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Zeitschrift für Naturforschung A

PCT-Invariance of Generalized de Broglie-Bargmann-Wigner Equations

Harald Stumpf

Published Oct 1, 2003• 1 citation
Publisher

Abstract

Generalized de Broglie-Bargmann-Wigner (GBBW)-equations are part of the derivation of effective theories for composite particles based on a nonlinear spinor field model with canonical quantization, relativistic invariant selfregularization and probability interpretation. Owing to the close connection between the model itself and its GBBW-equations the behavior of both is studied under PCT-transformations leading to the proof of invariance against such tranformation of the field Hamiltonian as well as of the GBBW-equations. Finally the algebraic formulation of the operators of these transformations in general state spaces is derived as the nonperturbative algebraic treatment of the model is referred to such spaces which are inequivalent to the Fock space representation of perturbation theory.

DOI
10.1515/zna-2003-9-1001

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Generalized de Broglie-Bargmann-Wigner (GBBW)-equations are part of the derivation of effective theories for composite particles based on a nonlinear spinor field model with canonical quantization, relativistic invariant selfregularization and probability interpretation. Owing to the close connection between the model itself and its GBBW-equations the behavior of both is studied under PCT-transformations leading to the proof of invariance against such tranformation of the field Hamiltonian as well as of the GBBW-equations. Finally the algebraic formulation of the operators of these transformations in general state spaces is derived as the nonperturbative algebraic treatment of the model is referred to such spaces which are inequivalent to the Fock space representation of perturbation theory.
A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in the presence of an electromagnetic field in phase space, which is spanned by position and kinetic momentum. This approach is independent of the chosen potentials, as it depends only on the electric and magnetic field variables. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field based solely on differential operators that simplify existing formulations. A gauge-dependent equation is derived first using Moyal's equation. A transformation of the Wigner function, introduced by Stratonovich, is then used to make the equation gauge-invariant, which gives us a strong formulation of the problem. This equation can be transformed into its weak form, which proves that both formulations are equivalent. An analysis of the different properties of the gauge-dependent and gauge-invariant formulations is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak formulations.
We introduce a gauge-invariant Wigner operator (GIWO) and a gauge-independent Wigner function (GIWF) that allow for both quantized and classical electromagnetic fields. If only classical fields are present, a Weyl transform analogous to the one associated with ordinary Wigner functions can be defined for any operator; in the case of quantized fields this is at least possible for Weyl-ordered functions of the position and kinetic momentum operators. We show how the evolution of the operators having a Weyl transform can be followed with the aid of the GIWF defined as the Heisenberg-picture expectation value of the GIWO, and derive for the GIWO the Heisenberg equation of motion which only involves the physical electric and magnetic fields. Some aspects of the case of entirely classical fields are discussed in more detail. (i) The GIWF in conjunction with the postulate that physical observables can be measured without referring to the gauge permits a quantum-mechanical treatment of a full experimental run without the problem of relating the measured values to the gauge-dependent density operator. (ii) A closed equation of motion for the GIWF is obtained. (iii) In the dipole approximation quantum features of the dynamics are lost. (iv) Quantum corrections to dynamics are associated with recoil effects.