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coherent states, Segal-Bargmann space, spheres, annihilation operators,
Euclidean group
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\begin{document}
\title{Coherent states on spheres}
\author{Brian C. Hall}
\address{Department of Mathematics\\
University of Notre Dame \\
Notre Dame, IN 46556, USA }
\email{bhall{@}nd.edu}
\author{Jeffrey J. Mitchell}
\address{Department of Mathematics\\
Baylor University \\
Waco, TX 76798, USA }
\email{jeffrey\_mitchell{@}baylor.edu}
\date{September 2001}
%\maketitle
\begin{abstract}
We describe a family of coherent states and an associated resolution of the
identity for a quantum particle whose classical configuration space is the $%
d $-dimensional sphere $S^{d}.$ The coherent states are labeled by points in
the associated phase space $T^{\ast }(S^{d}).$ These coherent states are
\textit{not} of Perelomov type but rather are constructed as the
eigenvectors of suitably defined annihilation operators.
We describe as well the Segal--Bargmann representation for the system, the
associated unitary Segal--Bargmann transform, and a natural inversion
formula. Although many of these results are in principle special cases of
the results of B. Hall and M. Stenzel, we give here a substantially
different description based on ideas of T. Thiemann and of K. Kowalski and
J. Rembieli\'{n}ski.
All of these results can be generalized to a system whose configuration
space is an arbitrary compact symmetric space. We focus on the sphere case
in order to be able to carry out the calculations in a self-contained and
explicit way.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction\label{intro.sec}}
In \cite{H1} B. Hall introduced a family of coherent states for a system whose
classical configuration space is the group manifold of a compact Lie group
$G.$ These coherent states are labeled by points in the associated phase
space, namely the cotangent bundle $T^{\ast}(G).$ The coherent states
themselves were originally defined in terms of the \textit{heat kernel} on
$G,$ although we will give a different perspective here. One may identify
\cite{H3} $T^{\ast}(G)$ with the complexified group $G_{\mathbb{C}}$, where,
for example, if $G=\mathrm{SU}(2)$ then $G_{\mathbb{C}}=\mathrm{SL}%
(2;\mathbb{C)}$. The paper \cite{H1} establishes a resolution of the identity
for these coherent states, and equivalently, a unitary Segal--Bargmann
transform. The Segal--Bargmann representation of this system is a certain
Hilbert space of holomorphic functions over the complex group $G_{\mathbb{C}%
}.$ Additional results may be found in \cite{H3,H2,H4} and the survey paper
\cite{H7}.
The coherent states for $G$ (in the form of the associated Segal--Bargmann
transform) have been applied to quantum gravity in \cite{A}, with proposed
generalizations due to Thiemann \cite{T1}. More recently the coherent states
themselves have been used by Thiemann and co-authors \cite{T2} in an attempt
to determine the classical limit of the quantum gravity theory proposed by
Thiemann in \cite{QSD}. In particular, the second entry in \cite{T2}
establishes good phase space localization properties (in several different
senses) for the coherent states associated to the configuration space
$G=\mathrm{SU}(2).$
In another direction, K. K. Wren \cite{Wr}, using a method proposed by N. P.
Landsman \cite{L1}, has shown that the coherent states for $G$ arise naturally
in the canonical quantization of $(1+1)$-dimensional Yang-Mills theory on a
spacetime cylinder. Here $G$ is the structure group of the theory and plays
the role of the reduced configuration space, that is, the space of connections
modulo based gauge transformation over the spatial circle. Wren considers
first the ordinary canonical coherent states for the unreduced
(infinite-dimensional) system. He then shows that after ``projecting'' them
into the gauge-invariant subspace (using a suitable regularization procedure)
these become precisely the generalized coherent states for $G,$ as originally
defined in \cite{H1}. Driver and Hall \cite{DH} elaborate on the results of
Wren and show in particular how the resolution of the identity for the
generalized coherent states can be obtained by projection (using a different
regularization scheme) from the resolution of the identity for the canonical
coherent states. See also \cite{AHS} for an appearance of the generalized
Segal--Bargmann transform in the setting of 2-dimensional Euclidean Yang-Mills theory.
Finally, the paper \cite{H9} shows that the generalized Segal--Bargmann
transform for $G$ can be obtained by means of geometric quantization (see also
\cite[Sect. 3.2]{H7}). This means that the associated coherent states for $G$
are of ``Rawnsley type'' \cite{Ra} and are thus in the spirit of Berezin's
approach to quantization.
We emphasize that the coherent states for $G$ are \textit{not of Perelomov
type} \cite{P}. Instead they are realized as the eigenvectors of certain
non-self-adjoint ``annihilation operators,'' as will be described in detail in
the present paper. (See Section \ref{conclusions.sec} for further comments.)
The coherent states and the resolution of the identity for $G$ ``descend'' is
a straightforward way to the case of a system whose configuration space is a
compact symmetric space $X$ \cite[Sect. 11]{H1}. Compact symmetric spaces are
manifolds of the form $G/K,$ where $G$ is a compact Lie group and $K$ is a
special sort of subgroup, namely, the fixed-point subgroup of an involution.
Examples include the spheres $S^{d}=\mathrm{SO}(d+1)\mathrm{/SO}(d)$ and the
complex projective spaces $\mathbb{C}P^{d}=\mathrm{SU}(d+1)/\mathrm{SU}(d).$
Compact Lie groups themselves can be thought of as symmetric spaces by
identifying $G$ with $\left( G\times G\right) /\Delta(G) $ where $\Delta(G)$
is the ``diagonal'' copy of $G$ inside $G\times G.$
We emphasize that in the case $X=S^{2}$ the 2-sphere is playing the role of
the \textit{configuration space} and thus the coherent states discussed here
are completely different from the spin coherent states in which the 2-sphere
plays the role of the phase space. Whereas the spin coherent states are
labeled by points in $S^{2}$ itself, our coherent states are labeled by points
in the cotangent bundle $T^{\ast}(S^{2}).$
Although the case of compact symmetric spaces can be treated by descent from
the group, it is preferable to give a direct treatment, and such a treatment
was given by Stenzel \cite{St}. Although Stenzel formulates things in terms of
a unitary Segal--Bargmann transform and does not explicitly mention the
coherent states, only a notational change is needed to re-express his results
as a resolution of the identity for the associated coherent states. In
particular Stenzel gives a much better description, in the symmetric space
case, of the measure that one uses to construct the resolution of the
identity. (See also \cite[Sect. 3.4]{H7}.)
More recently, the coherent states for the 2-sphere $S^{2}$ were independently
discovered, from a substantially different point of view, by Kowalski and
Rembieli\'{n}ski \cite{KR1}. These authors were unaware at the time of the
work of Hall and Stenzel. The paper \cite{KR2} then describes the resolution
of the identity for the coherent states on the 2-sphere, showing in a
different and more explicit way that the result of \cite[Thm. 3]{St} holds in
this case. (See Section VII of \cite{KR2} for comments on the relation of
their work to that of Stenzel.)
The purpose of this paper is to describe the coherent states for a compact
symmetric space using the points of view advocated by Thiemann and by Kowalski
and Rembieli\'{n}ski. For the sake of concreteness we concentrate in this
paper on the case $X=S^{d}.$ In \cite{H1} and \cite{St} the coherent states
are defined in terms of the heat kernel on the configuration space, which
takes the place of the Gaussian that enters into the description of the
canonical coherent states. Here by contrast the coherent states are defined to
be the eigenvectors of suitable annihilation operators, and only afterwards
does one discover the role of the heat kernel, in the position wave function
of the coherent states and in the reproducing kernel. The annihilation
operators, meanwhile, are defined by the ``complexifier'' method proposed by
Thiemann, which we will show is equivalent to (a generalization of) the
polar-decomposition method of Kowalski and Rembieli\'{n}ski. We emphasize,
though, that the approach described in this paper gives ultimately the same
results as the heat kernel approach of Hall and Stenzel.
