Conformally equivariant quantum Hamiltonians

Abstract.

Let (M, g) be a pseudo-Riemannian manifold and $\cal{F}_\lambda(M)$ the space of densities of degree $\lambda$ on M. We study the space $\cal{D}^2_{\lambda,\mu}(M)$ of second-order differential operators from $\cal{F}_\lambda(M)$ to $\cal{F}_\mu(M)$. If (M, g) is conformally flat with signature p - q, then $\cal{D}^2_{\lambda,\mu}(M)$ is viewed as a module over the group of conformal transformations of M. It turns out that, for almost all values of $\mu-\lambda$, the O(p+1, q+1)-modules $\cal{D}^2_{\lambda,\mu}(M)$ and the space of symbols (i.e., of second-order polynomials on $T^*M$) 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.<\P>