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Physical Review

Significance of Electromagnetic Potentials in the Quantum Theory

Yakir Aharonov, David Böhm

Published Aug 1, 1959• 6,995 citations
Publisher

Abstract

In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.

DOI
10.1103/physrev.115.485

Related papers

Further Considerations on Electromagnetic Potentials in the Quantum Theory

Yakir Aharonov, David Böhm

In this article, we discuss in further detail the significance of potentials in the quantum theory, and in so doing, we answer a number of arguments that have been raised against the conclusions of our first paper on the same subject. We then proceed to extend our treatment to include the sources of potentials quantum-mechanically, and we show that when this is done, the same results are obtained as those of our first paper, in which the potential was taken to be a specified function of space and time. In this way, we not only answer certain additional criticisms that have been made of the original treatment, but we also bring out more clearly the importance of the potential in the expression of the local character of the interaction of charged particles and the electromagnetic field.

Physical Review•Aug 15, 1961•403 citations

Self-Field Fluctuations and the Measurability of Electromagnetic Potentials in Quantum Mechanics

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Some earlier considerations of Mitler on the significance of electromagnetic potentials in quantum theory which attempt to take properly into account the fluctuations of the vacuum are re-examined in terms of the conventional formalism of quantum field theory. It is shown that, contrary to the conclusions reached by Mitler, the effects of self-field fluctuations are indeed small, in substantial agreement with remarks of Aharonov and Bohm on this subject.

Physical Review•Jan 11, 1965•0 citations

The quantum effects of electromagnetic fluxes

Silviu Olariu, I. Iovitzu Popescu

This paper is a review of the problem of the observable action of enclosed electromagnetic fluxes on the quantum-mechanical state of charged particles, known as the Aharonov-Bohm effect. The authors first describe the quantum effects of the fluxes in the quasiclassical approximation, and discuss their relation with basic quantum-mechanical principles. Then they examine the influence of modeling assumptions on the predicted effects of enclosed fluxes. They analyze the experiments demonstrating the reality of the quantum effects of electromagnetic and gravitational fluxes, and finally discuss the physical significance of these quantum effects, comparing the current interpretation, based on the concept of a nonintegrable phase factor, with alternative approaches.

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The electric Aharonov-Bohm effect

Ricardo Weder

The seminal paper of Aharonov and Bohm [Phys. Rev. 115, 485 (1959)]10.1103/PhysRev.115.485 is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, in spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been studied extensively. The experimental issues were settled by the remarkable experiments of Tonomura et al. [Phys. Rev. Lett. 48, 1443 (1982); Phys. Rev. Lett. 56, 792 (1986)] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez et al. [Phys. Rev. Lett. 99, 210401 (2007)]10.1103/PhysRevLett.99.210401 that shows that the results of the Tonomura et al. experiments cannot be explained by the action of a force. The theoretical issues were settled by Ballesteros and Weder [Commun. Math. Phys. 285, 345 (2009)10.1007/s00220-008-0579-1; J. Math. Phys. 50, 122108 (2009)10.1063/1.3266176; Commun. Math. Phys. 303, 175 (2011)]10.1007/s00220-010-1166-9 who rigorously proved that quantum mechanics predicts the experimental results of Tonomura et al. and of Caprez et al. The electric Aharonov-Bohm effect has been much less studied. Actually, its existence, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an ansatz for the solution to the Schrödinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will stimulate the experimental research on the electric Aharonov-Bohm effect.

arXiv:1006.1385•Journal of Mathematical Physics•May 1, 2011•16 citations

A study of Aharonov-Bohm effect: from quantum electrodynamics to one particle quantum mechanics

Benliang Li, Daniel William Hewak, Qi Jie Wang

In this article, we start our discussion on the Aharonov-Bohm effect then raise a plausible interpretation within the quantum electrodynamics (QED) framework. We provide a quantum treatment of the source of the electromagnetic potential and argue that the underlying mechanism in AB effect can be viewed as interactions between electrons described by QED theory where the interactions are mediated by virtue photons. On further analysis, we claim that the framework of one particle quantum mechanics (OPQM) can be shown, in general, as a mathematically approximated model which can be reformulated from QED theory. In addition, the classical Maxwell equations are derived from QED scattering process while both classical electromagnetic fields and potentials serve as mathematical tools that are constructed to approximate the interactions among elementary particles described by QED physics, i.e. neither classical fields nor potentials represent any real entities of nature. At the conclusion of this article, we make a few remarks on the hypnosis of the existence of magnetic monopoles.

