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arXiv:1105.0312

Aharonov-Bohm effect and classical Hamiltonian mechanics

A. I. Ershkovich, P. L. Israelevich

Published May 2, 2011• 1 citation
arXivHTMLPDF

Abstract

It is shown that Aharonov-Bohm effect may be derived from Hamilton-Jacobi equation. It might have been predicted well before the quantum mechanics creation by Heisenberg and Schroedinger.
DOI
10.48550/arxiv.1105.0312

Related papers

Electromagnetic potentials and Aharonov-Bohm effect

A. I. Ershkovich

Hamilton-Jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials (A,ϕ), similar to Schroedinger equation. We derived the Aharonov-Bohm effect from Hamilton-Jacobi equation thereby having proved that this effect is of classical origin. These facts enable us to arrive at the following conclusions: a) the very idea of special role of potentials (A,ϕ) in quantum mechanics (different from their role in classical physics) lost the ground, and becomes dubious, as this idea is based on the Aharonov-Bohm effect, b) failure to find any signs of a special role of these potentials in the appropriate experiments (Feinberg, 1963) is thereby explained, and c) discovery of classical analogues of the Aharonov-Bohm effect (Berry et al., 1980) is also explained by a classical nature of this effect. Elucidation of the "unlocal" interaction problem was made.

arXiv:1209.1078•Sep 4, 2012•2 citations

More variations on Aharonov-Bohm

A. I. Ershkovich

The article about Aharonov-Bohm effects is interesting and comprehensive. The primary and best-known effect shows that the vector potential A of the electromagnetic field is a physical reality rather than a mathematical artifice. That reality was implicit in the Schrödinger equation, as the Hamiltonian H depends on A instead of the electric field E and magnetic field B = ∇ × A. But the same statement also refers to the Hamilton-Jacobi equation in classical mechanics, so one may expect that a similar effect exists in classical physics as well.Indeed, classical mechanics is governed by the fundamental Hamilton-Jacobi equation for the action S: δS/δt + H = 0, which naturally follows from William Hamilton’s principle of least action. Both H and S depend on A (even with E = B = 0). Thus it is quite possible that the vector potential is also a physical reality in classical physics. Erwin Schrödinger arrived at his famous equation in 1926 by using a mechanics-optics analogy, the so-called eikonal equation. Hamilton in 1834 proved that eikonal and Hamilton-Jacobi equations are equivalent, so that the Schrödinger equation actually follows from Hamilton-Jacobi.Had Hamilton known that classical mechanics does not always hold, quantum mechanics might have appeared a century earlier. 1 1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . The analogy between the equations of classical and quantum mechanics provides a reason to search for experiments that might prove whether a classical analogue of the Aharonov-Bohm effect exists.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . Google Scholar© 2010 American Institute of Physics.

Physics Today•Apr 1, 2010•2 citations

Contrast Analysis on Two Theoretical Explanation of Aharonov-bohm Effect

Chen Lan-li

The paper introduces the Aharonov-Bohm effect in magnetic field with theories of classical electrodynamics and quantum mechanics.It expounds the limitation of classical electrodynamics,the stringency,integrity and scientific predictability of quantum mechanics via comparative analysis.

Journal of Chongqing University of Science and Technology•Jan 1, 2009•0 citations

More variations on Aharonov-Bohm

Herman Batelaan, Akira Tonomura

Batelaan and Tonomura reply: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 1 1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. as Peter Sturrock and Timothy Groves point out. We had included this reference in the early versions of our manuscript. However, limited space directed the focus of the paper to the “effect without a force” discussion, rather than a historic perspective.Alexander Ershkovich notes that in the Hamilton-Jacobi formulation of classical mechanics, both the action and the Hamiltonian depend on the vector potential; he ponders whether the AB effect might have a classical manifestation. Newton’s formulation of classical mechanics is equivalent to the Hamilton-Jacobi formulation. Because the absence of a field means the absence of a force in Newton’s formulation, classical trajectories are unaffected. That result is not expected to change in the equivalent Hamilton-Jacobi formulation. Thus the AB effect is usually considered to be a pure quantum effect. On the other hand, we may interpret “classical mechanics” in a broader sense, such as in general relativistic classical mechanics. In electrodynamics classical trajectories are not affected by a localized magnetic field through which they do not pass. However, considering that the energy content of a current-carrying solenoid is larger than that of one without current, the trajectory is clearly affected gravitationally, at least in principle. Though not due to the AB effect, that result elucidates that a generalized description may lead to other insights. The Hamilton-Jacobi equation may be an example of a theoretical vehicle by which to explore generalizations such as relativistic effects, separation of variables, multiple particle effects, or the classical limit of the de Broglie-Bohm theory. C. Alden Mead recollects interesting statements made by Bohm. We agree fully with Bohm’s statement that “it would be much more revolutionary for this effect to be wrong than for it to be right.” Attempts to disprove the AB effect should be seen for what they are, outright attempts at finding limits to the validity of quantum mechanics itself. And although quantum mechanics is unfinished with respect to, say, de-coherence theory and quantum gravity, the AB effect appears to be well within its validity range.We do not share Bohm’s astonishment that, as Mead relates, “certain physicists refused to accept the AB effect and even went to great lengths to try to disprove it.” Rather, to risk overusing a platitude, extraordinary phenomena should be exposed to extraordinary scrutiny. Failed attempts to disprove an idea often provide insight into its fundamental character. In that context, we reiterate the main message of our article. Many facets of the AB effect—for example, the electric version, the dispersionless nature, relativistic momentum conservation, the relation to the Mott-Schwinger effect, and the AB effects for other than electromagnetic gauge-invariant theories—need exploration. We predict a bright future for the AB effect, with many surprises to come.REFERENCESection:ChooseTop of pageREFERENCE <<1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. Google ScholarCrossref© 2010 American Institute of Physics.

