On the problem of ambiguity of electromagnetic potential

Physical Review

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 erst paper on the same subject. We then proceed to extend our treatment to include the sources of potentials quantummechanically, and we show that when this is done, the same results are obtained as those of our first paper, in which the potential was tal-en 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.

Philosophy of Science, 2001

Maudlin, Paul Teller; several referees for Philosophy of Science; and especially Harvey Brown, who introduced me to the work of Jeeva Anandan.

Physica Scripta, 2012

In this paper, we show that the use of the Helmholtz theorem enables the derivation of uniquely determined electromagnetic potentials without the necessity of using gauge transformation. We show that the electromagnetic field comprises two components, one of which is characterized by instantaneous action at a distance, whereas the other propagates in retarded form with the velocity of light. In our attempt to show the superiority of the new proposed method to the standard one, we argue that the action-at-a-distance components cannot be considered as a drawback of our method, because the recommended procedure for eliminating the action at a distance in the Coulomb gauge leads to theoretical subtleties that allow us to say that the needed gauge transformation is not guaranteed. One of the theoretical consequences of this new definition is that, in addition to the electric E and magnetic B fields, the electromagnetic potentials are real physical quantities. We show that this property of the electromagnetic potentials in quantum mechanics is also a property of the electromagnetic potentials in classical electrodynamics.

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 argues that gauge-invariant explanations, which attribute the phase to measurable quantities like magnetic flux Φ, 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 Φ(t) induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a critical analysis spanning quantum mechanics and quantum electrodynamics, I demonstrate that these explanations rely on nonlocal and discontinuous mechanisms, rendering them inadequate. Instead, the electromagnetic potentials A µ , 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 g µν may anchor spacetime's substantival reality via a generalized gravitational AB effect. Experimental proposals test this continuous accumulation, while a speculative massive photon scenario via the Proca equation deepens the potentials' ontological role. Reframing the AB effect as a linchpin for quantum reality, this study advocates a potential-centric ontology, challenging gauge symmetry's primacy and reshaping the foundations of quantum theory, particle physics, and gravitation with profound philosophical and empirical consequences.

Foundations of Physics, 2021

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.

The gauge freedom in the electromagnetic potentials indicates an underdeterminacy in Maxwell's theory. This underdeterminacy will be found in Maxwell's (1864) original set of equations by means of Helmholtz's (1858) decomposition theorem. Since it concerns only the longitudinal electric field, it is intimately related to charge conservation, on the one hand, and to the transversality of free electromagnetic waves, on the other hand (as will be discussed in Pt. II). Exploiting the concept of Newtonian and Laplacian vector fields, the role of the static longitudinal component of the vector potential being not determined by Maxwell's equations, but important in quantum mechanics (Aharonov-Bohm effect) is elucidated. These results will be exploited in Pt.III for formulating a manifest gauge invariant canonical formulation of Maxwell's theory as input for developing Dirac's (1949) approach to special-relativistic dynamics.

European Journal of Physics, 2020

In his recent paper (2020 Eur. J. Phys. 41 045202), Davis makes the claim that potentials and fields are ill-defined in the conventional treatment of electromagnetism. He argues that ‘the usual treatment is ambiguous, with that ambiguity being reflected in the gauge transformation equations’. He then proposes an approach based on two operational versions of Helmholtz’s theorem and claims that his approach does not exhibit gauge freedom and allows a rigourous definition of electromagnetic potentials. Here I argue that Davis’s approach does not provide a more rigours definition of potentials than that provided by the standard approach. Apparently, Davis does not realize that when applying an operational version of Helmholtz’s theorem to Maxwell’s equations, he is not avoiding gauge invariance but tacitly applying it by choosing the particular gauge-condition related to this version of the theorem. The application of the instantaneous Helmholtz’s theorem to Maxwell’s equations is equiv...

2005

In this work we substantiate the applying of the Tikhonov-Samarski vector decomposition theorem (TS-theorem) to vector fields in classical electrodynamics. We show that the standard Helmholtz vector decomposition theorem does not always apply to some vector fields in classical electrodynamics. Using the TS-theorem, within the framework of the so called {\it v-gauge}, we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light $c$ and another one (irrotational) with an arbitrary velocity $v$, however, we prove that the unique possible value of the "arbitrary" $v$ is exclusively the velocity of light $c$. I.e. we show that the so called {\it v-gauge} has neither mathematical nor physical meaning.

Annales De La Fondation Louis De Broglie, 2005

"Physical theories of fundamental significance tend to be gauge theories. These are theories in which the physical system being dealt with is described by more variables than there are physically independent degree of freedom. The physically meaningful degrees of freedom then reemerge as being those invariant under a transformation connecting the variables (gauge transformation). Thus, one introduces extra variables to make the description more transparent and brings in at the same time a gauge symmetry to extract the physically relevant content. It is a remarkable occurrence that the road to progress has invariably been towards enlarging the number of variables and introducing a more powerful symmetry rather than conversely aiming at reducing the number of variables and eliminating the symmetry" [1]. We claim that the potentials of Classical Electromagnetism are not indetermined with respect to the so-called gauge transformations. Indeed, these transformations raise paradoxes that imply their rejection. Nevertheless, the potentials are still indetermined up to a constant.

Richard Healey's talk was divided in two parts. In the first part he argued that we are not justified in believing that localized gauge potential properties are there, but we are in believing that holonomy properties are. In the second part, he conceded that the holonomy interpretation offers an incomplete local and causal account, but he maintained that the onus is on QM.