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arXiv:1609.03420 · Journal of Physics B Atomic Molecular and Optical Physics

Physical restrictions on the choice of electromagnetic gauge and their practical consequences

Howard R. Reiss

Published Feb 28, 2017• 9 citations
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Abstract

Abstract It is shown that electromagnetic potentials convey physical information beyond that supplied by electric and magnetic fields alone, and are thus more fundamental. Observable physical properties can impose conditions on the selection of electromagnetic gauge (i.e. sets of potentials) that are explicit and restrictive. This is true both classically and quantum mechanically. The implication that the choice of gauge carries physical information is confirmed by exhibiting a set of potentials that describes fields correctly, but that violates physical constraints. The basic conclusions are that physical requirements place limits on acceptable gauges; and that potentials are more fundamental than fields in both classical and quantum physics, representing a major generalization of the quantum-only Aharonov–Bohm effect. These important properties are obscured if the dipole approximation is employed. The properties demonstrated here relate directly to conditions that exist in strong-field laser applications.

DOI
10.1088/1361-6455/aa6375

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Abstract It is shown that electromagnetic potentials convey physical information beyond that supplied by electric and magnetic fields alone, and are thus more fundamental. Observable physical properties can impose conditions on the selection of electromagnetic gauge (i.e. sets of potentials) that are explicit and restrictive. This is true both classically and quantum mechanically. The implication that the choice of gauge carries physical information is confirmed by exhibiting a set of potentials that describes fields correctly, but that violates physical constraints. The basic conclusions are that physical requirements place limits on acceptable gauges; and that potentials are more fundamental than fields in both classical and quantum physics, representing a major generalization of the quantum-only Aharonov–Bohm effect. These important properties are obscured if the dipole approximation is employed. The properties demonstrated here relate directly to conditions that exist in strong-field laser applications.
Multiple bases are presented for the conclusion that potentials are fundamental in electrodynamics, with electric and magnetic fields as quantities auxiliary to the scalar and vector potentials -- opposite to the conventional ordering. One foundation for the concept of basic potentials and auxiliary fields consists of examples where two sets of gauge-related fields are such that one is physical and the other is erroneous, with the information for the proper choice supplied by the potentials. A major consequence is that a change of gauge is not a unitary transformation in quantum mechanics; a principle heretofore unchallenged. The primacy of potentials over fields leads to the concept of a hierarchy of physical quantities, where potentials and energies are primary, while fields and forces are secondary. Secondary quantities provide less information than do primary quantities. Some criteria by which strong laser fields are judged are based on secondary quantities, making it possible to arrive at inappropriate conclusions. This is exemplified by several field-related misconceptions as diverse as the behavior of charged particles in very low frequency propagating fields, and the fundamental problem of pair production at very high intensities. In each case, an approach based on potentials gives appropriate results, free of ambiguities. The examples encompass classical and quantum phenomena, in relativistic and nonrelativistic conditions. This is a major extension of the quantum-only Aharonov-Bohm effect, both in supporting the primacy of potentials over fields, and also in showing how field-based conceptions can lead to errors in basic applications.
A nominally legitimate set of electromagnetic potentials to represent electric and magnetic fields can cause violation of basic symmetries in both classical and quantum applications, creating errors in dynamical calculations. Preservation of symmetries places constraints on the choice of potentials and thus constitutes a robust reason to regard electromagnetic potentials as more fundamental than electromagnetic fields. This represents a major extension of the quantum-only example of the Aharonov-Bohm effect for establishing the primacy of potentials over fields. Symmetry violation caused by inappropriate gauge choice means that gauge transformations are not necessarily unitary, since they do not ensure preservation of the values of physical observables. Unitarity is examined in the Schr\odinger equation and in transition amplitudes.
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.
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.
