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John Wiley & Sons, Inc. eBooks

Quantum Electrodynamics: Potentials, Gauge Invariance, and Analogy to Classical Electrodynamics

Carl E. Baum

Published Apr 2, 2003• 0 citations
Publisher

Abstract

In this chapter the author explores the degree to which one can measure vector and scalar potentials, including the implications of gauge invariance. Assuming that there are sources (current and/or charge) in some region of space away from the observer, in what sense can the potentials be distinguished? In particular, one can compare static and dynamic conditions. Under static conditions, it is possible to have zero fields in the vicinity of the observer (away from the source region), while having nonzero potentials. These ideas are discussed in this chapter. Two different antennas, one emphasizing the vector potential and the other the scalar potential (Lorenz gauge), are discussed in which the same fields are away from the source region. The author then goes on to consider an alternate way of viewing the role of the vector potential in quantum electrodynamics (QED).

DOI
10.1002/0471231487.ch10

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In this chapter the author explores the degree to which one can measure vector and scalar potentials, including the implications of gauge invariance. Assuming that there are sources (current and/or charge) in some region of space away from the observer, in what sense can the potentials be distinguished? In particular, one can compare static and dynamic conditions. Under static conditions, it is possible to have zero fields in the vicinity of the observer (away from the source region), while having nonzero potentials. These ideas are discussed in this chapter. Two different antennas, one emphasizing the vector potential and the other the scalar potential (Lorenz gauge), are discussed in which the same fields are away from the source region. The author then goes on to consider an alternate way of viewing the role of the vector potential in quantum electrodynamics (QED).
Abstract The transitions from classical theories to quantum theories have attracted many interests. This paper demonstrates the analogy between the electromagnetic potentials and wave-like dynamic variables with their connections to quantum theory for audiences at advanced undergraduate level and above. In the first part, the counterpart relations in the classical electrodynamics (e.g. gauge transform and Lorenz condition) and classical mechanics (e.g. Legendre transform and free particle condition) are presented. These relations lead to similar governing equations of the field variables and dynamic variables. The Lorenz gauge, scalar potential and vector potential manifest a one-to-one similarity to the action, Hamiltonian and momentum, respectively. In the second part, the connections between the classical pictures of electromagnetic field and particle to quantum picture are presented. By characterising the states of electromagnetic field and particle via their (corresponding) variables, their evolution pictures manifest the same algebraic structure (isomorphic). Subsequently, pictures of the electromagnetic field and particle are compared to the quantum picture and their interconnections are given. A brief summary of the obtained results are presented at the end of the paper.
Reciprocity in wave propagation usually refers to the symmetry of the Green's function under the interchange of the source and the observer coordinates, but this condition is not gauge invariant in quantum mechanics, a problem that is particularly significant in the presence of a vector potential. Several possible alternative criteria are given and analyzed with reference to different examples with nonzero magnetic fields and/or vector potentials, including the case of a multiply connected spatial domain. It is shown that the appropriate reciprocity criterion allows for specific phase factors separable into functions of the source and observer coordinates and that this condition is robust with respect to the addition of any scalar potential. In the Aharonov-Bohm effect, reciprocity beyond monoenergetic experiments holds only because of subsidiary conditions satisfied in actual experiments: the test charge is in units of eee and the flux is produced by a condensate of particles with charge 2e2e2e.
When the electromagnetic potentials are expressed in the Coulomb gauge in terms of the electric and magnetic fields rather than the sources responsible for these fields they have a simple form that is nonlocal (i.e., the potentials depend on the fields at every point in space). It is this nonlocality of classical electrodynamics that is primarily responsible for the puzzle associated with the static Aharonov–Bohm effect: that its interference pattern is affected by fields in a region of space that the electron beam never enters.
It is shown that in semiclassical electrodynamics, which describes how electrically charged particles move according to the laws of quantum mechanics under the influence of a prescribed classical electromagnetic field, only a restricted class of gauge transformations is allowed. This lack of full gauge invariance, in contrast to the situation in classical and quantum electrodynamics which are fully gauge-invariant theories, is due to the requirement that the scalar potential in the Hamiltonian of wave mechanics should represent a physical potential. Probability amplitudes and energy differences are independent of gauge within this restricted class of gauge transformation.
The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, which results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, which reinforces the view that it only can be measured in closed paths.
The main notions of semiclassical scalar electrodynamics in different gauges (Hamiltonian, Couloumb, Lorentz) are discussed. These are semiclassical states, Poincare transformations, fields, observables, gauge equivalence. General properties of these objects are formulated as axioms of semiclassical theory; they are heuristically justified. In particular, a semiclassical state may be viewed as a set of classical background field and quantum state in the external background. Superpositions of these "elementary" states can be also considered. Set of all "elementary" semiclassical states forms a semiclassical bundle, with base being classical space and fibres being quantum states in the external background. Quantum symetry transformations (Poincare and gauge transformations) are viewed semiclassically as automorphisms of the semiclassical bundle. Specific features of electrodynamics are investigated for different gauges.