\documentstyle[12pt]{article}
\begin{document}
\setlength{\oddsidemargin}{.25in}
\setlength{\textwidth}{6.125in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.5in}
%\nonstopmode
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beaa}{\begin{eqnarray*}}
\newcommand{\eeaa}{\end{eqnarray*}}
\newcommand{\barr}{\begin{array}}
\newcommand{\earr}{\end{array}}
\newcommand{\benum}{\begin{enumerate}}
\newcommand{\eenum}{\end{enumerate}}
\newcommand{\blem}{\begin{lemma}}
\newcommand{\elem}{\end{lemma}}
\newcommand{\bthm}{\begin{theorem}}
\newcommand{\ethm}{\end{theorem}}
\newcommand{\bcor}{\begin{corollary}}
\newcommand{\ecor}{\end{corollary}}
\newcommand{\bpr}{\begin{prop}}
\newcommand{\epr}{\end{prop}}
\newcommand{\nn}{\nonumber}
\newcommand{\nind}{\noindent}
\newcommand{\Proof}{{\bf Proof}. }
\newcommand{\pr}{\nind{\bf Proof}. }
\newcommand{\re}{\nind{\bf Remark}. }
\newcommand{\res}{\nind{\bf Remarks}. }
\newcommand{\QED}{$\hfill\Box$}
\newcommand{\pa}{\partial}
\newcommand{\const}{{\rm const}}
\newcommand{\al}{\alpha}
\newcommand{\De}{\Delta}
\newcommand{\de}{\delta}
\newcommand{\g}{\gamma}
\newcommand{\G}{\Gamma}
\newcommand{\f}{\phi}
\newcommand{\F}{\Phi}
\newcommand{\z}{\zeta}
\newcommand{\r}{\rho}
\newcommand{\k}{\kappa}
\newcommand{\La}{\Lambda}
\newcommand{\la}{\lambda}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\Th}{\Theta}
\newcommand{\th}{\theta}
\newcommand{\ep}{\epsilon}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\cA}{{\cal A}}
\newcommand{\cB}{{\cal B}}
\newcommand{\cD}{{\cal D}}
\newcommand{\cZ}{{\cal Z}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cO}{{\cal O}}
\newcommand{\cI}{{\cal I}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cS}{{\cal S}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cR}{{\cal R}}
\newcommand{\cT}{{\cal T}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cK}{{\cal K}}
\newcommand{\cG}{{\cal G}}
\newcommand{\cM}{{\cal M}}
\newcommand{\bR}{{\bf R}}
\newcommand{\bh}{{\bf h}}
\newcommand{\bn}{{\bf n}}
\newcommand{\bZ}{{\bf Z}}
\newcommand{\bx}{{\bf x}}
\newcommand{\bz}{{\bf z}}
\newcommand{\bS}{{\bf S}}
\newcommand{\bM}{{\bf M}}
\newcommand{\bT}{{\bf T}}
\bibliographystyle{alpha}
\title{ Canonical Quantization of Yang Mills on a Circle}
\author{
J. Dimock\thanks{Research supported by NSF Grants PHY9200278 and PHY9400626}\\
Dept. of Mathematics \\
SUNY at Buffalo \\
Buffalo, NY 14214 }
\maketitle
\begin{abstract}
The canonical quantization of YM on a circle is developed in two different
formulations. In the first case the basic coordinates are the
gauge potentials and we give a complete construction, but find no observables.
In the second case the basic coordinates
are the holonomy operators (Wilson loop operators)
and here we also give a construction. A key ingredient is a
proof that the irreducible characters of the holonomy operators
are eigenvectors for the Hamiltonian.
\end{abstract}
\newpage
\section{Introduction}
Let $G$ be a compact connected semi-simple Lie group with
Lie algebra $\cG $.
In this paper we discuss the canonical quantization of the $\cG$-valued
Yang-Mills field theory on a circle. One begins with a certain
classical Hamiltonian system for the gauge fields
augmented by a family of constraints.
In carrying out the quantization
one has a choice of first imposing the constraint and then quantizing,
or else of first quantizing and then imposing the constraint. The two approaches
are not expected to be equivalent in general. In the first approach
the present problem has been discussed by Rajeev \cite{Raj88}
who shows that it all reduces to a
system with a finite number of degrees of freedom. In this paper
we consider the second approach and do obtain exactly the same structure
At first we need not restrict the discussion to one dimension. Consider the
spacetime $\bR \times \bM$ where $\bM = \bT ^d$ is the d-dimensional torus.
(One could also
take $\bM$ to be any compact Riemannian manifold).
A gauge potential $A_{\mu}$ on this
spacetime is a Lie algebra valued one form, and
has a field strength which is the 2-form
$F_{\mu\nu} = \pa_{\mu}A_{\nu} - \pa_{\nu}A_{\mu}
+[A_{\mu},A_{\nu}]$. The Yang Mills equations are
$\pa ^{\mu} F_{\mu\nu} - [A^{\nu}, F_{\mu\nu}]=0$ (summation convention).
We specialize to a
temporal gauge with $A_0= 0$; this is always possible. Then the the equations
become a pair of equations for a gauge potential $A_{k}$ on $\bM$
together with its time derivative $\Pi_k= \pa_0 A_k$.
With field strength $F_{ij} = \pa_{i}A_{j} - \pa_{i}A_{j}
+[A_{i},A_{j}]$ the equations are :
\bea \label{ym1} \pa_i \Pi_i + [A_i,\Pi_i] & =& 0 \\
\label{ym2}\pa_0 \Pi_k - \pa_j F_{jk} -[A_j, F_{jk}] &=& 0 \eea
We next give the problem a Hamiltonian formulation. Let $\Om^k =
\Om^k(M,\cG)$ be the smooth $\cG$-valued k-forms on $\bM$.
The phase
space is $\Om^1 \times \Om^1$ with a symplectic form defined
as follows. There is a canonical positive definite inner product $(\cdot, \cdot)$
on the Lie algebra which is the negative of the Killing form.
For any pair $A, \Pi$ in $\Om^1 \times \Om^1$ define an inner product
\be (A, \Pi) = \int_{\bM} (A_i, \Pi_i) \ee
and an associated norm $\|A\|$. There are similar inner products for
the other $\Om^k$. The symplectic form on the phase space is the
\be \si (A, \Pi;A', \Pi') = (A, \Pi') - (A', \Pi)
\ee
so $A$ and $\Pi$ are conjugate variables.
