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Philosophy of Science

On the Reality of Gauge Potentials

Richard Healey

Published Dec 1, 2001• 57 citations
Publisher

Abstract

Classically, a gauge potential was merely a convenient device for generating a corresponding gauge field. Quantum-mechanically, a gauge potential lays claim to independent status as a further feature of the physical situation. But whether this is a local or a global feature is not made any clearer by the variety of mathematical structures used to represent it. I argue that in the theory of electromagnetism (or a non-Abelian generalization) that describes quantum particles subject to a classical interaction, the gauge potential is best understood as a feature of the physical situation whose global character is most naturally represented by the holonomies of closed curves in space-time.

DOI
10.1086/392936

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Classically, a gauge potential was merely a convenient device for generating a corresponding gauge field. Quantum-mechanically, a gauge potential lays claim to independent status as a further feature of the physical situation. But whether this is a local or a global feature is not made any clearer by the variety of mathematical structures used to represent it. I argue that in the theory of electromagnetism (or a non-Abelian generalization) that describes quantum particles subject to a classical interaction, the gauge potential is best understood as a feature of the physical situation whose global character is most naturally represented by the holonomies of closed curves in space-time.
In this article, we discuss in further detail the significance of potentials in the quantum theory, and in so doing, we answer a number of arguments that have been raised against the conclusions of our first paper on the same subject. We then proceed to extend our treatment to include the sources of potentials quantum-mechanically, and we show that when this is done, the same results are obtained as those of our first paper, in which the potential was taken to be a specified function of space and time. In this way, we not only answer certain additional criticisms that have been made of the original treatment, but we also bring out more clearly the importance of the potential in the expression of the local character of the interaction of charged particles and the electromagnetic field.
The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials AμA_\muAμ​, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials gμνg_{\mu\nu}gμν​ may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.
It is frequently stated that the electromagnetic vector potential acquires a fundamental role in quantum physics, whereas classically it only represents a convenient, but by no means necessary, way of representing the electromagnetic field. Here we argue that this is a historical accident due to the fact that the electromagnetic field was discovered before photons, while the electron itself was discovered first as a particle, before it became clear that it must also be treated as a wave and therefore as an excitation of the underlying electron field. We illustrate the fact that the vector potential ought to play a fundamental role classically using the Aharonov-Bohm effect. This effect is considered as the strongest argument for the role the vector potential plays in quantum physics, however, here we offer a fully classical account of it. This is a consequence of the fact that any account, be it classical or quantum, must involve the vector potential in order to preserve the local nature of the Aharonov-Bohm (as well as all the other) phases.
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 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.