%-----------------------------------------------------------------------
% `Quantum Mechanics in Phase Space'
%
% Brian C. Hall, hallb@icarus.math.mcmaster.ca
%
% AMS-Latex
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\documentclass{conm-p-l}
\documentclass{amsart}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
% Absolute value notation
\newcommand{\abs}[1]{\lvert#1\rvert}
\let\Bbb = \mathbb
\let\frak = \mathfrak
\begin{document}
\title{Quantum Mechanics in Phase Space}
% author one information
\author{Brian C. Hall}
\address{Mcmaster University, Department of Mathematics and Statistics,
Hamilton, Ontario, Canada L8S-4K1}
\email{hallb@icarus.math.mcmaster.ca}
\thanks{Supported in part by NSERC}
\subjclass{Primary 22E30, 81S10; Secondary 53C55, 81R30, 58G11}
\date{November, 1996}
%-----------------------------------------------------------------------
% End of article.top
%-----------------------------------------------------------------------
\begin{abstract}
This paper concerns the generalized Segal-Bargmann transform for compact Lie
groups, which I introduced in \cite{H1}. Firstly, I wish to describe the
history of this transform and some related results by a variety of authors.
Secondly, I wish to discuss the physical interpretation of this transform as
providing a ``phase space Hilbert space'' for a quantum particle whose
configuration space is a compact Lie group. This interpretation is strongly
supported by the results of \cite{H2, H3}. Thirdly, I wish to announce some
new results concerning Toeplitz operators in the group setting.
In Section \ref{history.sec} I briefly recap the classical Segal-Bargmann
transform for $\Bbb{R}^n$, and then discuss the origins in the work of L.
Gross of the generalized Segal-Bargmann transform for compact groups. I
mention several results which clarify and extend my results and those of
Gross. In Section \ref{holo.sec} I set some notation and describe two
different versions of the generalized Segal-Bargmann transform. I discuss
the relative merits of each version of the transform.
In Section \ref{phase.sec} I explain how to identify the target manifold of
the transform with ``phase space,'' that is, with the cotangent bundle of
the compact group. This identification is essential to the desired
interpretation of the transform and involves some deep issues.
In Section \ref{inverse.sec} I describe an inversion formula for the
generalized Segal-Bargmann transform which says, roughly, that the position
wave function can be obtained from the phase space wave function by
integrating out the momentum variables. In Section \ref{uncertain.sec} I
discuss the general notion of a phase space probability density, and define
such a density in the context of a compact Lie group. I then give a uniform
bound on the phase space probability density. This bound limits the phase
space concentration of states and thus is a form of the uncertainty
principle.
In Section \ref{toeplitz.sec} I discuss Toeplitz operators, and announce
some new results. The main result is that the quantum mechanically important
operators, including Schr\"{o}dinger operators, can be represented in the
phase space Hilbert space as Toeplitz operators. Finally, in Section \ref
{geom.sec}, I compare the Hilbert space constructed in the generalized
Segal-Bargmann transform to the Hilbert space constructed by means of
geometric quantization.
\end{abstract}
\maketitle
1. History and background
2. The holomorphic function representation
3. The complex group as phase space
4. An inversion formula
5. An uncertainty principle
6. Toeplitz operators
7. Comparison with geometric quantization
\section{History and background\label{history.sec}}
In studying quantum mechanics of a particle moving in $\Bbb{R}^n$, one can
represent the Hilbert space in at least three ways: the ``position''
representation, the symmetric tensor representation, and the Segal-Bargmann
holomorphic function representation. The first Hilbert space is the space of
square-integrable functions on $\Bbb{R}^n$, with respect to either Lebesgue
measure or Gauss measure. (One can convert easily from one measure to the
other via the so-called ground state transform.) The second Hilbert space is
the completion, with respect to a suitable inner product, of the symmetric
tensor algebra over $\Bbb{R}^n$. The third Hilbert space is the space of
holomorphic functions on $\Bbb{C}^n$ which are square-integrable with
respect to Gauss measure. There is some confusion in the terminology--both
the second and third Hilbert spaces are commonly called ``Fock space.'' In
my opinion Fock's name is more properly attached to the symmetric tensor
space, and Segal and Bargmann's names to the holomorphic function space.
Each of these three Hilbert spaces has an intrinsically-defined set of
creation and annihilation operators, and a natural ground state (or vacuum
state). Between any two of the three spaces there is a unique unitary map
which takes ground state to ground state and which intertwines the creation
and annihilation operators. The intertwining map from the position
representation to the holomorphic function representation is called the
Segal-Bargmann transform \cite{B, S1, S2, S3}. All three spaces and the
associated operators and intertwining maps carry over to the
infinite-dimensional setting ($n\rightarrow \infty $) of quantum field
theory.
In \cite{Gr1} L. Gross developed an analog of the symmetric tensor
representation for a particle whose configuration space is an arbitrary
simply-connected compact Lie group $K$, and constructed a unitary
intertwining map between this space and the position Hilbert space. (See
also the partly expository paper \cite{Gr2}, and further results in \cite
{Gr3}.) As it turns out, the tensor space does not consist of symmetric
tensors; instead it is a completion of the universal enveloping algebra of
Lie$\left( K\right) $. The Gross isomorphism is the unique unitary map
taking ground state to ground state and intertwining the relevant creation
and annihilation operators. Gross' results came out of the study of the
infinite-dimensional set of paths in $K$, and the proof made use of
stochastic analysis.
Motivated by these results of Gross, I constructed in \cite{H1} a
holomorphic function representation of the Hilbert space, and an associated
unitary isomorphism of this space with the position Hilbert space. This
isomorphism is to be viewed as a generalization of the classical
Segal-Bargmann transform. The proof was in terms of Lie group theory and did
not involve stochastic analysis. Thus the whole ``triad'' of spaces from the
classical setting generalizes to the setting of a simply-connected compact
Lie group.
I should point out that there is at least one important physical system
whose configuration space is a compact Lie group, namely a rotor. That is,
if you consider the rotational degrees of freedom of a rigid body, then the
configuration space is the compact rotation group $\mathsf{SO}\left(
3\right) $. Of course $\mathsf{SO}\left( 3\right) $ is not simply-connected,
but much of the analysis is applicable anyway. Specifically, the generalized
Segal-Bargmann transform works perfectly well for non-simply-connected
groups.
O. Hijab \cite{Hi1, Hi2} gave a direct, non-probabilistic proof of the Gross
isomorphism. B. Driver \cite{D} gave a direct proof of the unitarity of the
map from the holomorphic function space to the completed universal
enveloping algebra, which was previously proved by combining my results with
those of Gross. Driver also gave a new proof of the unitarity of the
generalized Segal-Bargmann transform.
Going in the opposite direction, Gross and P. Malliavin \cite{GM} have
recently given a new probabilistic proof of the unitarity of the generalized
Segal-Bargmann transform for $K$. This proof is by the authors' admission
harder than either my proof or Driver's, but it gives an interesting
connection between the generalized Segal-Bargmann transform and the original
stochastic-analytical setting of Gross. Gross and Malliavin's proof makes
use of the \textit{infinite-dimensional classical} version of the
Segal-Bargmann transform. In fact, the whole triad of spaces for a compact
group can be embedded into the infinite-dimensional classical triad, using
the It\^{o} mapping. Here ``infinite-dimensional'' means that the underlying
configuration space is infinite-dimensional.
The role played by $\Bbb{C}^n$ in the classical theory is played in the
generalized theory by the ``complexification'' of the compact group $K$.
Driver and Gross \cite{DG} have further generalized one leg of the triad to
the setting of an arbitrary simply-connected complex Lie group $G$, one that
is not necessarily the complexification of a compact group. They give a
unitary map from a certain Hilbert space of holomorphic functions on $G$ to
a certain completion of the universal enveloping algebra of Lie$\left(
G\right) $, which as usual is the unique unitary map taking ground state to
ground state and intertwining the creation and annihilation operators. See
also \cite{Gr4}.
