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European Journal of Physics

Observable magnetic vector potential

P. Huguenin

Published Jul 1, 1981• 1 citation
Publisher

Abstract

The impulse of the electric field defined has all the properties of a vector potential. This observable vector field depends on the history of the electromagnetic field. The gauge invariance of the dynamics of charged particles ensures that the instantaneous values of the electromagnetic field govern the motion.
DOI
10.1088/0143-0807/2/3/008

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The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.
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The notion that the electromagnetic field is quantised is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantisation of this field are usually mathematically motivated and begin by introducing a vector potential, followed by the imposition of a gauge that allows the manipulation of the solutions of Maxwell's equations into a form that is amenable for the machinery of canonical quantisation. By contrast, here we quantise the electromagnetic field in a less mathematically and more physically-motivated way. Starting from a direct description of what one sees in experiments, we show that the usual expressions of the electric and magnetic field observables follow from Heisenberg's equation of motion. In our treatment, there is no need to invoke the vector potential in a specific gauge and we avoid the commonly-used notion of a fictitious cavity that applies boundary conditions to the field.
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Electromagnetic vector potential has physical significance in quantum mechanics as revealed by the Aharonov-Bohm effect for charged particles. However, till date it is thought that we cannot measure the vector potential directly as this is not a gauge invariant quantity. Contrary to this belief, here we show that one can indeed measure the electromagnetic vector potential using the notion of weak value. We show that it is simply the difference between the weak value of the canonical momentum of a charged particle in the presence and absence of magnetic field. This suggests that the vector potential is not only a physical entity but can be measured directly in the experiment.