  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 real line
or a unit circle. The non associativity arises when trying to lift the
translation group action on the 1-particle system to the second quantized
system.
