Non associative magnetic translations from parallel transport in projective Hilbert bundles

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.

arXiv admin note: substantial text overlap with arXiv:1905.01944