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A versus B! Topological nonseparability and the Aharonov-Bohm effect

Tim Oliver Eynck, Holger Lyre, Nicolai von Rummell

Published Sep 1, 2001• 4 citations
OpenAlex

Abstract

Since its discovery in 1959 the Aharonov-Bohm effect has continuously been the cause for controversial discussions of various topics in modern physics, e.g. the reality of gauge potentials, topological effects and nonlocalities. In the present paper we juxtapose the two rival interpretations of the Aharonov-Bohm effect. We show that the conception of nonlocality encountered in the Aharonov-Bohm effect is closely related to the nonseparability which is common in quantum mechanics albeit distinct from it due to its topological nature. We propose a third alternative interpretation based on the loop space of holonomies which serves to solve some of the problems and we trace back the topological nonlocality and thereby the Aharonov-Bohm effect to their quantum mechanical origin. All three discussed interpretations are, of course, empirically equivalent. In fact, they present us with an instructive case study for the thesis of theory underdetermination by empirical data.

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Topological effects of Aharonov-Bohm type and the role of the quantum entanglement

Hasan Özgür Çıldıroğlu, A. U. Yilmazer

The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.

AIP conference proceedings•Jan 1, 2018•4 citations

Time-symmetry and topology of the Aharonov–Bohm effect

Yakir Aharonov, Ismael L. Paiva, Zohar Schwartzman-Nowik +2 more

Abstract The Aharonov–Bohm (AB) effect has been highly influential in fundamental and applied physics. Its topological nature commonly implies that an electron encircling a magnetic flux source in a field-free region must close the loop in order to generate an observable effect. In this paper, we study a variant of the AB effect that apparently challenges this concept. The significance of weak values and nonlocal equations of motion is discussed as part of the analysis, shedding light on and connecting all these fundamental concepts.

arXiv:2303.00701•Journal of Physics A Mathematical and Theoretical•Oct 20, 2023•1 citation

Discussion: Healey on the Aharonov-Bohm Effect

Tim Maudlin

Richard Healey (1997) argues that the Aharonov-Bohm effect demands the recognition of either nonlocal or nonseparable physics in much the way that violations of Bell's inequality do. A careful examination of the effect and the arguments, though, shows that Healey's interpretation of the Aharonov-Bohm effect depends critically on his interpretation of gauge theories, and that the analogy with violations of Bell's inequalities fails.

Philosophy of Science•Jun 1, 1998•39 citations

Quantum Phases: 50 years of the Aharonov--Bohm effect and 25 years of the Berry phase

Mark R. Dennis, Sandu Popescu, Lev Vaidman A

This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm effect and the Berry phase. The issue includes work presented at two conferences, 50 Years of the Aharonov–Bohm Effect, 11–14 October 2009, Tel Aviv University, Israel, and the Aharonov–Bohm Effect and Berry Phase 50/25 Anniversary, 14–15 December 2009, Bristol University, UK. The Aharonov–Bohm (AB) effect and the Berry phase have many things in common, apart from the happy coincidence that they were discovered 25 years apart, making possible such a joint anniversary. First, they were both discovered in the same place, the H H Wills Physics Laboratory, University of Bristol, UK. Second, they both were there for everybody to see from day one, from the very beginning of quantum mechanics. They were, so to say, right on the surface; there was no need for sophisticated mathematics or involved constructions. Yet it took so long for somebody to see them. The explanation, we dare to speculate, is that they are so unusual, they challenge so much our everyday experience and ingrained intuitions about physics that it was difficult to recognise them. Third, both the AB effect and the Berry phase are quintessentially quantum; they both manifest themselves as topological/geometric quantum phases. The AB effect revolutionized our understanding of the role of potentials in physics. For the first time, it was shown that a particle moving in a field-free region could be affected by a field in a disjoint region. In doing so, it shattered one of the strongest held principles of physics. In the case of the Berry phase, the importance lies in its absolute generality—the topological/geometric phase named after Berry is acquired by all physical systems when the external parameters they depend upon undergo an adiabatic cyclic change. Remarkably this phase depends only on the geometry of the cyclic path in the abstract parameter space, a fact which deepens our appreciation of the underlying geometry of quantum mechanics. The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. First and foremost, they give a deeper understanding of the nature of quantum mechanics itself. Furthermore, we find them in fields as varied as cosmology, particle physics, non-abelian gauge theories, condensed matter, chemical and molecular physics, and laser dynamics. Generalizations of the AB effect to non-abelian gauge theories, such as the Wilson and `t Hooft loops, are important tools for studying confinement and spontaneous symmetry breaking. They explain charge quantization, the quantum Hall effect and the Josephson junction and also play a crucial role in electron microscope holography. Without them an understanding of many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behaviour, would be impossible. Like good wine, the AB effect and the Berry phase become finer and more appreciated with time. Even though they appear in textbooks and encyclopaedias, the number of researchers thinking and publishing papers on these two effects is increasing annually. The AB effect and the Berry phase keep being observed in new systems and with every day that passes novel applications are routinely found. This special issue is just a small snapshot of the state of the art 50 and 25 years, respectively, from their discoveries.

