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Constructive Quantum Field Theory, Indefinite Metric Gauge theories, Krein selfadjointness, Chern-Simons, Maxwell-Chern-Simons fields, Canonical Quantization 
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\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<A_\mu (x) A_\nu (y)\right> = } \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<A_\mu(x)A_\nu(y)\right>
\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 $|t|<t_{0}$.
\end{lemma}
To begin the proof notice that $\eta\,T_{j}\,\eta=T_{j}$ on vectors
in $D_{F}$ so by closure this relation is valid for
$\widetilde{T_{j}}$. Nelson's theorem on analytic vectors shows
$\eta\,e^{tT_{j}}\,\eta=e^{tT_{j}}$ proving the first statement.

For their Krein unitary property, consider
$\Phi,\Psi\in\,D(e^{z\widetilde{T_{j}}})$ for suitable complex
$z$ and sequences of vectors $\Phi_{n},\Psi_{n}$ which have compact
support with respect to the spectrum of $\widetilde{T_{j}}$ and
converge strongly to $\Phi,\Psi$.  Then notice
$T_{j}^{\dagger}=-T_{j}=\eta\,T_{j}^{*}\,\eta=\eta\,T_{j}\,\eta$ as
bounded operators on $\Phi_{n},\Psi_{n}$ so consequently
\begin{eqnarray*}
\{e^{z\widetilde{T_{j}}}\Phi,e^{z^{\prime}\widetilde{T_{j}}}\Psi\} &
= &
\lim_{n\rightarrow\infty}\{e^{z\widetilde{T_{j}}}\Phi_{n},e^{z^{\prime}
\widetilde{T_{j}}}\Psi_{n}\} \\
& = & \lim_{n\rightarrow\infty}(\Phi_{n},
e^{\overline{z}\widetilde{T_{j}}}\eta
e^{z^{\prime}\widetilde{T_{j}}}\Psi_{n}) \\
& = &  \lim_{n\rightarrow\infty}(\Phi_{n},\eta
e^{(z^{\prime}-\overline{z})\widetilde{T_{j}}}\Psi_{n}) \\
& = & \lim_{n\rightarrow\infty}\{\Phi_{n},
e^{(z^{\prime}-\overline{z})\widetilde{T_{j}}}\Psi_{n}\} \\
& = & \{\Phi,e^{(z^{\prime}-\overline{z})\widetilde{T_{j}}}\Psi\}
\end{eqnarray*}
For the commutator identities, set
$\Phi_{N}=\sum_{n=0}^{N}\frac{z^{n}T_{1}^{n}}{n!}\Phi$ which
converges strongly to $e^{z\widetilde{T_{1}}}\Phi$ for
$\Phi\,\in\,D_{F}$ and $|z|<z_{0}$.  Each $\Phi_{N}$ is a
finite particle vector belonging to the domain of $A_{\mu}(x)^{k}$
for any positive integer $k$ and $\mu=0,1,2$ so we may calculate the
expression
\begin{eqnarray*}
A_{0}(x)\Phi_{N} & = &
\sum_{n=0}^{N}\frac{z^{n}A_{0}(x)T_{1}^{n}}{n!}\Phi \\
& = & \sum_{n=0}^{N}\frac{z^{n}T_{1}^{n}}{n!}A_{0}(x)\Phi +
z\sum_{n=0}^{N-1}\frac{z^{n}T_{1}^{n}}{n!} [T_{1},A_{0}(x)]\Phi
\end{eqnarray*}
Taking the limit in $N$ shows the right-hand side converges for
$|z|<z_{0}$ with $e^{z\widetilde{T_{1}}}\Phi$ in the domain of the
closure of $A_{0}(x)$ and
$\widetilde{A_{0}(x)}e^{z\widetilde{T_{1}}}\Phi$ in the range of
$e^{z\widetilde{T_{1}}}$ leading to equation~(\ref{eq:A0prime})
since $T_{2}$ commutes with $A^{\prime}_{0}(x)$.  A similar
calculation results in equation~(\ref{eq:Aellprime}).  A short
calculation produces expressions for the gauge transformed fields
which for $t=1$ are
\begin{eqnarray}
A^{\prime}_{0}(x) & = & \frac{1}{\sqrt{|V|}}\ \sum_{k\in\Gamma}
\frac{\sqrt{2\omega}}{m}\,\left\{q\,^\mathrm{cs}_{1,\,0}(k)\cos(k\cdot x)+
q\,^\mathrm{cs}_{2,\,0}(k)\sin(k\cdot x)\right\} \\
A^{\prime}_{\ell}(x) & = & \frac{1}{\sqrt{|V|}}\
\sum_{k\in\Gamma}\,\left \{\left(\frac{2\epsilon_{\ell
\,n}k_{n}}{\omega\sqrt{2\omega}}q\,^\mathrm{cs}_{2,\,1}(k)-\frac{2\mu\,k_{\ell}}{m\omega\sqrt{2\mu}}
p\,^\mathrm{cs}_{2,\,2}(k)\right)\cos(k\cdot x) \right. \nonumber \\
& + & \left.\left(-\frac{2\epsilon_{\ell
\,n}k_{n}}{\omega\sqrt{2\omega}}q\,^\mathrm{cs}_{1,\,1}(k)+\frac{2\mu\,k_{\ell}}{m\omega\sqrt{2\mu}}
p\,^\mathrm{cs}_{1,\,2}(k)\right)\sin(k\cdot x) \right\} \nonumber \\
&   &
\end{eqnarray}
which are respectively skew-symmetric and symmetric.

