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More variations on Aharonov-Bohm Herman Batelaan, Akira Tonomura
Published Apr 1, 2010 • 1 citation
Abstract 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.
DOI 10.1063/1.4797322 Related papers More variations on Aharonov-Bohm
Peter Andrew Sturrock, T. R. Groves
Physics Today • Apr 1, 2010• 6 citations
More Variations on Aharonov-Bohm, Peter A. Sturock, Timothy R. Groves, Alexander Ershkovich, C. Alden Mead, Herman Batelaan and Akira Tonomura: Batelaan and Tonomura Reply
Herman Batelaan, Akira Tonomura
First paragraph: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 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.
• Jan 1, 2010• 0 citations
Aptly named Aharonov-Bohm effect has classical analogue, long history
Murray Peshkin
Physics Today • Aug 1, 2010• 1 citation
More variations on Aharonov-Bohm
A. I. Ershkovich
Physics Today • Apr 1, 2010• 2 citations
Aptly named Aharonov-Bohm effect has classical analogue, long history
Michael Berry
Physics Today • Jul 30, 2010• 12 citations
The electric Aharonov-Bohm effect
Ricardo Weder
arXiv:1006.1385 • Journal of Mathematical Physics • May 1, 2011• 16 citations
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.
Having read “The Aharonov-Bohm Effects: Variations on a Subtle Theme” by Herman Batelaan and Akira Tono-mura (Physics Today, September 2009, page 38), we note with some dismay that the principal effect, that related to magnetic flux, continues to be attributed to its re-discoverers rather than to the original discoverers. Batelaan and Tonomura write, “The AB effect was already implicit in the 1926 Schrödinger equation, but it would be another three decades before theorists Yakir Aharonov and David Bohm pointed it out.” 1 1. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). That is incorrect: It took only two decades before the effect was pointed out, not by Aharonov and Bohm but by Werner Ehrenberg and Raymond “Rory” Siday. 2 2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Aharonov and Bohm cited that prior publication in their second article. 3 3. Y. Aharonov, D. Bohm, Phys. Rev. 123, 1511(1961). According to Bohm’s biographer F. David Peat, “After their first paper had been published, Bohm learned that the effect had already been postulated by a maverick physicist called Rory E. Siday.” 4 4. F. D. Peat, Infinite Potential: The Life and Times of David Bohm, Addison-Wesley, Reading, MA (1997), p. It seems curious that the biographer should use the term “postulated” rather than “proposed” and that he should comment on Siday’s personality but not on Ehrenberg’s. One of us (Sturrock) knew them both: Raymond was very smart but somewhat brash; Werner was erudite and a model of propriety, who would never have subscribed his name to an article unless he was convinced it was correct.It seems that the main scientific, rather than sociological, reason that the AB proposal was taken more seriously than the ES proposal is that the former was presented in the context of quantum mechanics, the latter in the context of electron optics. Aharonov and Bohm made a rapid transition from quantum mechanics to wave theory. Ehrenberg and Siday made a more detailed transition from geometrical optics to wave theory. For the case of magnetic flux, the two articles presented identical theoretical predictions. Our view is that the ES article was tedious but conceptually sound, whereas the AB article was lively but involved a questionable conceptual leap—namely, a relativistic generalization based on a noncovariant analysis. Soon after Sturrock arrived at Stanford University in 1955, and before the Aharonov-Bohm publication, he asked Leonard Schiff, Stanford’s quantum mechanics expert, whether Schiff found the Ehrenberg-Siday proposal convincing. He did.In the interests of accuracy and of giving credit where credit is due, we think it would be appropriate to use “Ehrenberg-Siday effect” for the case of magnetic interference and to reserve “Aharonov-Bohm effect” for electrical interference.We are indebted to Elliott Bloom, Dieter Kern, Walt Harrison, Peter Hawkes, Garret Moddel, Fabian Pease, Jeff Scargle, and Lenny Susskind for helpful discussions on this matter.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). Google ScholarCrossref, ISI2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref3. Y. Aharonov, D. Bohm, Phys. Rev. 123, 1511(1961). Google ScholarCrossref, ISI4. F. D. Peat, Infinite Potential: The Life and Times of David Bohm, Addison-Wesley, Reading, MA (1997), p. Google Scholar© 2010 American Institute of Physics.
