Path integrals with a periodic constraint: The Aharonov–Bohm effect

Abstract

The Aharonov–Bohm effect is formulated in terms of a constrained path integral. The path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths. The interference terms are also calculated for an infinitely thin solenoid, which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment.

Green functions of the Dirac equation with magnetic-solenoid field

“…Thus, one can see that the Green functions for the nonrelativistic particle is irregular at rϭ0 when the contact interaction is attractive. We note that for the limiting case Bϭ0 ͑the uniform magnetic field is absent͒, S l (ϩ)(Ϫ /2) (x,xЈ) coincide with the known expression for the spinless particle, [15][16][17] which is natural in the case of a repulsive contact interaction. S l (ϩ)( /2) (x,xЈ) for Bϭ0 coincide with the corresponding expressions obtained in Ref.…”

“…The propagator of the spinless particle was found in Refs. 15,16, and 17 as a sum of partial propagators corresponding to homotopically different paths in the covering space of the physical background. The nonrelativistic propagator of the spin-1/2 particle in the AB field for a particular value of the self-adjoint extension parameter was discussed in Ref.…”

Aharonov-Bohm scattering of a localized wave packet: Analysis of the forward direction

“…We shall recast the propagator (10) in a way that is more convenient for performing the integration (12) (see also [2,7,9] for details). Let…”

“…It is also possible to evaluate the propagator using path-integral techniques over multiplyconnected spaces (see, for example, [9]). 1 We remark here that the propagator and, consequently, also ψ(r, t) depend upon the choice of the particular gauge for A(r) only through a phase factor.…”

On the diffusion of alpha-helical proteins in solvents

“…(2) deals with paths confined to a circular topology. The paths can be classified topologically and characterized by winding numbers [6][7][8][9][10][11]] n = 0, ±1, ±2, . .…”