A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour\~{a}o, and T. Thiemann \cite
{A} have given an infinite-dimensional extension of my generalized
Segal-Bargmann transform, in which the compact group $K$ is replaced by a
certain space of connections-modulo-gauge-transformations. Specifically,
given a manifold $M$ and a principal $K$-bundle over $M$ they describe a
unitary transform whose domain is $L^2\left( \text{connections/gauge}\right)
$ with respect to a suitable measure, and whose range is a certain $L^2$%
-space of holomorphic functions. According to the authors, this extension of
the generalized Segal-Bargmann transform ``is expected to play a key role in
a non-perturbative, canonical approach to quantum gravity in 4 dimensions.''
R. Loll \cite{L} has also used my generalized Segal-Bargmann transform in
the context of quantum gravity. Loll's approach is distinct from but
conceptually similar to that of Ashtekar \textit{et al}.
Finally, P. Biane \cite{Bi} has given an analog of the generalized
Segal-Bargmann transform in the setting of free probability in the sense of
Voiculescu.
\section{The holomorphic function representation\label{holo.sec}}
In passing from the Segal-Bargmann transform on $\Bbb{R}^n$ to the
generalized Segal-Bargmann transform on a compact Lie group $K$, one simply
replaces each of the Gaussians in the classical transform by its natural
geometric analog: a heat kernel. No fewer than three heat kernels appear in
the generalized transform: the heat kernel on the configuration space, the
heat kernel on the phase space, and the heat kernel on the ``momentum
space.''
The set-up is as follows \cite{H1}. Let $K$ be an arbitrary connected
compact Lie group. Unlike the Gross isomorphism, the generalized
Segal-Bargmann transform does not require that $K$ be simply-connected. In
particular the physically interesting case $K=\mathsf{SO}\left( 3\right) $
is permitted.
Let $\frak{k}$ be the real Lie algebra of $K$. Fix an inner product $%
\left\langle \ ,\ \right\rangle $ on $\frak{k}$ which is invariant under the
adjoint action of $K$. The choice of an invariant inner product on $\frak{k}$
gives rise to a bi-invariant Riemannian metric on $K$. Let $\Delta $ be the
Laplace-Beltrami operator for this metric. Algebraically, $\Delta $ is the
Casimir element of the universal enveloping algebra of $\frak{k}$. For
example, if
\[
K=\mathsf{SU}\left( 2\right) =\left\{ \left. \left(
\begin{array}{cc}
\alpha & \beta \\
-\overline{\beta } & \overline{\alpha }
\end{array}
\right) \right| \left| \alpha \right| ^2+\left| \beta \right| ^2=1\right\}
\text{,}
\]
then there is up to a constant exactly one invariant inner product on $\frak{%
k}$. With the associated bi-invariant metric, $\mathsf{SU}\left( 2\right) $
is isometric to a standard three-sphere. The operator $\Delta $ is then the
standard spherical Laplacian on $S^3$.
Let $K_{\Bbb{C}}$ be the complexification of $K$. This is defined by a
universal property (see \cite{H1}), and satisfies the following two
properties: 1) the Lie algebra $\frak{k}_{\Bbb{C}}$ of $K_{\Bbb{C}}$ is the
complexification of $\frak{k}$, and 2) $K_{\Bbb{C}}$ contains $K$ as a
subgroup. For example, if $K=\mathsf{SU}\left( n\right) $, then $K_{\Bbb{C}}=%
\mathsf{SL}\left( n;\Bbb{C}\right) $. Let $\rho _t\left( x\right) $ denote
the heat kernel on $K$, that is, the fundamental solution at the identity of
the heat equation
\[
\frac{d\rho }{dt}=\frac 12\Delta \rho _t\text{.}
\]
This is a smooth positive function on $K$ which can be accurately estimated
by means of the Poisson summation formula of Urakawa. (See \cite{H3, U}.) As
proved in \cite{H1}, $\rho _t$ has a unique analytic continuation from $K$
to $K_{\Bbb{C}}$.
The choice of an inner product on $\frak{k}$ determines a real-valued inner
product on $\frak{k}_{\Bbb{C}}$ given by $\left\langle
X_1+iY_1,X_2+iY_2\right\rangle =\left\langle X_1,X_2\right\rangle
+\left\langle Y_1,Y_2\right\rangle $, where $X_1,X_2,Y_1,Y_2$ are in $\frak{k%
}$. This inner product on $\frak{k}_{\Bbb{C}}$ is Ad-$K$-invariant but
\textit{not} Ad-$K_{\Bbb{C}}$-invariant, unless $K$ is commutative. This
inner product on $\frak{k}_{\Bbb{C}}$ gives rise to a left-invariant (but
not right-invariant) Riemannian metric on $K_{\Bbb{C}}$ and thus to a
left-invariant Laplace-Beltrami operator $\Delta _{K_{\Bbb{C}}}$. Let $\mu _t
$ denote the heat kernel on $K_{\Bbb{C}}$, that is, the fundamental solution
at the identity of the equation
\[
\frac{d\mu }{dt}=\frac 14\Delta _{K_{\Bbb{C}}}\mu _t\text{.}
\]
For convenience I have normalized the heat equation differently on $K$ and
on $K_{\Bbb{C}}$.
Finally, let $\nu _t$ be the function on $K_{\Bbb{C}}$ given by
\[
\nu _t\left( g\right) =\int_K\mu _t\left( gx\right) \,dx\text{,}\qquad g\in
K_{\Bbb{C}}\text{.}
\]
This function can be interpreted as the heat kernel on the Riemannian
symmetric space $K_{\Bbb{C}}/K$. The function $\nu _t$ (but unfortunately
not the function $\mu _t$) has an explicit formula, due to Gangolli. (See
\cite[Eq. 11, Ga]{H3}.)
So we have three different heat kernels on $K_{\Bbb{C}}$: the holomorphic
function $\rho _t$ which is the analytic continuation of the heat kernel on $%
K$; the real positive function $\mu _t$ which is the heat kernel on $K_{\Bbb{%
C}}$; and the real, positive, function $\nu _t$ which is the heat kernel on $%
K_{\Bbb{C}}/K$, viewed as a $K$-invariant function on $K_{\Bbb{C}}$. In the
context of the one-dimensional classical Segal-Bargman transform, these
would all be functions on $\Bbb{C}$, given by
\begin{eqnarray*}
\rho _t\left( z\right) &=&\frac 1{\sqrt{2\pi t}}e^{-z^2/2t}, \\
\mu _t\left( z\right) &=&\frac 1{\pi t}e^{-\left| z\right| ^2/t}, \\
\nu _t\left( z\right) &=&\frac 1{\sqrt{\pi t}}e^{-\left( \text{Im}z\right)
^2/t}\text{.}
\end{eqnarray*}
In all that we do the real positive number $t$ will be a parameter that
plays the role of Planck's constant ($\hbar $).
There are two versions of the generalized Segal-Bargmann transform for $K$.
The first is the ``Gaussian'' version. Let $dx$ denote Haar measure on $K$
and $\mathcal{H}\left( K_{\Bbb{C}}\right) $ the space of holomorphic
functions on $K_{\Bbb{C}}$. Define
\[
B_t:L^2\left( K,\rho _t\left( x\right) \,dx\right) \rightarrow \mathcal{H}%
\left( K_{\Bbb{C}}\right)
\]
by the formula
\[
B_tf\left( g\right) =\int_K\rho _t\left( gx^{-1}\right) f\left( x\right)
\,dx\qquad g\in K_{\Bbb{C}}\text{.}
\]
Here $\rho _t$ refers to the analytic continuation of $\rho _t$ from $K$ to $%
K_{\Bbb{C}}$. Since $\rho _t$ is the heat kernel, we see that
\[
B_tf=\text{ analytic continuation of }e^{t\Delta /2}f\text{.}
\]
\begin{theorem}
For each $t>0$, $B_t$ is an isometric isomorphism of $L^2\left( K,\rho
_t\left( x\right) \,dx\right) $ onto the space of holomorphic functions in $%
L^2\left( K_{\Bbb{C}},\mu _t\left( g\right) \,dg\right) $, where $dg$ is
Haar measure on $K_{\Bbb{C}}$.
