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\markboth{Quantization and irreducible representations }{Paul R. Chernoff }
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Mathematical Physics and Quantum Field Theory, \newline
Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 17--22.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Quantization and irreducible representations of
infinite-dimensional transformation groups and Lie algebras
\thanks{ {\em Mathematics Subject Classifications:} 22E65, 17B15.
\hfil\break\indent
{\em Key words:} Infinite dimensional Lie algebras, representations.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Published July 12, 2000. } }
\date{}
\author{ Paul R. Chernoff }
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\begin{abstract}
We present an analytic version of a theorem of Burnside and apply it to the
study of irreducible representations of doubly-transitive groups and Lie
algebras. Application to the Dirac quantization problem is given.
\end{abstract}
\section{Group actions}
Let $G$ be a group and let $M$ be a set. An {\em action} of $G$ on $M$ is a
map from $G$ to the permutations of $M$ such that, for $g,h \in G$ and $x
\in M$
$$\displaylines{
g \cdot (h \cdot x) = (gh) \cdot x \cr
e \cdot x = x\quad (e = \mbox{ identity}).
}$$
\paragraph{Example.} Let $G$ be a group and let $H$ be a subgroup of $G$. Let $M
= G/H$, the space of left cosets. Define
\[
g \cdot (aH) = (ga)H,
\]
the obvious ``translation'' action of $G$ on the coset space. (If $H =
\{e\}$ we have $M = G$ and the action is simply $G$ acting on itself by left
translation.)
Often $M$ has additional structure; for example, $M$ may be a manifold.
Then we want $G$ to act by diffeomorphisms (smooth mappings) of $M$. Or, if
$M$ carries a smooth measure, we may want $G$ to act via measure-preserving
diffeomorphisms.
\section{Burnside's theorem}
This is basically a nineteenth century theorem. See \cite{2}.
\paragraph{\bf Theorem.} {\em
Let $G$ be a discrete group acting on a discrete set $M$. Suppose that the
action of $G$ is doubly transitive: that is, if $x,y,x',y'$ are in $M$,
there exists $g \in G$ with $g \cdot x = x'$, $g \cdot y = y'$.
Then the natural unitary representation $U$ of $G$ on $l^2(M)$ is
(essentially) irreducible. That is, $U$ is irreducible if $|M| = \infty$,
while if $|M| < \infty$ there are just two irreducible components,
viz.~(scalars) and $l^2(M) \ominus$ (scalars), the orthogonal complement.}
\paragraph{Note.} The ``natural representation'' $U$ is just given by left
translation:
\[
(U_af)(x) = f(a^{-1} \cdot x),
\]
$a \in G$, $x \in M$, $f \in l^2(M)$.
\paragraph{Proof}. Let $T: l^2(M) \rightarrow l^2(M)$ be an intertwining operator
for $U$; that is, for all $a \in G$, $TU_a = U_aT$. Since $M$ is discrete,
the operator $T$ has a matrix kernel $K$ such that, for $f \in l^2$,
\[
(Tf)(x) = \sum_{y \in M} K(x,y)f(y).
\]
The intertwining condition readily implies the identity $K(a\cdot x,a\cdot
y) = K(x,y)$, which means that $K$ is constant on the $G$-orbits in $M
\times M$. But there are just two such orbits, namely the diagonal $\Delta$
and its complement.
Hence the space of intertwining operators is at most two-dimensional,
generated by the identity $I$ and projection onto the scalars $P$. But the
operator $P$ is $0$ if $|M|$ is infinite, so in the latter case the
representation is irreducible.\hfill$\Box$
\section{Main results}
Our main results are analogues of Burnside's theorem, but the analytic
details are more involved. For example, we use the Schwartz kernel theorem
to study the intertwining operators.
\subsection*{Transitive and doubly-transitive actions of Lie algebras}
Le $M$ be a smooth manifold, $\mbox{Vect}(M)$ the Lie algebra of smooth
vector fields on $M$, and $\frak{G}$ any Lie algebra. An action of
$\frak{G}$ on $M$ is just a homomorphism
\[
A: \frak{G} \rightarrow \mbox{Vect}(M)
\]
$X \in \frak{G} \mapsto A(X)$, a vector field on $M$ which is linear and
such that $A([X,Y]) = [A(X),A(Y)]$. (This is simply the ``infinitesimal
analogue'' of a group action.)
\paragraph{Definition.} 1. $\frak{G}$ acts transitively on $M$ provided
that, for each point $p \in M$, $\{A(X)_p: X \in \frak{G}\} = T_p(M)$,
the tangent space of $M$ at the point $p$.
2. $\frak{G}$ acts doubly-transitively on $M$ provided $\frak{G}$ acts
transitively on $M \times M{\backslash}\Delta$. That is, given $p \ne q \in
M$, $v \in T_p(M)$, $w \in T_q(M)$, there exists $X \in \frak{G}$ with
$A(X)_p = v$ and $A(X)_q = w$.
3. $n$-fold transitivity may be similarly defined.
