Aharonov–Bohm Effect in Phase Space

The Aharonov–Bohm effect is a genuine quantum effect typically characterized by a measurable phase shift in the wave function for a charged particle that encircles an electromagnetic field located in a region inaccessible to the particle. However, this definition is not possible in the majority of the phase-space descriptions since they are based on quasiprobability distributions. In this work, we characterize for the first time the Aharonov–Bohm effect within two different formalisms of quantum mechanics. One of them is the phase-space formalism relying on the canonical commutation relations and Weyl transform. In this framework, the aim is to obtain a consistent description of the quantum system by means of the quasiprobability Wigner function. The other one is the Segal–Bargmann formalism, which we mathematically describe and connect with quantum mechanics by means of the commutation relations of the creation and annihilation operators. After an introduction of both formalisms, we study the Aharonov–Bohm effect within them for two specific cases: one determined by a non-zero electric potential, and another determined by a non-zero magnetic vector potential. Subsequently, we obtain a more general description of the Aharonov–Bohm effect that encompasses the two previous cases and that we prove to be equivalent to the well-known description of this effect in the usual quantum mechanics formalism in configuration space. Finally, we delve into the Aharonov–Bohm effect, employing a density operator to depict states with positional and momentum uncertainty, showcasing its manifestation through distinctive interference patterns in the temporal evolution of Wigner functions under an electric potential, and emphasizing the intrinsically quantum nature of this phenomenon.