\end{theorem}
This is the Gaussian version of the transform, in which the measures on both
$K$ and $K_{\Bbb{C}}$ are as Gaussian as possible (that is, heat kernels).
The result is Theorem 1$^{\prime }$ of \cite{H1}. It is this version of the
transform that connects directly with the Gross isomorphism. Note that the
Hilbert spaces and the transform depend on the parameter $t$, which is to be
interpreted as Planck's constant.
The ``invariant'' version of the transform is the map
\[
C_t:L^2\left( K,dx\right) \rightarrow \mathcal{H}\left( K_{\Bbb{C}}\right)
\]
given by
\[
C_tf\left( g\right) =\int_K\rho _t\left( gx^{-1}\right) f\left( x\right)
\,dx\qquad g\in K_{\Bbb{C}}\text{.}
\]
Note that $C_t$ is the same map as $B_t$, except that we are using a
different inner product on the domain space.
\begin{theorem}
For each $t>0$, $C_t$ is an isometric isomorphism of $L^2\left( K,dx\right) $
onto the space of holomorphic functions in $L^2\left( K_{\Bbb{C}},\nu
_t\left( g\right) \,dg\right) $.
\end{theorem}
\noindent This result is Theorem 2 of \cite{H1}.
The chief advantage of this version of the generalized Segal-Bargmann
transform is that it is more invariant than the Gaussian version. The
measure $dx$ on $K$ is invariant under the left- and right-action of $K$, as
is the measure $\nu _t\left( g\right) \,dg$ on $K_{\Bbb{C}}$, and the
transform commutes with the left- and right-action of $K$. Of course,
sometimes one may not want to work invariantly; in the stochastic set-up of
Gross the identity is a distinguished point and it is natural to work with a
measure like $\rho _t\left( x\right) \,dx$ which is concentrated near the
identity. Nevertheless, if one thinks simply of quantum mechanics for a
particle with configuration space $K$, it is natural to make things as
invariant as possible. As explained in the appendix of \cite{H1}, all of
this makes little difference in the $\Bbb{R}^n$ case. But in the compact
group case there is a significant difference between the two versions of the
transform.
A more subtle advantage of the invariant version of the transform lies in my
desire to interpret $C_tf$ as the ``phase space wave function''
corresponding to the ``position wave function'' $f$. Looking at the $\Bbb{R}%
^n$ case, you will see that this interpretation is natural because the
``coherent state'' with parameter $z$ has mean position Re$z$ and mean
momentum Im$z$. By contrast, to get this interpretation in the $\Bbb{R}^n$
case for $B_t$, one needs to replace $z\in \Bbb{C}^n$ by $z/2$. (See the
appendix of \cite{H1}.) The analogous operation in $K_{\Bbb{C}}$ would be to
replace $g$ by $\sqrt{g}$, an unpleasant and ill-defined operation!
Finally, at a practical level, the invariant transform $C_t$ is simply
easier to calculate with. The reproducing kernel, for example, has a much
nicer form in the invariant case, and this leads to results which seem
unattainable in the Gaussian case.
All of this being said, the Gaussian version of the transform, and the
associated measure $\mu _t$, play a vital role in the results of Section \ref
{toeplitz.sec}.
A phase space Hilbert space is a natural setting for a semiclassical
analysis of quantum mechanics, simply because it brings quantum mechanics
closer to classical mechanics, which takes place in phase space. This idea
has been developed on a non-rigorous level by A. Voros \cite{V}, who gives a
simple and natural derivation of the Bohr-Sommerfeld quantization condition
by expressing the WKB method in the Segal-Bargmann representation.
Specifically, the Segal-Bargmann representation avoids the problem of
turning points in the WKB method. Rigorous developments along these lines
have been made by T. Paul and A. Uribe \cite{PU}, S. Graffi and Paul \cite
{GP}, and L. Thomas and S. Wassell \cite{TW}.
\section{The complex group as phase space\label{phase.sec}}
Physically, I wish to interpret the complex group $K_{\Bbb{C}}$ as the phase
space corresponding to the configuration space $K$. (See Section 3 of \cite
{H3}.) To this end, I wish to identify $K_{\Bbb{C}}$ with the customary
phase space, namely the cotangent bundle of $K$. We identify the cotangent
bundle $T^{*}\!\left( K\right) $ with $K\times \frak{k}^{*}$ via
left-translation, and then with $K\times \frak{k}$ via the inner product on $%
\frak{k}$. We then consider the map $\Phi :K\times \frak{k}\rightarrow K_{%
\Bbb{C}}$ given by
\[
\Phi \left( x,Y\right) =xe^{iY}\qquad x\in K,\text{ }Y\in \frak{k}\text{.}
\]
This map is a diffeomorphism of $K\times \frak{k}$ $=T^{*}\!\left( K\right) $
with $K_{\Bbb{C}}$. For example, if $K=\mathsf{SU}\left( n\right) $ and $K_{%
\Bbb{C}}=\mathsf{SL}\left( n;\Bbb{C}\right) $, then $Y\in \mathsf{su}\left(
n\right) $ is skew (and trace zero), so that $iY$ is self-adjoint and $e^{iY}
$ is self-adjoint and positive. Thus $xe^{iY}=UP$ is the ordinary polar
decomposition for a matrix in $\mathsf{SL}\left( n;\Bbb{C}\right) $.
Physically, $x$ represents position and $Y$ momentum.
Although the map between $T^{*}\!\left( K\right) $ and $K_{\Bbb{C}}$
apparently favors left over right, this is an illusion. You could just as
well identify $T^{*}\!\left( K\right) $ with $K\times \frak{k}^{*}$ via
right translation, then with $K\times \frak{k}$ via the inner product, and
then with $K_{\Bbb{C}}$ via the map $\left( x,Y\right) \rightarrow e^{iY}x$.
It is easy to check that the resulting map from $T^{*}\!\left( K\right) $ to
$K_{\Bbb{C}}$ is the same as the ``left'' one.
\begin{theorem}
If we identify the symplectic manifold $T^{*}\!\left( K\right) $ with the
complex manifold $K_{\Bbb{C}}$ as above, then we obtain a K\"{a}hler
manifold.
\end{theorem}
As explained in \cite{H3}, this result follows from the theory of adapted
complex structures, as developed by Lempert and Sz\"{o}ke \cite{LS, Sz} and
by Guillemin and Stenzel \cite{GS1, GS2}.
Let $\omega $ denote the symplectic 2-form on $T^{*}\left( K\right) $, and
let $J$ denote the complex structure on $K_{\Bbb{C}}$, which can be viewed
as living on $T^{*}\left( K\right) $ because of our identification. The
K\"{a}hler condition means that
\begin{eqnarray*}
\omega \left( JX,JY\right) &=&\omega \left( X,Y\right) , \\
\omega \left( X,JX\right) &\ge &0
\end{eqnarray*}
for all tangent vectors $X$ and $Y$. The first condition implies that the
Poisson bracket of two holomorphic functions is always zero.
Except in the case where $K$ is commutative, $K_{\Bbb{C}}$ is \textit{not} a
homogeneous K\"{a}hler manifold. In particular the symplectic structure on $%
K_{\Bbb{C}}$ obtained via identification with $T^{*}\!\left( K\right) $ is
neither left- nor right-invariant.
