Existence of Natural and Conformally Invariant Quantizations of Arbitrary Symbols

DOI

10.1142/S1402925110001057How to use a DOI?

Keywords

Invariant quantization; conformal structure; Cartan connection

Abstract

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection.

In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diffeomorphisms is taken into account. However, considering manifolds endowed with additional structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account.

The existence of such a quantization was proved recently in a series of papers in the context of projective geometry.

Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations.

Copyright

© 2010 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - P. Mathonet
AU  - F. Radoux
PY  - 2021
DA  - 2021/01/07
TI  - Existence of Natural and Conformally Invariant Quantizations of Arbitrary Symbols
JO  - Journal of Nonlinear Mathematical Physics
SP  - 539
EP  - 556
VL  - 17
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925110001057
DO  - 10.1142/S1402925110001057
ID  - Mathonet2021
ER  -