Quantum mechanics as a spontaneously broken gauge theory on a U(1) gerbe

TL;DR

This paper proposes that quantum mechanics can be viewed as a U(1) gauge theory on a gerbe, where spontaneous symmetry breaking via noncommutative space coordinates acts like a Higgs mechanism.

Contribution

It introduces a novel interpretation of quantum mechanics as a spontaneously broken U(1) gauge theory on a gerbe, linking noncommutativity to symmetry breaking.

Findings

Quantum mechanics has a natural U(1) gerbe structure on configuration space.

Spontaneous symmetry breaking in this gauge theory is caused by noncommutative space coordinates.

Noncommutativity acts as a Higgs mechanism in the quantum-mechanical U(1) gerbe.

Abstract

Any quantum-mechanical system possesses a U(1) gerbe naturally defined on configuration space. Acting on Feynman's kernel exp(iS/h), this U(1) symmetry allows one to arbitrarily pick the origin for the classical action S, on a point-by-point basis on configuration space. This is equivalent to the statement that quantum mechanics is a U(1) gauge theory. Unlike Yang-Mills theories, however, the geometry of this gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since this gauge symmetry is spontaneously broken, an analogue of the Higgs mechanism must be present. We prove that a Heisenberg-like noncommutativity for the space coordinates is responsible for the breaking. This allows to interpret the noncommutativity of space coordinates as a Higgs mechanism on the quantum-mechanical U(1) gerbe.