\section{An inversion formula\label{inverse.sec}}
In position-momentum coordinates, the measure $\nu _t\left( g\right) \,dg$
decomposes as follows
\[
\nu _t\left( g\right) \,dg=dx\,\widetilde{\nu }_t\left( Y\right) \,dY
\]
where $dx$ is Haar measure on $K$, $dY$ is Lebesgue measure on $\frak{k}$,
and $\widetilde{\nu }_t$ is an appropriate density on $\frak{k}$. See
Section \ref{geom.sec} for an explicit formula for $\widetilde{\nu }_t$. In
this notation we have the following inversion formula \cite[Theorem 1]{H2}.
\begin{theorem}
If $F=C_tf$, then
\begin{equation}
f\left( x\right) =\int_{\frak{k}}F\left( xe^{2iY}\right) \widetilde{\nu }%
_{t/2}\left( Y\right) \,dY\qquad x\in K\text{.} \label{inverse}
\end{equation}
\end{theorem}
\noindent Note the factors of two!
This formula says that one can recover the position wave function from the
phase space wave function by integrating out the momentum variables, with
respect to a suitable measure. Now, as explained in \cite{H2}, the integral (%
\ref{inverse}) may not be absolutely convergent. Strictly speaking, the
inversion should be accomplished by integrating over the set $\left|
Y\right| \le R$ and then taking a limit in $L^2\left( K,dx\right) $ as $%
R\rightarrow \infty $. In fact, the integral \textit{cannot} always be
convergent, because $f$ is an arbitrary $L^2$ function on $K$, which may be
infinite at certain points. However \cite[Theorem 3]{H2}, if $f$ is
sufficiently smooth, then (\ref{inverse}) is absolutely convergent and the
inversion formula may be taken literally.
Note also that because $B_t$ and $C_t$ are the same transform (but with
different inner products), (\ref{inverse}) may also be regarded as an
inversion formula for the Gaussian version of the transform. Since the
transform is just the heat operator, followed by analytic continuation, (\ref
{inverse}) may be regarded as a formula for the \textit{inverse} heat
operator for $K$.
In the $\Bbb{R}^n$ case, $\widetilde{\nu }_t$ is just a Gaussian, and (\ref
{inverse}) becomes
\[
f\left( x\right) =2^n\left( 2\pi t\right) ^{-n/2}\int_{\Bbb{R}^n}F\left(
x+2iy\right) e^{-2y^2/t}dy\text{.}
\]
Making the change of variable $\widetilde{y}=2y$ this is equivalent to
\begin{equation}
f\left( x\right) =\left( 2\pi t\right) ^{-n/2}\int_{\Bbb{R}^n}F\left( x+i%
\widetilde{y}\right) e^{-\widetilde{y}^2/2t}d\widetilde{y}\text{.}
\label{rn.inverse}
\end{equation}
We can now see how to verify the inversion formula in the $\Bbb{R}^n$ case.
Fix a holomorphic function $F$ on $\Bbb{C}^n$, and consider the right side
of (\ref{rn.inverse}). Note that $\left( 2\pi t\right) ^{-n/2}e^{-\widetilde{%
y}^2/2t}$ is the heat kernel on $\Bbb{R}^n$. Thus differentiating with
respect to $t$, integrating by parts, and using Cauchy-Riemann, we see that
the right side of (\ref{rn.inverse}) satisfies the \textit{inverse} heat
equation. Furthermore, letting $t\rightarrow 0$, the right side of (\ref
{rn.inverse}) converges to $F\left( x\right) $. So the right side of (\ref
{rn.inverse}) is nothing but the \textit{inverse} heat operator applied to $%
F\left( x\right) $. Since $C_t$ is just the forward heat operator (followed
by analytic continuation), the right side of (\ref{rn.inverse}) is $%
C_t^{-1}F $.
In the non-commutative case, the proof is largely the same, using the fact
that $\nu _t$ is the heat kernel for $K_{\Bbb{C}}/K$. One simply needs to
verify an identity \cite[Theorem 5]{H2} which relates the Laplacian for $K_{%
\Bbb{C}}/K$ to the Laplacian for $K$, using the Cauchy-Riemann equations.
This identity works only with the factors of two the way they are in the
theorem. The $\Bbb{R}^n$ case is special because in that case a re-scaling
of the space variable in the heat kernel can be absorbed by an appropriate
re-scaling of the time variable; no such result holds for the symmetric
space $K_{\Bbb{C}}/K$.
\section{An uncertainty principle\label{uncertain.sec}}
Let $\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right) $ denote the space of
holomorphic functions on $K_{\Bbb{C}}$ which are square-integrable with
respect to the measure $\nu _t\left( g\right) \,dg$. Take $F\in \mathcal{H}%
L^2\left( K_{\Bbb{C}},\nu _t\right) $ with $\left\| F\right\| =1$. This
means that
\begin{equation}
\int_{K_{\Bbb{C}}}\left| F\left( g\right) \right| ^2\nu _t\left( g\right)
\,dg=1\text{.} \label{normalize1}
\end{equation}
Now let $dL$ denote the Liouville ``phase volume'' measure on $T^{*}\!\left(
K\right) $, which we think of as a measure on $T^{*}\!\left( K\right) $ via
our identification of $T^{*}\!\left( K\right) $ and $K_{\Bbb{C}}$. Then (\ref
{normalize1}) can be rewritten as
\begin{equation}
\int_{K_{\Bbb{C}}}\left| F\left( g\right) \right| ^2\nu _t\left( g\right)
\sigma \left( g\right) \,dL=1\text{,} \label{normalize2}
\end{equation}
where $\sigma $ is the ``Jacobian'' of the map $\Phi $, given explicitly in
\cite[Lemma 5]{H3}.
I wish to think of $\left| F\left( g\right) \right| ^2\nu _t\left( g\right)
\sigma \left( g\right) $ as the phase space probability density associated
to the state $F$. Now, according to (\ref{normalize2}), $\left| F\left(
g\right) \right| ^2\nu _t\left( g\right) \sigma \left( g\right) $ is a
probability density, that is, a positive function which integrates to one.
What is less clear is the sense in which this is \textit{the} density
associated to the state $F$.
In general one should have a different definition of phase space probability
density for each quantization scheme. By a quantization scheme I mean a map $%
Q$ from functions on phase space to operators on a Hilbert space. In
specifying such a map some choice must be made (implicitly or explicitly) as
to how to order non-commuting operators. Given such a map $Q$ and a unit
vector $F$ in the corresponding Hilbert space, we can try to define a
probability density $p_F$ by the condition that
\begin{equation}
\int \phi \left( m\right) p_F\left( m\right) \,dL\left( m\right)
=\left\langle F,Q\left( \phi \right) F\right\rangle \label{prob.condition}
\end{equation}
for all functions $\phi $ on phase space. Here $\left\langle \ ,\
\right\rangle $ denotes the inner product in the quantum Hilbert space. The
condition (\ref{prob.condition}) serves to define $p_F$ as a distribution,
provided that $Q\left( \phi \right) $ is a bounded operator whenever $\phi $
is a $C^\infty $ function of compact support. In general there is no reason
that $p_F$ should be a positive function. For example, if $Q$ is the Weyl
quantization of $\Bbb{R}^{2n}$, then $p_F$ is the famous Wigner function.
(See Chap.\thinspace 1, Sec.\thinspace 8 and Chap.\thinspace 2,
Sec.\thinspace 1 of \cite{F}.) The Wigner function may or may not be
positive, depending on $F$.
As we shall see in the next section, the density $\left| F\left( g\right)
\right| ^2\nu _t\left( g\right) \sigma \left( g\right) $ satisfies the
condition (\ref{prob.condition}) if $Q$ is the Berezin-Toeplitz
quantization, which is a generalization of anti-Wick ordering. The fact that
this density is always positive reflects the fact that the Berezin-Toeplitz
quantization associates positive operators to positive functions. In the $%
\Bbb{R}^{2n}$ case, the Berezin-Toeplitz density can be obtained from the
Wigner function by convolving with an appropriate Gaussian
\cite[Chap. 2, Sec. 7]{F}. It is interesting that the Wigner function
convolved with this Gaussian is always positive, even when the Wigner
function itself is not.