Physics Today•Apr 1, 2010•1 citation

Sharpening the gravitational Aharonov–Bohm effect

Akshat Pandey

We study the recent gravitational analogue of the Aharonov–Bohm effect for a classical system, namely a complex scalar field. We use this example to demonstrate that the Aharonov–Bohm effect in principle has nothing to do with quantum-mechanics. We then discuss how this classical field description can be connected to the standard one particle quantum description of the Aharonov–Bohm effect.

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Aharonov–Bohm effect revisited

Gregory Eskin

Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.

arXiv:1504.04784•Reviews in Mathematical Physics•Mar 1, 2015•4 citations
Hamilton-Jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials (A,ϕ), similar to Schroedinger equation. We derived the Aharonov-Bohm effect from Hamilton-Jacobi equation thereby having proved that this effect is of classical origin. These facts enable us to arrive at the following conclusions: a) the very idea of special role of potentials (A,ϕ) in quantum mechanics (different from their role in classical physics) lost the ground, and becomes dubious, as this idea is based on the Aharonov-Bohm effect, b) failure to find any signs of a special role of these potentials in the appropriate experiments (Feinberg, 1963) is thereby explained, and c) discovery of classical analogues of the Aharonov-Bohm effect (Berry et al., 1980) is also explained by a classical nature of this effect. Elucidation of the "unlocal" interaction problem was made.
The article about Aharonov-Bohm effects is interesting and comprehensive. The primary and best-known effect shows that the vector potential A of the electromagnetic field is a physical reality rather than a mathematical artifice. That reality was implicit in the Schrödinger equation, as the Hamiltonian H depends on A instead of the electric field E and magnetic field B = ∇ × A. But the same statement also refers to the Hamilton-Jacobi equation in classical mechanics, so one may expect that a similar effect exists in classical physics as well.Indeed, classical mechanics is governed by the fundamental Hamilton-Jacobi equation for the action S: δS/δt + H = 0, which naturally follows from William Hamilton’s principle of least action. Both H and S depend on A (even with E = B = 0). Thus it is quite possible that the vector potential is also a physical reality in classical physics. Erwin Schrödinger arrived at his famous equation in 1926 by using a mechanics-optics analogy, the so-called eikonal equation. Hamilton in 1834 proved that eikonal and Hamilton-Jacobi equations are equivalent, so that the Schrödinger equation actually follows from Hamilton-Jacobi.Had Hamilton known that classical mechanics does not always hold, quantum mechanics might have appeared a century earlier. 1 1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . The analogy between the equations of classical and quantum mechanics provides a reason to search for experiments that might prove whether a classical analogue of the Aharonov-Bohm effect exists.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . Google Scholar© 2010 American Institute of Physics.
Batelaan and Tonomura reply: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 1 1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. as Peter Sturrock and Timothy Groves point out. We had included this reference in the early versions of our manuscript. However, limited space directed the focus of the paper to the “effect without a force” discussion, rather than a historic perspective.Alexander Ershkovich notes that in the Hamilton-Jacobi formulation of classical mechanics, both the action and the Hamiltonian depend on the vector potential; he ponders whether the AB effect might have a classical manifestation. Newton’s formulation of classical mechanics is equivalent to the Hamilton-Jacobi formulation. Because the absence of a field means the absence of a force in Newton’s formulation, classical trajectories are unaffected. That result is not expected to change in the equivalent Hamilton-Jacobi formulation. Thus the AB effect is usually considered to be a pure quantum effect. On the other hand, we may interpret “classical mechanics” in a broader sense, such as in general relativistic classical mechanics. In electrodynamics classical trajectories are not affected by a localized magnetic field through which they do not pass. However, considering that the energy content of a current-carrying solenoid is larger than that of one without current, the trajectory is clearly affected gravitationally, at least in principle. Though not due to the AB effect, that result elucidates that a generalized description may lead to other insights. The Hamilton-Jacobi equation may be an example of a theoretical vehicle by which to explore generalizations such as relativistic effects, separation of variables, multiple particle effects, or the classical limit of the de Broglie-Bohm theory. C. Alden Mead recollects interesting statements made by Bohm. We agree fully with Bohm’s statement that “it would be much more revolutionary for this effect to be wrong than for it to be right.” Attempts to disprove the AB effect should be seen for what they are, outright attempts at finding limits to the validity of quantum mechanics itself. And although quantum mechanics is unfinished with respect to, say, de-coherence theory and quantum gravity, the AB effect appears to be well within its validity range.We do not share Bohm’s astonishment that, as Mead relates, “certain physicists refused to accept the AB effect and even went to great lengths to try to disprove it.” Rather, to risk overusing a platitude, extraordinary phenomena should be exposed to extraordinary scrutiny. Failed attempts to disprove an idea often provide insight into its fundamental character. In that context, we reiterate the main message of our article. Many facets of the AB effect—for example, the electric version, the dispersionless nature, relativistic momentum conservation, the relation to the Mott-Schwinger effect, and the AB effects for other than electromagnetic gauge-invariant theories—need exploration. We predict a bright future for the AB effect, with many surprises to come.REFERENCESection:ChooseTop of pageREFERENCE <<1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. Google ScholarCrossref© 2010 American Institute of Physics.
Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.