Abstract
We show that the Aharonov–Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labelling charged superselection sectors. In the present paper, we show that this “topological” quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov–Bohm effect. To confirm these abstract results, we quantize the Dirac field in the presence of a background flat potential and show that the Aharonov–Bohm phase gives an irreducible representation of the fundamental group of the spacetime labelling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on spacetimes with a non-Abelian fundamental group.
Highlights
The analysis of superselection sectors is one of the central results of algebraic quantum field theory
We conclude that the Aharonov–Bohm effect manifests itself in sectors of AB type and, as we shall prove it is due to a presence of a background flat potential
Our final aim is to show that this is not merely an analogy but the real physical Aharonov–Bohm effect: we want to show that the quantization of the free Dirac field in the presence of a background flat potential gives rise to the same type of superselection sectors as those described in the abstract analysis of the previous section
Summary
The analysis of superselection sectors is one of the central results of algebraic quantum field theory. Superselection sectors with topological dimension 1 and “charge 1”, in a sense that shall be clarified later, are in one-to-one correspondence with twisted field nets These are representations of the field net of the Dirac field on flat Hilbert bundles over M with monodromy given by the “second quantization” of the monodromy of Lz (Theorem 4.4);. When no torsion appears in the first homology group of the spacetime, Lz is topologically trivial In this case, we are able to construct a twisted field solving the Dirac equation with interaction term Az, and the representation (1), with n = 1, takes the form (2) (Theorem 4.3). The non-trivial parallel transport carried by the family of localized endomorphisms defined by a sector is interpreted in terms of the background flat potential, shedding some light on the role that the corresponding interaction plays on quantum charges. Some of the results of the present paper have been presented in [39] in a simplified form, in particular by avoiding to discuss Haag duality and without going in details on the structure of twisted Dirac fields
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
