Abelian gerbes as a gauge theory of quantum mechanics on phase space

TL;DR

This paper constructs a U(1) gerbe with a connection over phase space, linking gauge theory, geometric structures, and quantum mechanics, and offers a coordinate-free interpretation of the uncertainty principle.

Contribution

It introduces a novel gauge-theoretic framework for quantum mechanics on phase space using Abelian gerbes and connections, connecting geometric structures with quantum principles.

Findings

Gerbe connection encodes quantum and classical structures.

Quantumness linked to gauge choice for the potential A.

Discretization of symplectic area relates to the uncertainty principle.

Abstract

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A,B,H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H=dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant h. U(1) gauge transformations acting on the triple A,B,H are also defined, parametrised either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness vs. classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2i\pi is an integral class in de Rham cohomology is related with the discretisation of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction with generalised complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.