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arXiv:quant-ph/0112122

Relativistic Wigner Function, Charge Variable and Structure of Position Operator

Bohdan I. Lev, А. А. Семенов, Constantin V. Usenko

Published Dec 20, 2001• 0 citations
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Abstract

The relativistic phase-space representation by means of the usual position and momentum operators for a class of observables with Weyl symbols independent of charge variable (i.e. with any combination of position and momentum) is proposed. The dynamical equation coincides with its analogue in the non-local theory (generalization of the Newton-Wigner position operator approach) under conditions when particles creation is impossible. Differences reveal themselves in specific constraints on possible initial conditions.

DOI
10.48550/arxiv.quant-ph/0112122

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The relativistic phase-space representation by means of the usual position and momentum operators for a class of observables with Weyl symbols independent of charge variable (i.e. with any combination of position and momentum) is proposed. The dynamical equation coincides with its analogue in the non-local theory (generalization of the Newton-Wigner position operator approach) under conditions when particles creation is impossible. Differences reveal themselves in specific constraints on possible initial conditions.
The thesis is devoted to the phase space representation of relativistic quantum mechanics. For a class of observables with matrix-valued Weyl symbols proportional to the identity matrix, the Weyl-Wigner-Moyal formalism is proposed. The evolution equations are found to coincide with their counterparts in relativistic quantum mechanics with non-local Hamiltonian. The difference between the theories is connected with peculiarities of the constraints on the initial conditions. Effective increase in coherence between eigenstates of the Hamiltonian is found. Relativistic coherent states that take into account a non-trivial charge structure of the position and momentum operators and satisfy the charge superselection rule are considered. On this basis, the entangled coherent states are developed.
Mathematical method of quantum phase space is very useful in physical applications like quantum optics and non-relativistic quantum mechanics. However, attempts to generalize it for the relativistic case lead to some difficulties. One of problems is band structure of energy spectrum for a relativistic particle. This corresponds to an internal degree of freedom, so-called charge variable. In physical problems we often deal with such of dynamical variables that do not depend on this degree of freedom. These are position, momentum, and any combination of them. Restricting our consideration to this kind of observables we propose the relativistic Weyl--Wigner--Moyal formalism that contains some surprising differences from its non-relativistic counterpart.
Description of scalar charged particles, based on the Feshbach - Villars formalism, is proposed. Particles are described by an object that is Wigner function in usual coordinates and momentums and density matrix in charge variable. It is possible to introduce the usual Wigner function for a large class of dynamical variables. Such an approach contains explicitly a measuring device frame. From our point of view it corresponds to the Copenhagen interpretation of quantum mechanics. It is shown how physical properties of such particles depend on the definition of operator of coordinate. Evolution equation for the Wigner function of a single-charge state in classical limit coincides with the Liouville equation. Localization peculiarities manifest themselves in specific constraints on possible initial conditions.
A description of scalar charged particles, based on the Feshbach-Villars formalism, is proposed. Particles are described by an object that is a Wigner function in usual coordinates and momenta and a density matrix in the charge variable. It is possible to introduce the usual Wigner function for a large class of dynamical variables. Such an approach explicitly contains a measuring device frame. From our point of view it corresponds to the Copenhagen interpretation of quantum mechanics. It is shown how physical properties of such particles depend on the definition of the coordinate operator. The evolution equation for the Wigner function of a single-charge state in the classical limit coincides with the Liouville equation. Localization peculiarities manifest themselves in specific constraints on possible initial conditions.
Some general assumptions based on the physical properties of elementary and of composite quantized systems lead in a natural way to general expressions for the position and spin operators of massive particles. All previously proposed position and spin operators, defined in the enveloping algebra of the Poincaré group, appear as special cases. By algebraic arguments a general local position operator is obtained which is proved to coincide under special conditions with the Pryce-Newton-Wigner position operator.