightman and G\aa rding~\cite{WiGa}. By using a representation of the canonically quantized Chern-Simons field in a Hilbert space with a Krein indefinite metric, we show the Chern-Simons field to be a Hilbert normal and Krein selfadjoint operator. Regularizing field operators with a finite periodic lattice allows construction of the Hamiltonian operator. Initially this operator is densely defined and closable but exhibits no immediate spectral properties which would suggest direct implementation of techniques from constructive quantum field theory in obtaining a euclidean path-space representation for a semigroup generated by this Hamiltonian. In this paper, we show the underlying structure of the Hamiltonian to be that of an accretive operator masked by unbounded Krein unitary `gauge' transformations. This leads to an unbounded semigroup which unlike Fr\"{o}hlich's theory of unbounded symmetric semigroups is not Hilbert selfadjoint~\cite{Frosemi}. In a second paper~\cite{CSOS}, we will obtain the corresponding euclidean theory and examine the relation between the physical states in the Wightman-G\aa rding framework and Osterwalder-Schrader positivity provided by this unbounded semigroup. In sec\-tion~\ref{s2}, we de\-scribe our rep\-re\-sen\-ta\-tion for the Maxwell-Chern-Simons field which differs from those in ~\cite{Hell,C4Cl} and establish regularity of the field that we use. It is convenient to represent these fields in terms of harmonic oscillator variables in section~\ref{s3} and the appendix since the Krein `gauge' transformations then become transparent as well as their properties as unbounded operators. In section~\ref{s4}, it is shown that the real part of the transformed Hamiltonian defines an essentially selfadjoint operator which is bounded below. The `gauge' transformed Hamiltonian is then shown to be an accretive operator in section~\ref{s5}. The bounds from section~\ref{s4}, allow us to prove a quadratic estimate for the transformed Hamiltonian by which it becomes a maximal accretive operator after closure and thus Krein essentially selfadjoint. \section{Maxwell-Chern-Simons Fields\label{s2}} We work in $d=s+1$ dimensional spacetime, where the number of spatial dimensions is $s=2$. The metric is given by $g_{00} = +1$ and $g_{ij} = -\delta _{ij}$ with the remaining components vanishing. As usual, Latin indices are spatial, ranging from $1$ to $s$, while Greek indices include the time component, $0$. We write the inner product of $d$-vectors $k$ and $x$ alternately as \[ kx = k_\mu x^\mu =k^\mu x_\mu= k_0 x_0 - \svec{k}\cdot \svec{x} = k_0 x_0 - k_i x_i \] where indices are raised or lowered by contraction with $g$ and repeated indices are summed over their range. It should always be clear from context whether $k$ and $x$ refer to $d$-vectors or $s$-vectors. We use analogous notation for $d$-divergences, e.g., $\d A \equiv \d_{\mu} A^{\mu}$. Our units are such that $\hbar =c=1$. Finally, we make use of the notation $\omega=\omega (k)= \sqrt{k\cdot k}$ and $\mu=\mu (k)=\sqrt{k\cdot k +m^2}=\sqrt{\omega^2 +m^2}$. Hopefully, our conventions are now clear. The Maxwell-Chern-Simons (MCS) Lagrangian density is given by \begin{equation} \L=-\frac{1}{2}\d_\mu A_\nu \d^\mu A^\mu +\frac{\lambda}{2}\left(\d A \right) ^2 +\frac{m}{2}\epsilon^{\mu\nu\rho}\d_\mu A_\nu A_\rho.\label{MCSlagrangian} \end{equation} The term $(\d A)^2$ fixes the gauge with parameter $\lambda$ \begin{equation} \lambda=\frac{2\xi}{2\xi -1}=-\frac{\gamma}{1-\gamma} \label{gaugefixdef} \end{equation} interpolating between the various covariant gauges. Here, $\gamma=\lambda =\xi=0$ corresponds to the Feynman gauge while $\gamma=1$, $\lambda\rightarrow \infty$, $\xi=1/2$ all correspond to Landau gauge. The parameters $\xi$ and $\gamma$ are more convenient than $\lambda$ for our representation below. The parameter $\gamma$ is introduced for comparison with other works that start with a different Lagrangian such as \cite{Hell} while $\xi$ is introduced for comparison with works such as \cite{C4Cl, c4wood}. The Euler-Lagrange equations corresponding to equation~(\ref{MCSlagrangian}) are \begin{equation} \Box A^\mu -\lambda\d^\mu(\d A) + m\epsilon^{\mu\alpha\beta}\d_\alpha A_\beta =0. \label{MCSeom} \end{equation} The homogeneous Green's function for (\ref{MCSeom}) can be written \begin{eqnarray} \lefteqn{\left = } \label{MCSprop} \\ &\frac{1}{(2\pi)^2}\int d\Omega_m (k) e^{-ik(x-y)} \left[ \frac{k_\mu k_\nu}{m^2} -g_{\mu\nu}-\frac{i}{m}\epsilon_{\mu\nu\rho} k^\rho\right] \nonumber\\ &+\frac{1}{(2\pi)^2}\int d\Omega_0 (k) e^{-ik(x-y)}\left[ -\frac{k_\mu k_\nu} {m^2} +\frac{i}{m}\epsilon_{\mu\nu\rho}k^\rho \right] \nonumber\\ &-\frac{1-2\xi}{(2\pi)^2}\int d\Omega_0 (k)e^{-ik(x-y)}\frac{1}{2\omega} \left[ \delta_{\mu 0}k_\nu +\delta_{\nu 0}k_\mu -i(x_0-y_0) k_\mu k_\nu -\frac{k_\mu k_\nu}{\omega}\right] \nonumber \end{eqnarray} which has support both on the lightcone and the mass $m$ hyperboloid. The canonical momenta associated to (\ref{MCSlagrangian}) are \begin{eqnarray} \pi^0 & \equiv & \frac{\delta\L}{\delta(\d_0 A_0)} = -\d_0 A^0+\lambda \d A \label{MCSpi0}\\ \pi^j & \equiv & \frac{\delta\L}{\delta(\d_0 A_j)} = -\d_0 A^j +\frac{m}{2} \epsilon^{0jn}A_n\label{MCSpij} \end{eqnarray} It is convenient to express (\ref{MCSeom}) in terms of differential forms. For this, we define a $1$-form $A\equiv A_\mu dx^\mu$. We let $*$ denote the Hodge duality operator associated with our metric $g$ and the orientation given by the volume form $dx^0\wedge dx^1\wedge dx^2$, so that $**=+1$. Furthermore, we define the codifferential $\delta$ in the usual way on $k$-forms by \begin{equation} \delta=-\mathrm{det}(g)(-1)^{d(k+1)}*d*=-(-1)^{3(k+1)}*d*=(-1)^k *d* \end{equation} so that, for a $k$-form $\beta$, we have \begin{eqnarray*} \delta\beta & = & \delta\left( \frac{1}{k!