\subsection*{Examples}
A. Let $(M,\mu)$ be a smooth manifold with a smooth measure $\mu$. Let
$\frak{G} = \mbox{Vect}_{\mu}(M)$, the Lie algebra of divergence-free
vector fields on $M$. If $\dim M \ge 2$, $\frak{G}$ acts $n$-fold
transitively on $M$ for all $n \ge 1$. (This is easy to see.)
\medskip
B. Let $\omega$ be a closed $2$-form on $M$, so that $(M,\omega)$ is a
symplectic manifold (= a ``phase space''). From $\omega$ we define a
Poisson bracket structure on $C^{\infty}(M)$:
\[
\{f,g\} = \omega(\xi_f,\xi_g)
\]
where $\xi_f$ is the Hamiltonian vector field corresponding to $f \in
C^{\infty}(M)$.
For example, take $M = \Bbb{R}^{2n}$ with canonical coordinates $q$'s and
$p$'s;
\begin{eqnarray*}
\omega &= &\sum_i dq_i \wedge dp_i \\
\{f,g\} &= &\sum_i \left( \frac {\partial f}{\partial q_i}
\frac {\partial g}{\partial p_i} - \frac {\partial f}{\partial p_i} \frac
{\partial g}{\partial q_i} \right) \\
\xi_f &= &\sum_i \left( \frac {\partial f}{\partial q_i} \frac
{\partial}{\partial p_i} - \frac {\partial f}{\partial p_i} \frac
{\partial}{\partial q_i} \right).
\end{eqnarray*}
$\xi_f$ may be viewed as a vector field, or as a first-order skew-symmetric
differential operator.
Then $A: f \mapsto \xi_f$ is $m$-fold transitive for all $m \ge 1$. (This
is via an easy ``patching'' argument using partitions of unity.)
\subsection*{Cocycles for a Lie algebra action}
Let $A: \frak{G} \rightarrow \mbox{Vect}(M)$ be an action of $\frak{G}$ on
$M$, by divergence-free vector fields for simplicity. Consider a $0$th
order perturbation of $A$:
\[
B(X) = A(X) + i\rho(X).
\]
Here $\rho(X) \in C^{\infty}(M)$ depends linearly on $X \in \frak{G}$.
$B(X)$ is a skew-symmetric first-order differential operator. We want
the mapping $X \mapsto B(X)$ to be a Lie algebra homomorphism:
\[
B([X,Y]) = [B(X),B(Y)].
\]
This leads to the following {\em cocycle identity}:
\[
\rho([X,Y]) = A(X) \cdot \rho(Y) - A(Y) \cdot \rho(X).
\]
\paragraph{Example. } (L.~van~Hove, 1951). Let $M = \Bbb{R}^{2n}$, ${\cal F} =
C^{\infty}(M) =$ the Poisson bracket Lie algebra over $M$. Let $A(f) =
\xi_f$, the Hamiltonian vector field corresponding to $f \in {\cal F}$. Set
\[
B(f) = \xi_f + i\theta(f)
\]
where $\theta: {\cal F} \rightarrow {\cal F}$ is linear and $\theta$
satisfies the cocycle identity (expressed in terms of Poisson brackets):
\[
\theta(\{f,g\}) = \{f,\theta(g)\} + \{\theta(f),g\}.
\]
But this just says that $\theta$ is a derivation of the Lie algebra ${\cal
F}$.
Thinking of $B(f)$, acting on $L^2(M)$, as a quantum operator corresponding
to the classical observable (= function) $f$, we impose the non-triviality
condition
\[
\theta(1) = 1
\]
so that $B(1) = I =$ the identity operator on $L^2(M)$.
The derivations of $C^{\infty}(M,\omega) = {\cal F}$ have been completely
determined for general symplectic manifolds $(M,\omega)$. For $M =
\Bbb{R}^{2n}$, van~Hove discovered the formula
\[
\theta(f) = f - \sum_{i=1}^n p_i\partial f/\partial p_i.
\]
Then, as required, $\theta(1) = 1$. Moreover $\theta$ is unique up to
an inner derivation.
\subsection*{Irreducibility theorems}
These are analogues of Burnside's theorem and theorems of Mackey and Shoda.
\paragraph{\bf Theorem 1.} {\em Let $(M,\mu)$ be a connected manifold with a smooth
measure $\mu$. Let $A: \frak{G} \rightarrow \mbox{Vect}_{\mu}(M)$ be an
action of the Lie algebra $\frak{G}$ via divergence-free skew-adjoint vector
fields on $M$. Assume that the action is doubly transitive.
Let $\rho$ be a cocycle for the action $A$. The representation $B$ is
defined by
\[
B(X) = A(X) + i\rho(X).
\]
Also, assume that the dimension of $M$ is $\ge 2$ or that $M = S^1$, so that
$M \times M{\backslash}\Delta$ is connected.
Then the representation $B$ on $L^2(M,\mu)$ has at most two irreducible
components.}
\paragraph{Sketch of Proof.} Consider $T: L^2 \rightarrow L^2$ an intertwining
operator for $B$ with kernel $K$ a distribution on $M \times M$. (Here we
use the Schwartz kernel theorem.) The intertwining condition leads to a
family of partial differential equations satisfied by the kernel $K$.