Turning our attention to the Hamiltonian we find a completely
analogous result in
\begin{lemma} \label{lemma:Hprime}
For all $\Phi\,\in\,D_{F}$ and complex t with $|t|<t_{0}$
\begin{eqnarray}
e^{t\widetilde{T_{2}}}e^{t\widetilde{T_{1}}}\,\widetilde{H}\,
e^{-t\widetilde{T_{1}}}e^{-t\widetilde{T_{2}}}\Phi & = &
H^{\prime}(V,\delta;t)\Phi \nonumber \\
H^{\prime}(V,\delta;t)& = &
H_{0}^\mathrm{boson}+H_{0}^\mathrm{cs}\,^{\prime}(t)+H^{\prime}_\mathrm{int}(t)
\label{eq:Hprime}
\end{eqnarray}
where $H^{\prime}_\mathrm{int}(t)$ is $H_\mathrm{int}(t)$ with $A_{\mu}(x)$
replaced by $A^{\prime}_{\mu}(x;t)$ and the gauge transformed
Chern-Simons part of the free Hamiltonian given by
\begin{eqnarray}
H_{0}^\mathrm{cs}\,^{\prime}(t)& = &
\sum_{k\,\in\Gamma}\left\{\frac{(\mu^{2}-t^{2}\,\omega^{2})}{2}
q\,^\mathrm{cs}_{\ell,\,2}(k)^{2}- \mu \right. \nonumber \\
& + &
\left.\frac{\omega}{2}\left(q\,^\mathrm{cs}_{\ell,\,1}(k)^{2}-4\right)\right.+
\left.\frac{\left(\sqrt{\mu}\,p\,^\mathrm{cs}_{\ell,\,2}(k)-
t\sqrt{\omega}\,p\,^\mathrm{cs}_{\ell,\,1}(k)\right)}{2}^{2}\right.
\nonumber
\\
& + &
\left.t\,\omega\,q\,^\mathrm{cs}_{\ell,\,0}(k)\left(q\,^\mathrm{cs}_{\ell,\,1}(k)
+\sqrt{\omega/\mu}q\,^\mathrm{cs}_{\ell,\,2}(k)\right)\right. \nonumber \\
& - &
\left.t\,\sqrt{\omega\,\mu}\,p\,^\mathrm{cs}_{\ell,\,2}(k)p\,^\mathrm{cs}_{\ell,\,0}(k)
+ \omega\,(1-t)\,p\,^\mathrm{cs}_{\ell,\,0}(k)p\,^\mathrm{cs}_{\ell,\,1}(k)\right.
\nonumber \\
& + &
\left.(1-t^{2})\frac{\omega}{2}\,\left(p\,^\mathrm{cs}_{\ell,\,0}(k)^{2}-
q^\mathrm{cs}_{\ell,\,0}(k)^{2}\right)\right. \nonumber \\
& + &
\left.\frac{\omega}{2}(1-t^{2})p\,^\mathrm{cs}_{\ell,\,1}(k)^{2}\right\}
\label{eq:H0csprime}
\end{eqnarray}
\end{lemma}
The statements for the domains of the unbounded operators are the
same as for Lemma~\ref{lemma:Aprime} while calculations use the
relation ${e^{t\widetilde{T_{2}}}e^{t\widetilde{T_{1}}}}\,S^{2}\,
e^{-t\widetilde{T_{1}}}e^{-t\widetilde{T_{2}}}\Phi=\left\{{e^{t\widetilde{T_{2}}}e^{t\widetilde{T_{1}}}}\,S\,
e^{-t\widetilde{T_{1}}}e^{-t\widetilde{T_{2}}}\right\}^{2}\Phi$ for
$|t|<t_{0}$ which is valid on $D_{F}$ after taking closures.  In
each of Lemmas~\ref{lemma:Aprime} and~\ref{lemma:Hprime} the
left-hand side is analytic in a disc $|t|<t_{0}$ while the
right-hand side is an entire function of t and provides a unique
analytic continuation of the operator on the left-hand side to the
entire complex plane.  However it is straight forward to find a
dense domain $D_{0}$ which is a common core for all these operators
and their powers on which both sides of
equations~(\ref{eq:A0prime}),~(\ref{eq:Aellprime}) and~(\ref{eq:Hprime})
are valid as entire functions in $t$.  Consider a k-finite particle
vector $\Phi_{k}$ realized as a normalized Hermite function
$\Phi_{k}(\{q\,^\mathrm{cs}_{\ell,\,2}\},\{q\,^\mathrm{cs}_{\ell,\,1}\},q^{\prime})$
in the q-harmonic oscillator variables with the coordinates
appearing in $\{T_{j}|j=1,2\}$ denoted explicitly. Let
$\chi\in\,C^{\infty}_{0}(R_{+})$ be positive, one for
$0\leq\,t\leq1/2$ and zero for $t\geq1$ say.  Form the cutoff
functions $\chi_{j}\,,_{N}(q)=\chi(\sum_{k\in\,
\Gamma,\,j}q\,^{cs}_{\ell,\,j}(k)^{2}/N^{2})$ for N=1,2,... which
have support inside a ball of radius $N$.  Then by taking products and
convolutions in different variables, we find
\begin{lemma}
Let $D_{0}$ denote the linear span of vectors formed by
\begin{equation}
\Phi_{k},_{N}(q)=\chi_{2},\,_{N}(q)\left(\chi_{1},\,_{N}*\,\Phi_{k}\right)(q)
\end{equation}
as k and N vary over the positive integers.  For each of the
operators and their powers appearing in Lemmas~\ref{lemma:Aprime}
and~\ref{lemma:Hprime}, $D_{0}$ forms a common dense core.  Moreover
each of the relations in
equations~(\ref{eq:A0prime}),~(\ref{eq:Aellprime}) and~(\ref{eq:Hprime})
are valid as entire analytic functions of t.  In particular the
corresponding operator relations are densely defined on $D_{0}$ for
t=1.
\end{lemma}
That $D_{0}$ is dense and a common core is clear as also the
analyticity.  Upon re-examining convergence of the exponential
series in equations~(\ref{eq:A0prime}),~(\ref{eq:Aellprime})
and~(\ref{eq:Hprime}) the $q\,^\mathrm{cs}_{\ell,\,2}(k)$ and
$p\,^\mathrm{cs}_{\ell,\,1}(k)$ operators are bounded by $N$ when acting on
vectors in $D_{0}$ while the $N$-particle estimates applied to the
operators $q\,^\mathrm{cs}_{\ell,\,0}(k)$ and $p\,^\mathrm{cs}_{\ell,\,0}(k)$
acting on $\Phi_{k}$ provide a bound $O((n+k+1)^{n/2})$. Vectors in
$D_{0}$ are now entire vectors for each of the Krein gauge
transformations.