}\beta_{\mu_1\cdots\mu_k}dx^{\mu_1} \wedge\cdots\wedge dx^{\mu_k}\right) \\ & = & -\frac{1}{(k-1)!}\d^{\mu_1}\beta_{\mu_1\cdots\mu_k}dx^{\mu_2}\wedge\cdots \wedge dx^{\mu_k} \end{eqnarray*} Then, we find that $\Box\equiv \d^\mu\d_\mu$ is given by \begin{equation} \Box=-(\delta d + d\delta) \end{equation} in analogy to the Laplace-Beltrami operator in the Riemannian case. With this notation, we may write (\ref{MCSeom}) as \begin{equation} \delta dA -m *dA=(\lambda-1)d\delta A \label{MCSdfeom} \end{equation} From this standpoint, it becomes easy to prove the following decomposition theorems which we shall utilize in the process of canonical quantization. \begin{thm}[MCS Decomposition]\label{MCSdecomp} Let $A$ be a $1$-form satisfying (\ref{MCSdfeom}). Then, there exist $1$-forms $V$ and $S$ satisfying the equations \begin{eqnarray} dV=m*V & & *dS=-\frac{(\lambda - 1)}{m}d\delta S \label{VSeom} \end{eqnarray} such that $A=V+S$. Conversely, given $1$-forms $V$ and $S$ satisfying (\ref{VSeom}), the $1$-form $A=V+S$ satisfies (\ref{MCSdfeom}). \end{thm} \proof For the first statement, define \[ V\equiv \frac{1}{m}*dA +\frac{1}{m^2}(\lambda -1)d\delta A \] so that \begin{eqnarray*} *dV &=& \frac{1}{m}*d*dA=\frac{1}{m}\delta dA\\ &=& \frac{1}{m}\left( m*dA+ (\lambda-1)d\delta A \right)\\ &=& m V \end{eqnarray*} where we have used (\ref{MCSdfeom}). Now, define the $1$-form $S\equiv A-V$. This satisfies \begin{eqnarray*} *dS &=& *dA-*dV=*dA-mV\\ &=& *dA-\left( *dA+\frac{\lambda-1}{m}d\delta A \right)\\ &=& -\frac{\lambda-1}{m}d\delta A \end{eqnarray*} Then, the result (\ref{VSeom}) follows by noting that \[ \delta A=\delta S +\delta V =\delta S + \delta \left( \frac{*dV}{m} \right) =\delta S \] For the converse, we note that equations~(\ref{VSeom}) imply \begin{eqnarray*} \delta d V=(-1)^2 *d* dV=+*d(mV)=m^2 V\\ \delta V = (-1)^1 *d*V = -*d* \left( \frac{*dV}{m} \right)=0\\ \delta d S=(-1)^2 *d* d S=+*d\left( -\frac{\lambda-1}{m}d\delta S \right)=0 \end{eqnarray*} Then, we simply substitute $A=V+S$ into equation~(\ref{MCSdfeom}) to find \begin{eqnarray*} \mathrm{LHS} &=& \delta d (V+S) - m *d(V+S)\\ &=& m^2 V - m*dV - m*dS\\ &=& -m \left( -\frac{\lambda-1}{m}d\delta S \right)\\ &=& (\lambda-1) d\delta S\\ \mathrm{RHS} &=& (\lambda-1)d\delta (V+S)=(\lambda-1) d\delta S \end{eqnarray*} and note that $\mathrm{RHS}=\mathrm{LHS}$. \qed Notice that the gauge dependence, as parameterized by $\lambda$, has been separated into the field $S$ in equations~(\ref{VSeom}). For this reason, we often refer to $V$ as the physical MCS field. We can say more about $V$ and $S$. It has long been known that there are certain necessary conditions that a valid decomposition of a solution of (\ref{MCSeom}) into the sum of a massive and a massless vector field must satisfy\cite{kimura, nakanishi}. These conditions follow easily from our theorem, so we state them as a corollary. \begin{cor}\label{VSprop} The vector field $V_\mu$ corresponding to the $1$-form $V$ satisfies \begin{eqnarray*} (\Box +m^2)V_\mu & = &0 \\ \d V & = & 0 \end{eqnarray*} which are the Proca field equations. The vector field $S_\mu$ corresponding to the $1$-form $S$ satisfies \begin{eqnarray*} \Box S_\mu & =& \d_\mu (\d S) \\ \Box (\d S) & = &0, \end{eqnarray*} provided $\lambda\neq 1$. \end{cor} One should note that, although $V_\mu$ satisfies the Proca field equations, it is not a true Proca field since it also satisfies the additional constraint (\ref{VSeom}). Thus, one expects the quantized version of $V_\mu$ to propagate one less physical mode than a $2+1$- dimensional Proca field. We shall see that this is indeed the case. In fact, even at the classical level, we have the following result. \begin{prop}[Proca Decomposition] \label{PtoMCS} Let $U$ be a $1$-form satisfying the Klein-Gordon equation \[ (\Box + m^2)U=0 \] for $m>0$. Then, it is both necessary and sufficient that $U$ satisfy the decomposition \begin{equation} U=V^+ + V^- + d\phi \end{equation} where $V^\pm$ satisfy $*dV^{\pm}=\pm mV^\pm$ and $\phi$ is a scalar solution to the Klein-Gordon equation. If $U$ is additionally a Proca field (i.e., $\delta U=0$), then we may choose $\phi\equiv 0$. \end{prop} \proof For necessity, define \begin{eqnarray} V^\pm \equiv -\frac{1}{2} \left(\frac{1}{m^2}d\delta U \mp\frac{1}{m}*dU -U \right) & \mathrm{\ \ and\ \ } & \phi\equiv \frac{1}{m^2}\delta U \end{eqnarray} The remainder then follows by straightforward calculations. \qed %Thus, intuitively, we can build a Proca field out