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Topology, Holes and Sources

Abstract

The Aharonov-Bohm effect is often called “topological.” But it seems no more topological than magnetostatics, electrostatics or Newton-Poisson gravity (or just about any radiation, propagation from a source). I distinguish between two senses of “topological.”

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Notes

  1. By “potential” I just mean primitive: the potential of the electromagnetic two-form F=dA is its primitive A↔(A,φ), the potential of the magnetic two-form B=d A is its primitive A, the potential of the electric one-form ∗E= is its primitive φ (the Hodge dual ∗ being taken in three dimensions), the potential of the three-form density ρ=d E is its primitive E.

  2. It is perhaps easiest to think of F as a purely magnetic field B produced by the current density J=dB in the solenoid.

  3. It will be convenient to view λ and ω as concentric disks.

  4. Ehrenberg & Siday [4], Aharonov & Bohm [2].

  5. Cf. Batterman [3, p. 555]: “Similarly, in the AB effect, it appears that we will need to refer to different nonseparable holonomy properties for each case in which there is a different flux running through the solenoid.”

  6. Aharonov & Bohm [2, p. 490]: “in a field-free multiply-connected region of space, the physical properties of the system still depend on the potentials.” Wu & Yang [11, p. 3845]: “The famous Bohm-Aharonov experiment […] showed that in a multiply connected region where f μν =0 everywhere there are physical experiments for which the outcome depends on the loop integral […] around an unshrinkable loop.” And p. 3856: “f μν underdescribes electromagnetism because of the Bohm-Aharonov experiment which involves a doubly connected space region.” Nash & Sen [8, p. 301]: “We […] consider the consequence of assuming the field F to be identically zero in some region Ω. At first one may think that there will be no physically measurable electromagnetic effects in such a region Ω. This is not so, effects may arise if the topology of Ω is non-trivial, e.g. if Ω is not simply connected. […] In terms of parallel transport one says that zero curvature does not imply trivial parallel transport if the region in which the curvature is zero is not simply connected. This underlies the fact that there is a sense in which the connection is a more fundamental object than the curvature, even though a connection is gauge dependent and not directly measurable.” Ryder [10, p. 101–104]: “the Bohm-Aharonov effect owes its existence to the non-trivial topology of the vacuum […]. The Bohm-Aharonov effect is the simplest illustration of the importance of topology in this branch of physics. […] The relevant space in this problem is the space of the vacuum, i.e. the space outside the solenoid, and that space is not simply connected. […] It is thus an essential condition for the Bohm-Aharonov effect to occur that the configuration space of the vacuum is not simply connected. […] in other words, it is because the gauge group of electromagnetism, U 1, is not simply connected that the Bohm-Aharonov effect is possible. […] The configuration space of the Bohm-Aharonov experiment is the plane ℝ2 […] with a hole in, and this is, topologically, the direct product of the line ℝ1 and the circle […]. There is, nevertheless, a positive effect on the interference fringes. The mathematical reason for this is that the configuration space of the null field (vacuum) is the plane with a hole in […].” Batterman [3, p. 544]: “We now have a U(1) bundle over a nonsimply connected base space: ℝ2−{origin}. This fact is responsible for the AB effect.” Ibid. pp. 552–553: “most discussions of the AB effect very quickly idealize the solenoid to an infinite line in space or spacetime. The flux, in this idealization, just is the abstract topological property of having space or spacetime be nonsimply connected. […] The issue is whether the idealizations—[…] and nonsimply connected space in the AB effect—do better explanatory work than some less idealized description. I believe that the idealized descriptions do, in fact, do a better job.” Ibid. p. 554: “It seems to me that for a full understanding of these anholonomies, one needs to appeal to the topology and geometry of the base space. […] If we take seriously the idea that topological features of various spaces […] can play an explanatory role […].” Footnote 29, same page: “it is most fruitful to treat the AB solenoid as an idealization that results in the multiple connectedness of the base space of a fiber bundle.” Ibid. p. 555: “The different cases are unified by the topological idealization of the solenoid as a string absent from spacetime which renders spacetime nonsimply connected. […] This topological feature enables us to understand the common behaviour in different AB experiments […]. […] how can it possibly be the case that appeal to an idealization such as the AB solenoid as a line missing from spacetime, provides a better explanation of genuine physical phenomena than can a less idealized, more “realistic” account where one does not idealize so severely? […] quite often […] appeal to highly idealized models does, in fact, provide better explanations.” Martin [7, p. 48]: “in the case of non-trivial spatial topologies, the gauge-invariant interpretation runs into potential complications. […] So-called holonomies […] encode physically significant information about the global features of the gauge field.” See also Lyre [5, pp. S377–S380], Nounou [9], Lyre [6, p. 659], Agricola & Friedrich [1, p. 275].

  7. Over and above any divergence-free electrical background that may or may not be present.

  8. It will be convenient to view Λ and Ω as concentric spheres.

  9. Or rather the total gravitational attraction radiated by a mass.

  10. For wherever A is closed it can be written locally as the gradient A= of a zero-form γ—just as E can be written locally, wherever it is closed, as the curl E= of a one-form ζ.

  11. Such a γ cannot be continuous everywhere; we can imagine a single discontinuity, say on the ray with values γ=2πnk, where the integer n is zero then one, k=C/2π being a constant.

  12. One should really say an odd number of times, as Jean-Philippe Nicolas has pointed out to me. Crossings in opposite directions cancel, and add nothing to the integral.

  13. Batterman [3, pp. 545–546]: “The phase or anholonomy depends continuously on the flux in the solenoid, but […] it depends discontinuously upon the shape of the circuit. For example, two loops around gives an anholonomy twice that of one loop around for constant magnetic flux.” And p. 555: “The different cases are unified by the topological idealization of the solenoid as a string absent from spacetime which renders spacetime nonsimply connected. In this way we can understand why, for a given fixed magnetic flux, a loop that goes n times around the solenoid will have an anholonomy that is n times that of a loop that goes around once.” A loop going around twice will cross each level curve twice. Alternatively, two different loops will also catch the same flux twice. Similar double-counting can be arranged in electrostatics too: a membrane enclosing the source twice, or two different membranes.

  14. Here I am indebted to Dennis Dieks, Éric Gourgoulhon and Jean-Philippe Nicolas.

  15. Here—returning to the point made by Jean-Philippe Nicolas—the number of perforations will be even: 0, 2, 4 etc.

  16. Deformations of ∂Ω can of course lead to the exclusion or inclusion of certain loops.

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Acknowledgements

I thank Nazim Bouatta, Dennis Dieks, Éric Gourgoulhon, Marc Lachièze-Rey and Jean-Philippe Nicolas for valuable clarifications and corrections.

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Afriat, A. Topology, Holes and Sources. Int J Theor Phys 52, 1007–1012 (2013). https://doi.org/10.1007/s10773-012-1413-2

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Keywords

  • Aharonov-Bohm effect
  • Topology
  • Electromagnetism
  • Quantum mechanics