By the way, it is worthwhile to contrast the $\Bbb{R}^n$ case of the
inversion formula of Section \ref{inverse.sec} with a standard result
concerning the Wigner function. It is well known that the position \textit{%
probability density} $\left| f\left( x\right) \right| ^2$ can be obtained
(in the $\Bbb{R}^n$ case) by taking the Wigner function and integrating out
the momentum variables. (This follows from the relationship between the
Wigner function and the Weyl quantization.) By contrast, our inversion
formula says that the position \textit{wave function} $f\left( x\right) $
can be obtained from the phase space wave function by suitably integrating
out the momentum variables.
In the $\Bbb{R}^n$ case, the probability density defined with respect to the
Segal-Bargmann transform has the disadvantage that it is defined in terms of
the Berezin-Toeplitz quantization instead of the more physically natural
Weyl quantization. However, the Segal-Bargmann density has several
advantages. First, it is always positive, while the Wigner function is not.
Second, it is defined in terms of a phase space wave function, which allows
you to take all of your operators and wave functions in phase space (as in
\cite{V}), and then simply to take the absolute value squared to get the
phase space probability density. Last, the Segal-Bargmann density can be
defined for systems with configuration space $K$, whereas (so far as I know)
the Wigner function can be defined only for $\Bbb{R}^n$. For small Planck's
constant (in $\Bbb{R}^n$), there is very little difference between the
Wigner function and the Segal-Bargmann probability density.
Any result which gives a bound on the concentration of a state in phase
space should be regarded as a form of the Heisenberg uncertainty principle.
In the context of the generalized Segal-Bargmann transform for $K$ we have
the following such result \cite[Theorem 1]{H3}.
\begin{theorem}
\label{phase.thm}For all $F\in \mathcal{H}L^2\left( K_{\Bbb{C}},\nu
_t\right) $ with $\left\| F\right\| =1$ the phase space probability density
satisfies
\[
\left| F\left( g\right) \right| ^2\nu _t\left( g\right) \sigma \left(
g\right) \le a_t\left( 2\pi t\right) ^{-n}\text{.}
\]
Here $n=\dim K$ and $a_t$ is a constant (independent of $F$ and $g$) which
tends to one exponentially fast as $t$ tends to zero.
\end{theorem}
This bound is sharp in a sense described in \cite{H3}. In the $\Bbb{R}^n$
case $\sigma \equiv 1$ and we may take $a_t=1$.
Recalling that $t=\hbar $ (Planck's constant), we can recognize the quantity
$\left( 2\pi t\right) ^n$ as the volume of a semiclassical cell in phase
space. Thus the above result says that if $E$ is a region of phase space
whose volume is small compared to the volume of a semiclassical cell, then
the probability of a particle being found in $E$ is small, with estimates
independent of the state $F$ of the particle. Thus Theorem \ref{phase.thm}
is a physically natural bound on the phase space concentration.
It is easy to see that there is a bound on $\left| F\left( g\right) \right|
^2\nu _t\left( g\right) \sigma \left( g\right) $ which is independent of $F$%
, for $F$ with $\left\| F\right\| =1$. However, since $K_{\Bbb{C}}$ is not a
homogeneous K\"{a}hler manifold, there is no \textit{a priori} reason that
there should exist bounds independent of $g$. In fact, amazing cancelations
occur to give this independence, suggesting that there is something
``right'' about this set-up.
\section{Toeplitz operators\label{toeplitz.sec}}
As on any reasonable $L^2$-space of holomorphic functions, we may define
Toeplitz operators on $\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right) $. Let
$\Pi _t$ denote the orthogonal projection operator from $L^2\left( K_{\Bbb{C}%
},\nu _t\right) $ onto the holomorphic subspace, which is a closed subspace.
Then for $\phi $ a bounded measurable function on $K_{\Bbb{C}}$ we may
define
\[
T_t\left( \phi \right) :\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right)
\rightarrow \mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right)
\]
by the formula
\[
T_t\left( \phi \right) F=\Pi _t\left( \phi F\right) \text{.}
\]
The projection is necessary because $\phi $ is not assumed holomorphic.
Because of the projection, two Toeplitz operators need not commute--this, of
course, is the whole point!
We can think of $T_t\left( \phi \right) $ as an operator on the full $L^2$%
-space $L^2\left( K_{\Bbb{C}},\nu _t\right) $ by making it zero on the
orthogonal complement of the holomorphic subspace. Then
\begin{equation}
T_t\left( \phi \right) =\Pi _tM_\phi \Pi _t \label{toeplitz2}
\end{equation}
where $M_\phi $ denotes multiplication by $\phi $. Writing things in this
way it is clear that
\begin{equation}
T_t\left( \overline{\phi }\right) =T_t\left( \phi \right) ^{*}
\label{adjoint}
\end{equation}
and that
\[
\left\| T_t\left( \phi \right) \right\| \le \left\| \phi \right\| _{L^\infty
}\text{.}
\]
In particular, if $\phi $ is real then $T_t\left( \phi \right) $ is
self-adjoint.
If $\phi $ is an unbounded but with moderate growth, then $T_t\left( \phi
\right) $ will be a densely-defined operator on $\mathcal{H}L^2\left( K_{%
\Bbb{C}},\nu _t\right) $. Because of domain issues, the adjoint identity (%
\ref{adjoint}) may not hold in the unbounded case. Some of these domain
issues will be addressed in certain cases in \cite{H4}.
In general,
\begin{eqnarray*}
T_t\left( \phi _1\right) \cdots T_t\left( \phi _n\right) =\Pi _tM_{\phi
_1}\Pi _t\cdots \Pi _tM_{\phi _n}\Pi _t\text{.}
\end{eqnarray*}
Now, if the $\phi $'s all happen to be holomorphic, then all the projections
except the first and last are unnecessary, so
\begin{eqnarray*}
T_t\left( \phi _1\right) \cdots T_t\left( \phi _n\right) &=&\Pi _tM_{\phi
_1}\cdots M_{\phi _n}\Pi _t \\
&=&T_t\left( \phi _1\cdots \phi _n\right) \text{.}
\end{eqnarray*}
(There are no non-constant bounded holomorphic functions on $K_{\Bbb{C}}$,
so there are some domain issues here, which I will ignore.) Taking the
adjoint of this identity gives a similar formula in the case where all the $%
\phi $'s are anti-holomorphic. Finally, if $\phi $ is anti-holomorphic and $%
\psi $ is holomorphic, then
\begin{eqnarray*}
T_t\left( \phi \right) T_t\left( \psi \right) &=&\Pi _tM_\phi \Pi _tM_\psi
\Pi _t \\
&=&\Pi _tM_\phi M_\psi \Pi _t \\
&=&T_t\left( \phi \psi \right) \text{.}
\end{eqnarray*}
Note that this argument does not work with the order of $\phi $ and $\psi $
reversed on the left! That is, the Toeplitz operator associated to the
holomorphic function must be on the right.
In the $\Bbb{C}^1$ case this means that
\begin{eqnarray*}
T_t\left( \overline{z}^nz^m\right) &=&T_t\left( \overline{z}^n\right)
T_t\left( z^m\right) \\
&=&T_t\left( \overline{z}\right) ^nT_t\left( z\right) ^m\text{.}
\end{eqnarray*}
Since $T_t\left( z\right) $ is (unitarily equivalent to) the creation
operator and $T_t\left( \overline{z}\right) $ is the annihilation operator,
this corresponds to ``anti-Wick ordering.'' Properties of the classical Wick
and anti-Wick orderings, and connections with Toeplitz operators, are given
in Chap.\thinspace 2, Sec.\thinspace 7 of \cite{F}.
