Gauge invariance of the Aharonov-Bohm effect in a quantum electrodynamics framework
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.
- W. Ehrenberg and R. E. Siday, The refractive index in electron optics and the principles of dynamics, Proc. Phys. Soc. B 62, 8 (1949).
- Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115, 485 (1959).
- R. G. Chambers, Shift of an electron interference pattern by enclosed magnetic flux, Phys. Rev. Lett. 5, 3 (1960).
- G. Matteucci and G. Pozzi, New diffraction experiment on the electrostatic Aharonov-Bohm effect, Phys. Rev. Lett. 54, 2469 (1985).
- P. L. Saldanha, Electrodynamic Aharonov-Bohm effect, Phys. Rev. A 108, 062218 (2023).
- M. Peshkin, The Aharonov-Bohm effect: Why it cannot be eliminated from quantum mechanics, Phys. Rep. 80, 375 (1981).
- P. L. Saldanha, Alternative expression for the electromagnetic Lagrangian, Braz. J. Phys. 46, 316 (2016).
- P. Pearle and A. Rizzi, Quantum-mechanical inclusion of the source in the Aharonov-Bohm effects, Phys. Rev. A 95, 052123 (2017).
- E. Santos and I. Gonzalo, Microscopic theory of the Aharonov-Bohm effect, EPL 45, 418 (1999).
- C. Marletto and V. Vedral, Aharonov-Bohm phase is locally generated like all other quantum phases, Phys. Rev. Lett. 125, 040401 (2020).
- P. L. Saldanha, Local description of the Aharonov–Bohm effect with a quantum electromagnetic field, Found. Phys. 51, 6 (2021a).
- P. L. Saldanha, Aharonov-Casher and shielded Aharonov-Bohm effects with a quantum electromagnetic field, Phys. Rev. A 104, 032219 (2021b).
- Y. Aharonov and A. Casher, Topological quantum effects for neutral particles, Phys. Rev. Lett. 53, 319 (1984).
- K. Kang, Gauge invariance of the local phase in the Aharonov-Bohm interference: Quantum electrodynamic approach, EPL 140, 46001 (2022).
- A. Hayashi, Gauge dependence of the Aharonov-Bohm phase in a quantum electrodynamics framework, Phys. Rev. A 108, 022212 (2023).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995).
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1999).
- L. Vaidman, Role of potentials in the Aharonov-Bohm effect, Phys. Rev. A 86, 040101(R) (2012).
- K.-H. Yang, Gauge transformations and quantum mechanics I. Gauge invariant interpretation of quantum mechanics, Ann. Phys. 101, 62 (1976).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.