Journal of Physics A Mathematical and Theoretical•Aug 12, 2010•4 citations

Gauge-Underdetermination and Shades of Locality in the Aharonov–Bohm Effect

Ruward A. Mulder

Abstract: I address the view that the classical electromagnetic potentials are shown by the Aharonov–Bohm effect to be physically real (which I dub: ‘the potentials view’). I give a historico-philosophical presentation of this view and assess its prospects, more precisely than has so far been done in the literature. Taking the potential as physically real runs prima facie into ‘gauge-underdetermination’: different gauge choices represent different physical states of affairs and hence different theories. This fact is usually not acknowledged in the literature (or in classrooms), neither by proponents nor by opponents of the potentials view. I then illustrate this theme by what I take to be the basic insight of the AB effect for the potentials view, namely that the gauge equivalence class that directly corresponds to the electric and magnetic fields (which I call the Wide Equivalence Class) is too wide, i.e., the Narrow Equivalence Class encodes additional physical degrees of freedom: these only play a distinct role in a multiply-connected space. There is a trade-off between explanatory power and gauge symmetries. On the one hand, this narrower equivalence class gives a local explanation of the AB effect in the sense that the phase is incrementally picked up along the path of the electron. On the other hand, locality is not satisfied in the sense of signal locality, viz. the finite speed of propagation exhibited by electric and magnetic fields. It is therefore intellectually mandatory to seek desiderata that will distinguish even within these narrower equivalence classes, i.e. will prefer some elements of such an equivalence class over others. I consider various formulations of locality, such as Bell locality, local interaction Hamiltonians, and signal locality. I show that Bell locality can only be evaluated if one fixes the gauge freedom completely. Yet, an explanation in terms of signal locality can be accommodated by the Lorenz gauge: the potentials propagate in waves at finite speed. I therefore suggest the Lorenz gauge potentials theory—an even narrower gauge equivalence relation—as the ontology of electrodynamics.

•Apr 1, 2020•9 citations

Can elementary quantum mechanics explain the Aharonov–Bohm effect?

C. Magni, Fausto Valz-Gris

It is shown that a satisfactory treatment of the Aharonov–Bohm problem requires no departure from the basic principles of ordinary quantum mechanics, a fact not entirely clear in the literature. The discussion ends up with the same formalism as originally proposed by Aharonov and Bohm, in agreement with experimental facts, although in a way that forces no alteration in the traditional interpretation of vector potentials.