\section{Main results\label{main.sec}}
In this section we briefly summarize the main results of the paper. All
results are explained in greater detail in the subsequent sections. Briefly,
our strategy is this. First, we construct complex-valued functions
$a_{1},\cdots,a_{d+1}$ on the classical phase space that serve to define a
complex structure on phase space. Second, we construct the quantum
counterparts of these functions, operators $A_{1},\cdots,A_{d+1}$ that we
regard as the annihilation operators. Third, we construct simultaneous
eigenvectors for the annihilation operators, which we regard as the coherent
states. Fourth, we construct a resolution of the identity for these coherent states.
We consider a system whose classical configuration space is the $d$%
-dimensional sphere $S^{d}$ of radius $r.$ We consider also the corresponding
phase space, the cotangent bundle $T^{\ast}(S^{d}),$ which we describe as
\[
T^{\ast}(S^{d})=\left\{ \left. (\mathbf{x},\mathbf{p})\in\mathbb{R}%
^{d+1}\times\mathbb{R}^{d+1}\right| x^{2}=r^{2},\,\mathbf{x}\cdot
\mathbf{p}=0\right\} ,
\]
where $\mathbf{p}$ is the \textit{linear} momentum.
In Section \ref{complex.sec} we consider the classical component of Thiemann's
method. To apply this method we must choose a constant $\omega$ with units of
frequency. The classical ``complexifier'' is then defined to be
\textit{kinetic energy function divided by }$\omega,$ which can be expressed
as
\[
\text{complexifier}=\frac{\text{kinetic energy}}{\omega}=\frac{j^{2}}{2m\omega
r^{2}},
\]
where $j^{2}$ is the total angular momentum. We construct complex-valued
functions $a_{1},\cdots,a_{d+1}$ by taking the position functions
$x_{1},\cdots,x_{d+1}$ and applying repeated Poisson brackets with the
complexifier. Specifically,
\begin{align}
a_{k} & =e^{i\left\{ \cdot,\text{complexifier}\right\} }x_{k}\nonumber\\
& =\sum_{n=0}^{\infty}\left( \frac{i}{2m\omega r^{2}}\right) ^{n}\frac
{1}{n!}\underset{n}{\underbrace{\left\{ \cdots\left\{ \left\{ x_{k}%
,j^{2}\right\} ,j^{2}\right\} ,\cdots,j^{2}\right\} }}\label{a.intro}%
\end{align}
If we let $\mathbf{a=}\left( a_{1},\cdots,a_{d+1}\right) $ then the
calculations in Section \ref{complex.sec} will give the following explicit
formula
\begin{equation}
\mathbf{a}=\cosh\left( \frac{j}{m\omega r^{2}}\right) \mathbf{x}%
+i\frac{r^{2}}{j}\sinh\left( \frac{j}{m\omega r^{2}}\right) \mathbf{p}%
.\label{a.intro2}%
\end{equation}
These complex-valued functions satisfy $a_{1}^{2}+\cdots+a_{d+1}^{2}=r^{2}$
and $\left\{ a_{k},a_{l}\right\} =0.$ In the case $d=2$ this agrees with Eq.
(6.1) of \cite{KR1}.
In Section \ref{annihilate.sec} we consider the quantum component of
Thiemann's method. We consider the quantum complexifier,
\[
\text{complexifier}=\frac{\text{kinetic energy}}{\omega}=\frac{J^{2}}{2m\omega
r^{2}},
\]
where $J^{2}$ is the total angular momentum operator. Then if $X_{1}%
,\cdots,X_{d+1}$ denote the position operators we define, by analogy with
(\ref{a.intro}),
\begin{align}
A_{k} & =\exp\left\{ i\frac{1}{i\hbar}\left[ \cdot,\text{ complexifier}%
\right] \right\} X_{k}\nonumber\\
& \sum_{n=0}^{\infty}\left( \frac{1}{2m\omega r^{2}\hbar}\right) ^{n}%
\frac{1}{n!}\underset{n}{\underbrace{\left[ \cdots\left[ \left[ X_{k}%
,J^{2}\right] ,J^{2}\right] ,\cdots,J^{2}\right] }}.\label{qa.intro}%
\end{align}
This may also be written as
\begin{equation}
A_{k}=e^{-J^{2}/2m\omega r^{2}\hbar}X_{k}e^{J^{2}/2m\omega r^{2}\hbar
}.\label{qa.intro2}%
\end{equation}
Equation (30) in Section \ref{annihilate.sec} gives the quantum counterpart of
(\ref{a.intro2}); it is slightly more complicated than (\ref{a.intro2})
because of quantum corrections. The annihilation operators satisfy $A_{1}%
^{2}+\cdots+A_{d+1}^{2}=r^{2}$ and $\left[ A_{k},A_{l}\right] =0.$ Applying
the same procedure in the $\mathbb{R}^{d}$ case produces the usual complex
coordinates on phase space and the usual annihilation operators (Section
\ref{rd.sec}).
One can easily deduce from (\ref{qa.intro2}) a ``polar decomposition'' for the
annihilation operator, given in (\ref{qpolar}) in Section \ref{annihilate.sec}%
. In the case $d=2$ this is essentially the same as what Kowalski and
Rembieli\'{n}ski take as the definition of the annihilation operators. This
shows that Thiemann's complexifier approach is equivalent to the polar
decomposition approach of Kowalski and Rembieli\'{n}ski.
In Section \ref{coherent.sec} we consider the coherent states, defined to be
the simultaneous eigenvectors of the annihilation operators. Using
(\ref{qa.intro2}) we may immediately write down some eigenvectors for the
$A_{k}$'s, namely, the vectors of the form
\begin{equation}
\left| \psi_{\mathbf{a}}\right\rangle =e^{-J^{2}/2m\omega r^{2}\hbar}\left|
\delta_{\mathbf{a}}\right\rangle ,\label{formula}%
\end{equation}
where $\left| \delta_{\mathbf{a}}\right\rangle $ is a simultaneous
eigenvector for the position operators corresponding to a point $\mathbf{a}$
in $S^{d}.$ A key result of Section \ref{coherent.sec} is that one can perform
an analytic continuation with respect to the parameter $\mathbf{a},$ thereby
obtaining coherent states $\left| \psi_{\mathbf{a}}\right\rangle $
corresponding to any point $\mathbf{a}$ in the \textit{complexified} sphere,
$S_{\mathbb{C}}^{d}=\left\{ \mathbf{a}\in\mathbb{C}^{d+1}|\,a^{2}%
=r^{2}\right\} .$ These vectors $\left| \psi_{\mathbf{a}}\right\rangle $ are
normalizable and satisfy
\[
A_{k}\left| \psi_{\mathbf{a}}\right\rangle =a_{k}\left| \psi_{\mathbf{a}%
}\right\rangle ,\quad\mathbf{a}\in S_{\mathbb{C}}^{d}.
\]
Equation (\ref{formula}) shows that the coherent states are expressible in
terms of the heat kernel on the sphere, thus demonstrating that Thiemann's
definition of the coherent states is equivalent to the definition in
\cite{H1,St} in terms of the heat kernel. The reproducing kernel for these
coherent states is also expressed in terms of the heat kernel on the sphere.
