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arXiv:math-ph/0510001

Some aspects about semiclassical electrodynamics and gauge transformations

Jorge Sánchez, Morales, J., D. E. Zambrano

Published Sep 30, 2005• 0 citations
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Abstract

We describe the behaviour of semiclassical electrodynamics under gauge transformations. For this purpose we observe the structure of Schrödinger equation and matricial elements under these transformations. We conclude this theory is not gauge invariant. As a consequence of this fact, we obtain a possible loss of predictability of physical results.

DOI
10.48550/arxiv.math-ph/0510001

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We describe the behaviour of semiclassical electrodynamics under gauge transformations. For this purpose we observe the structure of Schrödinger equation and matricial elements under these transformations. We conclude this theory is not gauge invariant. As a consequence of this fact, we obtain a possible loss of predictability of physical results.
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 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.
A gauge transformation of a simple electromagnetic system is analyzed. The Hamiltonian which is derived from the Dirac Lagrangian density is used for determining the state of an electron. The fact that this Hamiltonian is free of time differential operators plays a key role in the analysis and proves that this Hamiltonian is not invariant under a general gauge transformation. An application of a specific gauge transformation illustrates this fact. These results call for a further analysis of the role of gauge transformations in the theoretical structure of electrodynamics.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
The interpretation of gauge transformations in the semiclassical theory of radiation has been the subject of much discussion and controversy over the past ten years or so. A consistent formulation of this problem that involves a gauge-independent Heisenberg picture of the motion and related gauge-dependent Schrödinger pictures is presented. The role of the canonical momenta and their representation as differential operators on wave functions is clarified. Some discussion of the physical interpretation is also given.
In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.