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Journal of Mathematical Physics

The propagator in the generalized Aharonov–Bohm effect

C. H. Oh, Chopin Soo, C. H. Lai

Published May 1, 1988• 12 citations
Publisher

Abstract

The nonrelativistic propagator is derived by formulating the generalized Aharonov–Bohm effect, valid for any gauge group in a general multiply connected manifold, as a gauge artifact in the universal covering space. The loop phase factors and the free homotopy propagators arise naturally. An explicit expression for the propagator when there are two solenoids present is given.

DOI
10.1063/1.527957

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The Aharonov–Bohm effect on the noncommutative plane is considered. Developing the path integral formulation of quantum mechanics, we find the propagation amplitude for a particle in a noncommutative space. We show that the corresponding shift in the phase of the particle propagator due to the magnetic field of a thin solenoid receives certain gauge invariant corrections because of the noncommutativity. Evaluating the numerical value for this correction, an upper bound for the noncommutativity parameter is obtained.

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The nonrelativistic propagator is derived by formulating the generalized Aharonov–Bohm effect, valid for any gauge group in a general multiply connected manifold, as a gauge artifact in the universal covering space. The loop phase factors and the free homotopy propagators arise naturally. An explicit expression for the propagator when there are two solenoids present is given.
The Aharonov–Bohm effect in general gauge theories, for particles in gauge-curvature-free regions, is studied using the quantum mechanical propagator in the form of a Feynman sum over paths. Following Schulman [L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)], such paths are divided into their homotopy equivalence classes, and the contributions from each class of paths of the Feynman sum are identified with propagators of a wave equation in the universal covering manifold of M, resulting in a simple form for the propagator on M. A group homomorphism from ℋ, the fundamental homotopy group of M, to the gauge group G is shown to characterize possible Aharonov–Bohm effects, which can be divided into two types, Abelian and non-Abelian, according to whether ℋ*, the image of this homomorphism, is Abelian or non-Abelian. For a non-Abelian Aharonov–Bohm effect, it is necessary that both ℋ and G be non-Abelian. Simple examples illustrate the theory.
The Aharonov-Bohm effect is investigated in the Feynman path-integral formulation of quantum mechanics. We consider an idealized situation with an electron moving in a magnetic-field-free region outside a solenoid whose radius to length ratio is very small. The nonvanishing vector potential term in the Lagrangian is written as an angular-velocity-dependent potential. In order to account for a singularity due to the presence of the solenoid itself, a periodic constraint is imposed on the path integral. The propagator can then be evaluated using the polar-coordinate methods of Peak and Inomata. It is found that the propagator has the general form $K({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})=\ensuremath{\Sigma}{n}^{}{\ensuremath{\chi}}_{n}{K}_{n}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})$ where the sum is taken over all classes of homotopic paths and the ${\ensuremath{\chi}}_{n}$ are a one-dimensional representation of the homotopy group. This is the form of the propagator as conjectured by Schulman.
In the case of more than one solenoid, the computation of the Green’s function for the Aharonov-Bohm effect becomes rather complicated. In this paper, we discuss two possible approaches to this problem. We start with the Schulman’s approach based on the Feynman path integral. Applying this technique, recently having been generalized by several authors to non-Abelian gauge groups, we are able to derive some explicit formulae in the two-solenoid case. On the other hand, the interpretation of the problem in the framework of the theory of self-adjoint extensions allows us to apply the Krein’s formula. But in this case, some open problems still remain.
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.
The Aharonov–Bohm (AB) effect demonstrates that a charged particle can acquire an observable phase shift in a region where the local magnetic field identically vanishes provided the configuration space is multiply connected and the electromagnetic connection exhibits non-trivial holonomy. We present a gauge-covariant physico-mathematical framework that systematically disentangles three regimes: (i) the standard ideal AB effect governed by the holonomy of a flat but globally inexact connection on ΩR=R2∖DR with DR={r≤R}; (ii) the path-integral decomposition into homotopy sectors indexed by the winding number, which is constructed on the universal cover; and (iii) non-topological corrections arising from evanescent wave-function penetration when the solenoid is modeled as a finite radial barrier. The ideal AB phase is taken as the reference topological holonomy, while the finite-barrier extension developed here separates the measured phase, under explicit weak-penetration assumptions, as Δφtotal=ΔφABhol+δφfinite+O(Teff2). Here, ΔφABhol=qΦ/ħ is the standard topological holonomy, whereas δφfinite=O(e−2κℓb) is a small dynamical, microphysics-dependent correction controlled by the barrier height, particle energy, and effective forbidden length ℓb. Within Laskin’s fractional quantum mechanics, we use the fractional extension as a diagnostic test: the topological AB phase is independent of the Lévy index α, whereas the finite-barrier correction acquires an anomalous, α-dependent penetration length. The numerical material provides gauge-covariant diagnostics: a Peierls-substituted split-operator scheme, a sector-resolved Monte Carlo path integral benchmarked against the exact Aharonov–Bohm propagator of Gerry and Singh, a WKB/transfer-matrix finite-barrier estimate tracking δφfinite→0, and Berry-phase computations used as consistency checks.
The Aharonov–Bohm effect on the noncommutative plane is considered. Developing the path integral formulation of quantum mechanics, we find the propagation amplitude for a particle in a noncommutative space. We show that the corresponding shift in the phase of the particle propagator due to the magnetic field of a thin solenoid receives certain gauge invariant corrections because of the noncommutativity. Evaluating the numerical value for this correction, an upper bound for the noncommutativity parameter is obtained.