Interference in the Heisenberg Picture of Quantum Field Theory, Local Elements of Reality and Fermions

Abstract: We describe the quantum interference of a single photon in the Mach-Zehnder interferometer using the Heisenberg picture. Our purpose is to show that the description is local just like in the case of the classical electromagnetic field, the only difference being that the electric and the magnetic fields are, in the quantum case, operators (quantum observables). We then consider a single-electron Mach-Zehnder interferometer and explain what the appropriate Heisenberg picture treatment is in this case. Interestingly, the parity superselection rule forces us to treat the electron differently to the photon. A model using only local quantum observables of different fermionic modes, such as the current operator, is nevertheless still viable to describe phase acquisition. We discuss how to extend this local analysis to coupled fermionic and bosonic fields within the same local formalism of quantum electrodynamics as formulated in the Heisenberg picture.