The gauge transformations in Lemma~\ref{lemma:Hprime} are `natural'
for the Hamiltonian $H(V,\delta)$, as may be seen from
\begin{equation}
\widetilde{H}\,e^{itT_{j}}\Phi_{k,\,N}=e^{itT_{j}}\left\{H+it[H,T_{j}]
+(it)^{2}/2[[H,T_{j}],T_{j}]\right\}\Phi_{k,\,N}
\end{equation}
with $\Phi_{k,\,N}\in\,D_{0}$ and $-\infty<t<\infty$. By using
a Riemann approximation and then taking strong limits, the
right-hand side converges and defines convergence of the left-hand
side to the relation
\begin{eqnarray}
\widetilde{H}\int_{-\infty}^{\infty}F(\lambda)dE_{j}(\lambda)\Phi_{k,\,N}
& = & \int_{-\infty}^{\infty}\widetilde{F(t)}\widetilde{H}e^{itT_{j}}\Phi_{k,N}dt \nonumber \\
& = & \int_{-\infty}^{\infty}F(\lambda)dE_{j}(\lambda)H\Phi_{k,N}
\nonumber \\
& + &
\int_{-\infty}^{\infty}F^{\prime}(\lambda)dE_{j}(\lambda)[H,T_{j}]\Phi_{k,N}
\nonumber \\
& + & \int_{-\infty}^{\infty}F^{\prime\prime}(\lambda)/2
\,dE_{j}(\lambda)[[H,T_{j}],T_{j}]\Phi_{k,N}
\end{eqnarray}
where $\{E_{j}(\lambda) \}$ is the spectral measure associated with
the essentially self-adjoint operator $T_{j}$.  Choosing
$F\,\in\,C^{2}_{0}(R)$ shows that a large class of the bounded
functions of $\widetilde{T_{j}}$ belong to the domain of
$\widetilde{H}$. As we shall show in section~\ref{s5}, the Hamiltonians
$H(V,\delta)$ and $H^{\prime}(V,\delta;1)$ give Krein equivalent
dynamics.

