\documentclass[twoside]{article} \usepackage{amsfonts} \pagestyle{myheadings} \markboth{Quantization and irreducible representations }{Paul R. Chernoff } \begin{document} \setcounter{page}{17} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 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)} \vspace{\bigskipamount} \\ % 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 } \maketitle %\usepackage{amsmath,amssymb,} \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} \bibitem{1} A. Avez, {\em Symplectic group, quantum mechanics, and 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}