In Section \ref{res.sec} we describe a resolution of the identity for these
coherent states. In a suitable coordinate system this takes the form
\begin{equation}
I=\int_{\mathbf{x}\in S^{d}}\int_{\mathbf{p}\cdot\mathbf{x}=0}\left|
\psi_{\mathbf{a}}\right\rangle \left\langle \psi_{\mathbf{a}}\right|
\,\nu\left( 2\tau,2p\right) \left( \frac{\sinh2p}{2p}\right) ^{d-1}%
\,2^{d}d\mathbf{p}\,d\mathbf{x}\label{intro.res}%
\end{equation}
where $\mathbf{a}$ is a function of $\mathbf{x}$ and $\mathbf{p}$ as in
(\ref{a.intro2}). Here $\nu$ is the heat kernel for $d$-dimensional hyperbolic
space and $\tau$ is the dimensionless quantity given by $\tau=\hbar/m\omega
r^{2}.$ Explicit formulas for $\nu$ are found in Section \ref{res.sec}. The
resolution of the identity for the coherent states is obtained by a continuous
deformation of the resolution of the identity for the position eigenvectors.
In Section \ref{sb.sec} we discuss the Segal--Bargmann representation for this
system, namely, the space of holomorphic functions on the complexified sphere
that are square-integrable with respect to the density in (\ref{intro.res}).
We think of the Segal--Bargmann representation as giving a sort of
\textit{phase space wave function} for any state. There is an inversion
formula stating the position wave function can be obtained from the phase
space wave function by integrating out the momentum variables, specifically,
\[
\left\langle \left. \delta_{\mathbf{x}}\right| \phi\right\rangle
=\int_{\mathbf{p}\cdot\mathbf{x}=0}\left\langle \left. \psi_{\mathbf{a}%
(\mathbf{x},\mathbf{p})}\right| \phi\right\rangle \nu(\tau,p)\left(
\frac{\sinh p}{p}\right) ^{d-1}\,d\mathbf{p}%
\]
for any state $\left| \phi\right\rangle .$ Note that whereas the resolution
of the identity involves $\nu(2\tau,2p),$ the inversion formula involves
$\nu(\tau,p)$.
In Section \ref{rd.sec} we show that the complexifier method, when applied to
the $\mathbb{R}^{d}$ case, yields the usual canonical coherent states and
their resolution of the identity. In Section \ref{euclid.sec} we summarize
some of the relevant representation theory for the Euclidean group. Finally,
in Section \ref{conclusions.sec} we compare our construction to other
constructions of coherent states on spheres.
Although all of the results here generalize to arbitrary compact symmetric
spaces $X,$ we concentrate for the sake of explicitness on the case $X=S^{d}.
$ We will describe the general case in a forthcoming paper.
\section{Complex coordinates on phase space\label{complex.sec}}
In this section we define Poisson-commuting complex-valued functions
$a_{1},\cdots,a_{d+1}$ on the classical phase space. In Section
\ref{annihilate.sec} we will introduce the quantum counterparts of these
functions, commuting non-self-adjoint operators $A_{1},\cdots,A_{d+1}$ which
we regard as the annihilation operators for this system. In Section
\ref{coherent.sec} we will consider the coherent states, that is, the
simultaneous eigenvectors of the annihilation operators.
Consider the $d$-dimensional sphere of radius $r$ in $\mathbb{R}^{d+1},$
namely,
\[
S^{d}=\left\{ \left. \mathbf{x}\in\mathbb{R}^{d+1}\right| x_{1}^{2}%
+\cdots+x_{d+1}^{2}=r^{2}\right\} ,
\]
regarded as the \textit{configuration space} for a classical system ($d\geq1
$). Then consider the associated phase space, the cotangent bundle $T^{\ast
}(S^{d}),$ which we think of as
\[
T^{\ast}(S^{d})=\left\{ \left. \left( \mathbf{x},\mathbf{p}\right)
\in\mathbb{R}^{d+1}\times\mathbb{R}^{d+1}\right| \,x^{2}=r^{2},\,\mathbf{x}%
\cdot\mathbf{p}=0\right\} .
\]
Here $p$ is the \textit{linear} momentum, which must be tangent to $S^{d},$
i.e. perpendicular to $\mathbf{x}.$
We also have the \textbf{angular momentum} functions $j_{kl},$ $1\leq k,l\leq
d+1,$ given by
\[
j_{kl}=p_{k}x_{l}-p_{l}x_{k}.
\]
We may think of $j$ as a function on $T^{\ast}(S^{d})$ taking values in the
space of $\left( d+1\right) \times\left( d+1\right) $ skew-symmetric
matrices, that is, in the Lie algebra $\mathrm{so}(d+1)$. Thinking of $j$ as a
matrix we have
\[
\mathbf{j}\left( \mathbf{x},\mathbf{p}\right) =\left| \mathbf{p}%
\right\rangle \left\langle \mathbf{x}\right| -\left| \mathbf{x}\right\rangle
\left\langle \mathbf{p}\right| .
\]
For a particle constrained to the sphere it is possible and convenient to
express everything in terms of $\mathbf{x}$ and $\mathbf{j}$ instead of
$\mathbf{x}$ and $\mathbf{p}.$ We may alternatively describe $T^{\ast}(S^{d})
$ as the set of pairs $\left( \mathbf{x},\mathbf{j}\right) $ in which
$\mathbf{x}$ is a vector in $\mathbb{R}^{d+1},$ $\mathbf{j}$ is a
$(d+1)\times(d+1)$ skew-symmetric matrix, and $\mathbf{x}$ and $\mathbf{j}$
satisfy
\begin{equation}
x^{2}=r^{2}\label{constraint.1}%
\end{equation}
and
\begin{equation}
r^{2}\mathbf{j}=\left| \mathbf{jx}\right\rangle \left\langle \mathbf{x}%
\right| -\left| \mathbf{x}\right\rangle \left\langle \mathbf{jx}\right|
.\label{constraint.2}%
\end{equation}
This last condition says that if we \textit{define} $\mathbf{p}$ to be
$r^{-2}\mathbf{jx},$ then $\mathbf{j}=\left| \mathbf{p}\right\rangle
\left\langle \mathbf{x}\right| -\left| \mathbf{x}\right\rangle \left\langle
\mathbf{p}\right| .$ This relation reflects the constraint to the sphere and
does not hold for a general particle in $\mathbb{R}^{d+1}.$ On $T^{\ast}%
(S^{d})$ we have the relations
\begin{align}
\mathbf{jx} & =r^{2}\mathbf{p}\nonumber\\
\mathbf{jp} & =-p^{2}\mathbf{x}\label{jj2}%
\end{align}
In the case $d=2$ ($S^{2}$ sitting inside $\mathbb{R}^{3}$) a standard vector
identity shows that for any vector $\mathbf{v}\in\mathbb{R}^{3}$,
$\mathbf{jv}=(\mathbf{x}\times\mathbf{p})\times\mathbf{v},$ where $\times$ is
the cross-product and $\mathbf{x}\times\mathbf{p}$ is the usual angular
momentum vector $\mathbf{l}.$ So in the $\mathbb{R}^{3}$ case we may write
$\mathbf{l}\times\mathbf{v}$ instead of $\mathbf{jv}.$
The symplectic structure on $T^{\ast}(S^{d})$ may be characterized by the
Poisson bracket relations
\begin{align}
\left\{ j_{kl},j_{mn}\right\} & =\delta_{kn}j_{lm}+\delta_{lm}j_{kn}%
-\delta_{km}j_{l{}n}-\delta_{l{}n}j_{km}\nonumber\\
\left\{ x_{k},j_{lm}\right\} & =\delta_{kl}x_{m}-\delta_{km}x_{l}%
\label{xj}\\
\left\{ x_{k},x_{l}\right\} & =0.\nonumber
\end{align}
These are the commutation relations for the Euclidean Lie algebra, which is
the semidirect product $\mathrm{e}(d+1)\cong\mathrm{so}(d+1)\ltimes
\mathbb{R}^{d+1}.$
Poisson bracket relations involving $\mathbf{p}$ should be derived from
(\ref{xj}) using the relation $\mathbf{p}=r^{-2}\mathbf{jx}.$ Since the
constraint to the sphere alters the dynamics and hence the Poisson bracket
relations, we will not get the same formulas as in $\mathbb{R}^{d+1}.$ For
example, we have
\begin{equation}
\left\{ x_{k},p_{l}\right\} =\delta_{kl}-\frac{x_{k}x_{l}}{r^{2}%
}.\label{xp.bracket}%
\end{equation}
The complex coordinates on phase space will be constructed from the position
functions $x_{k}$ by means of repeated Poisson brackets with a multiple of the
kinetic energy function. In the sphere case it is convenient to express the
kinetic energy in terms of the total angular momentum $j^{2}$, given by
\begin{equation}
j^{2}=\sum_{k0.$ Next one selects any one point
in the sphere of radius $r$ and considers the ``little group,'' that is, the
stabilizer in $\mathrm{Spin}(d+1)$ of the point. For $r>0$ the little group is
simply $\mathrm{Spin}(d).$ The irreducible representations of $\mathrm{\tilde
{E}}(d+1)$ are then labeled by the value of $r$ and by an irreducible unitary
representation of the little group. In this paper we will consider only the
case in which the representation of the little group is trivial. Nevertheless
the definitions of the annihilation operators and of the coherent states make
sense in general.