\section{Real Part of the Hamiltonian\label{s4}}

Following the Krein gauge transformations in the previous section,
all operators in the remainder of this article will take their
physical values at $t=1$. The transformed Hamiltonian is thereby
split into two parts. One part defines a symmetric operator while
the other part defines a skew-symmetric operator.  We now show that
the symmetric part is bounded below and in fact is essentially
selfadjoint, hence the transformed Hamiltonian is accretive. The
transformed lattice Hamiltonian operator may be recast as
\begin{equation}
H'(V,\delta)=H_{0}^\mathrm{cs}\,'+H_{0}^\mathrm{boson}+
V(\phi_{1},\phi_{2})+H'_\mathrm{int}.
\end{equation}
To exploit positivity in the real part of the Hamiltonian, it is
convenient to remove the gradient terms from $H_{0}^\mathrm{boson}$ and
rewrite this as
\begin{equation}
H_{0}^\mathrm{boson}=H_{0}^\mathrm{boson}\,'
+\sum_{x\in{V}}\frac{\delta^{s}}{2}[\nabla\phi_{j}(x)]^{2}
\end{equation}
in which
\begin{equation}
H_{0}^\mathrm{boson}\,'=\sum_{k\in\Gamma}\frac{1}{2}[p_{j,\ell}^2(k)+
m_{0}^{2}q_{j,\ell}^{2}(k)-4\mu(k)]
\end{equation}
is again a free massive harmonic oscillator which on the lattice
acts in the same way as a number operator for determining domains.
By reincorporating the gradient terms into the covariant derivative,
the real part of $H'(V,\delta)$ takes the form
\begin{equation}
\mathrm{Re}H'(V,\delta)+c_{1}I=X+Y
\end{equation}
where $c_{1}$ is a positive constant to be chosen later and
\begin{equation}
X=H_{0}^\mathrm{cs}\,''+H_{0}^\mathrm{boson}\,'+c_{1}I,\ \
Y=Y_\mathrm{deg}+Y_{+}
\end{equation}
with
\begin{equation}
Y_\mathrm{deg}=\sum_{k\in\Gamma}[-\sqrt{\mu\omega}\,p_{\ell,1}^\mathrm{cs}(k)
p_{\ell,2}^\mathrm{cs}(k)]
\end{equation}
and
\begin{equation}
Y_{+}=\sum_{x\in{V}}\frac{\delta^{s}}{2}[\{\partial_{\ell}\phi_{1}(x)
-e\phi_{2}(x)A_{\ell}(x)\}^{2}+\{\partial_{\ell}\phi_{2}(x)
+e\phi_{1}(x)A_{\ell}(x)\}^{2}]+V(\phi_{1},\phi_{2}).
\end{equation}
The operator $H^\mathrm{cs}_{0}\,''$ denotes the real part of
$H^\mathrm{cs}_{0}\,'$ with the degeneracy producing term $Y_{deg}$
removed.  As a sum of squares of symmetric operators which have
dense analytic vectors, $Y_{+}$ clearly defines a symmetric operator
which is bounded below since the Higgs potential is taken to be
bounded below. The degenerate quadratic term $Y_\mathrm{deg}$ will
be treated as a perturbation which is of the same size as the
Chern-Simons number operator. Following the notation used in Reed
and Simon~\cite[pages 174 -- 176 and Theorem X.14 on page 164]{R&S},
it is a straight forward matter to use the Konrady trick for $Y_{+}$
with $Z=c_{2}X^{2}$ to obtain the quadratic estimate
\begin{equation}
Z^{2}\leq(X+Y+Z)^{2}+c_{3}I       \label{realquad}
\end{equation}
for some positive constant $c_{3}$. The degenerate quadratic piece
changes none of the estimates for the commutator $[X,[X,Y]]$ and is
easily bounded in the term $XY_\mathrm{deg}X$ by completing the square as
\begin{eqnarray}
Y_\mathrm{deg} & = &
\sum_{k\in\Gamma}\frac{1}{2}[-\mu\,p_{\ell,2}^\mathrm{cs}\,^{2}(k)
-\omega\,p_{\ell,1}^\mathrm{cs}\,^{2}(k)
+(\sqrt{\mu}p_{\ell,2}^\mathrm{cs}(k)-
\sqrt{\omega}p_{\ell,1}^\mathrm{cs}(k))^{2}]
\nonumber \\
& = & -Y_\mathrm{deg,1}+Y_\mathrm{deg,+}
\end{eqnarray}
whereupon $XY_\mathrm{deg,+}X\geq0$ and may be discarded with the other
positive terms in obtaining the quadratic estimate above.  The terms
in $Y_\mathrm{deg,1}$ are cancelled by terms that appear in
$H_{0}^\mathrm{cs}\,''$ with the result $H_{0}^\mathrm{cs}\,''-
Y_\mathrm{deg,1}\geq0$.
Now we find
\[2c_{2}X^{3}+2c_{2}XYX\geq2c_{2}X[X+Y_{+}+Y_\mathrm{deg,+}-Y_\mathrm{deg,1}]X
\geq2c_{2}X[H_{0}^\mathrm{boson,\,'}+c_{1}I]X\] so
\begin{equation}
2c_{2}X^{3}+2c_{2}XYX+c_{2}[X,[X,Y]]\geq\,c_{2}(2c_{1}-c_{3})X^2\geq0
\end{equation}
provided we choose $c_{1}\geq\,c_{3}/2$.  The quadratic estimate and
W\"{u}st's Theorem allow us to conclude that
$\mathrm{Re}H'(V,\delta)+c_{1}I=X+Y$ is essentially selfadjoint and
bounded below.  It is important to notice that the term
$2c_{2}XH_{0}^\mathrm{boson,\,'}X$ is a large positive operator and
only it's positivity is needed for the estimate~(\ref{realquad}).
In fact we have the sharper estimate
\begin{equation}
(X+Y+Z)^{2}\geq\,Z^{2}+c_{2}(2c_{1}-c_{3})X^{2}
+2c_{2}XH_{0}^\mathrm{boson,\,'}X\,\geq\,Z^{2}
\end{equation}
on $D_{F}\times D_{F}$ which is used in the next section.

