Conformally equivariant quantization

Abstract

Let $(M,g)$ be a pseudo-Riemannian manifold and $F_{\lambda }(M)$ the space of densities of degree $\lambda$ on $M$. We study the space $D_{\lambda ,\mu }^2(M)$ of second-order differential operators from $F_{\lambda }(M)$ to $F_{\mu }(M)$. If $(M,g)$ is conformally flat with signature $p-q$, then $D_{\lambda ,\mu }^2(M)$ is viewed as a module over the group of conformal transformations of $M$. We prove that, for almost all values of $\mu -\lambda$, the $O(p+1,q+1)$-modules $D_{\lambda ,\mu }^2(M)$ and the space of symbols (i.e., of second-order polynomials on $T^{\ast }M$) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudo-Riemannian manifolds and depends only on the conformal class $[g]$ of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on half-densities, as well as the well-known Yamabe Laplacian. We also recover in this framework the multi-dimensional Schwarzian derivative of conformal transformations.

32 pages, LaTeX, completely rewritten version, new results added