Gauge invariance is conventionally interpreted as a fundamental redundancy of description, implying that gauge potentials represent the same physical situation and that only gauge-invariant quantities, such as the electromagnetic field strengths and closed-loop holonomies, are physically observable. Within this orthodox view, the gauge potential AμA_\muAμ​ in quantum electrodynamics possesses no independent ontological status. In this paper we re-examine this interpretation by treating the electromagnetic source in the Aharonov--Bohm (AB) effect as a fully quantized dynamical system whose backreaction on the test particle is explicitly included. We show that, once source dynamics are incorporated, the standard proof of gauge invariance no longer applies. The observable AB phase acquires an additional contribution that can be expressed as the sum of two open-path integrals of the gauge potential and is generally gauge dependent. Since this contribution affects measurable interference, it must correspond to a unique physical quantity, suggesting that the gauge potential itself is physically real rather than merely a redundant mathematical representation. We then argue that the Lorenz condition, ∂μAμ=0\partial_\mu A^\mu = 0∂μ​Aμ=0, should be interpreted not as a gauge choice but as a fundamental physical constraint which, together with the appropriate initial and boundary conditions, uniquely determines the gauge potential. Within this framework, the open-path relative phase of a charged particle, ∫γ1Aμdxμ−∫γ2Aμdxμ\int_{\gamma_1} A_\mu dx^\mu - \int_{\gamma_2} A_\mu dx^\mu∫γ1​​Aμ​dxμ−∫γ2​​Aμ​dxμ, becomes a well-defined observable, even though it is formally gauge dependent under transformations that do not preserve the Lorenz condition or boundary conditions. To establish the operational significance of this quantity, we propose a Spatial-to-Internal Phase Transfer (SIPT) protocol that coherently transfers the open-path phase of a single fermion into localized internal quantum memories, allowing the phase to be reconstructed through local correlation measurements without recombining the original spatial branches. We discuss possible implementations based on atomic electron capture and semiconductor--superconductor hybrid systems, and argue that such an experiment would constitute an \emph{experimentum crucis}, analogous to the Michelson--Morley experiment, capable of distinguishing experimentally between gauge redundancy and gauge-potential realism. Finally, we show that the conventional gauge-fixed formulation of quantum electrodynamics is mathematically equivalent to a formulation in which the Lorenz condition is imposed as a fundamental law. We further argue that local gauge invariance of the interaction follows from current conservation and the structure of the matter–field coupling rather than representing an independent physical symmetry, and that the gauge-parameter independence of physical amplitudes does not uniquely establish gauge redundancy. These results call for a reexamination of the conceptual foundations of quantum gauge theories and open the door to experimental tests that could transform our understanding of the ontological status of gauge potentials.
The Aharonov--Bohm (AB) phase for a closed interferometric loop is widely regarded as gauge invariant. This conclusion is usually obtained by treating the electromagnetic source as a fixed background that is unaffected by the test charge. We investigate whether this conclusion remains valid when backreaction is included within the standard gauge-fixed formulation of quantum electrodynamics (QED). Our analysis indicates that, when the interfering branches become entangled with distinct electromagnetic field configurations, changes of the gauge-fixing condition can induce a non-vanishing shift of the total phase. This result gives rise to an apparent paradox: the total phase predicted by the QED formalism appears gauge dependent, whereas experiments robustly exhibit a definite interference shift. We examine two possible resolutions of this paradox. The standard approach attributes the observed interference only to a gauge-invariant component of the total phase and treats the remaining gauge-dependent contribution as physically irrelevant. We argue, however, that the QED formalism itself predicts the total phase governing the interference outcome and does not provide a dynamical prescription that extracts a gauge-invariant component while discarding the remaining terms. We then explore an alternative perspective in which the electromagnetic potential AμA_\muAμ​ is regarded as a physical field whose configuration is determined by the Lorenz condition together with appropriate causal and boundary conditions. From this viewpoint, the total phase can be interpreted as a well-defined physical quantity within the QED formalism. We conclude by discussing a possible experimental test based on direct measurements of open-path relative phases, which may help clarify the ontological status of electromagnetic potentials.