The Hamiltonian for the system is
\be H= \| {\Pi} \|^2/2 + \| F \|^2 /4 \ee
The dynamics generated by this Hamiltonian are $d A_k/ dt = \Pi_k $
and $d \Pi_k /dt = \pa_j F_{jk} +[A_j, F_{jk}]$ just as in
equation (\ref{ym2}) above. Equation (\ref{ym1}) is not dynamical but is a constraint
equation preserved by the time evolution.
If $J = \pa_i \Pi_i + [A_i,\Pi_i] \in \Om^0$ the constraint is
that $(J,h)=0$ for all $h \in \Om^0$.
One can also write
\be (J,h) = - (\Pi ,dh) + (A,[\Pi, h]) \ee
where $dh $ is the exterior derivative.
Here is the formal quantization procedure.
Consider the functions
$(A,u)$ and $(\Pi ,v)$ for $u,v \in \Om^1$. We ask for corresponding families of
operators denoted $A(u)$ and $\Pi (v)$ on some complex
Hilbert space which commute with themselves and statisfy the
canonical commutation relations:
\be \label{lie0} [A(u),\Pi (v)] = i(u,v) \ee
Then $(J,h)$ is also an operator denoted $J(h)$, as
is the Hamiltonian $H$. The constraint is imposed as a condition
on state vectors: the physical states $\Psi$ are those
satisfying $J(h)\Psi = 0$. Observables are
self-adjoint operators that commute with $J(h)$ and they act on the space of physical
states.
The Hamiltonian $H$ should be a positive self-adjoint
operator satisfiying $[H,J(h)]=0$. The Hamiltonian generates
time evolution through the unitary group $\exp(-iHt)$. Our goal is to see how much
of this can actually be realized.
Let us ignore the difficult question of time evolution for the moment, and
see what structure we have. One computes formally
\bea
\label{lie1} \ [ J(h),A(u)] & = &i A([h,u]) +i (u,dh) \\
\label{lie2} \ [J(h),\Pi (v)] & = & i\Pi ([h,v]) \\
\label{lie3} \ [J(h), J(h')] & = &iJ([h,h']) \eea
These equations say that we have a representation of a certain infinite
dimensional Lie algebra. The first two say that the $-iJ(h)$ generates
gauge tranformations and the last says $-iJ(h)$ is
a representation of the gauge algebra.
(Because one has to Wick order to
define the $J(h)$ there is the possibility that there will be extra terms on the right
side of (\ref{lie3}). This turns out not to be the case. The extra terms, if present,
would be called anomalies or central extensions,
and for $d=1$ we would be talking about Kac-Moody algebras.)
It will be more useful to consider unitary representations
of the Lie group obtained by exponentiating the Lie algebra.
This avoids problems with specifying domains of
unbounded operators. The representation is a collection of unitary operators
$W(u,v)$ defined for $u,v \in \Om^1$
and $U(g)$ defined for $g$ in the
gauge group, the smooth functions from $\bM$ to $G$. These are required to satisfy
\bea \label{g1} W(u,v)W(u',v') &= &W(u+u',v+v')\exp (-i\sigma(u,v;u',v')/2)\\
\label{g2} U(g)W(u,v)U(g^{-1}) &= &W(gug^{-1},gvg^{-1}) \exp(i(v,g^{-1}dg))\\
\label{g3} U(g)U(g') &=& U(gg') \eea
Our group contains the Heisenberg group (Weyl form of the canonical commutation
relations) and gauge group as subgroups. One recovers operators $A, \Pi, J$ satisying
(\ref{lie0}),(\ref{lie1}),(\ref{lie2}),(\ref{lie3}) by
\bea i( A(v)-\Pi(u)) &=& d/dt \ W(tu,tv) |_{t=0}\\
-iJ(h) &=& d/dt U(e^{th}) |_{t=0} \eea
The physical states are those which are invariant
under all the $U(g)$ and the observables are operators which commute with all the $U(g)$
Next we show that examples of
(\ref{g1}),(\ref{g2}),(\ref{g3}) exist.
We give two examples. The first is based
on a $\Pi$ representation of the Weyl relations, that is $\Pi$ is represented by
a multiplication operator in a standard manner.
Let $\cH_r(\bM, \cG)$ be the Sobolev space
obtained by completing $\Om^1(\bM, \cG)$ in the norm $\|u\|_r = \| (- \De + 1)^{r/2}u\|$
Then $\cH_r$ is decreasing in $r$
and one can identity the dual space of $\cH_r$ with $\cH_{-r}$.
Let ${\hat \cG}=\cH_{-r}$ with $r$ sufficiently large so that the injection
from $\cH_0 \to \cH_r$ is trace
class. Then for every $s>0$ there is a Gaussian measure $\mu_s$ on ${\hat \cG}$
such that for all $u \in \Om^1 \subset \cH_r$
\be \int_{{\hat \cG}} exp(i(\Pi,u)) d\mu_s(\Pi) = exp(-s \|u\|^2/2) \ee
The Hilbert space for the representation is then $L^2({\hat \cG},d\mu_s )$.
The unitary representation is based on the facts that $\mu_s$ is
is invariant under $\Pi \to g^{-1} \Pi g$ (since $\|u\|$ is invariant),
and quasi-invariant under
$\Pi \to \Pi +v$. The latter means that
the induced measure is absolutely continuous with respect
to the original measure. One can explicitly compute the derivative
\be \frac {d\mu_s (\Pi +v)}{d\mu_s (\Pi )} = \exp(-\|v\|^2/2s \ -(\Pi,v)/s) \ee
The representation is defined by
\bea \label{pi1}(W(u,v)\Psi )(\Pi ) &=& \exp(-i(\Pi ,u) +i(u,v)/2)
\sqrt{\frac {d\mu_s (\Pi -v)}{d\mu_s (\Pi )}}\Psi(\Pi -v)\\
\label{pi2} (U(g)\Psi)(\Pi ) &= & \exp(i(\Pi,g^{-1}dg))\Psi (g^{-1}\Pi g) \eea
In checking that this is a representation one needs to use
the fact that $\beta (g) = g^{-1}dg$ is a cocycle,
that is $\beta (gg') = \beta (g) +g\beta (g')g^{-1}$.
This representation of the gauge group has been studied by many authors
and is known to be irreducible in dimension $d \geq 3$ \cite{AHT81} \cite{GGV77}.
This means that in this case there are no physical states,
and the only observables are multiples of the identity (by
Schur's lemma).
There is also an A-representation. This is also defined on $L^2({\hat \cG},d\mu )$
but now $A$ is given as a multiplication operator. We have
\bea \label{a1} (W(u,v)\Psi)(A)& =
&\exp (i(A,v) -i(u,v)/2)\sqrt{\frac{d\mu_s (A -u)}{d\mu_s (A)}} \Psi(A -u) \\
\label{a2} (U(g) \Psi )(A)& = &
\sqrt{\frac{d\mu_s (A^g)}{d\mu_s(A)}}
\Psi (A^g) \eea
where $A^g=g^{-1} Ag + g^{-1} dg$
The A-representation of the gauge group is equivalent to the $\Pi$ representation
\cite{GGV77} and suffers from the same difficulties.