%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{thm}[lemma]{Theorem}


\section{Skew-Symmetric Part of the Hamiltonian\label{s5}}

The Krein transformations in section~\ref{s3} were chosen to obtain a
Hamiltonian of the form
\begin{equation}
H'(V,\delta)=\mathrm{Re}H'(V,\delta)+S
\end{equation}
in which S is a densely defined skew-symmetric operator given by
\[S = S_{1}+S_{2}\]
where
\begin{eqnarray}
S_{1} & = &
\sum_{x\in{V}}\delta^{s}e\,[\pi_{1}\phi_{2}-\pi_{2}\phi_{1}](x)
A_{0}^{\prime}(x)\\
S_{2} & = &
\sum_{k\in\Gamma}\left[\omega(k)q_{\ell,\,0}^\mathrm{cs}(k)\left\{
q_{\ell,\,1}^\mathrm{cs}(k) +\sqrt{\frac{\omega}{\mu}}
\,q_{\ell,\,2}^\mathrm{cs}(k)\right\}\right.\nonumber
\\
 & - &
\left.\sqrt{\omega\,\mu}\,p_{\ell,\,0}^\mathrm{cs}(k)
p_{\ell,\,2}^\mathrm{cs}(k)\right]
\end{eqnarray}
As $\mathrm{Re}H'(V,\delta)$ is essentially selfadjoint and bounded
below, Chernoff's extension of W\"{u}st's theorem~\cite{Gold1985}
will show that $H'(V,\delta)$ is essentially maximal accretive given
the estimate
\[\parallel Zu\parallel\leq\,\parallel(X+Y+Z+S)u\parallel+c_{5}I\parallel u\parallel\]
for vectors $u$ in the core domain $D_{F}$ or equivalently
\begin{equation}
(Z)^{2}\leq(X+Y+Z)^{2}+[X+Y+Z,S]+S^{*}S+c_{5}I
\end{equation}
as a quadratic form on $D_{F}\times D_{F}$. Using the estimate of
the last section, it will then be sufficient to verify
\begin{equation}
c_{2}(2c_{1}-c_{3})X^{2}+2c_{2}XH_{0}^\mathrm{boson}\,'X+
[X+Y+Z,S]+S^{*}S+c_{5}I\geq 0
\end{equation}
which follows from:
\begin{lemma}
On the domain $D_{F}$, the commutator $[X+Y+Z,S]$ is Kato-tiny
relative to $XH_{0}^\mathrm{boson}\,'X$.
\end{lemma}

