STM-D-0738Paper2024Published and peer-reviewed

Gauge invariance of the Aharonov-Bohm effect in a quantum electrodynamics framework

Pablo L. Saldanha

Abstract and summary · read the original at the source

In one page

The Aharonov-Bohm effect is the cleanest evidence that the electromagnetic potentials are physical: an electron passing outside a solenoid, in a region where the magnetic field is zero, still shifts its interference fringes by an amount fixed by the magnetic flux its two paths enclose. Pablo Saldanha, at the Federal University of Minas Gerais, asks what the effect looks like when the electromagnetic field is treated quantum-mechanically as well, rather than as a fixed classical backdrop. He solves exactly for the ground-state energy of the quantum field in the presence of the classical charges and currents that make the potentials, then uses first-order perturbation theory to find the extra shift the electron itself causes. That shift depends on which arm of the interferometer the electron takes, and the difference between the two is the Aharonov-Bohm phase — acquired locally, through photon exchange with the solenoid, rather than at a distance. He then changes gauge and shows the closed-loop phase does not move. The phase along a single open path does move, which is why only closed loops are measurable.

Why it matters hereChapter 10 rests on the vector potential being a real and local handle on phase, and the site’s standing rule is to say the potential controls phase rather than that energy is stored in it — precisely because the gauge question always gets asked. Saldanha answers it in the strongest available framework: quantise the field too, change the gauge, and the measurable phase does not budge.

What it claims

01In the magnetic Aharonov-Bohm effect the interference pattern depends on the enclosed magnetic flux even though the particle propagates only where the magnetic field is null and the vector potential is not; in the electric version it depends on the scalar potential difference between the paths with no electric field present; and in the recently proposed electrodynamic version a nonzero phase difference appears even when the paths enclose no magnetic flux and see no scalar potential difference.Section I, opening paragraphs; Section II with Figure 1
Settled physics
02An exact solution can be given for the electromagnetic ground energy arising from the interaction of the quantum electromagnetic field with the classical charges and currents that act as the sources of the potentials, worked in the Lorenz gauge.Abstract; Section III
Published and peer-reviewed
03First-order perturbation theory then gives the extra change in that ground energy caused by a quantum charged particle of known wave function, and the result agrees with earlier second-order treatments in the Lorenz gauge; because the energy depends on which path the particle takes, the Aharonov-Bohm phase difference follows, and it can be read as a local exchange of photons between the particle and the sources rather than as action at a distance.Abstract; Section I, closing paragraph; Section III
Published and peer-reviewed
04Performing a gauge change on the scalar and vector potential operators of the quantum field leaves the Aharonov-Bohm phase difference unchanged for closed paths, and this gauge invariance is demonstrated for the magnetic, electric and electrodynamic versions alike.Section IV; Equation 35 and the closed-loop result following it
Published and peer-reviewed
05The phase accumulated along a single nonclosed path does depend on the chosen gauge, and the value predicted from the effective Hamiltonian differs from the value predicted from the particle’s influence on the electromagnetic ground energy — so a gauge-invariant Aharonov-Bohm phase requires the possible trajectories to form a closed path, and measuring the phase on nonclosed paths should not be possible.Section V, first paragraph; Discussion and conclusion
Published and peer-reviewed
06The intermediate Aharonov-Bohm phase of the electrodynamic effect is itself gauge-invariant, because those configurations are closed paths with time-varying fields: the phase difference can be written as contributions from a magnetic flux and an electric flux through a spacetime surface whose boundaries are the two possible trajectories.Section V, third paragraph
Published and peer-reviewed

Read it · abstract

Abstract

The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, what results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, what reinforces the view that it only can be measured in closed paths.

Pablo L. Saldanha, Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil. Physical Review A 109, 062205 (2024). Preprint: arXiv:2405.08536 [quant-ph].

(Abstract only. The Mach-Zehnder scheme of Figure 1, the quantum-field derivation of Sections III and IV and the discussion of the intermediate phase are at the source — see the rights note above. The preprint is free to read at arXiv.)

The way in

https://doi.org/10.1103/PhysRevA.109.062205Licence checked on the source itself: the arXiv posting 2405.08536 carries the arXiv non-exclusive distribution licence and no Creative Commons statement, and the published version is under the APS default licence. This page therefore carries the summary, the claims and the author’s own abstract, and sends the reader to the source. The preprint is free to read at arXiv.

How to cite it

Pablo L. Saldanha (2024) Gauge invariance of the Aharonov-Bohm effect in a quantum electrodynamics framework. doi:10.1103/PhysRevA.109.062205

Where it sits in the curriculum

Scalar waves and the field behind the fields