Journal of Mathematical Physics•Jan 1, 1995•25 citations
Since its discovery in 1959 the Aharonov-Bohm effect has continuously been the cause for controversial discussions of various topics in modern physics, e.g. the reality of gauge potentials, topological effects and nonlocalities. In the present paper we juxtapose the two rival interpretations of the Aharonov-Bohm effect. We show that the conception of nonlocality encountered in the Aharonov-Bohm effect is closely related to the nonseparability which is common in quantum mechanics albeit distinct from it due to its topological nature. We propose a third alternative interpretation based on the loop space of holonomies which serves to solve some of the problems and we trace back the topological nonlocality and thereby the Aharonov-Bohm effect to their quantum mechanical origin. All three discussed interpretations are, of course, empirically equivalent. In fact, they present us with an instructive case study for the thesis of theory underdetermination by empirical data.
The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.
Abstract The Aharonov–Bohm (AB) effect has been highly influential in fundamental and applied physics. Its topological nature commonly implies that an electron encircling a magnetic flux source in a field-free region must close the loop in order to generate an observable effect. In this paper, we study a variant of the AB effect that apparently challenges this concept. The significance of weak values and nonlocal equations of motion is discussed as part of the analysis, shedding light on and connecting all these fundamental concepts.
This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm effect and the Berry phase. The issue includes work presented at two conferences, 50 Years of the Aharonov–Bohm Effect, 11–14 October 2009, Tel Aviv University, Israel, and the Aharonov–Bohm Effect and Berry Phase 50/25 Anniversary, 14–15 December 2009, Bristol University, UK. The Aharonov–Bohm (AB) effect and the Berry phase have many things in common, apart from the happy coincidence that they were discovered 25 years apart, making possible such a joint anniversary. First, they were both discovered in the same place, the H H Wills Physics Laboratory, University of Bristol, UK. Second, they both were there for everybody to see from day one, from the very beginning of quantum mechanics. They were, so to say, right on the surface; there was no need for sophisticated mathematics or involved constructions. Yet it took so long for somebody to see them. The explanation, we dare to speculate, is that they are so unusual, they challenge so much our everyday experience and ingrained intuitions about physics that it was difficult to recognise them. Third, both the AB effect and the Berry phase are quintessentially quantum; they both manifest themselves as topological/geometric quantum phases. The AB effect revolutionized our understanding of the role of potentials in physics. For the first time, it was shown that a particle moving in a field-free region could be affected by a field in a disjoint region. In doing so, it shattered one of the strongest held principles of physics. In the case of the Berry phase, the importance lies in its absolute generality—the topological/geometric phase named after Berry is acquired by all physical systems when the external parameters they depend upon undergo an adiabatic cyclic change. Remarkably this phase depends only on the geometry of the cyclic path in the abstract parameter space, a fact which deepens our appreciation of the underlying geometry of quantum mechanics. The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. First and foremost, they give a deeper understanding of the nature of quantum mechanics itself. Furthermore, we find them in fields as varied as cosmology, particle physics, non-abelian gauge theories, condensed matter, chemical and molecular physics, and laser dynamics. Generalizations of the AB effect to non-abelian gauge theories, such as the Wilson and `t Hooft loops, are important tools for studying confinement and spontaneous symmetry breaking. They explain charge quantization, the quantum Hall effect and the Josephson junction and also play a crucial role in electron microscope holography. Without them an understanding of many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behaviour, would be impossible. Like good wine, the AB effect and the Berry phase become finer and more appreciated with time. Even though they appear in textbooks and encyclopaedias, the number of researchers thinking and publishing papers on these two effects is increasing annually. The AB effect and the Berry phase keep being observed in new systems and with every day that passes novel applications are routinely found. This special issue is just a small snapshot of the state of the art 50 and 25 years, respectively, from their discoveries.
Abstract: I address the view that the classical electromagnetic potentials are shown by the Aharonov–Bohm effect to be physically real (which I dub: ‘the potentials view’). I give a historico-philosophical presentation of this view and assess its prospects, more precisely than has so far been done in the literature. Taking the potential as physically real runs prima facie into ‘gauge-underdetermination’: different gauge choices represent different physical states of affairs and hence different theories. This fact is usually not acknowledged in the literature (or in classrooms), neither by proponents nor by opponents of the potentials view. I then illustrate this theme by what I take to be the basic insight of the AB effect for the potentials view, namely that the gauge equivalence class that directly corresponds to the electric and magnetic fields (which I call the Wide Equivalence Class) is too wide, i.e., the Narrow Equivalence Class encodes additional physical degrees of freedom: these only play a distinct role in a multiply-connected space. There is a trade-off between explanatory power and gauge symmetries. On the one hand, this narrower equivalence class gives a local explanation of the AB effect in the sense that the phase is incrementally picked up along the path of the electron. On the other hand, locality is not satisfied in the sense of signal locality, viz. the finite speed of propagation exhibited by electric and magnetic fields. It is therefore intellectually mandatory to seek desiderata that will distinguish even within these narrower equivalence classes, i.e. will prefer some elements of such an equivalence class over others. I consider various formulations of locality, such as Bell locality, local interaction Hamiltonians, and signal locality. I show that Bell locality can only be evaluated if one fixes the gauge freedom completely. Yet, an explanation in terms of signal locality can be accommodated by the Lorenz gauge: the potentials propagate in waves at finite speed. I therefore suggest the Lorenz gauge potentials theory—an even narrower gauge equivalence relation—as the ontology of electrodynamics.