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The reality of loops

Alexander Afriat

Published Jan 3, 2013• 0 citations
OpenAlex

Abstract

The transformation properties of the electromagnetic four-potential can make one wonder whether it has any physical reality. But it is felt (with the Aharonov-Bohm effect in mind) that the reality in question has to be \emph{somewhere}, 'thereabouts'; if it is not in the potential itself, might it not be in the loops around which it is integrated? I argue that no ontological solidity is gained by retreating to the loops, which are just as flimsy as the potential.

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Aptly named Aharonov-Bohm effect has classical analogue, long history

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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.

Physics Today•Aug 1, 2010•1 citation

Quantum Evidence for the Reality of Electromagnetic Potentials and the Gravitational Field

Shan Gao

The ontological status of the fundamental fields appearing in gauge theories and general relativity has been one of the deepest conceptual questions in modern physics. In electromagnetism, the central issue concerns whether the electromagnetic potential \(A_\mu\), rather than only the gauge-invariant field strength \(F_{\mu\nu}\), represents a physically real structure. In general relativity, the analogous question concerns whether the gravitational field \(g_{\mu\nu}\), despite its dependence on coordinate choices, should be regarded as a physically real field or merely as a mathematical representation of invariant spacetime relations. In this paper, we present a comprehensive argument for the reality of both \(A_\mu\) and \(g_{\mu\nu}\). For the electromagnetic case, we show that when the source is treated as a fully quantized dynamical system with backreaction, the closed-loop Aharonov--Bohm phase acquires a gauge-dependent contribution that is the sum of two open-path integrals of \(A_\mu\). Since this phase contributes to observed interference patterns, it must correspond to a unique physical quantity, forcing the conclusion that \(A_\mu\) is a real physical field. We further argue that the Lorenz condition is the additonal physical law that, together with boundary conditions, determines \(A_\mu\) uniquely. For the gravitational case, we extend this analysis to both closed-loop and open-path phases. First, we show that closed-loop gravitational phases, such as those measured in the Colella-Overhauser-Werner (COW) experiment and the Overstreet et al. experiment, become gauge dependent under diffeomorphisms when source backreaction is properly accounted for. Second, we show that existing quantum optical experiments using local homodyne tomography already measure open-path relative phases of photons in spatially separated modes, which contain a gravitational contribution—the open-path gravitational phase. Since both closed-loop and open-path gravitational phases are gauge dependent under diffeomorphisms, their measurability entails the reality of the gravitational field \(g_{\mu\nu}\). We further argue that the harmonic (de Donder) gauge is the additional physical law that, together with boundary conditions, determines \(g_{\mu\nu}\) uniquely. These results establish that both electromagnetic potentials and the gravitational field are real physical fields, not mere mathematical auxiliaries. They challenge the conventional identification of gauge invariance with physical reality and have profound implications for non-Abelian gauge theories and quantum gravity.

•Jul 29, 2026•0 citations
The transformation properties of the electromagnetic four-potential can make one wonder whether it has any physical reality. But it is felt (with the Aharonov-Bohm effect in mind) that the reality in question has to be \emph{somewhere}, 'thereabouts'; if it is not in the potential itself, might it not be in the loops around which it is integrated? I argue that no ontological solidity is gained by retreating to the loops, which are just as flimsy as the potential.
It is often said that the Aharonov-Bohm effect shows that the vector potential enjoys more ontological significance than we previously realized. But how can a quantum-mechanical effect teach us something about the interpretation of Maxwell's theory—let alone about the ontological structure of the world—when both theories are false? I present a rational reconstruction of the interpretative repercussions of the Aharonov-Bohm effect, and suggest some morals for our conception of the interpretative enterprise.
I argue that the metaphysical import of the Aharonov-Bohm effect has been overstated: correctly understood, it does not require either rejection of gauge invariance or any novel form of nonlocality. The conclusion that it does require one or the other follows from a failure to keep track, in the analysis, of the complex scalar field to which the magnetic vector potential is coupled. Once this is recognised, the way is clear to a local account of the ontology of electrodynamics (or at least, to an account no more nonlocal than quantum theory in general requires); I sketch a possible such account.
Quantum physics demands some radical revision of our fundamental beliefs about physical reality. We know that because there are certain verified physical phenomena-two-slit interference, the disappearance of interference upon monitoring, violations of Bell's inequality-that have no classical analogs. But the exact nature of that revision has been under dispute since the foundation of quantum theory. I offer a method of clarifying what the commitments of a clearly formulated physical theory are, and apply it to a discussion of some options available to account for another non-classical phenomenon: the Aharonov-Bohm effect.
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 ontological status of the fundamental fields appearing in gauge theories and general relativity has been one of the deepest conceptual questions in modern physics. In electromagnetism, the central issue concerns whether the electromagnetic potential AμA_\muAμ​, rather than only the gauge-invariant field strength FμνF_{\mu\nu}Fμν​, represents a physically real structure. In general relativity, the analogous question concerns whether the gravitational field gμνg_{\mu\nu}gμν​, despite its dependence on coordinate choices, should be regarded as a physically real field or merely as a mathematical representation of invariant spacetime relations. In this paper, we present a comprehensive argument for the reality of both AμA_\muAμ​ and gμνg_{\mu\nu}gμν​. For the electromagnetic case, we show that when the source is treated as a fully quantized dynamical system with backreaction, the closed-loop Aharonov--Bohm phase acquires a gauge-dependent contribution that is the sum of two open-path integrals of AμA_\muAμ​. Since this phase contributes to observed interference patterns, it must correspond to a unique physical quantity, forcing the conclusion that AμA_\muAμ​ is a real physical field. We further argue that the Lorenz condition is the additonal physical law that, together with boundary conditions, determines AμA_\muAμ​ uniquely. For the gravitational case, we extend this analysis to both closed-loop and open-path phases. First, we show that closed-loop gravitational phases, such as those measured in the Colella-Overhauser-Werner (COW) experiment and the Overstreet et al. experiment, become gauge dependent under diffeomorphisms when source backreaction is properly accounted for. Second, we show that existing quantum optical experiments using local homodyne tomography already measure open-path relative phases of photons in spatially separated modes, which contain a gravitational contribution—the open-path gravitational phase. Since both closed-loop and open-path gravitational phases are gauge dependent under diffeomorphisms, their measurability entails the reality of the gravitational field gμνg_{\mu\nu}gμν​. We further argue that the harmonic (de Donder) gauge is the additional physical law that, together with boundary conditions, determines gμνg_{\mu\nu}gμν​ uniquely. These results establish that both electromagnetic potentials and the gravitational field are real physical fields, not mere mathematical auxiliaries. They challenge the conventional identification of gauge invariance with physical reality and have profound implications for non-Abelian gauge theories and quantum gravity.