of two distinct %physical MCS fields, or ``project'' from a Proca field onto a physical %MCS field. This will be made more precise in the quantized theory, %where we will see $V$ propagates a certain combination of the two %physical modes $\wb$ and $\wc$ defined in %\ref{dubyas} %that %span the Proca physical one-particle space $\HR'^{(1)}$. The well-known bosonization of the MCS field exhibiting its single, massive, physical degree of freedom (see, e.g., \cite{DJT}) also follows easily from theorem~\ref{MCSdecomp}. Thus, we state it as a corollary as well. \begin{cor} A vector field satisfying \begin{equation} \epsilon_{\mu\nu\rho}\d^\nu V^\rho = m V_\mu \label{Veom} \end{equation} (i.e., $*dV=m V$ as per (\ref{VSeom})) must necessarily be of the form \begin{eqnarray*} V_0 &=& \frac{\sqrt{-\lap}}{m}\phi \\ V_j &=& - \left[ \frac{\d_0\d_j}{m\sqrt{-\lap}}-\frac{\epsilon_{jn}\d_n} {\sqrt{-\lap}} \right]\phi \end{eqnarray*} for some scalar field $\phi$ \end{cor} \proof Taking the time derivative of (\ref{Veom}) for $\mu=j$, we find \begin{eqnarray*} m\d_0 V_j &=& -\epsilon_{jn}\d_n\d_0 V_0 +\epsilon_{jn}\d_0\d_0 V_n \\ &=& -\epsilon_{jn}\d_n\d_0 V_0 +\epsilon_{jn}(\lap-m^2)V_n \end{eqnarray*} and noting that \[ m^2 V_n = m\epsilon_{nk}(\d_0 V_k-\d_k V_0) \] we obtain the necessary condition \[ \lap V_j = m\epsilon_{jn}\d_n V_0-\d_0\d_j V_0 \] from which the result follows by defining \[ \phi\equiv\frac{m}{\sqrt{-\lap}}V_0 \] \qed We proceed now to the quantization. Using the intuition gained from theorem \ref{MCSdecomp}, we seek a massive vector field $V_\mu(x)$ satisfying \begin{eqnarray} \lefteqn{\hmr{\Omega_R}{V_\mu(x)V_\nu(y)\Omega_R} =} \label{V2pt} \\ & \frac{1}{(2\pi)^2}\int d\Omega_m(k) \left[ \frac{k_\mu k_\nu}{m^2}-g_{\mu\nu}- i\frac{\epsilon_{\mu\nu\lambda} k^\lambda}{m} \right]e^{-ik(x-y)} \nonumber \end{eqnarray} and a commuting massless vector field $S_\mu(x)$ satisfying \begin{eqnarray} \lefteqn{\hmr{\Omega_R}{S_\mu(x)S_\nu(y)\Omega_R}=} \label{S2pt}\\ &\frac{1}{(2\pi)^2}\int d\Omega_0 (k) e^{-ik(x-y)}\left[ -\frac{k_\mu k_\nu} {m^2} +\frac{i}{m}\epsilon_{\mu\nu\rho}k^\rho \right]\nonumber \\ &\mbox{}-\frac{1-2\xi}{(2\pi)^2}\int d\Omega_0 (k)e^{-ik(x-y)}\frac{1}{2\omega} \left[ \delta_{\mu 0}k_\nu +\delta_{\nu 0}k_\mu -i(x_0-y_0) k_\mu k_\nu -\frac{k_\mu k_\nu}{\omega}\right]\nonumber \end{eqnarray} so that \begin{equation} \hmr{\Omega_R}{V_\mu(x)V_\nu(y)\Omega_R} + \hmr{\Omega_R}{S_\mu(x)S_\nu(y)\Omega_R} =\left \end{equation} given in equation~(\ref{MCSprop}). First, we consider the massive field $V_\mu (x)$. Guided by proposition \ref{PtoMCS}, we define \begin{eqnarray} V_\mu (x) &=& \frac{1}{\sqrt{2}}\left(- \frac{\d_\mu\d_\alpha}{m^2}- g_{\mu\alpha}+\frac{1}{m}\epsilon_{\mu\alpha\beta}\d^\beta \right) U^{\alpha} (x) \\ & = & \frac{1}{2\pi\sqrt{2}}\int d\Omega_m (k)\left[ b^\alpha (k) \left( \frac{k_\alpha k_\mu}{m^2}-g_{\mu\alpha}-\frac{i}{m} \epsilon_{\alpha\mu\beta}k^\beta \right)e^{-ikx}\right. \nonumber\\ && + \left. b^{\dag\,\alpha}(k)\left(\frac{k_\alpha k_\mu}{m^2}- g_{\mu\alpha}+\frac{i}{m}\epsilon_{\alpha\mu\beta}k^\beta \right)e^{ikx} \right]\label{MCSfield} \end{eqnarray} where $U$ is a massive vector field given by \begin{equation} U_\mu (x) = \frac{1}{2\pi} \int d\Omega_m (k) [e^{-ikx}b_\mu (k) + e^{ikx} b^\dag _\mu (k)] \label{Pfield} \end{equation} with $d\Omega_m (k)$ denoting the invariant measure on the positive mass $m$ hyperboloid and $b,\, b^\dag$ satisfying the commutation relations \begin{equation} \left[ b_\mu(k),b^\dag_\nu(k') \right]=-g_{\mu\nu}\,2\mu(k) \delta(k-k')\label{PCCR} \end{equation} This is clearly a covariant field that, by construction, satisfies the field equations $\epsilon_{\mu\nu\rho}\d^\nu V^\rho(x)=m V_\mu(x)$. Furthermore, one checks that (\ref{V2pt}) is satisfied under the usual assumption $b_\mu(\svec{k})\Omega_R=0$. In fact, all is made rigorous by realizing the expressions as bilinear forms on the Proca Fock space $\HR^P$ given by \[ \HR^P\equiv\Gamma_\mathrm{sym}(L^2(\R^d,d\Omega_m(k))\otimes\C^d), \] i.e., the symmetric Fock space over $\HR^{P(1)}\equiv L^2(\R^d,d\Omega_m(k))\otimes\C^d$. %Explicitly, Fock space is given by %\begin{eqnarray} %\HR & = & \bigoplus_{n=0}^{\infty}\HR^{(n)} \label{Fockspace} \\ %\HR^{(0)}\equiv\C & & \HR^{(n)}\equiv\bigotimes_{n}\HR^{(1)} \nonumber %\end{eqnarray} %where $\bigotimes$ here denotes a symmetric tensor product. Given $\Phi\in\HR^P$, we denote its projection onto the $n$-particle subspace $\HR^{P(n)}$ by $\Phi^{(n)}$. With this notation, the inner product on $\HR^P$ is given by \[ \hmr{\Phi}{\Psi} = \sum_{n=0}^{\infty} \hmr{\Phi^{(n)}}{\Psi^{(n)}}^{(n)} \] with \begin{eqnarray} \lefteqn{\hmr{f}{g}^{(n)} =} \\ & & \prod_{j=1}^{n}\left[\int d\Omega_m(k_j)\right] \overline{f_{\mu_1\cdots\mu_n}(k_1,\cdots,k_n)} g_{\mu_1\cdots\mu_n}(k_{1},\cdots,k_{n}) \nonumber \end{eqnarray} for $f,g\in\HR^{P(n)}$ with $n>0$ and $\hmr{f}{g}^{(0)}=\overline{f}g$ for $f,g\in\HR^{P(0)}$. We denote adjoints with respect to $\hmr{\cdot}{\cdot}$ by a superscript~$*$ and refer to them as Hilbert adjoints. We introduce the usual Fock creation and annihilation bilinear forms on $\HR^P$ by \begin{eqnarray} \lefteqn{[c_\mu^*(\svec{k})\Phi]^{(n)}_{\mu_1\cdots\mu_n}(k_1,\cdots,k_n)=} \label{fockrep} \\ & \frac{1}{\sqrt{n}}\sum_{j=1}^{n}\delta_{\mu\mu_j}\delta(\svec{k}-\svec{k}_j) 2\mu(\svec{k}_j) \Phi^{(n-1)}_{\mu_1\cdots\widehat{\mu_j}\cdots\mu_n} (k_1,\cdots,\widehat{k_j},\cdots,k_n) \nonumber \\ \lefteqn{[c_\mu(\svec{k})\Phi]^{(n)}_{\mu_1\cdots\mu_n}(k_1,\cdots,k_n)=}\\ & \sqrt{n+1} \Phi^{(n+1)}_{\mu\mu_1\cdots\mu_n}(k,k_1,\cdots,k_n) \nonumber \end{eqnarray} where a hat over a variable or an index means to omit it. These are well-defined (as bilinear forms) on vectors $\Phi\in D(N^{1/2})\subset\HR$ where $N$ is the selfadjoint number operator. % defined by % \begin{equation} % [N\Phi]^{(n)}_{\mu_1\cdots\mu_n}(k_1,\cdots,k_n)= % n\Phi^{(n)}_{\mu_1\cdots\mu_n}(k_1,\cdots,k_n). \label{N} % \end{equation} In fact, $c_\mu(\svec{k})$ defines an operator on this domain\cite{R&S}. These formally satisfy the canonical commutation relations \begin{equation} [c_\mu(\svec{k}),c_\nu^*(\svec{k'})]=\delta_{\mu\nu}\ 2\mu(\svec{k})\ \delta(\svec{k}-\svec{k'}). \end{equation} We also need to introduce the indefinite metric on $\HR^P$ which is given by $\kmr{\Phi}{\Psi} = \hmr{\Phi}{\eta \Psi}$. The action of $\eta$ on the $n$-particle subspace is given by \begin{eqnarray} \lefteqn{\hmr{f}{\eta g}^{(n)}=} \\ & & \prod_{j=1}^{n}\left[ \int d\Omega_m(k_j)\ (-g)_{\mu_j\nu_j}\right] \overline{f_{\mu_1\cdots\mu_n}(k_1,\cdots,k_n)} g_{\nu_1\cdots\nu_n}(k_{1},\cdots,k_{n}) \nonumber \end{eqnarray} which in turn determines its action on $\HR^P$. In short, $\eta$ is the second quantization of the operator \[ f_\mu (k) \mapsto (-g_{\mu\nu} f_\nu)(k) \] on $\HR^{P(1)}$ \cite{R&S, phi2}. With respect to the indefinite inner product $\kmr{\cdot}{\cdot}$, $\HR^P$ is a Krein space \cite{bognar}. This follows from the observations that $\eta = \eta^*$ and $\eta^2 = 1$. We denote adjoints with respect to the Krein metric by a superscript~$\dag$. Clearly, it is also true that $\eta = \eta^\dag$. Finally, we can define $b,\ b^\dag$ as Krein creation and annihilation forms: \begin{eqnarray} b_\mu^\dag (\svec{k}) = g_{\mu\nu}c_{\nu}^{*} (\svec{k}) & & b_\mu(\svec{k})= - c_\mu (\svec{k}) \label{kreinrep} \end{eqnarray} These, then, formally satisfy equation~(\ref{PCCR}). Furthermore, as bilinear forms, we have the relations \begin{eqnarray*} N & = & \int d\Omega_m (k) c_\mu^*(\svec{k})c_\mu(\svec{k}) \\ & = & \int d\Omega_m (k) b_\mu^\dag(\svec{k})(-g_{\mu\nu})b_\nu(\svec{k}) \end{eqnarray*} Calculating the commutator, we obtain \begin{equation} \left[ V_\mu(x),V_\nu(y) \right]=-i\left( \d_\mu\d_\nu+g_{\mu\nu}- \frac{\epsilon_{\mu\nu\lambda} \d^\lambda}{m}\right)\Delta_m (x-y) \end{equation} Note that this is not a local expression, despite the facts that the Pauli-Jordan function $\Delta_m (x)$ given by %\ref{paulijordan} \begin{equation} \Delta_m (x) = \frac{-i}{(2\pi)^s}\int d^d k\,\sgn{k_0}\delta(k^2-m^2) e^{-ikx}. \label{paulijordan} \end{equation} vanishes for spacelike separations and differential operators are, in general, local. This subtlety arises from the fact that a time derivative of $\Delta_m(x)$ evaluated at time zero yields a delta function (see, e.g., appendix I of \cite{BS}). Thus, we have the following non-vanishing equal-time commutators: \begin{eqnarray} \left[ V_0 (t,\svec{x}), V_n(t,\svec{y}) \right]=\frac{i \d_n}{m^2} \delta(\svec{x}-\svec{y}) \label{V0n}\\ \left[ V_j (t,\svec{x}),V_n(t,\svec{y}) \right]=-i\epsilon_{jn}\delta(\svec{x} -\svec{y})\label{Vjn} \end{eqnarray} This is analogous to the situation that arises in canonical quantization of the Proca field without use of an indefinite metric\cite{BS}. %However, the ``physical'' %MCS field $V_\mu(x)$ still contains unphysical modes since, by proposition %\ref{MCSbose}, there is only one physical degree of freedom. In order to exhibit the bosonization (and, hence, the single physical mode of \cite{DJT,C4Cl}) within this framework, we note that (\ref{MCSfield}) can be written \begin{equation} V_\mu(x)=\frac{1}{2\pi\sqrt{2}}\int d\Omega_m (k)\left[ (Lgb)_\mu(k) e^{-ikx} + (\bar{L}gb^\dag)_\mu(k) e^{ikx}\right] \end{equation} where \begin{eqnarray} L_{\mu\nu} & \equiv & \frac{k_\mu k_\nu}{m^2}-g_{\mu\nu}-\frac{i}{m} \epsilon_{\mu\nu\lambda}k^\lambda \\ & = & \left(\frac{\sqrt{\mu^2 + \omega^2}}{m}\wc-\wb\right)\otimes \left(\frac{\sqrt{\mu^2 +\omega^2}}{m}\wc-\wb\right)^* \end{eqnarray} using the basis \begin{eqnarray} \wa= \frac{1}{\sqrt{\mu^2 +\omega^2}}\left(\begin{array}{c}\mu\\-k_1\\-k_2 \end{array} \right) \nonumber \\ \wb=\frac{i}{\omega}\left(\begin{array}{c}0\\k_2\\-k_1 \end{array} \right) \label{dubyas} \\ \wc=\frac{1}{\omega\sqrt{\mu^2+\omega^2}}\left(\begin{array}{c}\omega^2\\ \mu k_1 \\ \mu k_2 \end{array} \right) \nonumber \end{eqnarray} which is orthonormal in the $\C^3$ Euclidean inner product. Then, we can define \begin{eqnarray} a(\svec{k})=\frac{1}{\sqrt{2}}\left(\frac{\sqrt{\mu^2 +\omega^2}}{m}\wc -\wb\right)^*_\alpha g_{\alpha\beta} b_\beta(\svec{k})\\ a^\dag (\svec{k}) = \frac{1}{\sqrt{2}}\left(\frac{\sqrt{\mu^2 + \omega^2}}{m}\wc+\wb\right)^*_\alpha g_{\alpha\beta} b^\dag_\beta(\svec{k}) \end{eqnarray} where we've used $\bar{\wb}=-\wb$. Hence, \begin{equation} \left[ a(\svec{k}),a^\dag(\svec{k}') \right]=2\mu\delta(\svec{k}-\svec{k}') \end{equation} and suppressing the vector index we have the representation \begin{eqnarray} V(x)&=&\frac{1}{2\pi}\int d\Omega_m (k)\left[ \left( \frac{\sqrt{\omega^2 + \mu^2}}{m}\wc -\wb\right)a(\svec{k}) e^{-ikx}\right.