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Non-Symplectic Deformations of Geometric Quantisation

Kerr Maxwell

TL;DR

This work develops geometric pseudo-quantisation by relaxing the curvature constraint in geometric quantisation, either via a curvature form $\Omega$ or through pullbacks by non-symplectic maps. It derives equations of motion for simple pseudo-quantisations using the BKS pairing and demonstrates how deformed commutators arise in concrete settings, including pullback constructions and folded/b^m-symplectic contexts. By analyzing polarisations and admissible observables, it identifies when pseudo-quantisations yield well-defined dynamics and when flows are obstructed, revealing how nonconstant curvature terms modify the Schrödinger evolution. Overall, the framework provides a geometric route to generalized uncertainty principles and noncanonical commutators, linking mathematical deformations of symplectic structure to physically motivated quantum-modification scenarios.

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.

Non-Symplectic Deformations of Geometric Quantisation

TL;DR

This work develops geometric pseudo-quantisation by relaxing the curvature constraint in geometric quantisation, either via a curvature form or through pullbacks by non-symplectic maps. It derives equations of motion for simple pseudo-quantisations using the BKS pairing and demonstrates how deformed commutators arise in concrete settings, including pullback constructions and folded/b^m-symplectic contexts. By analyzing polarisations and admissible observables, it identifies when pseudo-quantisations yield well-defined dynamics and when flows are obstructed, revealing how nonconstant curvature terms modify the Schrödinger evolution. Overall, the framework provides a geometric route to generalized uncertainty principles and noncanonical commutators, linking mathematical deformations of symplectic structure to physically motivated quantum-modification scenarios.

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.

Paper Structure

This paper contains 14 sections, 6 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Pseudo-prequantisation
  3. Standard Prequantisation
  4. Pseudo-Prequantisation
  5. Pseudo-Prequantisation via Pullback
  6. Pullback of Prequantisation
  7. Local Coordinates
  8. Pseudo-Quantisation
  9. Polarisation
  10. Polarisation Preservation
  11. Standard Case
  12. Polarised Scaled Case
  13. General Scaled Case
  14. BKS Pairings of Pseudo-quantisations

Key Result

Theorem 1.1

Let $(M,\omega)$ be pseudo-quantised with connection form $\Theta = (1+f)\theta$ and polarisation $P$ where $f$ is a monomial of order $n$. Let $\psi\in\breve{\mathcal{H}}_P$ be an element of the pseudo-quantum Hilbert space. Then, if $f$ is polarised, the flow of the quadratic momentum observable i

Figures (2)

  • Figure 1:
  • Figure 2: The integral points of the moment map for the folded$S^2$ (left) and standard$S^2$ (right).

Theorems & Definitions (25)

  • Theorem 1.1: Main Result
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8: Quantisation in a folded neighbourhood
  • Theorem 2.9
  • ...and 15 more