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arXiv:1110.1843 · Modern Physics Letters B

NEW APPLICATION OF DECOMPOSITION OF U (1) GAUGE POTENTIAL: AHARONOV–BOHM EFFECT AND ANDERSON–HIGGS MECHANISM

JIAN-FENG LI, YU JIANG, WEI-MIN SUN, HONG-SHI ZONG, FAN WANG

Published Jun 27, 2012• 7 citations
arXivHTMLPDF

Abstract

In this paper, we study the Aharonov–Bohm (A–B) effect and Anderson–Higgs mechanism in Ginzburg–Landau model of superconductors from the perspective of the decomposition of U (1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A⊥ in A–B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A‖, which reflects the Anderson–Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A‖, while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring.

DOI
10.1142/s0217984912501242

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In this paper, we study the Aharonov–Bohm (A–B) effect and Anderson–Higgs mechanism in Ginzburg–Landau model of superconductors from the perspective of the decomposition of U (1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A⊥ in A–B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A‖, which reflects the Anderson–Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A‖, while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring.
The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts; the one is the so-called transverse part represented by a divergenceless vector and the other the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when we use the Helmholtz theorem are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, we choose here the Aharonov-Bohm effect. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. We show that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is uniquely determined by the magnetic field and does not alter by a gauge transformation, on the other hand, the longitudinal part depends on the choice of special gauge. We show that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.
In the original setting of the Aharonov-Bohm, the gauge invariant physical longitudinal mode of the vector potential, which is expressed by the gauge invariant physical current, gives the desired contribution to the Aharonov-Bohm effect. While the scalar mode of the vector potential, which changes under the gauge transformation so that it is the unphysical mode, give no contribution to the Aharonov-Bohm effect. Then Aharonov-Bohm effect really occurs by the physical longitudinal mode in the original Aharonov-Bohm’s setting. In the setting of Tonomura et al., where the magnet is shielded with the superconducting material, not only the magnetic field but also the longitudinal mode of the vector potential become massive by the Meissner effect. Then not only the magnetic field but also the physical longitudinal mode does not come out to the region where the electron travels. In such setting, only the scalar mode of the vector potential exists in the region where the electron travels, but there is no contribution to the Aharonov-Bohm effect from that mode.
The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials AμA_\muAμ​, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials gμνg_{\mu\nu}gμν​ may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.
Abstract The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts: 1) the so‐called transverse part represented by a divergenceless vector; and 2) the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when the Helmholtz theorem is used are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, the Aharonov‐Bohm effect is chosen here. As for the Aharonov‐Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z‐axis. It is shown that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is determined uniquely by the magnetic field and does not alter by a gauge transformation. On the other hand, the longitudinal part depends on the choice of special gauge. It is shown that the Aharonov‐Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.
Previous time-dependent Ginzburg–Landau simulations demonstrated that Abrikosov vortex dynamics in superconductors can be influenced by Aharonov–Bohm-type curl-less vector potentials even in simply connected geometries, without closed-loop trajectories enclosing magnetic flux [A. M. Gulian et al., Phys. Lett. A 382 (2018) 1058]. The origin of this effect has remained unexplained. Here we investigate further nonequilibrium effects in which the relation between a curl-less vector potential and an evolving quantum state produces observable consequences. During transient nonequilibrium state evolution, collective quantum systems may develop measurable responses to curl-less gauge fields despite a simply connected final equilibrium configuration with no persistent current. Using superconducting condensates, we formulate protocols based on the gauge-invariant coupling between quantum phase and the curl-less vector potential, analyze their dynamics using time-dependent Ginzburg–Landau theory and interpret rapid phase formation within the Kibble–Zurek picture. The predicted transient supercurrents and voltage signals provide a physical interpretation of nonequilibrium sensitivity to curl-less gauge fields and motivate experimental tests.