\\ &&+ \left.\left(\frac{\sqrt{\omega^2+\mu^2}}{m}\wc+\wb \right)a^\dag(\svec{k}) e^{ikx}\right]\nonumber \end{eqnarray} for which the time evolution is implemented by \begin{equation} H^V_0 = \int d\Omega_m(k) \mu(\svec{k}) a^\dag(\svec{k})a(\svec{k}) \label{Vham} \end{equation} From this expression for $V_\mu$, it is clear that the physical field is both Krein and Hilbert symmetric. It also has a dense set of analytic vectors, thereby making its closure selfadjoint. Next, we turn to the ghost field $S_\mu$. We wish to realize $S_\mu$ on the Maxwell Fock space $\HR^M$ given as the symmetric Fock space over the single particle space \begin{equation} \HR^{M(1)}\equiv L^2 (\R^d, d\Omega_0)\otimes\C^d \label{Maxfock} \end{equation} where $d\Omega_0$ is the invariant measure on the forward light cone. As before, we equip $\HR^M$ with an indefinite metric $\kmr{\cdot}{\cdot}\equiv\hmr{\cdot}{\eta\cdot}$ where $\eta=-g$ on $\HR^{M(1)}$. %\ref{Maxfock}. Then, we will have a realization of the Maxwell-Chern-Simons field $A_\mu\equiv V_\mu\otimes 1 +1\otimes S_\mu\equiv V_\mu + S_\mu$ on $\HR=\HR^P\otimes\HR^M$. For this, we first choose the following $\C^3$-orthonormal vectors \begin{eqnarray} \va & = &\frac{1}{\omega\sqrt{2}} \left( \begin{array}{c} \omega \\ k_1 \\ k_2 \end{array} \right) \\ %= \frac{1}{2}[(1+i)\hat{e}_+ + (1-i)\hat{e}_-] \\ \vb &= &\frac{i}{\omega}\left( \begin{array}{c} 0\\k_2\\-k_1\end{array}\right)\\ \vc &= & \frac{1}{\omega\sqrt{2}} \left( \begin{array}{c} \omega \\ -k_1 \\ -k_2 \end{array} \right) %=\frac{1}{2}[(1-i)\hat{e}_+ + (1+i)\hat{e}_-] \end{eqnarray} %where $\hat{e}_\pm$ are the $s=2$ versions of %\ref{eplusminus}. %Notice %that the projection $Q^T$ onto the physical subspace of the Maxwell field is %given by $\vb\otimes\vb^*$. Next, we define bilinear forms \begin{eqnarray} a_R(\svec{k})=(\vc)_\alpha g^{\alpha\beta} a_\beta (\svec{k}) \\ a_Q(\svec{k})=(\va)_\alpha g^{\alpha\beta} a_\beta (\svec{k}) \\ a_Q^\dag (\svec{k}) =(\va)_\alpha g^{\alpha\beta} a^\dag_\beta (\svec{k}) \\ a_R^\dag (\svec{k}) =(\vc)_\alpha g^{\alpha\beta} a^\dag_\beta (\svec{k}) \end{eqnarray} in terms of the Maxwell Krein forms $a,\,a^\dag$ satisfying %\ref{MCCR}. \begin{equation} \left[ a_\mu (\svec{k}),a_\nu^\dag (\svec{k}') \right]=-g_{\mu\nu}2 \omega(\svec{k})\delta(\svec{k}-\svec{k}'). \label{MCCR} \end{equation} These differ slightly from analogous forms given in \cite{Hell,C4Cl}, but are more convenient for our purposes. They satisfy the algebra \begin{equation} \left[ a_Q(\svec{k}),a_R^\dag(\svec{k}') \right]=-2\omega\delta(\svec{k}-\svec{k}') =\left[ a_R(\svec{k}),a_Q^\dag(\svec{k}') \right] \end{equation} with the remaining commutators vanishing. Then, we can define the field \begin{eqnarray} S(x;\gamma)&=& \frac{1}{2\pi}\int d\Omega_0 (k)\left\{ e^{-ikx}\left[ \frac{\omega}{m}\va a_R(\svec{k}) \right.\right. \label{Sfield}\\ &+& \left.\left.\left( \frac{\omega}{m}\va -\sqrt{2}\vb +(1-\gamma) \left( \frac{m}{2\omega}\vc -ix_0 m\va \right)\right) a_Q(\svec{k})\right]\right. \nonumber\\ &+& \left. e^{ikx}\left[ \frac{\omega}{m}\va^* a^\dag_R(\svec{k}) + \left( \frac{\omega}{m}\va^* -\sqrt{2}\vb^* \right.\right.\right.\nonumber\\ &+& \left.\left.\left. (1-\gamma) \left( \frac{m}{2\omega}\vc^*+i x_0 m \va^* \right)\right)a_Q^\dag(\svec{k}) \right]\right\}\nonumber \end{eqnarray} where we have again suppressed the vector index. Clearly, this expression simplifies considerably in Landau gauge $\gamma=1$. One can check that (\ref{Sfield}) yields the correct two-point expression (\ref{S2pt}) on $\HR^M$. The time evolution of $S_\mu$ is implemented by the Hamiltonian \begin{eqnarray} \lefteqn{H^S_0 (\gamma)=} \label{Sham}\\ &\int d\Omega_0 (k)\omega\left[ -a_Q^\dag(\svec{k})a_R(\svec{k}) -a_Q^\dag(\svec{k})a_R(\svec{k})-(1-\gamma)\frac{m^2}{\omega^2} a_Q^\dag(\svec{k})a_Q(\svec{k})\right] \nonumber\\ &=\int d\Omega_0 (k)\omega a_\mu^\dag(\svec{k})\left( -g^{\mu\nu} -(1-\gamma)\frac{m^2k^\mu k^\nu}{2\omega^4}\right)a_\nu(\svec{k})\nonumber\\ &=\int d\Omega_0 (k) \omega c_\mu^* (\svec{k})\left( \delta_\mu^\nu +(1-\gamma)\frac{m^2k_\mu k^\nu}{2\omega^4}\right)c_\nu(\svec{k})\nonumber \end{eqnarray} It is Krein symmetric, but not Hilbert symmetric unless one chooses Landau gauge $\gamma=1$. However, standard arguments show its closure to be Hilbert normal and Krein selfadjoint. By techniques similar to the Maxwell case~\cite{c4wood}, we find \begin{thm}\ \begin{enumerate} \item $[0,\infty)\subset\sigma(H_0^S)$ \item $H_0^S$ is accretive \end{enumerate} \end{thm} We note that the field $S_\mu$ is also non-local. We have, in fact, the non-vanishing equal-time commutation relations \begin{eqnarray} \left[ S_0 (t, \svec{x}),S_n (t,\svec{y}) \right]=\frac{-i}{m^2}\d_n \delta(\svec{x}-\svec{y}) \\ \left[ S_j (t,\svec{x}),S_n (t,\svec{y}) \right]=\frac{i}{m}\epsilon_{jn} \delta(\svec{x}-\svec{y}) \end{eqnarray} However, the full MCS field $A_\mu=V_\mu+S_\mu$ is local since \begin{eqnarray} \left[ A_0 (t, \svec{x}),A_n(t,\svec{y}) \right]= \left[ V_0(t,\svec{x}),V_n(t,\svec{y}) \right]+ \left[ S_0(t,\svec{x}),S_n(t,\svec{y}) \right]=0\\ \left[ A_j(t,\svec{x}),A_n(t,\svec{y}) \right]= \left[ V_j(t,\svec{x}),V_n(t,\svec{y}) \right]+ \left[ S_j(t,\svec{x}),S_n(t,\svec{y}) \right]=0 \end{eqnarray} where we have used equations~(\ref{V0n}) and~(\ref{Vjn}). We regularize the above fields by restricting them to a finite spatial volume $V\subset (\delta \mathbb{Z})^s$ with periodic boundary conditions (thus, $V$ is a product of finite cyclic groups). We denote the volume of $V$ by $|V|$. We denote the dual group of $V$ by $\Gamma$. The measure on $V$ is taken to be \begin{equation} \frac{1}{\sqrt{|V|}}\sum_{\svec{x}\in V} \delta^s \end{equation} while that on $\Gamma$ is \begin{equation} \frac{1}{\sqrt{|V|}}\sum_{\svec{k}\in \Gamma} \end{equation} with Fourier transform \begin{equation} \tilde{f} (\svec{k})=\frac{1}{\sqrt{|V|}}\sum_{\svec{x}\in V}\delta^s e^{-i\svec{k}\cdot\svec{x}}f(\svec{x}) \end{equation} We choose to use midpoint derivatives on the lattice so that \begin{equation} k_n = \frac{2}{\delta}\sin \left( \frac{k_n \delta}{2} \right) \end{equation} and \begin{equation} \omega^2 = k_n k_n = \sum_{n=1}^s \frac{4}{\delta^2} \sin^2\left( \frac{k_n \delta}{2} \right) \end{equation} With these conventions, the regularized MCS field becomes \begin{eqnarray} A(x) & = &\frac{1}{\sqrt{|V|}}\sum_{k\in \Gamma_0} \frac{1}{2\mu}\left[ \left( \frac{\sqrt{\omega^2 + \mu^2}}{m}\wc -\wb\right)a(\svec{k}) e^{-ikx}\right. \nonumber\\ &+& \left.\left(\frac{\sqrt{\omega^2+\mu^2}}{m}\wc+\wb \right)a^\dag(\svec{k}) e^{ikx}\right]_{k_0=\mu} \nonumber\\ &+&\frac{1}{\sqrt{|V|}}\sum_{k\in \Gamma_0} \frac{1}{2\omega}\left\{ e^{-ikx}\left[ \frac{\omega}{m}\va a_R(\svec{k}) \right.\right. \label{Afield}\\ &+& \left.\left.\left( \frac{\omega}{m}\va -\sqrt{2}\vb +(1-\gamma) \left( \frac{m}{2\omega}\vc -ix_0 m\va \right)\right) a_Q(\svec{k})\right]\right. \nonumber\\ &+& \left. e^{ikx}\left[ \frac{\omega}{m}\va^* a^\dag_R(\svec{k}) + \left( \frac{\omega}{m}\va^* -\sqrt{2}\vb^* \right.\right.\right.\nonumber\\ &+& \left.\left.\left. (1-\gamma) \left( \frac{m}{2\omega}\vc^*+i x_0 m \va^* \right)\right)a_Q^\dag(\svec{k}) \right]_{k_0=\omega}\right\}\nonumber \end{eqnarray} in which our infrared regularization is to sum over $\Gamma_0\equiv \Gamma\backslash \{0\}$. This field is defined as a distribution-valued unbounded operator on the lattice Fock space $\HR=\HR^P\otimes\HR^M$. Here, $\HR^P$ is the symmetric Fock space over $L^2(\Gamma_0)\otimes\C^d$ with measure \begin{equation} \frac{1}{\sqrt{|V|}}\sum_{k\in\Gamma_0} \frac{1}{2\mu} \end{equation} while $\HR^M$ is the symmetric Fock space over $L^2(\Gamma_0)\otimes\C^d$ with measure \begin{equation} \frac{1}{\sqrt{|V|}}\sum_{k\in\Gamma_0} \frac{1}{2\omega} \end{equation} In the following sections, we shall loosely use $\Gamma$ to indicate whichever momentum sums are appropriate. \section{Krein Gauge Transformations\label{s3}} The formal Lagrangian density for the MCS-Higgs model will be \begin{equation} \mathcal{L}=\mathcal{L}\,^\mathrm{cs}+\mathcal{L}\,^\mathrm{boson} \end{equation} where $\mathcal{L}\,^\mathrm{cs}$ is given by equation~(\ref{MCSlagrangian}) and the boson term is \begin{equation} \mathcal{L}\,^\mathrm{boson}=\sum_{j=1}^{2}\left[D_{\mu}(\phi_{j})D^{\mu}(\phi_{j}) -m_{0}^{2}(\phi_{j})^{2}\right] -V(\phi_{1}(x),\phi_{2}(x)) \end{equation} for two neutral scalar fields $\phi_{j},j=1,2$ with a potential $V(\phi_{1}(x),\phi_{2}(x))=\lambda_{4}(\phi_{1}^{2}(x)+\phi_{2}^{2}(x))^{2} +\lambda_{2}(\phi_{1}^{2}(x)+\phi_{2}^{2}(x))$ with $m_{0}\geq0,\lambda_{4}> 0$ and covariant derivative $D_{\mu}(\phi)=(\partial_{\mu}+ieA_{\mu})(\phi)$. The usual canonical construction produces a Hamiltonian operator on the lattice of the form \begin{equation} H(V,\delta)=H_{0}\,^\mathrm{cs}+H_{0}\,^\mathrm{boson}+H\,_{int}+V(\phi_{1},\phi_{2}) \end{equation} with the free MCS-Hamiltonian in equations~(\ref{Vham}) and~(\ref{Sham}) and a free boson-Hamiltonian with mass $m_{0}$. The interaction terms contain the MCS field as \begin{eqnarray} H\,_\mathrm{int} & = & H\,_\mathrm{el}+H\,_\mathrm{mag} \nonumber \\ H\,_\mathrm{el} & = & \sum_{x\in{V}}\delta^{s}\left[e\left\{\pi_{1}(x)\phi_{2}(x) -\pi_{2}(x)\phi_{1}(x)\right\}A_{0}(x)\right] \nonumber \\ H\,_\mathrm{mag} & = & \sum_{x\in{V}}\delta^{s}\left[e\left\{\partial_{\ell}\phi_{2}(x)\phi_{1}(x) -\partial_{\ell}\phi_{1}(x)\phi_{2}(x)\right\}A_{\ell}(x)\right. \nonumber \\ & + & \left.