How to work with singular gauge transformations in QFT [closed]

I was recently considering a problem analogous to the Aharonov-Bohm (AB) effect but in the context of quantum field theory. Consider then Dirac electrons minimally coupled to an AB flux and described by the (Euclidean) Lagrangian density

$L= \bar{\psi} \, \gamma^{\mu}(\partial_{\mu}+i \, \phi \, \partial_{\mu}\eta)\psi$

Here $\bar{\psi}$ and $\psi$ are fermionic (grassmanian) fields, $\phi$ is the strength of the flux and $\eta$ is a scalar field. As in the standard AB case, it seems that I can gauge $\eta$ away. However, since $\eta$ contains the flux, it is not a smooth field and the gauge transformation will be "singular". I would then expect to obtain some extra contribution from the path integral measure over non homotopic field configurations. Does anybody know how to treat this problem? I have found a great deal about quantum mechanical path integrals over non simply connected spaces in the literature, but nothing about the QFT extension. Any suggestion, references etc is welcome!!!

closed as too broad by ACuriousMind♦, Kyle Kanos, JamalS, Brandon Enright, FlorisDec 4 '14 at 4:17

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I suggest you put in more equations in here. – Your Majesty Dec 3 '14 at 10:40
• Have you heard of instantons? Since your "AB theory" is the special case of a U(1) gauge theory where the gauge field is (for what reason I'm unsure) pure gauge, that is exactly what we're dealing with. Answering this question would require a complete introduction into gauge theory for someone unfamiliar with it, so I'm voting to close as too broad. – ACuriousMind Dec 3 '14 at 18:00
• Thanks for your answer... I am familiar with instantons in quantum mechanics. Here the problem is related but not exactly the same since this is not the classic example of the instanton in the non-abelian gauge theory. Actually, the problem is closely related to the Fujikawa's treatment of anomalies. Can you at least give me a good reference on the connection between path integral measure and instantons? – b_line Dec 4 '14 at 6:09