•Jul 12, 2016•2 citations

Measuring Electromagnetic Vector Potential via Weak Value

Arun Kumar Pati

Electromagnetic vector potential has physical significance in quantum mechanics as revealed by the Aharonov-Bohm effect for charged particles. However, till date it is thought that we cannot measure the vector potential directly as this is not a gauge invariant quantity. Contrary to this belief, here we show that one can indeed measure the electromagnetic vector potential using the notion of weak value. We show that it is simply the difference between the weak value of the canonical momentum of a charged particle in the presence and absence of magnetic field. This suggests that the vector potential is not only a physical entity but can be measured directly in the experiment.

arXiv:1602.04140•Feb 12, 2016•2 citations
In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.
In this article, we discuss in further detail the significance of potentials in the quantum theory, and in so doing, we answer a number of arguments that have been raised against the conclusions of our first paper on the same subject. We then proceed to extend our treatment to include the sources of potentials quantum-mechanically, and we show that when this is done, the same results are obtained as those of our first paper, in which the potential was taken to be a specified function of space and time. In this way, we not only answer certain additional criticisms that have been made of the original treatment, but we also bring out more clearly the importance of the potential in the expression of the local character of the interaction of charged particles and the electromagnetic field.
This paper is a review of the problem of the observable action of enclosed electromagnetic fluxes on the quantum-mechanical state of charged particles, known as the Aharonov-Bohm effect. The authors first describe the quantum effects of the fluxes in the quasiclassical approximation, and discuss their relation with basic quantum-mechanical principles. Then they examine the influence of modeling assumptions on the predicted effects of enclosed fluxes. They analyze the experiments demonstrating the reality of the quantum effects of electromagnetic and gravitational fluxes, and finally discuss the physical significance of these quantum effects, comparing the current interpretation, based on the concept of a nonintegrable phase factor, with alternative approaches.
The seminal paper of Aharonov and Bohm [Phys. Rev. 115, 485 (1959)]10.1103/PhysRev.115.485 is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, in spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been studied extensively. The experimental issues were settled by the remarkable experiments of Tonomura et al. [Phys. Rev. Lett. 48, 1443 (1982); Phys. Rev. Lett. 56, 792 (1986)] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez et al. [Phys. Rev. Lett. 99, 210401 (2007)]10.1103/PhysRevLett.99.210401 that shows that the results of the Tonomura et al. experiments cannot be explained by the action of a force. The theoretical issues were settled by Ballesteros and Weder [Commun. Math. Phys. 285, 345 (2009)10.1007/s00220-008-0579-1; J. Math. Phys. 50, 122108 (2009)10.1063/1.3266176; Commun. Math. Phys. 303, 175 (2011)]10.1007/s00220-010-1166-9 who rigorously proved that quantum mechanics predicts the experimental results of Tonomura et al. and of Caprez et al. The electric Aharonov-Bohm effect has been much less studied. Actually, its existence, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an ansatz for the solution to the Schrödinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will stimulate the experimental research on the electric Aharonov-Bohm effect.
In this article, we start our discussion on the Aharonov-Bohm effect then raise a plausible interpretation within the quantum electrodynamics (QED) framework. We provide a quantum treatment of the source of the electromagnetic potential and argue that the underlying mechanism in AB effect can be viewed as interactions between electrons described by QED theory where the interactions are mediated by virtue photons. On further analysis, we claim that the framework of one particle quantum mechanics (OPQM) can be shown, in general, as a mathematically approximated model which can be reformulated from QED theory. In addition, the classical Maxwell equations are derived from QED scattering process while both classical electromagnetic fields and potentials serve as mathematical tools that are constructed to approximate the interactions among elementary particles described by QED physics, i.e. neither classical fields nor potentials represent any real entities of nature. At the conclusion of this article, we make a few remarks on the hypnosis of the existence of magnetic monopoles.
Electromagnetic vector potential has physical significance in quantum mechanics as revealed by the Aharonov-Bohm effect for charged particles. However, till date it is thought that we cannot measure the vector potential directly as this is not a gauge invariant quantity. Contrary to this belief, here we show that one can indeed measure the electromagnetic vector potential using the notion of weak value. We show that it is simply the difference between the weak value of the canonical momentum of a charged particle in the presence and absence of magnetic field. This suggests that the vector potential is not only a physical entity but can be measured directly in the experiment.