Conformally equivariant quantization

Details Conformally equivariant quantization Conformally equivariant quantization

1998-01-27

arxiv.org/abs/math/9801122v3

Mathematics

Differential Geometry

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

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Conformally equivariant quantization does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Conformally equivariant quantization, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Conformally equivariant quantization will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-162689