The representation we actually want to consider is the $s \to \infty$ limit of
the $A$-representation (for d=1). This corresponds to taking Lesbesgue measure on the
function space. In order to accomodate this singular Hilbert space we
need to formulate the problem in terms of $C^*$-algebras. We digress to review this
approach as it applies to our problem
\bigskip
\nind{\bf An algebraic approach.} A general reference is \cite{BrRo81}.
Let $W(u,v)$ be a representation of the canonical commution relations (\ref{g1}).
We define the Weyl algebra
$ \cF$ to be the $ C^*$ algebra generated by the $ W(u,v)$ .
This algebra is independent of the particular
representation. Every element is a norm limit of finite sums $\sum_i c_iW(u_i,v_i)$
for some complex numbers $c_i$.
States are now positive linear functionals on the $C^*$ algebra, for example the
expectation with respect to a unit vector in the representation space.
By the GNS reconstruction theorem
any state determines a representation of the algebra. The theorem
states that given a state $\om$ on $\cF$ there is a Hilbert space
$\cH_{\om}$, a representation $\pi_{\om}$ of $\cF$ by operators on $\cH_{\om}$,
and a cyclic vector $\Om_{\om} \in \cH_{\om}$
such that $\om (F) = (\Om_{\om}, \pi_{\om}(F) \Om_{\om})$.
for any $F \in \cF$. Any two such representations are unitarily equivalent.
The operators $U(g)$ are not necessarily in $\cF$, but they can
still be accomodated as automorphisms $\beta _g$ of the algebra.
In any representation of (\ref{g1}),(\ref{g2}),(\ref{g3})
define $\beta _g$ on $\cF$ by
\be \label{beta1}\beta _g(F) = U(g)FU(g)^{-1} \ee
We have $\beta_{g'}\beta_{g} = \beta_{g'g}$ and since
\be \label{beta2}\beta _g(W(u,v)) = W(gug^{-1},gvg^{-1})\exp(i(v,g^{-1}dg)) \ee
it follows that each $\beta _g$ defines
an automorphism of the algebra. We say that the
automorphism is unitarily implementable in a particular
representation of the algebra if there is a unitary
representation $U(g)$ of the gauge group such that
(\ref{beta1}) is satisfied.
A physical state ( gauge invariant state) is now a state $\om$ on
$\cF$ such that $\om \circ \beta_g = \om$.
After reconstruction in such a
state the gauge automorphisms will be unitarily implementable
by operators $U_{\om}(g)$ satisfying $U_{\om}(g)\Om_{\om} = \Om_{\om}$.
Observables are now elements of the algebra left invariant by $\beta_g$.
These form a subalgebra of $\cF$ and
one can take the attitude that one only needs a state on the observables.
However in this case one has to do something further to show that the
constraint is satisfied by the state.
One would like to treat time evolution in a similar way by asking
for a one paramenter group of automorphisms $\al_t$ so that in some
sense $\al_t (F)= e^{iHt}F e^{-iHt}$. There is not much hope that this
is true in general: the quantum Hamiltonian is just too singular.
However in the case $d=1$ it does work out this way as we now show.
\section{YM on a circle: fields}
Now suppose that the spacetime is ${\bf R} \times {\bf S}^1$ where $\bS ^1$ is
the interval $[0,L]$ with the endpoints identified. In this case we can carry out
the program.
The main simplification is that on $\bS$ the one forms are just functions and
there are no two forms. Thus the classical Hamiltionian is just
$ H= \| {\Pi} \|^2/2 $ and the dynamical equations are $dA/dt = \Pi$ and
$d\Pi /dt =0$. These have solutions $(A_t,\Pi_t) = T(t)(A,\Pi)= (A + t\Pi, \Pi)$.
Since the equations are linear this is also the solution
for the quantum problem. Taking into account
that $W(u,v)$ is formally $\exp(i\sigma(A,\Pi;u,v))$
and that $T(t)$ is sympectic (i.e. leaves
$\sigma$ invariant), a time evolution automorphism $\al_t$
should satisfy
\be \label{auto} \al_t W(u,v) = W(T(-t)(u,v)) =W(u-tv, v) \ee
We define the automorphism by this equation: since $T(t)$ is symplectic,
$ W(T(-t)(u,v))$ is a new representation of (\ref{g1}) generating $\cF$, and
hence there is a
a unique automorphism $\al_t$ of $\cF$ satisfying (\ref{auto}).
Furthermore since $T(t)$ is a
one parameter group of operators we get a one parameter group of automorphisms
$\al_t$.
Note that the $\al_t$
commute with the gauge automorphisms $\beta_g$.
We seek a state invariant under both.
We could go directly to the answer, but
at the same time we want to develop a more regular approximation to these dynamics.
Instead of adding a gauge
fixing term as is usual for gauge theories, we add a mass term to the Hamiltonian.
Thus we consider
\be \label{cham} H_s= 1/2( \|\Pi\|^2 + \|A\|^2/4s^2) \ee
which has a mass $m= 1/2s$.
This generates a classical time evolution given by
\[ T_s(t) =\left (
\barr {cc} \cos (t/2s) & 2s \sin ( t/2s) \\
-1/2s \sin( t/2s) & \cos(t/2s) \earr \right ) \]
This operator is also symplectic and so for the quantum problem we
may define a one-parameter group of time evolution automorphisms by
\be \al^s_t W(u,v) = W(T_s(-t)(u,v)) \ee
Note that this reduces to the true time evolution as $s \to \infty$, and that
it does not commute with gauge transformations.
Now suppose $\cF$ is generated by the representation (\ref{a1})
of $W(u,v)$ on $L^2({\hat \cG},d\mu_s )$. Let $\Om_s = 1$ and define a
state $\om_s$ on $\cF$ by $\om_s(F) = (\Om_s,F\Om_s)_s$.
Our main result states that $\om_s$ is
the physical vacuum for approximate dynamics,
that these states have a limit $\om$ as
$s \to \infty$, and that $\om$ is the physical vacuum for the true
dynamics.