\frac{e^{2}}{2}\left\{(\phi_{1}^{2}(x)+\phi_{2}^{2}(x)\right\}A_{\ell}^{2}(x)\right] \nonumber \\ V(\phi_{1},\phi_{2}) & = & \sum_{x\in{V}}\delta^{s}V(\phi_{1}(x),\phi_{2}(x)) \end{eqnarray} Since in our representation the MCS-field is a normal operator, the `magnetic' part of the Hamiltonian has lost the positivity which would normally arise from the spatial covariant derivatives while the `electric' part has both real and imaginary terms in its numerical range. Usually, to maintain positivity of the Hamiltonian operator for a gauge theory of this type, quantization would be carried out in either the Coulomb or axial gauges. However, such gauges are not particularly amenable for renormalization issues arising when taking the continuum limit of the lattice theory. In order to understand the structure of $H(V,\delta)$ above and make transparent the Krein transformations that we use below, we resort to transforming the annihilation and creation operators into harmonic oscillator coordinates and momenta. Our particular conventions are standard for the boson terms and about which we shall say little while those for the MCS-field are given in more detail. Both are described in the appendix. From equation~(\ref{CSHOcoord}) in appendix~A, the Fourier components of $A_{0}(x)$ contain both skew-symmetric and symmetric terms. The first Krein transformation removes the symmetric terms in $A_{0}(x)$ by means of \begin{equation} T_{1}=\sum_{p\in\Gamma\,'}(-i)\sqrt{\frac{\omega}{\mu}} q_{\ell,\,2}^\mathrm{cs}(p)p_{\ell,\,0}^\mathrm{cs}(p). \end{equation} while the second Krein transformation \begin{equation} T_{2}=\sum_{p\in\Gamma\,'}(-i)q_{\ell,\,0}^\mathrm{cs}(p)p_{\ell,\,1}^\mathrm{cs}(p). \end{equation} removes the skew-symmetric terms from $A^{\prime}_{\ell}(x)$ after a Krein unitary transformation using $T_{1}$. Each of the expressions above is realized as a sum of commuting essentially selfadjoint operators for which the finite particle vectors $D_{F}$ are analytic vectors. Each is also skew-symmetric with respect to the Krein metric so will be the infinitesimal generator of a Krein unitary but unbounded operator. The number of particle estimate for each $T_{j}$ on a $k$-particle vector increases as $O((n+k+1)^{n})$ in the analytic vector calculation which leads to geometric convergence in $t$ as below. \begin{lemma} \label{lemma:Aprime} The operators $\widetilde{T_{j}}, j=1,2$ are self-adjoint and \dag -skew-symmetric. Further each $exp[t\widetilde{T_{j}}]$ is selfadjoint and \dag-unitary for real t and satisfy \begin{equation} e^{t\widetilde{T_{2}}}e^{t\widetilde{T_{1}}}\,\widetilde{A_{0}(x)}\, e^{-t\widetilde{T_{1}}}e^{-t\widetilde{T_{2}}}\Phi =\left(A_{0}(x)+t[T_{1},A_{0}(x)]\right)=A^{\prime}_{0}(x;t)\Phi \label{eq:A0prime} \end{equation} \begin{equation} e^{t\widetilde{T_{2}}}e^{t\widetilde{T_{1}}}\,\widetilde{A_{\ell}(x)}\, e^{-t\widetilde{T_{1}}}e^{-t\widetilde{T_{2}}}\Phi =\left(A_{\ell}(x)+t[T_{1},A_{\ell}(x)]+t[T_{2},A_{\ell}(x)]\right)\Phi =A^{\prime}_{\ell}(x;t)\Phi \label{eq:Aellprime} \end{equation} for all $\Phi\in\,D_{F}$ and complex t with $|t|0$ provided $c_{2}$ is large enough and $c_{1}>c_{3}/2$. As $(X+Y+Z)^{*}$ and $Z^{*}$ are densely defined on the finite particle vectors, Chernoff's theorem applied to $X+Y+S=X+Y+Z+S - Z$ produces: \begin{thm} After Krein gauge transformations, the transformed Hamiltonian $H'(V,\delta)=\mathrm{Re}H'(V,\delta)+S$ is maximal accretive after closure on the finite particle domain $D_{F}$ and hence Krein essentially selfadjoint. \end{thm} Krein selfadjointness follows by a simple argument which can be found in~\cite[Lemma 2.2, page 5]{c4ym}. \section*{Acknowledgement} One of us, JLC, would like to thank J\"{u}rg Fr\"{o}hlich for the hospitality of the Institute for Theoretical Physics, ETH, Z\"{u}rich, during the winter of 1998 and discussions on an indefinite metric approach to the Higgs-Maxwell-Chern-Simons model. \appendix \newcommand{\qb}[2]{\ensuremath{q_{#2,#1}}} \newcommand{\pb}[2]{\ensuremath{p_{#2,#1}}} \newcommand{\qcs}[2]{\ensuremath{q^\mathrm{cs}_{#2,#1}}} \newcommand{\pcs}[2]{\ensuremath{p^\mathrm{cs}_{#2,#1}}} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection\arabic{equation}} \section{Harmonic Oscillator Coordinates} We choose standard harmonic oscillator coordinates for the time zero boson field and its conjugate momentum, yielding (see, e.g.,~\cite[chapter~II, section~B]{jaffethesis}): \begin{eqnarray} \phi_j(x)&=&\frac{1}{\sqrt{|V|}}\sum_{k\in \Gamma}\frac{1}{2\mu}\left[ e^{ik\cdot x}a_j(k)+e^{-ik\cdot x}a_j^*(k)\right] \nonumber \\ &=&\frac{1}{\sqrt{2|V|}}\sum_{k\in\Gamma} \frac{1}{2\mu}\left[ \qb{j}{1}(k) \cos (k\cdot x) + \qb{j}{2}(k) \sin (k\cdot x) \right] \\ \pi_j(x)&=&\frac{1}{\sqrt{|V|}}\sum_{k\in \Gamma}\frac{-i}{2}\left[ e^{ik\cdot x}a_j(k)-e^{-ik\cdot x}a_j^*(k)\right] \nonumber \\ &=&\frac{1}{\sqrt{2|V|}}\sum_{k\in\Gamma} \frac{1}{2\mu}\left[ \pb{j}{1}(k) \cos (k\cdot x) + \pb{j}{2}(k) \sin (k\cdot x) \right] \end{eqnarray} for $j=1,2$. The free boson Hamiltonian with Wick ordering then becomes \begin{eqnarray} H_0^\mathrm{boson}&=&\sum_{k\in \Gamma} \frac{1}{2} a_j^*(k)a_j(k) = \sum_{k\in \Gamma} \frac{1}{2}\left[ \pb{j}{\ell}^2 + \mu^2 \qb{j}{\ell}^2\right] \end{eqnarray} In order to define harmonic oscillator coordinates for the MCS field $A_\mu=V_\mu+S_\mu$ in Landau gauge ($\gamma=1$), we first express the MCS field in terms of Fock operators $b_\mu$ defined by \begin{eqnarray} b_0(k)=\frac{c_0(k)}{\sqrt{2\omega}}, & \displaystyle b_1(k)=\frac{k_n c_n(k)}{\omega \sqrt{2\omega}}, & b_2(k)=\frac{a(k)}{\sqrt{2\mu}} \end{eqnarray} so that $b_0,b_1$ represent the ghost degrees of freedom while $b_2$ corresponds to the single physical mode. In terms of these, we have for the time zero MCS field \begin{equation} A_\mu=\frac{1}{\sqrt{|V|}}\sum_{k\in \Gamma} \left[ M_{\mu\nu}b_\nu (k) e^{ik\cdot x} + \overline{M}_{\mu\alpha}(-g)_{\alpha\beta}b^*_\beta(k) e^{-ik\cdot x}\right]\label{fullMCSfockexp} \end{equation} with \begin{equation} M=\left( \begin{array}{ccc} -\frac{\omega}{m\sqrt{2\omega}} & 0 & -\frac{\omega}{m\sqrt{2\mu}} \\ -\frac{1}{\sqrt{2\omega}}\left( \frac{k_1}{m}-\frac{ik_2}{\omega} \right) & \frac{-ik_2}{\omega\sqrt{2\omega}} & -\frac{1}{\sqrt{2\mu}}\left( \frac{\mu k_1}{m\omega}-\frac{ik_2}{\omega}\right)\\ -\frac{1}{\sqrt{2\omega}}\left( \frac{k_2}{m}+\frac{ik_1}{\omega} \right) & \frac{ik_1}{\omega\sqrt{2\omega}} & -\frac{1}{\sqrt{2\mu}}\left( \frac{\mu k_2}{m\omega} +\frac{ik_1}{\omega}\right) \end{array}\right) \end{equation} Note the appearance of the indefinite metric in the second term of~(\ref{fullMCSfockexp}). We define Krein operators $a_\mu=-b_\mu$ and $a^\dag_\mu=g_{\mu\beta}b^*_\beta$. It is in terms of these %Krein operators that we define our harmonic oscillator coordinates for the MCS field: \begin{eqnarray} \qcs{\mu}{1} (k) &=&\frac{1}{2}\left[ a_\mu (k) + a^\dag_\mu (k) + a_\mu (-k) + a_\mu^\dag (-k) \right]\\ \qcs{\mu}{2} (k) &=&\frac{i}{2}\left[ a_\mu (k) - a^\dag_\mu (k) - a_\mu (-k) + a_\mu^\dag (-k) \right]\\ \pcs{\mu}{1} (k) &=&\frac{i}{2}\left[ a_\mu (k) - a^\dag_\mu (k) + a_\mu (-k) - a_\mu^\dag (-k) \right]\\ \pcs{\mu}{2} (k) &=&\frac{1}{2}\left[-a_\mu (k) - a^\dag_\mu (k) + a_\mu (-k) + a_\mu^\dag (-k) \right] \end{eqnarray} These satisfy the relations \begin{eqnarray} \qcs{\mu}{\ell}(-k)=(-1)^{\ell+1}\qcs{\mu}{\ell}(k) &\mathrm{\ and\ } & \pcs{\mu}{\ell}(-k)=(-1)^{\ell+1}\pcs{\mu}{\ell}(k) \end{eqnarray} Thus, upon restriction to ``allowed'' momenta $k,k'\in\Gamma'$, we have the commutation relations \begin{equation} [\qcs{\mu}{\ell}(k),\pcs{\nu}{n}(k')]=ig_{\mu\nu}\delta_{\ell n}\delta_{k,k'} \end{equation} This leads to the following expressions for the time zero MCS field %and its conjugate momentum in Landau gauge \begin{eqnarray} A_0 (x) &=& \frac{1}{\sqrt{|V|}}\sum_{k\in\Gamma'} \left\{ \cos(k\cdot x) \left[ \frac{2\omega}{m\sqrt{2\omega}}\qcs{0}{1}(k) + \frac{2\omega}{m\sqrt{2\mu}}\qcs{2}{1}(k) \right] \right. \label{CSHOcoord} \\ &+& \left. \sin(k\cdot x) \left[ \frac{2\omega}{m\sqrt{2\omega}} \qcs{0}{2}(k) +\frac{2\omega}{m\sqrt{2\mu}}\qcs{2}{2}(k)\right] \right\} \nonumber \\ A_\ell (x) &=& \frac{1}{\sqrt{|V|}}\sum_{k\in\Gamma'} \left\{ \cos(k\cdot x) \left[ -\frac{2k_\ell}{m\sqrt{2\omega}}\pcs{0}{2}(k)- \frac{2\mu k_\ell}{m\omega\sqrt{2\mu}}\pcs{2}{2}(k)\right.\right. \nonumber\\ &-& \left.\frac{2\epsilon_{\ell n}k_n}{\omega} \left( \frac{\qcs{0}{2}(k)}{\sqrt{2\omega}}- \frac{\qcs{1}{2}(k)}{\sqrt{2\omega}}+ \frac{\qcs{2}{2}(k)}{\sqrt{2\mu}} \right)\right] \nonumber \\ &+& \sin(k\cdot x) \left[ \frac{2k_\ell}{m\sqrt{2\omega}}\pcs{0}{1}(k) +\frac{2\mu k_\ell}{m\omega\sqrt{2\mu}}\pcs{2}{1}(k) \right. \nonumber \\ &+& \left.\left. \frac{2\epsilon_{\ell n}k_n}{\omega} \left( \frac{\qcs{0}{1}(k)}{\sqrt{2\omega}} - \frac{\qcs{1}{1}(k)}{\sqrt{2\omega}} + \frac{\qcs{2}{1}(k)}{\sqrt{2\mu}} \right) \right] \right\} \nonumber \end{eqnarray} For the free MCS Hamiltonian in Landau gauge with Wick ordering, this leads to \begin{eqnarray} H_0^\mathrm{cs} &=& \sum_{k\in\Gamma'}\left\{ -\frac{\omega}{2}\left( \pcs{0}{\ell}(k)\pcs{0}{\ell}(k) +\qcs{0}{\ell}(k)\qcs{0}{\ell}(k) \right) \right. \nonumber \\ &+& \frac{\omega}{2} \left( \pcs{1}{\ell}(k)\pcs{1}{\ell}(k) + \qcs{1}{\ell}(k)\qcs{1}{\ell}(k) \right) \\ &+& \left. \frac{\mu}{2}\left( \pcs{2}{\ell}(k)\pcs{2}{\ell}(k) + \qcs{2}{\ell}(k)\qcs{2}{\ell}(k) \right)\right\}\nonumber \end{eqnarray} %\bibliography{RefKrein(1)} \begin{thebibliography}{10} \bibitem{BFSI} D.Brydges, J.~Fr{\"{o}}hlich, and E.Seiler. \newblock On the quantization of gauge fields. {I}. \newblock {\em Commun.Math.Phys.}, 71:227--284, 1979. \bibitem{FroMarch} J.Fr{\"{o}}hlich and P.A.Marchetti. \newblock Quantum {F}ield {T}heory of {V}ortices and {A}nyons. \newblock {\em Commun.Math.Phys.}, 21:177 -- 223, 1989. \bibitem{Sei1982} E.Seiler. \newblock {\em Gauge {T}heories as a {P}roblem in {C}onstructive {Q}uantum {F}ield {T}heory and {S}tatistical {M}echanics}. \newblock Lecture Notes in Physics, vol.159, Springer-Verlag, 1982. \bibitem{DJT} S.Deser, R.Jackiw, and S.Templeton. \newblock Topologically {M}assive {G}auge {T}heories. \newblock {\em Ann.Phys.}, 140:372--411, 1982. \bibitem{Dunn1995} G.V.Dunne. \newblock {\em Self-{D}ual {C}hern-{S}imons {T}heory}. \newblock Lecture Notes in Physics, vol.36, Springer-Verlag, 1995. \bibitem{Chen:1996} L.Chen, G.Dunne, K.Haller, and E.~Lim-Lombridas. \newblock Canonical quantization of spontaneously broken topological massive gauge theory. \newblock {\em J.Math.Phys.}, 37:2602--2627, 1996. \bibitem{Dunn1998} G.V.Dunne. \newblock Aspects of {C}hern-{S}imons {T}heory. \newblock In A.Comtet et~al., editors, {\em Topological aspects of low dimensional systems, Les Houches 1998}. 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