If you compute a matrix entry of a Toeplitz operator between two holomorphic
functions then the projections in (\ref{toeplitz2}) are unnecessary. Thus
\[
\left\langle F,T_t\left( \phi \right) F\right\rangle =\int_{K_{\Bbb{C}}}\phi
\left( g\right) \left| F\left( g\right) \right| ^2\nu _t\left( g\right) \,dg%
\text{,}
\]
or, in the notation of Section \ref{uncertain.sec},
\[
\left\langle F,T_t\left( \phi \right) F\right\rangle =\int_{K_{\Bbb{C}}}\phi
\left( g\right) \left| F\left( g\right) \right| ^2\nu _t\left( g\right)
\sigma \left( g\right) \,dL\text{.}
\]
Thus the expectation value of $T_t\left( \phi \right) $ in the state $F$ is
equal to the integral of $\phi $ against the probability density $\left|
F\left( g\right) \right| ^2\nu _t\left( g\right) \sigma \left( g\right) $.
It is natural to regard the map $\phi \rightarrow T_t\left( \phi \right) $
as a quantization map, that is, a map from functions on the classical phase
space $K_{\Bbb{C}}=T^{*}\!\left( K\right) $ to operators on a Hilbert space,
with the parameter $t$ playing the role of Planck's constant. In light of
the above properties, this quantization should be thought of as a
generalization of the anti-Wick ordering. The idea of using Toeplitz
operators as a means of quantization began with Berezin. Toeplitz operators
have been studied as a quantization map for various symplectic manifolds by
various authors, including: Klimek and Lesniewski \cite{KL1, KL2}, for the
disk and other Riemann surfaces; Coburn \cite{Co}, for $\Bbb{C}^n$;
Borthwick, Lesniewski, and Upmeier \cite{BLU}, for bounded symmetric
domains; and Bordemann, Meinrenken, and Schlichenmaier \cite{BMS}, for
arbitrary compact K\"{a}hler manifolds. In all of these examples the
K\"{a}hler manifolds studied are either compact or homogeneous. The case of $%
K_{\Bbb{C}}=T^{*}\!\left( K\right) $ is thus of interest because it is
non-compact and (unless $K$ is commutative) non-homogeneous.
I am not going to discuss general properties of Toeplitz operators on $%
\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right) $, beyond the simple remarks
above. Instead I am going to describe how certain quantum-mechanically
important operators on $L^2\left( K,dx\right) $, including Schr\"{o}dinger
operators, can be represented as Toeplitz operators. I hope that this will
lead to a semiclassical analysis of Schr\"{o}dinger operators on $L^2\left(
K,dx\right) $, using the phase space Hilbert space $\mathcal{H}L^2\left( K_{%
\Bbb{C}},\nu _t\right) $. The results of this section are from the
forthcoming paper \cite{H4}.
>From the point of view of quantum mechanics the most important operators on $%
L^2\left( K,dx\right) $ are the momentum operators, the kinetic energy
operator, and the multiplication (potential energy) operators. I want to
conjugate each of these operators by the generalized Segal-Bargmann
transform $C_t$ to obtain operators on $\mathcal{H}L^2\left( K_{\Bbb{C}},\nu
_t\right) $, and to express these new operators as Toeplitz operators.
Let $\left\{ X_k\right\} $ be an orthonormal basis for $\frak{k}$ with
respect to our chosen inner product. Define momentum operators $\left\{
P_k\right\} $ by the formula
\[
P_kf\left( x\right) =it\left. \frac d{ds}\right| _{s=0}f\left(
xe^{sX_k}\right) \text{.}
\]
These are left-invariant first-order differential operators, the natural
analogs of the operators $it\frac \partial {\partial x_k}$ on $L^2\left(
\Bbb{R}^n,dx\right) $. Define the kinetic energy operator by the formula
\[
\frac 12\sum_kP_k^2=-\frac{t^2}2\Delta \text{.}
\]
This is a bi-invariant operator on $K$. The momentum and kinetic energy
operators are essentially self-adjoint on $C^\infty \left( K\right) $.
Finally, define multiplication operators $M_Vf=Vf$ for any function $V$ on $%
K $. Each of these operators, when conjugated by $C_t$ produces an operator
on $\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right) $.
\begin{theorem}
\label{momentum.thm}For all $t>0$,
\[
C_tP_kC_t^{-1}=T_t\left( \phi _k\right)
\]
where
\[
\phi _k\left( xe^{iY}\right) =\left\langle Y,X_k\right\rangle +tf_k\left(
Y\right)
\]
and where $f_k\left( Y\right) $ is a bounded function which is independent
of $t$. If $K$ is commutative then $f_k\equiv 0$.
For all $t>0$,
\[
C_t\left( -\frac{t^2}2\Delta \right) C_t=T_t\left( \phi \right)
\]
where
\[
\phi \left( xe^{iY}\right) =\frac{\left| Y\right| ^2}2-\frac n4t-\frac{t^2}%
2\left| \rho \right| ^2\text{.}
\]
Here $n=\dim K$ and $\rho $ is half the sum of the positive roots.
\end{theorem}
See \cite{H3} for information on the roots. I have been a bit vague about
domain issues here. In the case of the kinetic energy operator, the natural
domain of the Toeplitz operator $T_t\left( \phi \right) $ coincides with the
image under $C_t$ of the self-adjoint domain of $\Delta $. However, the
corresponding statement for the momentum operators seems to be false. So a
more precise statement is that $C_tP_kC_t^{-1}$ is equal to the closure of $%
T_t\left( \phi _k\right) $, where $P_k$ is the self-adjoint version of the
momentum operator.
Note that the function $\phi _k$ (the ``Toeplitz symbol'' of the operator $%
P_k$) is equal to the $k$th component of the classical momentum $Y$, plus a
bounded term of order $t$. Similarly, the Toeplitz symbol of the kinetic
energy operator is the classical kinetic energy function plus order $t$
corrections, which happen to be constants. These corrections are to be
expected because we are mixing two different quantization schemes.
For example, in the $\Bbb{R}^n$ case, the Toeplitz symbol of the kinetic
energy operator is
\[
\frac{\left| Y\right| ^2}2-\frac n4t\text{,}
\]
because $\rho =0$ in that case. Now, $-\left( t^2/2\right) \Delta $ is the
Weyl quantization of the classical kinetic energy function, and we are then
taking not the Weyl symbol but the Toeplitz (anti-Wick) symbol of $-\left(
t^2/2\right) \Delta $. But the Toeplitz symbol is obtained from the Weyl
symbol by applying $e^{-t\Delta /4}$, which gives precisely the correction
term $-nt/4$. (See \cite[Prop. 2.96, 2.97]{F}.)
\begin{theorem}
\label{position.thm}Suppose $V$ is of the form $V=e^{t\Delta /2}\widetilde{V}
$, for some function $\widetilde{V}$. Then
\[
C_tM_VC_t^{-1}=T_t\left( \phi _V\right)
\]
where
\[
\phi _V\left( g\right) =\frac{\int_K\mu _t\left( gy^{-1}\right) \widetilde{V}%
\left( y\right) \,dy}{\nu _t\left( g\right) }\text{.}
\]
\end{theorem}
Recall that $\mu _t$ is the full heat kernel on $K_{\Bbb{C}}$, $\nu _t$ is
the heat kernel for $K_{\Bbb{C}}/K$, and that the two functions are related
by the formula $\nu _t\left( g\right) =\int_K\mu _t\left( gx\right) \,dx$.