\bthm \
\benum
\item The state $\om_s$ satisfies
\be \label {oms}
\om_s(W(u,v)) = \exp ( -\|u\|^2/8s - \|v\|^2s/2) \ee
and $\om_s \circ \al^s_t = \om_s$. After reconstruction
(or on $L^2({\hat \cG},d\mu_s )$
time evolution is unitarily implemented with positive energy; i.e. there is
a positive self-adjoint
operator $H_s$ so
\be \al_t^s(F)= \exp(iH_st)F \exp(-iH_st) \ee
\item $\om(F) = lim_{s \to \infty}\om_s(F)$ exists for every $F \in \cF$ and satisfies
\be \label{om} \om (W(u,v)) = \left\{ \barr{cc} 1 & v=0 \\
0 & v \neq 0 \earr \right. \ee
The state $\om$ satisfies $\om \circ \beta_g = \om$
and $\om \circ \al_t = \om$. After reconstruction
gauge tranformations are unitarily implemented and
time evolution is unitarily implemented with positive energy.
\eenum \ethm
\re The state $\om$ is not as trivial as it might look.
The reconstruction
theorem still generates a representation of the Weyl algebra. Furthermore
it is known that the
Weyl algebra is simple and so every representation is faithful.
What is lost is that the representation is not regular in
that the map $(u,v) \to
W(u,v)$ is not continuous. This means one cannot define field operators $A,\Pi$
in this representation, but only their exponentials.
\bigskip
\pr \
\nind (1.) The first part is standard. The evaluation of $\om_s$ on $W(u,v)$
is a computation and
these values determine the state completely.
It then suffices to check the invariance on the $W(u,v)$
where it is a simple computation. The unitary implementability of time translations
follows. To see the positive energy one can check that
$\om_s(W(u',v')\al^s_tW(u,v))$ has a (distribution) Fourier transform in $t$
with non-negative support. Since this function is equal to
$(\Om_s,W(u',v')\exp (iH_st)W(u,v) \Om_s)$ and since linear combinations of
the $W(u,v)\Om_s$ are dense the result follows.
(An alternative is to make a Fock space construction of the state $\om_s$. Then
$H_s$ is a Wick ordered version of the classical Hamiltonian and
the positivity of $H_s$ is manifest. This is just the standard construction of a
free scalar field except that there is no term $\|\pa A/ \pa x \|^2$ in the Hamiltonian.
This means $H_s$ is just a number operator that counts each particle $m=1/2s$.)
\nind (2) The fact that $\om_s(W(u,v))$ has a limit equal to (\ref{om}) is immediate.
The limit exists on general $F$ by making a uniform approximation by linear
combinations of the $W(u,v)$'s. The property of being a positive linear
functional carries over to the limit.
The invariance of the state $\om$ under time translation and gauge transformations
is easy to check on $W(u,v)$ using (\ref{om}) and follows generally.
For the positivity of the energy we check that
$ \om (W(u',v') \al_t W(u,v))$ has a (distribution) Fourier transform
with non-negative support. Indeed if $v' \neq -v$ the function is zero
while if $v' = -v$ it is
\beaa && \om (W(u',-v)W(u-tv,v) \\ &=&\om (W(u'+u -tv,0))
\exp (-i\sigma (u',-v;u-tv,v)/2) \\
&=& \const \exp (it\|v\|^2/2)
\eeaa
This completes the proof.
\bigskip
\nind{\bf More on the Hamiltonian $H_s$:} $H_s$ regarded as an operator on
$L^2({\hat \cG},d\mu_s )$
is an infinite dimensional
differential operator, a generalization of the
finite dimensional operator
\be L =1/2(-\De +s^{-1} x \cdot \nabla ) \ee
This is well-known, but we develop a version suited
to our purposes.
Start with the finite dimensional case. Consider the Hilbert space $L^2(\bR^k,d\mu_s)$
with $d\mu_s(x) = \exp(- |x|^2/2s)dx$, at first for $k=1$. The operator $L$ is symmetric
and has a complete set of eigenfunctions. These are the
Hermite polynomials defined
by
\be He_n(x) = (-s)^n e^{x^2/2s} d^n/dx^n e^{-x^2/2s} \ee
with eigenvalues $n/2s$. Normalized eigenfunctions are
$(2\pi s)^{-1/4}(n!)^{-1/2}He_n$. The eigenfunction expansion
defines $L$ as a self-adjoint operator in a standard way,
and if a function in the domain of
$L$ is sufficiently differentiable this definition agrees with the
definition as a differential operator.
Similarly we have for any $k$ that $L$ has a complete set of eigenfunctions
$ He_{n_1} \otimes ... \otimes He_{n_k}$ with eigenvalues
$\sum_j n_j/2s$, and this defines $L$ as a self-adjoint operator in
this case.
We need the technical result:
\blem \label{core} Let $\cD $ be finite linear combinations of exponentials $\exp(i\la x)$
in $L^2(\bR^k,d\mu_s)$.
Then $\cD$ is a core for $L$, i.e. $L$ is essentially self-adjoint on $\cD$.
\elem
\pr Take $k=1$ for notational simplicity.
$\cD$ is a dense set in the domain of $L$. To prove it is a core
it suffices to show that it is
invariant under the action of $e^{-Lt}$, this is a corollary of Stone's theorem.
The Hermite polynomials
are related to the exponentials by the generating functional
\be \exp(i\la x- \la^2 s/2) = \sum_n \la^n He_n(x) / n! \ee
with convergence also in $L^2$.
>From this we have
\[ e^{-Lt}\exp(i\la x - \la^2) = \exp(i(e^{-t}\la) x - (e^{-t}\la)^2) \]
and hence the invariance.
\blem \label{exp} $e^{i(A,v)}$ is in $D(H_s)$ and
\[ H_se^{i(A,v)} =(\|v\|^2/2 + i(A,v)/2s ) e^{i(A,v)} \] \elem
\pr Since $\Om_s =1$ and $H_s \Om_s =0$ we have
\beaa \exp(-iH_st)e^{i(A,v)} &=& \al_{-t}^s( W(0,v))\Om_s \\
&=& W(u_t,v_t)\Om_s \\
&=& \exp (i(A,v_t) -i(u_t,v_t)/2 - \|u_t \|^2 -(A,u_t)/2s) \eeaa
where $u_t = 2s \sin(t/2s)v$ and $v_t = \cos(t/2s)v$. The last expression
is strongly differentiable in $t$ hence $e^{i(A,v)}$ is in $D(H_s)$.
Computing the derivative at $t=0$ gives the result.
\blem \label{hs} Let $f \in D(L)$ in $L^2(\bR^k,d\mu_s)$. and let
\[ F(A) = f( (A, \f_1), ..., (A,\f_k)) \]
where $\f_1, ..., \f_k$ is an orthonormal set in
$L^2(\bS, \cG)$. Then $F \in D(H_s)$ and
\[ (H_sF)(A) = (L f)( (A, \f_1), ..., (A,\f_k)) \]
\elem
\pr By lemma \ref{exp} the result
holds for $f \in \cD$.
