An explicit formula for the natural and conformally invariant quantization

TL;DR

This paper provides an explicit formula for a natural, conformally invariant quantization of trace-free symbols, building on previous proofs and methods involving Cartan connections and projective invariance.

Contribution

It introduces a concrete formula for conformally invariant quantization, extending prior theoretical results with explicit computational tools.

Findings

Explicit formula for conformally invariant quantization

Method adapts Cartan connection techniques from projective setting

Enhances understanding of natural invariance in geometric quantization

Abstract

In [5], P. Lecomte conjectured the existence of a natural and conformally invariant quantization. In [7], we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in [7] and to tools already used in [8] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in [8].