Choosing a sphere of radius $r$ amounts to requiring that the operators in
(\ref{qxj}) satisfy
\begin{equation}
X^{2}=r^{2}.\label{qconstraint.1}%
\end{equation}
We will shortly impose an additional condition among the $X$'s and $J$'s that
forces the representation of the little group to be trivial. For now, however,
we will assume only the $\mathrm{e}(d+1)$ relations (\ref{qxj}) and the
condition (\ref{qconstraint.1}).
We define the total angular momentum $J^{2}$ as in the classical case by
\begin{equation}
J^{2}=\sum_{k0$), and that
the formula (\ref{qpolar}) is valid in this generality. However, to compute
$\mathbf{A}$ more explicitly than this we need to further specify the
irreducible representation of $\mathrm{\tilde{E}}(d+1).$ We limit ourselves to
the case in which the representation of the little group $\mathrm{Spin}(d) $
is trivial. This corresponds to a quantum particle on the sphere with no
internal degrees of freedom. In the case the case of $S^{2},$ this corresponds
to taking the ``twist'' (in the notation of Kowalski and Rembieli\'{n}ski) to
be zero. We show in Section \ref{euclid.sec} that the little group acts
trivially if and only if the following relation holds:
\begin{equation}
X^{2}J_{kl}=J_{km}X_{m}X_{l}-J_{lm}X_{m}X_{k}.\label{qconstraint.2}%
\end{equation}
This is the quantum counterpart of the classical constraint
(\ref{constraint.2}).
In computing $\mathbf{A}$ it is convenient to introduce ``momentum'' operators
$P_{k}$ given by
\[
P_{k}:=\frac{J_{kl}X_{l}}{r^{2}}.
\]
These operators are \textit{not} self-adjoint and we have chosen to put the
$J$'s to the left of the $X$'s (because we have put $\mathbf{J}$ to the left
of $\mathbf{X}$ in (\ref{qbracket}) and (\ref{qpolar})). We may re-write
(\ref{qconstraint.2}) in terms of the $P_{k}$'s as
\begin{equation}
J_{kl}=P_{k}X_{l}-P_{l}X_{k}.\label{qconstraint.3}%
\end{equation}
The position and momentum operators satisfy
\begin{equation}
\frac{1}{i\hbar}\left[ X_{k},P_{l}\right] =\delta_{kl}I-\frac{X_{k}X_{l}%
}{r^{2}}.\label{qxp.bracket}%
\end{equation}
(Compare (\ref{xp.bracket}).) We may also compute using (\ref{qxj}) the
quantum counterpart of $\mathbf{x}\cdot\mathbf{p}=0,$ which is really two
relations on the quantum side:
\begin{align*}
\mathbf{P}\cdot\mathbf{X} & =0\\
\mathbf{X}\cdot\mathbf{P} & =i\hbar dI.
\end{align*}
We now write down the formulas that allow us to compute $\mathbf{A}$ in terms
of $\mathbf{X}$ and $\mathbf{P}$:
\begin{align}
\mathbf{JX} & =r^{2}\mathbf{P}\nonumber\\
\mathbf{JP} & =-P^{2}\mathbf{X}+i\hbar(d-1)\mathbf{P}.\label{jj22}%
\end{align}
The first line is simply the definition of $\mathbf{P}$. The second line comes
from (\ref{qconstraint.2}) or (\ref{qconstraint.3}) and is essential to the
explicit calculation of the annihilation operators in terms of $\mathbf{X}$
and $\mathbf{P}.$ Note that there is an additional quantum correction here. To
verify the second line of (\ref{jj22}), write $\mathbf{J} $ in terms of
$\mathbf{P}$ using (\ref{qconstraint.3}) and then use (\ref{qxp.bracket}).
We now treat $\mathbf{J}$ as a $2\times2$ matrix acting on the ``basis''
$\mathbf{X}$ and $\mathbf{P},$ as given in (\ref{jj22}). Since all the entries
of this $2\times2$ matrix commute, we can just treat $P^{2}$ as a scalar and
compute an ordinary $2\times2$ matrix exponential. So effectively we have
\[
\mathbf{J}=\left(
\begin{array}
[c]{cc}%
0 & -P^{2}\\
r^{2} & i\hbar(d-1)
\end{array}
\right) .
\]
One can then compute the exponential of this matrix either by hand or using a
computer algebra program. A calculation shows that $P^{2}=r^{-2}J^{2}$ as in
the classical case. It is convenient to express things in terms of the scalar
operator
\[
J:=\sqrt{J^{2}+\hbar^{2}(d-1)^{2}/4}.
\]
Then after exponentiating $\mathbf{J,}$ (\ref{qpolar}) becomes
\begin{align}
\mathbf{A} & =e^{\hbar/2m\omega r^{2}}\cosh\left( \frac{J}{m\omega r^{2}%
}\right) \mathbf{X}+e^{\hbar/2m\omega r^{2}}\frac{\hbar(d-1)}{2J}\sinh\left(
\frac{J}{m\omega r^{2}}\right) \mathbf{X}\nonumber\\
& +ie^{\hbar/2m\omega r^{2}}\frac{r^{2}}{J}\sinh\left( \frac{J}{m\omega
r^{2}}\right) \mathbf{P}.\label{qa.explicit}%
\end{align}
This expression is similar to the corresponding classical expression
(\ref{a.explicit}), with only the following differences: 1) there is an
overall factor of $\exp(\hbar/2m\omega r^{2}),$ 2) the quantity $j$ in
(\ref{a.explicit}) is replaced by $(J^{2}+\hbar^{2}(d-1)^{2}/4)^{1/2},$ and 3)
there is an extra $\sinh$ term in the coefficient of $\mathbf{X}$ that does
not occur in the classical formula. Note that the above expression formally
coincides with the classical one in the limit $\hbar\rightarrow0.$ In the case
$d=2$ with $r=m\omega=\hbar=1$ (\ref{qa.explicit}) agrees with Eq. (4.16) in
\cite{KR1}.