By lemma \ref{core} we can find for any $f \in D(L)$
a sequence $f_n \in \cD$ so $f_n \to f$
and $L f_n \to L f$. If we define define
$F_n(A) = f_n( (A, \f_1), ..., (A,\f_k))$
it follows that $(H_s F_n)(A) =(L f_n)( (A, \f_1), ..., (A,\f_k))$.
The convergence of $f_n$ and $Lf_n$ implies that $F_n \to F$
and $H_sF_n \to (Lf)( (A, \f_1), ..., (A,\f_k))$.
Since $H_s$ is closed $F \in D(H_s)$ and $H_sF$ is as claimed.
\section{YM on a circle: observables}
The defect with the theory so far is that we have no observables.
The natural candidate for an observable would be the Wilson loop operator or
holonomy operator for parallel transport around the circle.\footnote
{One might also consider multiple loops around the circle. For this paper we
focus on the single loop}
We cannot construct this operator in the $C^*$ algebra $\cF$. The second choice
would be identify the operator in the vonNeumann algebra $\pi _{\om}(\cF) ''$
generated by $\cF$ in the physical representation. This also seems difficult.
Instead we abandon the fields
and let the holonomy operators themselves
play the fundamental role; in some sense they are new coordinates for the problem.
Having made this decision one might proceed by attempting to construct a
$C^*$ algebra based on the holonomy operators, see \cite{AsIs92b}. However we
take a somewhat different route.
The strategy is as follows. We know that
the physical theory can be approximated
by the more regular theory on $L^2({\hat \cG},d\mu_s )$ with
Hamiltonian $H_s$. We first construct the holonomy operators
on this space and give them the time evolution generated by $H_s$.
Then we show that their vacuum correlation functions have a limit when $s \to \infty$.
Finally we use a Wightman type reconstruction theorem to obtain the physical theory.
The last step is similar to a construction for Euclidean field theories due to to
Frohlich, Seiler, and collaborators \cite{Fro80} \cite{Sei82}.
We remark that
there are a number
of covariant/path-space/imaginary time treatments of YM in two-dimensions
in which the holonomy operators
also play a key role \cite{AHH84},
\cite{AHH86}, \cite{GKS89}, \cite{Dri89}, \cite{Fin90}, \cite{Wit91},
\cite{Wit92}. Since the loops live in two dimensions these theories are
somewhat richer.
They do have some of the same flavor as our canonical/operator/real time treatment,
and it would be interesting to understand the connection better.
Let us assume that the group $G$ is a closed subgroup of $U(n)$ so that
the Lie algebra $\cG$ is a subalgebra of the skew-adjoint $n \times n$ matrices.
Consider the parallel transport operator for a connection $A$ on $\bS^1$, at first
as given by a smooth $\cG$- valued function on [0,L] with $A(0)=A(L)$. The
parallel transport $U(x,A) \in G$
from 0 to $x$ is the unique solution of the equation
\be \label{ptrans} dU/dx = A U \ \ \ \ \ \ U(0) = 1 \ee
One way to construct the solution is as follows. For any interval $I$ define
$A(I) = \int_I A(x) dx$. Let $I_k = [(k-1)L/2^n,kL/2^n]$ and let
$I_k(x) = [(k-1)L/2^n,x]$. Then define an approximate solution
$U_N(x,A)$ by defining for $x \in I_k$
\be \label{approx} U_N(x,A) = \exp(A(I_k(x)))\exp (A(I_{k-1}))... \exp(A(I_0)) \ee
The actual solution is then given by
\be \label{ulim} U(x,A) = \lim _{N \to \infty} U_N(x, A) \ee
Of course $U(L,A) \neq U(0,A)$ in general.
Now let $A$ be a $\cG$-valued white noise random variable
on $\bS^1$ with variance $s$.
That is suppose for each $u \in L^2(\bS,\cG) = L^2([0,L],\cG) $
there is a measurable function $A(u)$
on some measure space so $\int A(u)A(v) = s(u,v)$. For us the measure space is
$({\hat \cG},\mu_s )$ and $A(u) = (A,u)$ if $u$ is smooth, and is defined by $L^2$
limits in general.
Parallel transport with connection $A$ is now defined by the stochastic
differential equation
\be \label{sptrans} dU/dx = (A - sC/2)U \ \ \ \ \ \ U(0) = 1 \ee
Here $C$
is the Casimir operator defined as the matrix
\be \label{cas} C = -\sum_{\al} e_{\al}e_{\al} \ee
where $\{ e_{\al} \}$ is any orthonormal basis for $\cG$.
This extra term is necessary to keep the solution in $G$.
An approximate solution $U_N(x,A)$ can still be defined by
(\ref{approx}) provided we interpret
\be A(I)= \sum_{\al} e_{\al} A(e_{\al}\chi_I) \ee
where $\chi_I$ is the characteristic function of $I$.
As first shown by McKean \cite {McK69}
(see also \cite{Ibe76}, \cite{Fre84})
the limit $ U(x,A) = \lim _{N \to \infty} U_N(x, A)$ still exists almost everywhere
and is the unique solution to (\ref{sptrans}). Furthermore the
$G$-valued random variables $U(x,A)$ are a realization of
Brownian motion with variance $s$ on the group G.
We expand on this last statement.
Brownian motion on $G$ (starting at the identity) can be
characterized as the Markov process with
infinitesimal generator $s\De $, where $\De $ is
the Laplacian on the group, i.e. it is minus the Casimir operator (\ref{cas})
but with the basis vectors $\{ e_{\al}\}$ interpreted as left-invariant
vector fields on $G$. Our
statement that $U(x,A)$ is a Brownian motion
means that for functions $f_1 ,..., f_n $ on G
and for $0 \leq x_1 \leq x_2 \leq ... \leq L$:
\bea \label{fk} & & \int f_1 (U(x_1,A)) ... f_n(U(x_n,A)) d\mu_s(A) \nn \\
&=&( \exp(x_1s\De /2 )f_1 \exp ((x_2 -x_1)s\De /2) f_2 ...f_n)(1) \eea
Here $\exp(xs\De)f$ is the solution of the heat equation
$\pa u/\pa x = s \De u$ with initial condition $u(0)=f$.
The parallel transport operator has the covariance
property
\[ U(x,A^g) = g(x)^{-1}U(x,A) g(0) \]
This follows from the uniqueness of solutions.
The holonomy operator itself is defined as parallel transport once around the
full circle:
it is the $G$-valued random
variable $ U(L, A)$. If $g$ is a smooth gauge function on
$\bf S^1$ then $g(L) = g(0)$
and
$U(L,A^g) = g(0)^{-1}U(L,A) g(0)$.
