A purely geometrical Aharonov-Bohm effect

Abstract

We present an application of the affine covariant integral quantization (ACIQ) (Adv. Oper. Theory, 5, 2020; Adv. Oper. Theory, 7, 2022) to quantum mechanics on the punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), which comprises translations, rotations, and dilations of the plane. Due to the topology of the punctured plane, our quantization procedure gives rise to an affine vector potential. This potential can be interpreted as the Aharonov-Bohm (AB) gauge field produced by an infinite solenoid. This observation supports a reinterpretation of the AB effect: it emerges from the topological constraint imposed by the impenetrable coil rather than from an externally applied classical gauge field. In addition to this gauge structure, ACIQ also generates a repulsive, centrifugal-like scalar potential, a feature already encountered when applying ACIQ to motion on the half-line, whose phase space is the open half-plane. These results provide a new perspective on the AB effect, highlighting the central roles of topology and symmetry in quantum mechanics.