Conformally equivariant quantization
| dc.creator | Duval, C. | |
| dc.creator | Ovsienko, V. | |
| dc.date | 1998-01-27 | |
| dc.date | 2000-02-25 | |
| dc.date.accessioned | 2026-07-07T05:23:41Z | |
| dc.date.available | 2026-07-07T05:23:41Z | |
| dc.description | Let $(M,g)$ be a pseudo-Riemannian manifold and $F_λ(M)$ the space of densities of degree $λ$ on $M$. We study the space $D^2_{λ,μ}(M)$ of second-order differential operators from $F_λ(M)$ to $F_μ(M)$. If $(M,g)$ is conformally flat with signature $p-q$, then $D^2_{λ,μ}(M)$ is viewed as a module over the group of conformal transformations of $M$. We prove that, for almost all values of $μ-λ$, the $O(p+1,q+1)$-modules $D^2_{λ,μ}(M)$ 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. | |
| dc.description | 32 pages, LaTeX, completely rewritten version, new results added | |
| dc.identifier | https://arxiv.org/abs/math/9801122 | |
| dc.identifier | http://arxiv.org/abs/math/9801122 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76539 | |
| dc.subject | Differential Geometry | |
| dc.subject | Quantum Algebra | |
| dc.subject | 58B99;17B56;17B65;47E05 | |
| dc.title | Conformally equivariant quantization | |
| dc.type | text |