Thus if $\chi$ is a function on $G$ invariant under conjugation, i.e.
a class function, we have that
\be Y(\chi, A) = \chi (U(L,A)) \ee is gauge invariant.
If $\chi$ is also real then we have an observable, that is a bounded
self-adjoint operator on $L^2({\hat \cG},d\mu_s )$ commuting
with the gauge transformation
$U(g)$ defined in (\ref{a2}).
We study the expectation values of these objects in the
vacuum $\Om_s =1$.
\bpr \label{tzero} Let $\chi_i$ be smooth class functions. Then
\be \lim_{s \to \infty} (\Om_s,Y(\chi_1)...Y(\chi_n)\Om_s)
= \int_G \chi_1 (g)... \chi_n (g) dg \ee
where $dg$ is Haar measure on $G$.
\epr
\pr by (\ref{fk}) we have
\[ (\Om_s,Y(\chi_1)...Y(\chi_n)\Om_s)
= (e^{sL\De/2}\chi_1... \chi_n)(1) \]
The proposition now follows since for any $h \in G$ and any smooth function $f$ on $G$
we have
\be \label{heat} \lim_{s \to \infty} (e^{s\De}f)(h) = \int f(g)dg \ee
This result holds on any compact Riemannian manifold. The limit (\ref{heat})
is equivalent to the
statement that $ \lim_{s \to \infty} e^{s\De}(f - (f,1)) = 0$ at the
at the point. The convergence holds in $L^2(G,dg)$ since 0 is a simple eigenvalue
of the Laplacian with eigenvector 1,
and all other eigenvalues are negative. The convergence also holds in any
Sobolev space since these spaces may be defined by
the Laplacian.
The result now follows by a Sobolev inequality.
(For Sobolev spaces on a manifold see for example \cite {Gil74}).
\bigskip
\nind{\bf Time evolution.} The time evolved
holonomy operators should also be observables. For any class function $\chi$
we define
\be Y^s(t,\chi) = e^{ iH_s t } Y(\chi)e^{ -iH_s t } \ee
Since $H_s$
does not commute with gauge transformations this is not yet an observable, however
we expect to recover the invariance in the $s \to \infty$ limit, at least
formally.
It will be convenient to restrict the choice of class functions to
irreducible characters.
Let the irreducible representations of the group
G be indexed by $\la$ and let $\chi_{\la}$ be the character in the
representation $\la$.
Then $\chi_{\la}$ is an eigenvector of $-\De$ on $G$
with some eigenvalue $c_{\la}$.
The next result says that $Y(\la) \equiv Y(\chi_{\la})$ is an
approximate eigenvector of $H_s$ with eigenvalue
\be E_{\la} =Lc_{\la}/2 \ee
\blem \label{major}
\[ \|(e^{-it H_s}- e^{-itE_{\la}}) Y(\la) \| \leq \cO(s^{-1/2}) \]
\elem
\nind The proof is given in the next section.
Assuming this result we can now prove our second main result.
\newpage
\bthm \
\benum \item The following limit exists:
\[ W(t_1,\la_1,...,t_n, \la_n) = lim_{s \to \infty}
(\Om_s, Y^s(t_1,\la_1) ... Y^s(t_n, \la_n)\Om_s) \]
\item There exists a Hilbert space $\cH$ with operators $Y(t, \la)$
and a cyclic vacuum $\Om$ so that
\[ W(t_1,\la_1,...,t_n, \la_n) =
(\Om, Y(t_1,\la_1) ... Y(t_n, \la_n)\Om) \]
There is a self adjoint operator $H$ on $\cH$ so that $H \geq 0$, $H\Om =0$, and
\[ Y(\la,t) = e^{iHt}Y(\la) e^{-iHt} \]
where $Y(\la) = Y(0, \la)$ are the time zero fields.
\item $Y(\la ) \Om$ is an eigenvector of $H$ with eigenvalue $E_{\la}$.
\item The vacuum is cyclic for the time zero fields.
\eenum
\ethm
\re The fact that the holonomy operator is an eigenvector of the Hamiltonian
was anticipated by Witten \cite{Wit91}.
\bigskip
\pr
The proof of (1) is by induction on n. Since $H_s\Om_s = 0$ there is no time
dependence for $n=1$ and the result follows from
Proposition \ref{tzero}. In the general case we write
\bea && (\Om_s, Y^s(t_1, \la_1) ... Y^s(t_n ,\la_n)\Om_s) \nn \\
&=&(\Om_s, Y( \la_1) e^{-iH_s(t_1 -t_2)}Y(\la_2) e^{-iH_s(t_2 -t_3)}
... Y(\la_n)\Om_s) \nn \\
&=&(\Om_s, Y( \la_1)( e^{-iH_s(t_1 -t_2)}-e^{-iE_{\la}(t_2 -t_1)})Y(\la_2)
e^{-iH_s(t_2 -t_3)} ... Y(\la_n)\Om_s) \nn \\
& + & e^{-iE_{\la}(t_2 -t_1)}(\Om_s, Y( \la_1)Y(\la_2)
e^{-iH_s(t_2 -t_3)} ... Y(\la_n)\Om_s)
\eea
By lemma \ref{major}
$ \|( e^{-iH_st}-e^{-iE_{\la}t})Y(\la)\Om_s \|$
converges to zero as $s \to \infty$, and
since the other factors in the first term are bounded
uniformly in $s$ this implies that the first term converges to zero.
For the second term we use
the fact that the product of two characters is a finite sum of characters.
This means that
\be \label{prod} Y(\la_1) Y(\la_2) = \sum_{\la} C(\la_1,\la_2; \la) Y(\la). \ee
for some constants $ C(\la_1,\la_2; \la)$.
Then the second term can be reduced to a correlation function with $n-1$ fields, and hence
converges by the inductive hypothesis.
The proof of (2) is a version of the standard Wightman
reconstruction theorem. We omit the details.
For (3) note that
\[ (( e^{-iHt}-e^{-iE_{\la}t})Y(\la)\Om,
Y(t_1, \la_1) ... Y(t_n ,\la_n)\Om) = O \]
since it is the limit of the same expression with $s$ and as already noted such a
limit is zero. Thus $( e^{-iHt}-e^{-iE_{\la}t})Y(\la)\Om$ is
orthogonal to a dense set and hence is zero.
For (4) we observe that states $ Y(t_1, \la_1) ... Y(t_n ,\la_n)\Om$
can be written as finite combinations of time zero fields . This follows
from part (3) of the theorem and the fact that (\ref{prod}) holds also at $s = \infty$.