Moreover this family is elliptic. Hence $K$ is smooth off the diagonal
$\Delta$, and the double transitivity of $A$ may be used to show that there
exists at most a two-dimensional family of intertwining
operators.\hfill$\Box$
\paragraph{Theorem 2.} {\em Let $(M,\mu)$ be a connected manifold with smooth
measure $\mu$. Let the action $A$ of $\frak{G}$ and the cocycle $\rho$
satisfy the hypotheses of Theorem~1. In particular, $A$ is assumed to be
doubly transitive.
Also assume that the cocycle $\rho$ satisfies the following condition:
Given a point $p \in M$ denote by $\rho_p$ the character of the stabilizer
algebra $\frak{G}_p = \{X \in \frak{G}: A(X)_p = 0\}$, determined by
restricting the character $\rho$ to $\frak {G}_p$.
Finally, assume that there are two points $p,q \in M$ such that $\rho_p$ and
$\rho_q$ restrict to distinct characters of $\frak{G}_p \cap \frak{G}_q$.}
\paragraph{Conclusion.} The representation $B = A + \rho$ is irreducible on
$L^2(M,\mu)$.
\medskip
(N.B.~In this theorem we do not need to assume that $M \times
M{\backslash}\Delta$ is connected. So the theorem holds for $M = \Bbb{R}$,
e.g.)
\paragraph{Proof.} Theorem~2 is basically an application of Theorem~1. The
condition on the character $\rho$ is used to show that the intertwining
kernel $K(x,y)$ must vanish off the diagonal, from which it follows that the
intertwining operators are just scalar multiples of the identity
$I$.\hfill$\Box$
\subsection*{Applications}
1. Van~Hove's prequantization representations are irreducible:
Here $M = \Bbb{R}^{2n}$, $\frak{G} = {\cal F} =
C_{\mbox{comp}}^{\infty}(\Bbb{R}^n,\omega)$, the Poisson bracket Lie algebra;
$A(f) = \xi_f =$ The Hamiltonian vector field generated by $f \in {\cal F}$;
$\rho = \lambda\theta$, where $\lambda$ is a real non-$0$ scalar;
$\theta(f) =$ van Hove's derivation $= f - \sum_{i=1}^n p_i\theta f/\partial
p_i$.
${\cal F}_q \cap {\cal F}_b = \{f \in {\cal F}: \nabla f$ (or $\xi_f$)
vanishes at the points $a$ and $b\}$.
If $f \in {\cal F}_a \cap {\cal F}_b$, $\rho_a(f) = \lambda f(a)$ and
$\rho_b(f) = \lambda f(b)$. But $f(a)$ and $f(b)$ can be anything at all,
so $\rho_a \ne \rho_b$. Therefore Theorem~2 applies to show that the
representation on $L^2(\Bbb{R}^n)$ given by
\[
B_{\lambda}(f) = \xi_f + i\lambda\rho(f)
\]
is irreducible.
2. The above generalizes to the case of any non-compact symplectic manifold
$(M,\omega)$ with $\omega$ exact.
3. For compact $(M,\omega)$, A.~Avez defines
\[
\theta(f) = \mbox{mean value of $f$ on $M$.}
\]
Then $B_{\lambda}(f) = \xi_f + i\lambda\theta(f)$ has two irreducible
components, namely the scalars and their orthogonal complement.
4. The prequantization representations of Souriau, Kostant, and Urwin are
all (essentially) irreducible.
\begin{thebibliography}{0}
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Anosov's systems}, in ``Dynamical Systems and Microphysics'' (A.~Blaquiere
et al., Eds.), pp.~301--324, Springer--Verlag, New York, 1980.
\bibitem{2} W. S. Burnside, ``Theory of Groups of Finite Order'', 2nd
ed., p.~249, Dover, New York, 1911 (reprint 1955).
\bibitem{3} P. R. Chernoff, {\em Mathematical obstructions to
quantization}, Hadronic J. {\bf 4} (1981), 879--898.
\bibitem{4} P. R. Chernoff, {\em Irreducible representations of
infinite-dimensional transformation groups and Lie algebras}, I.,
J.~Functional Analysis {\bf 130} (1995), 255--282.
\bibitem{5} A. A. Kirillov, {\em Unitary representations of the group of
diffeomorphisms and of some of its subgroups}, Selecta Math. Soviet {\bf 1}
(1981), 351--372.
\bibitem{6} J. M. Souriau, {\em Quantization g\'eom\'etrique},
Comm. Math. Phys. {\bf 1} (1966), 374--398.
\bibitem{7} L. van Hove, {\em Sur certaines repr\'esentations unitaires
d'un groupe infini de transformations}, Acad. Roy. Belg. Cl. Sci. M\'em.
Collect. 80(2) {\bf 29} (1951), 1--102.
\end{thebibliography}
\noindent{\sc Paul R. Chernoff} \\
Department of Mathematics \\
University of California \\
Berkeley, CA 94720, USA \\
e-mail: chernoff@math.berkeley.edu
\end{document}