It is clear from (\ref{qa.explicit}) that the $A_{k}$'s are unbounded
operators, as expected since the $a_{k}$'s are unbounded functions. This means
that the $A_{k}$'s cannot be defined on the whole Hilbert space, but only on
some dense subspace, which should be specified. We take the expression
(\ref{qa.commutator}) as our definition of the annihilation operators. We
first define the $A_{k}$'s on what we will call the ``minimal domain,''
namely, the space of finite linear combinations of spherical harmonics (that
is, of eigenvectors for $J^{2}$). The expression (\ref{qa.commutator}) makes
sense on the minimal domain, since each of the three factors making up $A_{k}$
preserve this space. We consider also a ``maximal domain'' for the $A_{k}$'s,
defined as follows. Given any vector $\left| \phi\right\rangle $ in the
Hilbert space, we expand $\left| \phi\right\rangle $ in a series expansion in
terms of spherical harmonics. Then we apply $A_{k}$ term-by-term, that is, by
formally interchanging $A_{k}$ with the sum. The result will then be a formal
series of spherical harmonics. If this formal series converges in the Hilbert
space then we say that $\left| \phi\right\rangle $ is in the maximal domain
of $A_{k}$ and that the value of $A_{k}\left| \phi\right\rangle $ is the sum
of this series. (It can be shown that the product of $x_{k}$ and a spherical
harmonic of degree $n$ is the sum of a spherical harmonic of degree $n+1$ and
a spherical harmonic of degree $n-1.$ It follows that the degree $l$ term in
the expansion of $A_{k}\left| \phi\right\rangle $ involves only the degree
$n-1$ and degree $n+1$ terms of $\left| \phi\right\rangle .$ So each term in
the formal series for $A_{k}\left| \phi\right\rangle $ can be computed by
means of a finite sum.)
It can be shown that if one starts with the operator $A_{k}$ on its minimal
domain and then takes its closure (in the functional analytic sense) the
result is the operator $A_{k}$ on its maximal domain. Thus if we want $A_{k}$
to be a closed operator there is only one reasonable choice for its domain.
The coherent states will not be finite linear combinations of spherical
harmonics but will be in the maximal domain of all the $A_{k}$'s.
\section{The coherent states\label{coherent.sec}}
We are now ready to introduce the coherent states, which we define to be the
simultaneous eigenvectors of the annihilation operators. These coherent states
are \textit{not of Perelomov type}. Although we have described the quantum
Hilbert space as an irreducible representation of $\mathrm{\tilde{E}}(d+1),$
the coherent states are not obtained from one fixed vector by the action of
$\mathrm{\tilde{E}}(d+1).$ Indeed the only elements of $\mathrm{\tilde{E}%
}(d+1)$ that preserve the set of coherent states are the rotations. See
Section \ref{conclusions.sec} for a comparison of these coherent states to the
generalized Perelomov-type coherent states for $\mathrm{\tilde{E}}(d+1),$ as
constructed either by De Bi\`{e}vre or by Isham and Klauder.
The coherent states will be simultaneous eigenvectors of the annihilation
operators $A_{k}$, and thus can be thought of as the quantum counterparts of a
classical state with definite values for the complex coordinates $a_{k}.$
Note, however, that the coherent states are not eigenvectors of the creation
operators $A_{k}^{\dagger}.$
We use the formula (\ref{qa.commutator}) for the annihilation operators. If we
introduce the dimensionless form of the total angular momentum,
\[
\tilde{J}^{2}=\frac{1}{\hbar^{2}}J^{2},
\]
then this may be expressed as
\begin{equation}
\mathbf{A}=e^{-\tau\tilde{J}^{2}/2}\mathbf{X}e^{\tau\tilde{J}^{2}%
/2},\label{qa.dimensionless}%
\end{equation}
where $\tau$ is the dimensionless quantity given by
\[
\tau=\frac{\hbar}{m\omega r^{2}}.
\]
The parameter $\tau$ is a new feature of the sphere case; no such
dimensionless quantity arises in the $\mathbb{R}^{d}$ case. The significance
of $\tau$ for the coherent states is that it controls the ratio of the spatial
width of the coherent states to the radius of the sphere. Specifically, we
expect the spatial width $\Delta X$ of a coherent state to be approximately
$\sqrt{\hbar/2m\omega},$ at least if this quantity is small compared to $r.$
In that case
\[
\frac{\Delta X}{r}\approx\frac{\sqrt{\hbar/2m\omega}}{r}=\sqrt{\frac{\tau}{2}%
}.
\]
So if $\tau\ll1$ we expect the coherent states to be concentrated in a small
portion of the sphere, and to look, in appropriate coordinates, approximately
Gaussian. This has been proved \cite{T2} for the case of $S^{3}=\mathrm{SU}(2).$
Kowalski and Rembieli\'{n}ski implicitly take $\tau=1$ in their treatment of
the $d=2$ case, since they choose units with $m=r=\hbar=1,$ and since they do
not have the parameter $\omega.$ (See our comments in Section
\ref{complex.sec} about the parameter $\omega.$) This seems a needless loss of
generality, although it is easy to insert $\tau$ in the appropriate places in
their formulas.
We now proceed with the construction of the coherent states. For each
$\mathbf{a}$ in the \textit{real} sphere $S^{d},$ let $\left| \delta
_{\mathbf{a}}\right\rangle $ be the (generalized) position eigenfunction with
$X_{k}\left| \delta_{\mathbf{a}}\right\rangle =a_{k}\left| \delta
_{\mathbf{a}}\right\rangle .$ Since we assume that the little group acts
trivially these position eigenfunctions are unique up to a constant and we may
normalize them so that the action of the rotation group takes $\left|
\delta_{\mathbf{a}}\right\rangle $ to $\left| \delta_{R\mathbf{a}%
}\right\rangle ,$ $R\in\mathrm{SO}(d+1).$ If we let
\begin{equation}
\left| \psi_{\mathbf{a}}\right\rangle =e^{-\tau\tilde{J}^{2}/2}\left|
\delta_{\mathbf{a}}\right\rangle .\label{first.states}%
\end{equation}
then it follows immediately from (\ref{qa.dimensionless}) that $\left|
\psi_{\mathbf{a}}\right\rangle $ is a simultaneous eigenvector for each
$A_{k}$ with eigenvalue $a_{k}.$ Although $\left| \delta_{\mathbf{a}%
}\right\rangle $ is non-normalizable, the smoothing nature of the operator
$\exp(-\tau\tilde{J}^{2}/2)$ guarantees that $\left| \psi_{\mathbf{a}%
}\right\rangle $ has finite norm for all $\mathbf{a}\in S^{d}.$ A key result
of this section is the following proposition, which asserts that we can
analytically continue the coherent states $\left| \psi_{\mathbf{a}%
}\right\rangle $ with respect to $\mathbf{a}$ so as to obtain states labeled
by points $\mathbf{a}$ in the \textit{complex} sphere $S_{\mathbb{C}}^{d}.$
\begin{proposition}
\label{continue.prop}There exists a unique family of states $\left|
\psi_{\mathbf{a}}\right\rangle $ parameterized by $\mathbf{a}\in
S_{\mathbb{C}}^{d} $ such that 1) the states depend holomorphically on
$\mathbf{a},$ and 2) for $\mathbf{a}\in S^{d},$ they agree with the states in
(\ref{first.states}). These are normalizable states and satisfy
\[
A_{k}\left| \psi_{\mathbf{a}}\right\rangle =a_{k}\left| \psi_{\mathbf{a}%
}\right\rangle ,\quad\mathbf{a}\in S_{\mathbb{C}}^{d}.
\]
\end{proposition}
We call these states the \textit{coherent states}. Note that we have then one
coherent state for each point in $S_{\mathbb{C}}^{d},$ that is, one coherent
state for each point in the classical phase space. It can be shown that these
are (up to a constant) the \textit{only} simultaneous eigenvectors of the
annihilation operators. These coherent states are \textit{not} normalized to
be unit vectors. The proof of Proposition \ref{continue.prop} is at the end of
this section.