The latter is established by limits from $s < \infty$. This completes the proof.
\bigskip
\nind{\bf Remarks}
One can actually compute all these correlation functions by reducing them
to time zero fields as indicated above and then using Proposition \ref{tzero}.
Now we can prove the following result which shows the
equivalence with Rajeev \cite{Raj88}
who constrains before quantizing. He finds that the Hilbert space is
$L^2_{inv}(G, dg)$, the square integrable class functions
on $G$, and the Hamiltonian is $L/2(-\De)$
\bcor There is a unitary operator $U:L^2_{inv}(G, dg) \to \cH$ such that
$U \cdot 1 = \Om$ and $U^{-1} Y(\la) U = \chi_{\la}$ and
$U^{-1} e^{-iHt} U = e^{iL\De t/2}$
\ecor
\Proof The $\chi_{\la}$ form an orthonormal basis for $L^2_{inv}(G, dg)$ by the
Peter-Weyl theorem.
We define $U$ on finite sums by
\[ U(\sum_{\la} a_{\la}\chi_{\la}) = \sum_{\la} a_{\la}Y(\la) \Om \]
This is norm preserving by Propostion \ref{tzero}
and has dense range by the theorem. Thus it extends to
a unitary operator. The identities are easily checked on vectors
$\sum_{\la} a_{\la}\chi_{\la}$ and since the operators are bounded they hold generally.
\section{Proof of lemma 3.2}
Fix an unitary irreducible representation of $G$ which associates to each $g \in G$
a unitary matrix $g' \in U(r)$. This
induces a representation of the Lie algebra $\cG$
by skew-symmetric $r \times r$ matrices. If $X \in \cG$ is represented by $X'$
then
$ (e^X)' = e^{ X'}$. Let $e_{\al}$ be an orthonormal basis for $\cG$.
Any $X \in \cG$ is written $X= \sum_{\al} x_{\al}e_{\al}$
and $X'= \sum_{\al} x_{\al}e_{\al}'$. We have $|X|^2=|x|^2 =
\sum_{\al}|x_{\al}|^2$ and the matrix $X'$ has an operator
norm satisfying $|X'| \leq \cO(1)|x|$. The Casimir operator in this representation
is $-\sum_{\al} e_{\al}'e_{\al}'$ and
must be a multiple of the identity $c'I$.
\blem For a constants $\cO(1)$ independent of $x$:
\be \label{x1} \sum_{\al} x_{\al} \frac{ \pa ( e^{X'})}{ \pa x_{\al}} = e^{X'} X' \ee
\be \label{x2} | \frac{ \pa (e^{X'})}{ \pa x_{\al}} -e^{X'} e_{\al}'| \leq
\cO(1)|x|e^{\cO(1)|x|} \ee
\be \label{x3} |(\sum_{\al}\frac{ \pa^2 }{ \pa x^2_{\al}} + c' )e^{X'}| \leq
\cO(1)|x|e^{\cO(1)|x|} \ee
\elem
\pr
We compute (see for example \cite{Mil72} p.160)
\beaa \frac{\pa ( e^{X'}) }{ \pa x_{\al}}
& = & e^{X'}\sum _{j=0}^{\infty} \frac{(-1)^j}{(j+1)!} (Ad X')^j
\frac{\pa X'}{ \pa x_{\al}} \nn \\
&=& e^{X'}(e_{\al}' + \sum _{j=1}^{\infty} \frac{(-1)^j}{(j+1)!} (Ad X')^j e_{\al}')
\eeaa
If we multiply by $x_{\al}$ and sum over $\al$
all the terms in the sum are zero and this gives (\ref{x1}).
Next use $ |Ad X' (Y') | \leq \cO(1)|x||y| $ to estimate
\beaa |\frac{ \pa ( e^{X'})}{ \pa x_{\al}} - e^{X'} e_{\al}|
& \leq & \cO(1) \sum _{j=1}^{\infty} \frac{1}{(j+1)!} (\cO(1)|x|)^j \nn \\
& \leq & \cO(1)|x| e^{\cO(1)|x|} \eeaa
This proves (\ref{x2}).
Finally we compute
\beaa \sum_{\al} \frac { \pa^2 ( e^{X'})}{ \pa x^2_{\al}}
&=& e^{X'} \sum_{\al} (e_{\al}' +
\sum _{j=1}^{\infty} \frac{(-1)^j}{(j+1)!} (Ad X')^j e_{\al}')^2 \\
&+& e^{X'} \sum_{\al} \sum _{j=2}^{\infty} \sum_{k=0}^{j-1}
\frac{(-1)^j}{(j+1)!}
(Ad X')^{j-k-1}( Ad e_{\al}')
(Ad X')^k e_{\al}'
\eeaa
Identify the leading term $-e^{X'}c'$ and bound the remainder
as before. This gives (\ref{x3}) and completes the proof.
\bigskip
The next result concerns $U_N'=U_N(L,A)'$.
We regard it as an element of $ L^2({\hat \cG},M_r,d\mu_s )$
where $M_r$ is all $r \times r$ matrices, and the space is all
square integrable $M_r$-valued functions $\psi$ on ${\hat \cG}$ with norm
\bea \| \psi \|^2 & = &\int Tr(\psi(A)^*\psi(A)) d\mu_s(A)\\
&=& \int \sum_{\beta,\beta ' = 1}^r |\psi_{\beta,\beta '}(A)|^2 d\mu_s(A)
\eea
Since $U_N'$ is unitary $\|U_N'\| = \sqrt{r}$.
The operator $H_s$ operates on this space by acting on each entry.
\blem \label{bound} $U_N'$ is in the domain of $H_s$. With $E' = Lc'/2$ and $2^N \geq s$
\[ \| (H_s - E') U_N' \| \leq \cO(2^{-N/2}s^{1/2})
+ \cO(s^{-1/2}) \]
\elem
\pr We write
\[ U_N' = \exp (A_{2^N})...\exp (A_1) \]
where
\[ A_j = \sum_{\al}L^{1/2} 2^{-N/2} e_{\al}' A(\f_{\al,j}) \]
and where
\[ \f_{\al,j} =L^{-1/2} 2^{N/2}e_{\al}\chi_{I_j} \]
is an orthonormal set in $L^2(\bS,\cG)$.
