Conformally equivariant quantization

TL;DR

This paper develops a conformally equivariant quantization framework for second-order differential operators on pseudo-Riemannian manifolds, establishing canonical isomorphisms and extending to arbitrary conformal classes, with applications to Laplacians and conformal invariants.

Contribution

It introduces a conformally equivariant quantization for quadratic Hamiltonians on pseudo-Riemannian manifolds, extending previous results to arbitrary conformal classes and providing new geometric operators.

Findings

Isomorphism between differential operators and symbols for almost all parameter values

Extension of quantization to all pseudo-Riemannian manifolds based on conformal class

Construction of conformally invariant Laplace operators and Schwarzian derivatives

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 $D^2_{\lambda,\mu}(M)$ of second-order differential operators from $F_\lambda(M)$ to $F_\mu(M)$. If $(M,g)$ is conformally flat with signature $p-q$, then $D^2_{\lambda,\mu}(M)$ 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 $D^2_{\lambda,\mu}(M)$ 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.