Abstract
I argue that the Holonomy Interpretation, at least as it has been presented in Richard Healey’s Gauging What’s Real, faces serious problems. These problems are revealed when certain approximations and idealizations that are innate in the original formulation of the Aharonov-Bohm effect are thrust aside; in particular, when the temporal dimension is taken into account. There are two ways in which time re-appears in the picture: by considering complete solutions to the original problem, where the magnetic flux is static, and by examining the effects of time dependent magnetic fluxes. Both cases expose explanatory gaps in the Interpretation, as well as conflicts between it and customary ideas about relativistic locality and local action on which the Interpretation depends.
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Notes
As opposed to fully quantized.
According to Healey, a property of an object is qualitative (as opposed to individual) if it does not depend on the existence of any particular individual, and it is intrinsic just in case the object has that property in and of itself, and without regard to any other thing (pp. 46–47). Thus, Healey argues, in the case at hand holonomy properties are qualitative because although they take particular values for particular solenoids, they do not depend on the existence of any specific solenoid; and they are intrinsic because they are possessed by the space-time regions outside a solenoid whether quantum particles are present or not.
In particular, if the magnetic flux were quantized, different phase differences would result in the same phase factors which, in turn, would represent the same effects.
At least by Healey’s definition—see pp.70–71 and 109–110.
And he continues: What these [properties] are is not determined by any qualitative intrinsic physical properties that attach at, or in arbitrarily small neighborhoods of, space-time location on the loop. These properties may be represented by the generalized phase difference of the particle’s wave-function around the loop. Accordingly, they may be referred to as phase difference properties (p. 110).
In Sect. 4 of their (1959).
Since the time at the point of origin can be set to 0, it is omitted in the exprssions for K henceforth.
In the case of the A-B effect, the point of origin is typically thought to be the source of electrons and the final point the detector.
Surely, one could come up with several paths connecting point (r′, 0) to point (r″, τ) that contain only combinations of space-like and light-like segments. But the point of fact is that one cannot exclude time-like paths or paths with time-like segments from the calculations.
The classic reference here is Feynman (1948).
Following Schulman (1971), Bernido and Inomata solve the equation in the so-called covering space. The reason for doing so is explained by Schulman: On the [...] coordinate space the Hamiltonian is not essentially self-adjoint; it does possess this property, though, on a covering space. The propagator on the covering space is therefore well defined. On the original space, the propagator is some linear combination of covering space propagators with the coefficients in this linear combination selecting some self-adjoint extension of the original Hamiltonian. Nonetheless, the paths Bernido and Inomata refer to in the first part of their analysis are in space-time. And the parametrization that appears in their final expression for the partial propagator is also spatiotemporal, despite the fact that some of the calculations have been performed in covering space. Also spatiotemporal are the parameters in the expressions of the partial propagators I use in my discussion.
Modulo the approximation \(r^{\prime} r^{{\prime}\prime} \mu\,{\gg}\,{\hbar}{\tau},\) which is used only because the exact expression is difficult to calculate.
“Hidden” in cos {2π (m − n)ξ ...}.
Note that ‘holonomies with winding number 0’ refers to ‘holonomies or phase differences associated with loops that encircle the solenoid just once’.
n = 0 corresponds to a path that passes above the solenoid without encircling it, and n = −1 corresponds to a similar path that passes below.
The language used throughout follows Healey’s. One could readily point out that any talk about properties possessed at a time will, inevitably, cause problems related to the troublesome notion of simultaneity and its relativity. In what follows I will ignore the possible implications of this, as I intend to examine the problems for the Holonomy Interpretation from a different perspective.
In this discussion I am assuming that the purported physical objects (loops with holonomy properties distributed non-separably over them) have precedence over their mathematical representations (holonomies) for the following reason. The very possibility of mathematical representations of physical objects and causes is due to the physically recorded presence of these objects and causes. Hence, if the Holonomy Interpretation got it right, holonomies represent holonomy properties, but it is the latter that propagate, act physically, and give rise to the measurable effects which are recorded by the holonomies. This stance is also in concert with the views expressed in Gauging What’s Real, I believe, where holonomies are taken to act mathematically but also to represent physically instantiated QIPs.
The deformation I am alluding to is necessary because only holonomies of twisted loops can combine to give holonomies with higher winding numbers. Hence the same goes for the loops over which the properties they represent are distributed.
Whose integral is over an open path.
For reasons of simplicity only. The discussion that follows could have drawn from the results of Bernido and Inomata as well.
See Comay (1987).
Given the cylindrical symmetry of the physical configuration, we may bring the problem down to two spatial and one temporal dimensions.
In this discussion I follow Brown and Home (1992).
Note that the solution that gives this phase shift is approximate. Also, note that the time-like character of the integration paths is explicit since r = r(t).
Assumed to be associated with a holonomy and the properties it represents.
Note that this problem would persist even if one insisted that the influence originates from the non-separable holonomy properties that attach on a space-like loop that is some backwards projection of C 2. The furthest back in time one may go is t′ + dt, where dt is very small but not zero in order to wear off the effects of the induced magnetic field. Even at that time, however, only the non-separable holonomy properties which are distributed over the loop that almost touches the solenoid (presumed cylindrical throughout) would not violate relativistic locality.
References
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Acknowledgments
In the process of writing this paper, I incurred numerous intellectual debts which I hereby acknowledge. To Daniel Arovas, Georgios O. Papadopoulos, Lawrence S. Schulman, Andromahi Spanou and Alexander A. Voronov, who generously offered advise and counsel on formal matters. To Richard Healey, John Norton and Paul Teller, who engaged in long conversations and email exchanges on the philosophical issues involved. Last but not least, to the two anonymous referees of this journal whose questions, comments and criticisms resulted in clarifications and improvements. Needless to say, I am the only one responsible for errors, mistakes and confusions that may still be present in this work. Finally, I would also like to thank the review editor of this journal, Ralf Busse, for encouraging me to turn a book review into a critical remark.
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Nounou, A.M. Holonomy Interpretation and Time: An Incompatible Match? A Critical Discussion of R. Healey’s Gauging What’s Real: The Conceptual Foundations of Contemporary Gauge Theories. Erkenn 72, 387–409 (2010). https://doi.org/10.1007/s10670-010-9212-8
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DOI: https://doi.org/10.1007/s10670-010-9212-8