To prove the Lemma we calculate individual terms in the commutator.
From $[Z,S]=c_{2}[X^{2},S]=c_{2}\left(X[X,S]+[X,S]X\right)$ since X
acts as a free Hamiltonian, $[X,S]$ is bounded by
$H_{0}^\mathrm{boson}\,' X^{1/2}$ while [Z,S] is bounded by
$H_{0}^\mathrm{boson}\,' X^{3/2}$. Both of these are Kato-tiny
relative to $2c_{2}XH_{0}^\mathrm{boson}\,'X$. In X+Y let us
recombine the gradient boson terms in Y with the boson terms in X to
replace $H_{0}^\mathrm{boson}\,'$ by $H_{0}^\mathrm{boson}$ leading
to
$X+Y=H_{0}^\mathrm{cs}\,'+H_{0}^\mathrm{boson}+c_{1}I+Y_\mathrm{deg}+Y_{0}$
where the `magnetic part' of the Chern-Simons interaction now
appears with the boson potential in
\begin{eqnarray}
Y_{0}& = &
\sum_{x\in{V}}\frac{\delta^{s}}{2}\left[-e\left\{\partial_{\ell}\phi_{1}(x)
\phi_{2}(x)-\partial_{\ell}\phi_{2}(x)\phi_{1}(x)\right\}
A_{\ell}^{\prime}(x)\right.
\nonumber \\
& + & \left.\frac{e^{2}}{2}\left\{
\phi_{1}^{2}(x)+\phi_{2}^{2}(x)\right\}A_{\ell}^{\prime}(x)\,^{2}
\right]\nonumber \\
& + & V(\phi_{1},\phi_{2})\nonumber \\
& = & Y_\mathrm{mag} + V(\phi_{1},\phi_{2}).
\end{eqnarray}
Clearly $[Y_\mathrm{deg},S_{1}]=0$ with the remaining terms for this part
of the commutator given by
\begin{eqnarray}
[Y_\mathrm{deg},S_{2}] & = &
\sum_{k\in\Gamma}\left[-i\sqrt{\mu}\omega^{3/2}q_{\ell,\,0}^\mathrm{cs}(k)
p_{\ell,\,2}^\mathrm{cs}(k)\right. \nonumber \\ & - &
\left.i\omega^{2}q_{\ell,\,0}^\mathrm{cs}(k)
p_{\ell,\,1}^\mathrm{cs}(k)\right]
\end{eqnarray}
Consequently on the core domain, $-[Y_\mathrm{deg},S]\leq c_{6}X$
for $c_{1}$ large enough as the commutator is quadratic in
Chern-Simons harmonic oscillators. It is useful to look at the
commutators in $[Y_\mathrm{mag},S]$ separately.  For
$[Y_\mathrm{mag},S_{1}]$ the fields $A_{0}\,'$ and $A_{\ell}\,'$
commute so it is only necessary to calculate the boson field
commutators. Lattice quantization preserves enough locality for the
boson fields that for terms quadratic in the individual boson
fields, the relation
$[\phi_{j}^{2}(x),\pi_{j}(y)]=2i\delta_{P,\,V}(x-y)\phi_{j}(x)$ with
the periodic lattice $\delta_{P,\,V}$ shows their contribution to
$[Y_\mathrm{mag},S_{1}]$ vanishes.  As we use a symmetric
$k\leftrightarrow -k$ summation range, it is possible to effect an
integration by parts of sorts, nevertheless the remaining terms in
this commutator are not zero and we arrive at
\begin{eqnarray}
