Non-associative magnetic translations from parallel transport in projective Hilbert bundles

Abstract

The non-associativity of translations in a quantum system with magnetic field back-ground has received renewed interest in association with topologically trivial gerbes over Rn.The non-associativity is described by a 3-cocycle of the groupRnwith values inthe unit circleS1.The gerbes over a space Mare topologically classified by the Dixmier–Douady class which is an element of H3(M,Z). However, there is a finer description interms of local differential forms of degreesd=0,1,2,3 and the case of the magnetic translations forn=3 the 2-form part is the magnetic fieldBwith non zero divergence.In this paper we study a quantum field theoretic construction in terms of n-component fermions on a circle.The nonassociativity arises when trying to lift the translation group action on the 1-particle system to the second quantized system.