Existence of Natural and Conformally Invariant Quantizations of Arbitrary Symbols
- 10.1142/S1402925110001057How to use a DOI?
- Invariant quantization; conformal structure; Cartan connection
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.
- © 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/).
Cite this article
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 -