[Y_\mathrm{mag},S_{1}] & = & \sum_{x\in{V}}\delta^{s}i e^{2}
[\partial_{\ell}\phi_{1}(x)\phi_{1}(x)
+\partial_{\ell}\phi_{2}(x)\phi_{2}(x)]A_{\ell}^{\prime}(x)
A_{0}^{\prime}(x)
\nonumber \\
& + &
\sum_{x\in{V}}\delta^{2s}(-ie^{2})[\partial_{x,\,\ell}\delta_{P,\,V}(x-y)
\phi_{1}(x)\phi_{1}(y)]A_{\ell}^{\prime}(x)A_{0}^{\prime}(x)
\nonumber  \\
& + &
\sum_{x\in{V}}\delta^{2s}(-ie^{2})[\partial_{x,\,\ell}\delta_{P,\,V}(x-y)
\phi_{2}(x)\phi_{2}(y)]A_{\ell}^{\prime}(x)A_{0}^{\prime}(x).
\nonumber \\
&   &
\end{eqnarray}
Using $H_{0}^\mathrm{boson}\,'$ to bound $\phi_{j}(x)\phi_{j}(y)$
and $X$ to bound $A_{\ell}^{\prime}(x)A_{0}^{\prime}(x)$ the
commutator $-[Y_\mathrm{mag},S_{1}] \,\leq
c_{7}H_{0}^\mathrm{boson}\,'X$, which is also Kato-tiny compared to
$H_{0}^\mathrm{boson}\,'X^{2}$. Up to multiplicative terms quadratic
in the boson fields, the size of the commutator
$[Y_\mathrm{mag},S_{2}]$ is determined by $[A_{\ell}^{\prime}
\,,S_{2}]$. From the the Landau gauge expression for
$A_{\ell}^{\prime}(x)$ in section~\ref{s3}, it is straight forward to
arrive at the expression
\begin{eqnarray}
[A_{\ell}^{\prime}(x),S_{2}] & = &
\frac{1}{\sqrt{|V|}}\sum_{k\in\Gamma}\left[
\left\{-i\frac{\sqrt{2\omega}}{m}k_{\ell}\,q_{2,\,0}^\mathrm{cs}(k)\right\}
\cos(k\cdot x)\right. \nonumber \\
& + & \left.\left\{i\frac{\sqrt{2\omega}}{m}
k_{\ell}\,q_{1,\,0}^\mathrm{cs}(k)\right\}\sin(k\cdot x)\right]
\end{eqnarray}
which is dominated by $H_{0}^\mathrm{boson}\,'X^{1/2}$.  Hence the
quadratic term $[A_{\ell}^{\prime}\,^{2}(x),S_{2}]$ requires an
additional factor $X^{1/2}$ to produce the estimate
$-[Y_\mathrm{mag},S_{2}]\leq c_{8}H_{0}^\mathrm{boson}\,'X$ which is
also Kato-tiny relative to $H_{0}^\mathrm{boson}\,'X^{2}$.

Finally $[V(\phi_{1},\phi_{2}),S]=0$.  This is immediate for
$[V(\phi_{1},\phi_{2}),S_{2}]=0$ while the commutator identities
\begin{eqnarray}
[\phi_{j}(x)^{4},\pi_{j}(y)] & = &
4i\delta_{P,\,V}(x-y)\phi_{j}(x)^{3} \nonumber \end{eqnarray}
\begin{eqnarray}
[\phi_{1}^{2}(x)\phi_{2}^{2}(x),\pi_{1}(y)] & = &
2i\delta_{P,\,V}(x-y)\phi_{1}(x)\phi_{2}^{2}(x) \nonumber
\end{eqnarray}
readily show that $[V(\phi_{1},\phi_{2}),S_{1}]=0$ as well.
Collecting together these estimates for the various commutators
leads to the inequality
\begin{equation}
(X+Y+Z)^{2}+[X+Y+Z,S] \geq Z^{2}-c_{5}I
\end{equation}
for some constant $c_{5}>0$ 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}. Springer, 1999.

