Quantization of (volume-preserving) actions on R^d

TL;DR

This paper develops a framework for quantizing volume-preserving group actions on R^d, linking formal representations with unitary operators and establishing existence and rigidity results.

Contribution

It introduces a novel approach to quantizing (volume-preserving) actions on R^d using formal power series and Fourier integral operators, with new existence and rigidity theorems.

Findings

Quantizations can be realized as unitary representations via Fourier integral operators.

Existence of quantizations is established under certain conditions.

Rigidity results show the uniqueness of these quantizations.

Abstract

We associate a space of (formal) representations on the space of h-formal power series with coefficients in the space of smooth functions on Rd (which we call quantizations) with an action of a group on Rd by smooth diffeomorphisms. If the action is further volume preserving, these quantizations can be realized as unitary representations on square summable functions on Rd by bounded h-dependent Fourier integral operators, the formal case corresponding to the asymptotics in the limit h going to zero. We construct DGAs controlling these quantizations and prove existence and rigidity results for them.