Gerbes and Heisenberg's Uncertainty Principle

TL;DR

This paper demonstrates that gerbes with connections on classical phase space can encode quantum uncertainty principles, linking geometric structures to fundamental quantum mechanics via path integrals.

Contribution

It introduces a novel geometric framework using gerbes to represent quantum phase and derives the uncertainty principle from a quantisation condition on a 3-form field.

Findings

Gerbes with connections can be defined on classical phase space.

A quantisation condition on the 3-form field strength is equivalent to Heisenberg's uncertainty principle.

The U(1)-valued phase of Feynman path integrals can be described using Cech 2-cocycles.

Abstract

We prove that a gerbe with a connection can be defined on classical phase space, taking the U(1)-valued phase of certain Feynman path integrals as Cech 2-cocycles. A quantisation condition on the corresponding 3-form field strength is proved to be equivalent to Heisenberg's uncertainty principle.