 Background gauge field

A background gauge field is a nondynamical source for a conserved current, rather than a field integrated over in the path integral. For a Dirac current with source term $+B_{\mu }{j}^{\mu }$, the combined transformations $\psi \mapsto e^{i\alpha }\psi$ and $B_{\mu }\mapsto B_{\mu }+{\partial }_{\mu }\alpha$ preserve the action. Whether this also preserves the generating functional depends on the absence of the corresponding gauge anomaly.