\bibitem{WiGa}
A.Wightman and L.G{\aa}rding.
\newblock Fields as operator-valued distributions in relativistic quantum field
  theory.
\newblock {\em Archiv.Fysik}, 28:129--184, 1965.

\bibitem{Frosemi}
J.Fr{\"{o}}hlich.
\newblock Unbounded, symmetric semigroups on a separable {H}ilbert space are
  essentially selfadjoint.
\newblock {\em Adv. Appl. Math.}, 1:237--256, 1980.

\bibitem{CSOS}
D.A.Bowman and J.L.Challifour.
\newblock Canonical quantization of lattice {H}iggs-{M}axwell-{C}hern-{S}imons
  fields: {O}sterwalder-{S}chrader positivity.
\newblock In preparation.

\bibitem{Hell}
K.Haller and E.Lim-Lombridas.
\newblock {M}axwell-{C}hern-{S}imons theory in covariant and {C}oulomb gauges.
\newblock {\em Ann.Phys.}, 246:1--48, 1996.

\bibitem{C4Cl}
J.L.Challifour and J.P.Clancy.
\newblock A path space formula for {G}auss vectors in {C}hern-{S}imons
  {Q}{E}{D}.
\newblock {\em J.Math.Phys.}, 40:5318--5340, 1999.

\bibitem{c4wood}
J.L.Challifour and W.A.Wood.
\newblock Path-space formulas for covariant gauges in {K}rein spaces.
\newblock {\em J.Math.Phys.}, 28:2763--2774, 1987.

\bibitem{kimura}
T.Kimura.
\newblock Manifestly {C}ovariant {C}anonical {Q}uantization of 3-d
  {Q}{E}{D}with {C}hern-{S}imons {T}erm.
\newblock {\em Prog.Theor.Phys.}, 81:1109--1114, 1989.

\bibitem{nakanishi}
N.Nakanishi.
\newblock Asymptotic {C}ompleteness and the {T}hree-{D}imensional {G}auge
  {T}heory having the {C}hern-{S}imons {T}erm.
\newblock {\em Int.J.Mod.Phys.A}, 4:1055--1064, 1989.

\bibitem{R&S}
M.Reed and B.Simon.
\newblock {\em Modern {M}ethods of {M}athematical {P}hysics, vol.2}.
\newblock Academic Press, 1975.

\bibitem{phi2}
B.Simon.
\newblock {\em The $P(\phi)_{2}$ {E}uclidean ({Q}uantum) {F}ield {T}heory}.
\newblock Princeton University Press, 1974.

\bibitem{bognar}
J.Bognar.
\newblock {\em Indefinite {I}nner {P}roduct {S}paces}.
\newblock Springer-Verlag, 1974.

\bibitem{BS}
N.N.Bogoliubov and D.V.Shirkov.
\newblock {\em Introduction to the {T}heory of {Q}uantized {F}ields}.
\newblock Wiley, 1959.

\bibitem{Gold1985}
J.A.Goldstein.
\newblock {\em Semigroups of {L}inear {O}perators and {A}pplications}.
\newblock Oxford Mathematical Monographs, Oxford University Press, 1985.

\bibitem{c4ym}
J.L.Challifour.
\newblock Self-adjointness of {L}attice {Y}ang-{M}ills {H}amiltonians and
  {K}ato's {I}nequality with {i}ndefinite {M}etric.
\newblock {\em Ann.Inst.Henri Poincar\'{e}}, 42:1--15, 1985.

\bibitem{jaffethesis}
A.M.Jaffe.
\newblock {\em {D}ynamics of a {C}ut-{O}ff $\lambda\phi^4$ {F}ield {T}heory}.
\newblock PhD thesis, Princeton University, October 1965.

\end{thebibliography}
\bibliographystyle{unsrt}

\end{document}
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