I am an author of the 1961 paper that first called the Aharonov-Bohm effect by that name. 1 1. M. Peshkin, I. Talmi, L. J. Tassie, Ann. Phys. 12, 426 (1961). The term has since appeared in more than 1000 published papers. However, it has been suggested a few times, most recently by Peter Sturrock and Timothy Groves, that the name does not do justice to Werner Ehrenberg and Raymond Siday, who found in 1949 that the motion of electrons can be influenced by magnetic fields confined to regions that the electrons do not enter. 2 2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). I argue that the name Aharonov-Bohm effect is appropriate, although for a reason that I did not fully appreciate in 1961.The 1959 Aharonov-Bohm paper profoundly changed the way we think about electromagnetic fields in quantum mechanics. 3 3. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). The gauge-invariant part of the vector potential was promoted to a real physical field, not just a convenient device for summarizing certain information about the electric and magnetic fields. In the words of C. N. Yang, “The electromagnetic field strength fµv in quantum mechanics underdescribes electromagnetism, as the Bohm-Aharonov effect demonstrates … information about the phase factor exp[ie∮(A⋅dx)/ℏc] for all closed loops correctly describes electromagnetism.” 4 4. C. N. Yang, in Proceedings of the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1983, S. Kamefuchi et al. , eds., Physical Society of Japan, Tokyo (1984), p. 5.See also T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). A century earlier, James Clerk Maxwell changed the way we think about action at a distance by identifying what we now call the Maxwell fields as real physical things that contain energy and momentum and that enable microscopic conservation laws; they are not just mathematical functions that summarize the necessary information about the past motions of charges. Later those physical fields had to be quantized. Yakir Aharonov and David Bohm found that in quantum mechanics the Maxwell fields in a multiply connected region do not contain all the physics; the vector potential must also be endowed with reality to make sense of the subtler interactions in quantum mechanics. Neither Aharonov and Bohm nor Ehrenberg and Siday were the first to observe that magnetic fields in places where a charged particle’s wave function vanishes may influence the motion of that particle. 5 5. P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931) F. London, Nature 140, 793 (1937). Paul Dirac, in his 1931 paper on magnetic monopoles, noted that the electron’s wavefunction must vanish on singular flux lines but said nothing about that raising any question of nonlocality. Fritz London pointed out in the 1930s that the motion of electrons in a superconducting ring depends on an external magnetic flux through the hole in the ring, where the electrons cannot go. He wrote in his 1937 paper, “The most stable state of a ring has no current, unless an external magnetic field is applied.” Like Dirac, he said nothing about a nonlocal action of the magnetic field being surprising or unusual. Of course, London, like Dirac, was focusing on something else.Ehrenberg and Siday were also focusing on something else—electron optics—when they found in 1949 that the motion of an electron can depend on the magnetic field in a region from which the electron is excluded. They chose not to mention that curious phenomenon in the abstract of their paper, although they did explicitly say elsewhere that it was “curious.”None of those earlier authors went on to conclude that such a phenomenon implies that the vector potential has to be seen as a real physical field in quantum mechanics. Only Aharonov and Bohm did that.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Peshkin, I. Talmi, L. J. Tassie, Ann. Phys. 12, 426 (1961). Google ScholarCrossref, ISI2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref3. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). Google ScholarCrossref, ISI4. C. N. Yang, in Proceedings of the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1983, S. Kamefuchi et al. , eds., Physical Society of Japan, Tokyo (1984), p. 5. Google ScholarSee also T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). Google ScholarCrossref, ISI5. P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931) Google ScholarCrossref F. London, Nature 140, 793 (1937). , Google ScholarCrossref© 2010 American Institute of Physics.
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.
Writing about what is usually termed the Aharonov-Bohm (AB) effect, Peter Sturrock and Timothy Groves argue (Physics Today, April 2010, page 8) that the same physics was discovered a decade earlier 1 1. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). and should rightly be called the Ehrenberg-Siday effect. I agree that Werner Ehrenberg and Raymond Siday deserve recognition for their anticipation of AB. Indeed, in recent talks celebrating the 50th anniversary of AB, I began by describing the unfairly neglected paper by Ehrenberg and Siday. Nevertheless, I have come to a different conclusion from Sturrock and Groves: The expression “Aharonov-Bohm effect” is justified, for two reasons.First, although there is no doubt that the work by Ehrenberg and Siday anticipated how inaccessible magnetic flux can influence electron interference, that was as a curiosity, at the end of a paper whose main emphasis was the Hamiltonian analysis of electron optics. By contrast, Yakir Aharonov and David Bohm emphasized from the start, as an essential and general aspect of quantum mechanics, the physical influence of inaccessible fields that act nonlocally through the vector potential.Second, Ehrenberg and Siday’s semi-classical approximation—essentially applying the Dirac magnetic phase factor to electrons traveling on either side of the flux—implies a wavefunction that is multivalued and therefore not the correct solution of Schrödinger’s equation. The lack of a single-valued wavefunction leaves their prediction open to doubt. By contrast, Aharonov and Bohm derived the exact single-valued solution for waves scattered by a flux line. Recently, it was shown that the Ehrenberg-Siday approximation corresponds to the first terms in a “many-whirls” representation that treats the exact AB wavefunction as a topological sum over paths circling the flux. 2 2. M. V. Berry, J. Phys. A (in press). Attribution of credit is a delicate matter. It tends to excite strong feelings, and I write about it reluctantly. But the Ehrenberg-Siday paper does seem to exemplify the unfortunate phenomenon identified by Alfred North White-head in a 1916 address to the British Association for the Advancement of Science: “Everything of importance has been said before, by someone who did not discover it.”In a companion letter in the April 2010 issue, Alexander Ershkovich correctly points out that the AB effect is present in classical Hamiltonian mechanics, even though remote magnetic fields cannot influence Newtonian trajectories. He advocates a “search for experiments that might prove … a classical analogue of the Aharonov-Bohm effect.” Such an experiment exists already. In the classical physics of waves on a moving medium, the flow velocity acts like the vector potential in quantum mechanics, so the flow vorticity acts like the magnetic field; the analogy is precise. Fine details of the AB wavefunction were observed in ripples on the surface of water swirling irrotationally into a bathtub vortex, 3 3. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, J. C. Walmsley, Eur. J. Phys. 1, 154 (1980). whose core is the analogue of inaccessible magnetic flux. REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref2. M. V. Berry, J. Phys. A (in press). Google Scholar3. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, J. C. Walmsley, Eur. J. Phys. 1, 154 (1980). Google ScholarCrossref© 2010 American Institute of Physics.
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.