The matrix elements of $U_N'$ are smooth functions of $ A(\f_{\al,j})$, bounded
by one,
with derivatives which
are integrable for a Gaussian measure. Hence by
lemma \ref{hs}, $U_N'$ is in the domain of $H_s$ and
\bea \label {hs2} (H_s-E')U_N'
&=& 1/2\sum_{j=1}^{2^N}[ \prod_{k >j} e^{A_k}]\
( -\sum_{\al}\frac{\pa^2 }{ \pa x_{\al,j}^2} -2^{-N}Lc')e^{A_j}
[ \prod_{k j} e^{A_k}]\
\sum_{\al} x_{\al,j} \frac{\pa ( e^{A_j})}{ \pa x_{\al,j}}
[ \prod_{k j} e^{A_k}]e^{A_j}A_j
[ \prod_{k j'$, then the only dependence on $A(\f_{\al,j}) = x_{\al,j}$
is as a factor $x_{\al,j}$. Since $\int x_{\al,j} d\mu_s(x) = 0$ the term vanishes.
Thus we have
\[ \|II\|^2 = (2s)^{-2} \sum_{j} \int Tr(A_j^*A_j) d \mu_s(A) \]
Since $|A_j| \leq \cO(1) 2^{-N/2}|x_j| $
this yields
\[ \|II\|^2 \leq \cO(1) 2^{-N}s^{-2} \sum_{j=1}^{2^N} \int |x_j|^2 d \mu_s(x)
\leq \cO(1) s^{-1} \]
and hence $\|II\| \leq \cO(1) s^{-1/2}$. This completes the proof.
\bigskip
{\bf Proof of lemma 3.2:}
Start with the representation
\[ (e^{-iH_st} -e^{-iE't}) U_N' = -i
\int_0^t e^{-iH_s u}(H_s - E') e^{-iE'(t-u)} U_N' du \]
By lemma \ref{bound} for $2^N \geq s$ and fixed $t$:
\[ \| (e^{-iH_st} -e^{-iE't}) U_N' \| \leq \cO(2^{-N/2}s^{1/2})
+ \cO(s^{-1/2}) \]
Now let $N \to \infty$. We have $U_N \to U$ a.e., hence $U_N' \to U'$ a.e.
by the continuity of the representation, and hence $\| U_N'- U'\| \to 0$.
It follows that
\[ \| (e^{-iH_s} -e^{-iE't}) U' \| \leq
\cO(s^{-1/2}) \]
Then $\chi '(U) = Tr U'$ satisfies the same bound in
$L^2({\hat \cG},d\mu_s )$ which is our result.
\section{Discussion}
By working exclusively with "observables" in section 3 we
have given up gauge transformations. This makes it difficult
to establish that the "observables" and the vacuum really
are gauge invariant, although this is formally true.
To really establish the invariance it would be nice to combine the observable
construction of section 3 with the field construction of section 2.
The most straightforward way to combine the two approaches
would be to show that vacuum expectation
values of arbitrary products of Weyl operators $W$ and loop operators $Y$ have
a limit as $s \to \infty$. Then a reconstruction theorem
would give a structure in which our two theories could be embedded.
It is not particularly easy to obtain this limit. The separate proofs
we have given for the $W$'s and the $Y$'s are not very compatible.
Since we have plenty of boundedness one might hope that a convergent
subsequence could be found. But even this requires a little uniformity in
$s$ which is hard to find.
Perhaps one should not be too optimistic. For a discussion of related
difficulties in the abelian case see \cite{Fro80}, \cite{AsIs92a}.
\begin{thebibliography}{99}
\bibitem{AHT81} S. Albeverio, R. Hoegh-Krohn, D. Testard, Irreducibility and
reducibility of the group of mappings of a Riemannian manifold into a compact
semisimple Lie group, J. Func. Anal. 41,378-396, (1981).
\bibitem{AHH84} S. Albeverio, R. Hoegh-Krohn, H. Holden, Markov cosurfaces
and gauge fields,Acta Phys. Austr. Suppl. XXVI, 211-231, (1984).
\bibitem{AHH86} S. Albeverio, R. Hoegh-Krohn, H. Holden, Random fields with
values in Lie groups and Higgs fields, in Stochastic Processes in Classical
and Quantum Systems, S. Albeverio, C. Casati, D. Merlini, eds. Springer, Berlin
(1986).
\bibitem{AsIs92a} A.Ashtekar, C. Isham, Inequivalent observable algebras.
Another ambiguity in field quantization, Physics Letters B 274, 393-398, (1992).
\bibitem{AsIs92b} A. Ashtekar, C. Isham, Representations of the holonomy algebras
of gravity and non-Abelian gauge theories, Class. Quantum Gravity 9, 1433-67 (1992).
\bibitem{BrRo81} O. Bratteli, D. Robinson,
Operator Algebras and Quantum Statistical Mechanics II,
Springer, New York (1981).
\bibitem{Dri89} B. Driver, $YM_2$: Continuum expectations, lattice convergence, and
lassos, Commun. Math. Phys. 123, 575-616 (1989).
\bibitem{Fin90} D. Fine, Quantum Yang-Mills on the two-sphere,
Commun. Math. Phys. 134,273-292 (1990)
\bibitem{Fre84} I. Frenkel, Orbital Theory for affine Lie algebras, Invent.
Math., 301-352 (1984)
\bibitem{Fro80} J. Frohlich, Some results and comments on quantized gauge
fields. in Recent Developments in Gauge Theories, G. t'Hooft et.al. eds.,
Plenum,(1980).
\bibitem{GGV77} I. Gelfand, M. Graev, A. Versik, Representations of the group of
smooth mappings from a manifold X into a compact Lie group, Compositio Mathematica 35,
299-334, (1977).
\bibitem{Gil74} P. Gilkey, The Index theorem and the Heat Equation, Publish or
Perish, Boston (1974).
\bibitem{GKS89} L.Gross, C.King, A.Sengupta, Two dimensional Yang-Mills theory
via stochastic differential equations, Ann. Phys. 194, 65- (1989)
\bibitem{Ibe76} M. Ibero, Integral stochastique multiplicatives et construction
de diffusions sur un groupe de Lie, Bull. Sc. Math. 100, 175-191 (1976).
\bibitem{McK69} H. McKean, Stochastic Integrals, Academic Press, New York (1969)
\bibitem{Mil72} W. Miller, Symmetry groups and their applications, Academic Press,
New York, (1972).
\bibitem{Raj88} S. Rajeev, Yang-Mills theory on a cylinder,
Physics Letters B 212, 203-205, (1988).
\bibitem{Sei82} E. Seiler, Gauge Theories as a Problem of Constructive
Quantum Field Theory and Statistical Mechanics, Springer, New York (1982)
\bibitem{Wit91} E. Witten, On gauge theories in two dimensions,
Commun. Math. Phys. 141,153-209 (1991)
\bibitem{Wit92} E. Witten, Two dimensional gauge theories revisited, J. of Geom.
and Phys. 9,303-368 (1992)
\end{thebibliography}
\end{document}