Conformally equivariant quantum Hamiltonians (Q1612517)

The space \( {\mathcal D}^{2}_{\lambda \mu }(M)\) of second-order differential operators from the space of densities of degree \(\lambda\) to the space of densities of degree \(\mu\) on a pseudo-Riemannian manifold \(M\) is studied. If \(M\) is conformally flat with signature \(p-q\) then \( {\mathcal D}^{2}_{\lambda \mu }(M)\) is viewed as a module over the group of conformal transformations of \(M\). Then, for almost all values of \(\mu - \lambda\) the \(O(p+1,q+1)\)-modules \( {\mathcal D}^{2}_{\lambda \mu }(M)\) and the space of symbols are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. It is also proved that this quantization map extends to arbitrary pseudo-Riemannian manifolds and depends only on the conformal class 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 a Yamabe Laplacian.