Non-Symplectic Deformations of Geometric Quantisation

TL;DR

This paper introduces geometric pseudo-quantisation, a deformation of traditional geometric quantisation that relaxes curvature conditions, and explores its equations of motion and examples in symplectic geometry.

Contribution

It presents the concept of geometric pseudo-quantisation arising from non-symplectic deformations, expanding the framework of geometric quantisation.

Findings

Derived equations of motion for pseudo-quantisations

Computed pseudo-quantisation for examples in symplectic geometry

Described the deformed canonical commutator

Abstract

We introduce the notion of geometric pseudo-quantisation based on geometric quantisation with a weakened curvature condition. We show how such a structure arises naturally from simple deformations of the symplectic structure and pullbacks of prequantum data by non-symplectic diffeomorphisms. Our main result is deriving the equations of motion for some simple pseudo-quantisations. We also compute the pseudo-quantisation of several simple examples from symplectic and almost-symplectic geometry, as well as the general form of the resulting deformed canonical commutator.