Note that since the operator $\tilde{J}^{2}$ commutes with rotations, the
action of the rotation subgroup $\mathrm{SO}(d+1)$ of $\mathrm{E}(d+1)$ will
take $\left| \psi_{\mathbf{a}}\right\rangle $ to $\left| \psi_{R\mathbf{a}%
}\right\rangle $ for any $R\in\mathrm{SO}(d+1).$ On sufficiently regular
states we can analytically continue the action of $\mathrm{SO}(d+1)$ to an
action of $\mathrm{SO}(d+1;\mathbb{C}),$ which will take $\left|
\psi_{\mathbf{a}}\right\rangle $ to $\left| \psi_{R\mathbf{a}}\right\rangle $
for any $R\in\mathrm{SO}(d+1;\mathbb{C}).$ Then any coherent state can be
obtained from any other by the action of $\mathrm{SO}(d+1;\mathbb{C}).$ Since,
however, the action of $\mathrm{SO}(d+1;\mathbb{C})$ is neither unitary nor
irreducible, this observation still does not bring the coherent states into
the Perelomov framework.
We can give an explicit formula for the coherent states in the position
representation in terms of the \textit{heat kernel} on $S^{d}.$ The heat
kernel is the function on $S^{d}\times S^{d}$ given by $\rho_{\tau}%
(\mathbf{x},\mathbf{y})=\left\langle \delta_{\mathbf{x}}\right|
e^{-\tau\tilde{J}^{2}/2}\left| \delta_{\mathbf{y}}\right\rangle .$ It can be
shown (see \cite{H1} or the formulas below) the $\rho_{\tau}$ extends
(uniquely) to a holomorphic function on $S_{\mathbb{C}}^{d}\times
S_{\mathbb{C}}^{d},$ also denoted $\rho_{\tau}.$ In terms of the analytically
continued heat kernel the coherent states are given by
\[
\left\langle \delta_{\mathbf{x}}|\psi_{\mathbf{a}}\right\rangle =\rho_{\tau
}(\mathbf{a},\mathbf{x}),\quad\mathbf{a}\in S_{\mathbb{C}}^{d},\,\mathbf{x}\in
S^{d}.
\]
Meanwhile, explicit formulas for the heat kernel may be found, for example, in
\cite{Ta,Ca}. For $\mathbf{x}$ and $\mathbf{y}$ in the real sphere,
$\rho_{\tau}(\mathbf{x},\mathbf{y})$ depends only on the angle $\theta$
between $\mathbf{x}$ and $\mathbf{y},$ where $\theta=\cos^{-1}(\mathbf{x}%
\cdot\mathbf{y}/r^{2}).$ This remains true for $\rho_{\tau}(\mathbf{a}%
,\mathbf{x}),$ with $\mathbf{a}\in S_{\mathbb{C}}^{d}$, except now
$\theta=\cos^{-1}(\mathbf{a}\cdot\mathbf{x}/r^{2})$ is complex-valued. Of
course the inverse cosine function is multiple-valued, but because the heat
kernel is an even, $2\pi$-periodic function of $\theta,$ it does not matter
which value of $\theta$ we use, provided that $\cos\theta=\mathbf{a}%
\cdot\mathbf{x}/r^{2}.$
We now record the formulas, writing $\rho_{\tau}^{d}$ to indicate the
dependence on the dimension. For $d=1,2,3$ we have
\begin{align*}
\rho_{\tau}^{1}(\mathbf{a},\mathbf{x}) & =(2\pi\tau)^{-1/2}\sum_{n=-\infty
}^{\infty}e^{-(\theta-2\pi n)^{2}/2\tau}\\
\rho_{\tau}^{2}(\mathbf{a},\mathbf{x}) & =(2\pi\tau)^{-1}e^{\tau/8}\frac
{1}{\sqrt{\pi\tau}}\int_{\theta}^{\pi}\frac{1}{\sqrt{\cos\theta-\cos\phi}}%
\sum_{n=-\infty}^{\infty}(-1)^{n}(\phi-2\pi n)e^{-(\phi-2\pi n)^{2}/2\tau
}\,d\phi\\
\rho_{\tau}^{3}(\mathbf{a},\mathbf{x}) & =(2\pi\tau)^{-3/2}e^{\tau/2}\frac
{1}{\sin\theta}\sum_{n=-\infty}^{\infty}(\theta-2\pi n)e^{-(\theta-2\pi
n)^{2}/2\tau}.
\end{align*}
In the formula for $\rho_{\tau}^{2}$ we may without loss of generality take
$\theta$ with $0\leq\operatorname{Re}\theta\leq\pi,$ in which case the
integral is to be interpreted as a contour integral in the strip
$0\leq\operatorname{Re}\phi\leq\pi.$ The relatively simple formula for the
heat kernel on $S^{3}=\mathrm{SU}(2)$ allows for detailed calculations for the
coherent states in this case, as carried out in \cite{T2}. To find the formula
in higher dimensions we use the inductive formula
\[
\rho_{\tau}^{d+2}(\mathbf{a},\mathbf{x})=-e^{d\tau/2}\frac{1}{2\pi\sin\theta
}\frac{d}{d\theta}\rho_{\tau}^{d}(\mathbf{a},\mathbf{x}).
\]
There is also an expression for the heat kernel in terms of spherical
harmonics. For example, when $d=2$ we have
\begin{equation}
\rho_{\tau}^{2}(\mathbf{a},\mathbf{x})=\sum_{l=0}^{\infty}e^{-\tau
l(l+1)/2}\sqrt{2l+1}P_{l}(\cos\theta),\label{coherent.expand}%
\end{equation}
where the $P_{l}$ is the Legendre polynomial of degree $l.$ (Compare Eq. (5.3)
of \cite{KR1}.) The earlier expression for $\rho_{\tau}^{2}$ is a sort of
Poisson-summed version of (\ref{coherent.expand})--see \cite{Ca}.
We also consider the reproducing kernel, defined by
\begin{equation}
R_{\tau}(\mathbf{a,b})=\left\langle \left. \psi_{\mathbf{b}}\right|
\psi_{\mathbf{a}}\right\rangle ,\quad\mathbf{a},\mathbf{b}\in S_{\mathbb{C}%
}^{d}.\label{repro.def}%
\end{equation}
In terms of the analytically continued heat kernel the reproducing kernel is
given by
\[
R_{\tau}(\mathbf{a,b})=\rho_{2\tau}(\mathbf{a},\mathbf{\bar{b}}),\quad
\mathbf{a},\mathbf{b}\in S_{\mathbb{C}}^{d}.
