Non associative magnetic translations from parallel transport in projective Hilbert bundles

TL;DR

This paper investigates the non-associativity of magnetic translations in quantum systems, using a geometric and cohomological framework involving projective Hilbert bundles and gerbes, with implications for quantum field theory.

Contribution

It introduces a quantum field theoretic approach to magnetic translations, linking non-associativity to geometric structures like gerbes and differential forms, especially in three-dimensional cases.

Findings

Non-associativity characterized by a 3-cocycle in magnetic translations.

Relation between gerbes, Dixmier-Douady class, and magnetic field divergence.

Lifting translation group actions to second quantization reveals non-associativity.

Abstract

The non-associativity of translations in a quantum system with magnetic field background has received renewed interest in association with topologically trivial gerbes over $\mathbb{R}^n.$ The non-associativity is described by a 3-cocycle of the group $\mathbb{R}^n$ with values in the unit circle $S^1.$ The gerbes over a space $M$ are topologically classified by the Dixmier-Douady class which is an element of $\mathrm{H}^3(M,\mathbb{Z}).$ However, there is a finer description in terms of local differential forms of degrees $d=0,1,2,3$ and the case of the magnetic translations for $n=3$ the 2-form part is the magnetic field $B$ with non zero divergence. In this paper we study a quantum field theoretic construction in terms of $n$-component fermions on a circle. The non associativity arises when trying to lift the translation group action on the 1-particle system to the second quantized system.