In particular, if $V\equiv 1$ then $\phi _V\equiv 1$. If $K$ is commutative,
then the heat kernel $\mu _t$ factors as $\mu _t\left( xe^{iY}\right) =\rho
_{t/2}\left( x\right) \nu _t\left( e^{iY}\right) $. The factor of two is a
result of the different normalizations of the heat kernel on $K$ and on $K_{%
\Bbb{C}}$. Thus in the commutative case the above reduces to
\begin{eqnarray*}
\phi _V\left( xe^{iY}\right) &=&\int_K\rho _{t/2}\left( xy^{-1}\right)
\widetilde{V}\left( y\right) \,dy \\
&=&e^{-t\Delta /4}V\left( x\right) \text{,}
\end{eqnarray*}
provided that $e^{-t\Delta /4}V$ exists. In the non-commutative case, $\phi
_V\left( xe^{iY}\right) $ will depend non-trivially on both $x$ and $Y$.
Note that the theorem assumes that $V$ can be expressed in the form $%
e^{t\Delta /2}\widetilde{V}$. This is more restrictive than necessary;
according to \cite{H4}, $\phi _V$ exists provided only that $V$ is of the
form $e^{t\Delta /4}\widehat{V}$. However, I believe that in general $%
C_tM_VC_t^{-1}$ simply cannot be expressed as a Toeplitz operator, not even
if you allowed $\phi _V$ to be a distribution.
Since Toeplitz quantization is a generalization of anti-Wick ordering, the
map $V\rightarrow \phi _V$ could be viewed as a generalization of Wick
ordering. For example, in the $\Bbb{R}^1$ case, if $V\left( x\right) =x^4$,
then $\phi _V\left( x+iy\right) =x^4-3tx^2+\frac 34t^2$, which is just the
Wick ordering of $x^4$. Note that if $K$ is non-commutative then the map $%
V\rightarrow \phi _V$ takes a function of $x$ only and produces a function
of both $x$ and $Y$.
\section{Comparison with geometric quantization\label{geom.sec}}
As we have noted the identification of $T^{*}\!\left( K\right) $ with $K_{%
\Bbb{C}}$ makes $K_{\Bbb{C}}$ into a K\"{a}hler manifold. This K\"{a}hler
manifold admits a global K\"{a}hler potential given by
\[
f\left( xe^{iY}\right) =\left| Y\right| ^2\text{.}
\]
(See \cite[Theorem 5.6, Sz]{LS} or \cite{GS1, GS2}.) The theory of geometric
quantization thus provides a way of constructing a Hilbert space, using the
``K\"{a}hler polarization.'' After suitable identifications, and writing $t$
for $\hbar $, the Hilbert space is
\[
\mathcal{H}L^2\left( K_{\Bbb{C}},\gamma _t\right) \text{,}
\]
where $\gamma _t$ is the measure given by
\[
d\gamma _t=b_te^{-f/t}dL\text{.}
\]
(See \cite[Eq. 5.7.11]{W}.) Here $b_t$ is a constant which is conventionally
taken to be $\left( 2\pi t\right) ^{-n}$. Now in terms of $x,Y$ coordinates,
the Liouville measure is just Haar measure $dx$ times Lebesgue measure $dY$.
(See \cite[Lemma 4]{H3}.) So
\[
d\gamma _t\left( x,Y\right) =b_te^{-\left| Y\right| ^2/t}\,dx\,dY\text{.}
\]
Let us compare this to the Hilbert space $\mathcal{H}L^2\left( K_{\Bbb{C}%
},\nu _t\right) $. According to \cite[Eq. 11, Lemma 5]{H3}, the measure $\nu
_t\left( g\right) \,dg$ satisfies
\begin{equation}
\nu _t\left( g\right) \,dg=c_t\,u\left( Y\right) e^{-\left| Y\right|
^2/t}\,dx\,dY\text{.} \label{explicit}
\end{equation}
Here $c_t$ is a constant (given explicitly in \cite{H3}) and $u$ is a
function independent of $x$ and $t$. Note that (\ref{explicit}) reflects
both the formula for $\nu _t$ and the conversion factor $\sigma $ between
Haar measure on $K_{\Bbb{C}}$ and Liouville measure. In the case $K=\mathsf{%
SU}\left( 2\right) $, and with a suitable inner product on $\frak{k}$, $u$
is given by the formula
\[
u\left( Y\right) =\frac{\sinh \left| Y\right| }{\left| Y\right| }\text{.}
\]
In general, $u\left( Y\right) $ is Ad-$K$-invariant and is given explicitly
on a maximal abelian subabalgebra of $\frak{k}$ by the formula
\[
u\left( Y\right) =\prod_{\alpha \in R^{+}}\frac{\sinh \alpha \left( Y\right)
}{\alpha \left( Y\right) }\text{.}
\]
Here $R^{+}$ is the set of positive roots.
If $K$ is commutative, then $u\equiv 1$, and the two Hilbert spaces are the
same, except for an irrelevant constant. If $K$ is non-commutative, then the
two Hilbert spaces are not the same, although they differ only by a
relatively small factor, which is $t$-independent.
I do not fully understand at present the difference between these two
spaces. However, from a certain point of view ``my'' space $\mathcal{H}%
L^2\left( K_{\Bbb{C}},\nu _t\right) $ is preferable to the geometric
quantization space $\mathcal{H}L^2\left( K_{\Bbb{C}},\gamma _t\right) $. My
point of view is that a phase space Hilbert space is not so much a new way
of quantizing $T^{*}\left( K\right) $ as it is a handy unitary transform
that may be useful in studying operators in the conventional position
Hilbert space $L^2\left( K,dx\right) $. From this point of view the
``goodness'' of a phase space Hilbert space is measured by how well it is
related to the position Hilbert space and by how easy it is to calculate
with.
The generalized Segal-Bargmann space seems to be better related to the
position Hilbert space than the geometric quantization space. While there is
a unitary transform, say $D_t$, from $L^2\left( K,dx\right) $ onto $\mathcal{%
H}L^2\left( K_{\Bbb{C}},\gamma _t\right) $ (see \cite[Sec. 10]{H1}), this
transform will not be given in terms of the heat operator, and it seems
difficult to understand its kernel. Moreover, it seems unlikely that there
could be an inversion formula for $D_t$ of the sort we have for the
generalized Segal-Bargmann transform. Finally, many of the results about
Toeplitz operators would fail if $D_t$ replaces $C_t$. In particular, I do
not see how to express $D_tM_VD_t^{-1}$ as a Toeplitz operator.
The generalized Segal-Bargmann space also seems easier to calculate with
than the geometric quantization space. For example, the reproducing kernel
for $\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right) $ is known fairly
explicitly \cite[Theorem 6, H3]{H1}, whereas the reproducing kernel for $%
\mathcal{H}L^2\left( K_{\Bbb{C}},\gamma _t\right) $ would be given by a
complicated series expansion. Our knowledge of the reproducing kernel in $%
\mathcal{H}L^2\left( K_{\Bbb{C}},\nu _t\right) $ leads to sharp pointwise
bounds and to the uncertainty principle of Section \ref{uncertain.sec},
results which seem unattainable for $\mathcal{H}L^2\left( K_{\Bbb{C}},\gamma
_t\right) $.
Further study is merited to understand the difference between the
generalized Segal-Bargmann space and the geometric quantization space.
\begin{thebibliography}{BMS}
\bibitem[A]{A} A. Ashtekar, J. Lewandowski, D. Marolf, and J. Mour\~{a}o,
T. Thiemann, Coherent state transforms for spaces of connections, \textit{J.
Funct. Anal.} \textbf{135} (1996), 519-551.
\bibitem[B]{B} V. Bargmann, On a Hilbert space of analytic functions and an
associated integral transform, Part I, \textit{Comm. Pure Appl. Math.}
\textbf{14} (1961), 187-214.
\bibitem[Bi]{Bi} P. Biane, Segal-Bargmann transform, functional calculus on
matrix spaces and the theory of semi-circular and circular systems,
\textit{J. Funct. Anal.} \textbf{144} (1997), 232-286.