\]
Note that $R_{\tau}(\mathbf{a},\mathbf{b})$ depends holomorphically on
$\mathbf{a}$ and anti-holomorphically on $\mathbf{b}.$
\textit{Proof of Proposition \ref{continue.prop}}. There are two ways to prove
this proposition. The simplest way is to use the expression for $\psi
_{\mathbf{a}}$ in terms of the heat kernel $\rho_{\tau}$ and the explicit
formulas above for $\rho_{\tau}.$ It is easily seen that $\rho_{\tau}$ extends
to an entire holomorphic function of $\theta.$ Thus the expression
$\left\langle \delta_{\mathbf{x}}|\psi_{\mathbf{a}}\right\rangle =\rho_{\tau
}(\mathbf{a},\mathbf{x})$ makes sense for any $\mathbf{a}$ in $S_{\mathbb{C}%
}^{d},$ with $\cos\theta$ and thus also $\theta$ taking complex values. It is
not hard to see that the $\left| \psi_{\mathbf{a}}\right\rangle $, so
defined, is in the (maximal) domain of the annihilation operators and that it
depends holomorphically on $\mathbf{a}\in S_{\mathbb{C}}^{d}.$ Since
$A_{k}\left| \psi_{\mathbf{a}}\right\rangle =a_{k}\left| \psi_{\mathbf{a}%
}\right\rangle $ for $\mathbf{a}\in S^{d},$ an analytic continuation argument
will show that this equation remains true for all $\mathbf{a}\in
S_{\mathbb{C}}^{d}.$ Alternatively we may use the expansion of the coherent
states in terms of spherical harmonics as in (\ref{coherent.expand}) and show
that this expression can be analytically continued term-by-term in
$\mathbf{a}$. (Compare Section 4 of \cite{H1}.) $\square$
\section{The resolution of the identity\label{res.sec}}
We now choose a coordinate system in which $r=1$ and $m\omega=1$. This amounts
to using the normalized position $\mathbf{x}/r$ and normalized momentum
$\mathbf{p}/m\omega r.$ Since these choices set our position and momentum
scales we cannot also take $\hbar=1.$ Note that the dimensionless parameter
$\tau=\hbar/m\omega r^{2}$ equals $\hbar$ in such a coordinate system. We now
write $\left| \psi_{\mathbf{x},\mathbf{p}}\right\rangle $ for $\left|
\psi_{\mathbf{a}(\mathbf{x},\mathbf{p})}\right\rangle $.
\begin{theorem}
\label{res.thm}The coherent states have a resolution of the identity of the
form
\begin{equation}
I=\int_{\mathbf{x}\in S^{d}}\int_{\mathbf{p}\cdot\mathbf{x}=0}\left|
\psi_{\mathbf{x},\mathbf{p}}\right\rangle \left\langle \psi_{\mathbf{x}%
,\mathbf{p}}\right| \,\nu\left( 2\tau,2p\right) \left( \frac{\sinh2p}%
{2p}\right) ^{d-1}\,2^{d}d\mathbf{p}\,d\mathbf{x}\label{res.int}%
\end{equation}
where $\nu(s,R)$ is the solution to the differential equation
\[
\frac{d\nu(s,R)}{ds}=\frac{1}{2}\left[ \frac{\partial^{2}\nu}{\partial R^{2}%
}-(d-1)\frac{\cosh R}{\sinh R}\frac{\partial\nu}{\partial R}\right]
\]
subject to the initial condition
\[
\lim_{s\downarrow0}\,c_{d}\int_{0}^{\infty}f(R)\nu(s,R)(\sinh R)^{d-1}%
\,dR=f(0)
\]
for all continuous functions $f$ on $\left[ 0,\infty\right) $ with at most
exponential growth at infinity. Here $c_{d}$ is the volume of the unit sphere
in $\mathbb{R}^{d}$, $d\mathbf{x}$ is the surface area measure on $S^{d},$ and
$\tau$ is the dimensionless quantity $\tau=\hbar/m\omega r^{2}.$
\end{theorem}
The operator on the right side of the equation for $\nu$ is just the radial
part of the Laplacian for $d$-dimensional hyperbolic space \cite[Sect.
5.7]{Da}. This means that $\nu(s,R)$ is the heat kernel for hyperbolic space,
that is, the fundamental solution of the heat equation. Hyperbolic space is
the non-compact, negatively curved ``dual'' of the compact, positively curved
symmetric space $S^{d}.$ Note that the function $\nu$ is evaluated at ``time''
$2\tau$ and radius $2p.$ The inversion formula for the Segal--Bargmann
transform, described in Section \ref{sb.sec}, involves the function $\nu$
evaluated at time $\tau$ and radius $p.$
The resolution of the identity for the coherent states will be obtained by
continuously varying the dimensionless parameter $\tau.$ When $\tau=0$ the
coherent states are simply the position eigenvectors, which have a resolution
of the identity because the position operators are self-adjoint. We will show
that the function $\nu$ satisfies the correct differential equation to make
the resolution of the identity remain true as we move to non-zero $\tau.$
Theorem \ref{res.thm} is a special case of Theorem 3 of \cite{St}, written out
more explicitly and re-stated in terms of coherent states instead of the
Segal--Bargmann transform. However we give below a self-contained and
elementary proof. The case $d=2$ is also described (with a different proof) in
\cite{KR2}. Since $S^{3}=\mathrm{SU}(2),$ the $d=3$ case belongs to the group
case, which is found in \cite{H1}. See also Section 4.4 of the second entry in
\cite{T2} for another proof in the $\mathrm{SU}(2)$ case.
We report here the formulas for the function $\nu(s,R),$ which may be found,
for example, in \cite[Sect. 5.7]{Da} or \cite[Eq. (8.73)]{Ca}. Writing
$\nu_{d}(s,R)$ to make explicit the dependence on the dimension we have
\begin{align*}
\nu_{1}(s,R) & =(2\pi s)^{-1/2}\,e^{-R^{2}/2s}\\
\nu_{2}(s,R) & =(2\pi s)^{-1}e^{-s/8}\frac{1}{\sqrt{\pi s}}\int_{R}^{\infty
}\frac{\rho e^{-\rho^{2}/2s}}{\left( \cosh\rho-\cosh R\right) ^{1/2}}%
\,d\rho\\
\nu_{3}(s,R) & =(2\pi s)^{-3/2}e^{-s/2}\frac{R}{\sinh R}e^{-R^{2}/2s}.
\end{align*}
and the recursion relation
\[
\nu_{d+2}(s,R)=-\frac{e^{-ds/2}}{2\pi\sinh R}\frac{\partial}{\partial R}%
\nu_{d}(s,R).
\]
Estimates on the behavior as $R\rightarrow\infty$ of $\nu$ may be found in
\cite[Sect 5.7]{Da} and in \cite{DM}. Note the similarities between the
formulas for $\nu$ and the formulas for the heat kernel $\rho_{\tau}$ on the sphere.
Some care must be taken in the interpretation of the integral (\ref{res.int}).
Even in the $\mathbb{R}^{d}$ case this integral is not absolutely convergent
in the operator norm sense. Rather the appropriate sense of convergence is the
weak sense. This means that for all vectors $\phi_{1},\phi_{2}$ in the Hilbert
space we have
\begin{equation}
\left\langle \phi_{1}|\phi_{2}\right\rangle =\int_{\mathbf{x}\in S^{d}}%
\int_{\mathbf{p}\cdot\mathbf{x}=0}\left\langle \phi_{1}|\psi_{\mathbf{x}%
,\mathbf{p}}\right\rangle \left\langle \psi_{\mathbf{x},\mathbf{p}}|\phi
_{2}\right\rangle \,\nu\left( 2\tau,2p\right) \left( \frac{\sinh2p}%
{2p}\right) ^{d-1}\,2^{d}d\mathbf{p}\,d\mathbf{x}\label{res.int2}%
\end{equation}
where the integral (\ref{res.int2}) is an absolutely convergent complex-valued
integral. This of course is formally equivalent to (\ref{res.int}). We will
prove Theorem \ref{res.thm} at first without worrying about convergence or
other similar technicalities. Then at the end we will explain how such matters
can be dealt with.
\textit{Proof of Theorem \ref{res.thm}}. We now write the coherent states as
$\left| \psi_{\mathbf{a}}^{\tau}\right\rangle $ to emphasize the dependence
on the dimensionless quantity $\tau=\hbar/m\omega r^{2}.$ We regard the
coherent states $\left| \psi_{\mathbf{a}}^{\tau}\right\rangle $ as living in
some fixed ($\tau$-independent) Hilbert space (for example, $L^{2}(S^{d})$)
and given heuristically by
\begin{equation}
\left| \psi_{\mathbf{a}}^{\tau}\right\rangle =e^{-\tau\tilde{J}^{2}/2}\left|
\delta_{\mathbf{a}}\right\rangle ,\quad\mathbf{a}\in S_{\mathbb{C}}%
^{d},\label{psia}%
\end{equation}
where $\left| \delta_{\mathbf{a}}\right\rangle $ is a position eigenvector.