\bibitem[BMS]{BMS} M. Bordemann, E. Meinrenken, and M. Schlichenmaier,
Toeplitz quantization of K\"{a}hler manifolds and $gl\left( N\right) $, $%
N\rightarrow \infty $ limits, \textit{Comm. Math. Phys}. \textbf{165}
(1994), 281-296.
\bibitem[BLU]{BLU} D. Borthwick, A. Lesniewski, and H. Upmeier,
Non-perturbative deformation quantization of Cartan domains, \textit{J.
Funct. Anal.} \textbf{113} (1993), 153-176.
\bibitem[Co]{Co} L. Coburn, Deformation estimates for the Berezin-Toeplitz
quantization, \textit{Comm. Math. Phys.} \textbf{149} (1992), 415-424.
\bibitem[D]{D} B. Driver, On the Kakutani-It\^{o}-Segal-Gross and
Segal-Bargmann-Hall isomorphisms, \textit{J. Funct. Anal.} \textbf{133}
(1995), 69-128.
\bibitem[DG]{DG} B. Driver and L. Gross, Hilbert spaces of holomorphic
functions on complex Lie groups, \textit{to appear in} ``Proceedings of the
1994 Taniguchi Symposium.''
\bibitem[F]{F} G. Folland, ``Harmonic Analysis in Phase Space,'' Princeton
Univ. Press, Princeton, NJ, 1989.
\bibitem[Ga]{Ga} R. Gangolli, Asymptotic behaviour of spectra of compact
quotients of certain symmetric spaces, \textit{Acta Math.} \textbf{121}
(1968), 151-192.
\bibitem[GP]{GP} S. Graffi and T. Paul, The Schr\"{o}dinger equation and
canonical perturbation theory, \textit{Comm. Math. Phys.} \textbf{108}
(1987), 25-40.
\bibitem[Gr1]{Gr1} L. Gross, Uniqueness of ground states for
Schr\"{o}dinger operators over loop groups, \textit{J. Funct. Anal.} \textbf{%
121} (1993), 373-441.
\bibitem[Gr2]{Gr2} L. Gross, The homogeneous chaos over compact Lie groups,
\textit{in} ``Stochastic Processes: A Festschrift in Honour of Gopinath
Kallianpur'' (S. Cambanis \textit{et al}., Eds.), Springer-Verlag, New
York/Berlin, 1993.
\bibitem[Gr3]{Gr3} L. Gross, Harmonic analysis for the heat kernel measure
on compact homogeneous spaces, \textit{\ in} ``Stochastic Analysis on
Infinite Dimensional Spaces: Proceedings of the U.S.-Japan Bilateral
Seminar, January 4-8, 1994, Baton Rouge, Louisiana'' (H. Kunita and H.-H.
Kuo, Eds.), Longman House, Essex, England, 1994, pp. 99-110.
\bibitem[Gr4]{Gr4} L. Gross, A local Peter-Weyl theorem, preprint.
\bibitem[GM]{GM} L. Gross and P. Malliavin, Hall's transform and the
Segal-Bargmann map,\textit{\ in ``}It\^{o}'s Stochastic Calculus and
Probability Theory'' (M. Fukushima, N. Ikeda, H. Kunita, and S. Watanabe,
Eds.), Springer-Verlag, New York/Berlin, 1996.
\bibitem[GS1]{GS1} V. Guillemin, M. Stenzel, Grauert tubes and the
homogeneous Monge-Amp\`{e}re equation, \textit{J. Diff. Geom.} \textbf{34}
(1991), 561-570.
\bibitem[GS2]{GS2} V. Guillemin, M. Stenzel, Grauert tubes and the
homogeneous Monge-Amp\`{e}re equation, II, \textit{J. Diff. Geom.} \textbf{35%
} (1992), 627-641.
\bibitem[H1]{H1} B. Hall, The Segal-Bargmann ``coherent state'' transform
for compact Lie groups, \textit{J. Funct. Anal.} \textbf{122} (1994),
103-151.
\bibitem[H2]{H2} B. Hall, The inverse Segal-Bargmann transform for compact
Lie groups, \textit{J. Funct. Anal.} \textbf{143} (1997), 98-116.
\bibitem[H3]{H3} B. Hall, Phase space bounds for quantum mechanics on a
compact Lie group, \textit{Comm. Math. Phys.}, to appear.
\bibitem[H4]{H4} B. Hall, Examples of Toeplitz operators for compact Lie
groups, in preparation.
\bibitem[Hi2]{Hi1} O. Hijab, Hermite functions on compact Lie groups, I,
\textit{J. Funct. Anal.} \textbf{125} (1994), 480-492.
\bibitem[Hi2]{Hi2} O. Hijab, Hermite functions on compact Lie groups, II,
\textit{J. Funct. Anal}. \textbf{133} (1995), 41-49.
\bibitem[KL1]{KL1} S. Klimek and A. Lesniewski, Quantum Riemann surfaces I.
The unit disc, \textit{Comm. Math. Phys.} \textbf{146} (1992), 103-122.
\bibitem[KL2]{KL2} S. Klimek and A. Lesniewski, Quantum Riemann surfaces:
II. The discrete series, \textit{Lett. Math. Phys.} \textbf{24} (1992),
125-139.
\bibitem[LS]{LS} L. Lempert and R. Sz\"{o}ke, Global solutions of the
homogeneous complex Monge-Amp\`{e}re equation and complex structures on the
tangent bundle of Riemannian manifolds, \textit{Math. Ann.} \textbf{290}
(1991), 689-712.
\bibitem[L]{L} R. Loll, Non-perturbative solutions for lattice quantum
gravity, \textit{Nuclear Phys. B} \textbf{444} (1995), 619-639.
\bibitem[PU]{PU} T. Paul and A. Uribe, A construction of quasimodes using
coherent states, \textit{Ann. Inst. Henri Poincar\'{e}} \textbf{59} (1993),
357-381.
\bibitem[S1]{S1} I. Segal, Mathematical problems of relativistic physics,
Chap.\thinspace VI, \textit{in} ``Proceedings of the Summer Seminar,
Boulder, Colorado, 1960, Vol. II.'' (M. Kac, Ed.). Lectures in Applied
Mathematics, American Math. Soc., Providence, Rhode Island, 1963.
\bibitem[S2]{S2} I. Segal, Mathematical characterization of the physical
vacuum for a linear Bose-Einstein field, Illinois J. Math. \textbf{6}
(1962), 500-523.
\bibitem[S3]{S3} I. Segal, The complex wave representation of the free
Boson field, \textit{in} ``Topics in functional analysis: Essays dedicated
to M.G. Krein on the occasion of his 70th birthday'' (I. Gohberg and M. Kac,
Eds). Advances in Mathematics Supplementary Studies, Vol. 3, pp. 321-343.
Academic Press, New York, 1978.
\bibitem[Sz]{Sz} R. Sz\"{o}ke, Automorphisms of certain Stein manifolds,
\textit{Math. Z.} \textbf{219} (1995), 357-385.
\bibitem[TW]{TW} L. Thomas and S. Wassell, Semiclassical approximation for
Schr\"{o}dinger operators on a two-sphere at high energy, \textit{J. Math.
Phys.} \textbf{36} (1995), 5480-5505.
\bibitem[U]{U} H. Urakawa, The heat equation on compact Lie group, \textit{%
Osaka J. Math.} \textbf{12} (1975), 285-297.
\bibitem[V]{V} A. Voros, Wentzel-Kramers-Brillouin method in the Bargmann
representation, \textit{Phys. Rev. A} \textbf{40} (1989), 6814-6825.
\bibitem[W]{W} N. Woodhouse, ``Geometric Quantization,'' Oxford Univ.
Press, Oxford, New York, 1980.
\end{thebibliography}
\end{document}