Our strategy is essentially the one proposed by T. Thiemann in \cite[Sect.
2.3]{T1}. We begin with two lemmas that allow us to carry out this strategy
explicitly in this situation. The proofs of these lemmas are given at the end
of the proof of Theorem \ref{res.thm}.
\begin{lemma}
\label{volume.lem}The measure
\[
\left( \frac{\sinh2p}{2p}\right) ^{d-1}\,2^{d}d\mathbf{p}\,d\mathbf{x}%
\]
is invariant under the action of $\mathrm{SO}(d+1;\mathbb{C})$ on
$S_{\mathbb{C}}^{d}\cong T^{\ast}(S^{d}).$
\end{lemma}
\begin{lemma}
\label{laplace.lem}Let $J_{\mathbf{a}}^{2}$ and $J_{\mathbf{\bar{a}}}^{2}$
denote the differential operators on $S_{\mathbb{C}}^{d}$ given by
\begin{align*}
J_{\mathbf{a}}^{2} & =-\sum_{k0,$ in which case the little group is $\mathrm{Spin}(d).$ Fixing a
value for $r$ amounts to assuming that the operators $X_{k}$ satisfy $\sum
X_{k}^{2}=r^{2}.$
The purpose of this section is to show that the little group acts trivially if
and if the following relation holds for all $k$ and $l$:
\begin{equation}
X^{2}J_{kl}=J_{km}X_{m}X_{l}-J_{lm}X_{m}X_{k}\label{q.constr}%
\end{equation}
(sum convention). This is equivalent to the relation
\begin{equation}
J_{kl}=P_{k}X_{l}-P_{l}X_{k}\label{qjpx}%
\end{equation}
where by definition $P_{k}=r^{-2}J_{kl}X_{l}.$
Note that (\ref{q.constr}) is the quantum counterpart of the constraint to the
sphere (\ref{constraint.2}) and therefore representations of $\mathrm{\tilde
{E}}(d+1)$ satisfying it are closest to the classical motion on a sphere.
Nevertheless, other representations are of interest, and describe a quantum
particle on a sphere with internal degrees of freedom. We will consider the
general case in a future work.
Suppose now that (\ref{q.constr}) holds. We wish to show that this implies
that the representation of the little group is trivial. So we consider the
space of generalized eigenvectors for the operators $X_{k}$ satisfying
\begin{align}
X_{k}\left| \psi\right\rangle & =0,\quad k=1,\cdots,d\nonumber\\
X_{d+1}\left| \psi\right\rangle & =r.\label{x.evector}%
\end{align}
This is the space on which the little group acts, where the Lie algebra of the
little group is given by the operators $J_{kl}$ with $1\leq k,l\leq d.$ But
now if (\ref{q.constr}) holds then for $k,l\leq d$ we have
\[
r^{2}J_{kl}\left| \psi\right\rangle =0
\]
since in that case $X_{k}\left| \psi\right\rangle =X_{l}\left|
\psi\right\rangle =0.$ This shows (for $r>0$) that if (\ref{q.constr}) holds
then the little group acts trivially.
Consider now the quantity
\begin{equation}
W_{kl}:=X^{2}J_{kl}-J_{km}X_{m}X_{l}+J_{lm}X_{m}X_{k},\label{wkl.def}%
\end{equation}
which satisfies $W_{lk}=-W_{kl}.$ The condition (\ref{q.constr}) is equivalent
to $W_{kl}=0.$ Consider also the quantity
\begin{equation}
C:=\sum_{k0.$ Thus these constructions are inequivalent to the one considered in
this paper. Furthermore these constructions do not generalize to higher-rank
symmetric spaces \cite{Sz2}.
Besides these, there have been to our knowledge two other proposed
constructions of coherent states on spheres (and other homogeneous spaces).
These constructions, inequivalent to \cite{H1,St} and to each other, are those
of S. De Bi\`{e}vre \cite{De} and of C. Isham and J. Klauder \cite{IK}. Both
\cite{De} and \cite{IK} are based on extensions of the Perelomov approach, in
that their coherent states are all obtained from one fixed vector $\psi_{0}$
by the action of the Euclidean group. As explained in those papers, the
ordinary Perelomov approach is not applicable in this case, because the
irreducible representations of the Euclidean group are not square-integrable.
Non-square-integrability means that the usual Perelomov-type integral, which
should be a multiple of the identity operator, is in this case divergent.
De Bi\`{e}vre's approach to this problem is to apply to the fiducial vector
$\psi_{0}$ only a part of the Euclidean group. We describe just the simplest
case of \cite{De}. (This special case was worked out independently in a more
elementary way by Torresani \cite{To}.) Specifically, if we work in
$L^{2}(S^{d})$ then start with a basic coherent state $\psi_{0}$ such that a)
$\psi_{0}$ is invariant under rotations about the north pole $\mathbf{n}$ and
b) $\psi_{0}$ is supported in the northern half-sphere with a certain rate of
decay at the equator. One may think of $\psi_{0}$ being concentrated near the
north pole and approximating a state whose position is at the north pole and
whose momentum is zero. The other coherent states are then of the form
\[
\exp(i\mathbf{k}\cdot\mathbf{x})\psi_{0}(R^{-1}\mathbf{x})
\]
where we consider only pairs $(\mathbf{k},R)$ satisfying $\mathbf{k}\cdot
R\mathbf{n}=0.$ This last restriction is crucial. Since $\psi_{0}$ is
invariant under rotations about the north pole, the coherent states are
determined by the values of $\mathbf{k}$ and $R\mathbf{n}$ and are thus
labeled by points in the cotangent bundle of $S^{d}.$ The resolution of the
identity for these coherent states follows from the general procedure in
\cite{De} but can also be proved in this case by an elementary application of
the Plancherel formula. The condition that $\psi_{0}$ be supported in the
northern half-sphere is crucial to the proof.
It is clear that the coherent states considered in this paper are quite
different from those in \cite{De}. First, De Bi\`{e}vre's coherent states do
not depend holomorphically on the parameters. Second, each coherent state must
be supported in a half-sphere, hence cannot be real-analytic in the space
variable. Third, there does not seem to be any preferred choice for $\psi_{0}$
in \cite{De}, whereas for the coherent states considered here the only choice
one has to make is the value of the parameter $\omega.$
Meanwhile, Isham and Klauder use a different method of working around the
non-square-integrability of the irreducible representations of $\mathrm{E}%
(d+1).$ The use reducible representations, corresponding to integration over
some small range $\left[ r,r+\varepsilon\right] $ of radii. This allows for
a family of coherent states invariant under the full Euclidean group and
allows a more general basic coherent state $\psi_{0},$ without any support
conditions. On the other hand it seems natural to get back to an irreducible
representation by letting $\varepsilon$ tend to zero, so that the particle is
constrained to a sphere with one fixed radius. Unfortunately, although the
representation itself does behave well under this limit (becoming irreducible)
the coherent states themselves do not have a limit as $\varepsilon$ tends to
zero. (See the remarks at the bottom of the first column on p.609 in
\cite{IK}.) This seems to be a drawback of this approach.
Finally, we mention that in the group case, the coherent states described in
this paper can be obtained by means of geometric quantization, as shown in
\cite{H9}. This means that in the group case the coherent states are of
``Rawnsley type'' \cite{Ra}. However, this result does not carry over to the
case of general compact symmetric spaces. In particular the results of
\cite{H9} apply only to those spheres that are also groups, namely,
$S^{1}=\mathrm{U}(1)$ and $S^{3}=\mathrm{SU}(2).$
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\end{document}
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