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Journal of Physics Conference Series

Thermodynamics of relativistic Newton—Wigner particle in external potential field

Alexander S. Larkin, Vladimir S. Filinov

Published Nov 11, 2015• 2 citations
Publisher

Abstract

Thermodynamic properties of relativistic spinless particle described by the Klein–Gordon equation have been studied using the Newton–Wigner theory of particle in external potential field. Concept of Wiener path integral was extended on relativistic case. A new path integral Monte–Carlo method was developed for relativistic particle in external potential field. The bounds of applicability of available analytical approaches and related results have been specified by comparison with Monte–Carlo calculations. Developed path integral formalism can be directly extended on systems of many identical Newton–Wigner particles, which interact with external field and each other.

DOI
10.1088/1742-6596/653/1/012114

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Thermodynamic properties of relativistic spinless particle described by the Klein–Gordon equation have been studied using the Newton–Wigner theory of particle in external potential field. Concept of Wiener path integral was extended on relativistic case. A new path integral Monte–Carlo method was developed for relativistic particle in external potential field. The bounds of applicability of available analytical approaches and related results have been specified by comparison with Monte–Carlo calculations. Developed path integral formalism can be directly extended on systems of many identical Newton–Wigner particles, which interact with external field and each other.
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is discussed. A new method is proposed for approximations of the corresponding path integral. It is shown that calculation of this integral does not give the Green's function of the Klein-Gordon equation but leads to the propagator of a scalar field in the twocomponent Feshbach-Villars formalism. The difficulties in calculating path integrals in the presence of nontrivial external fields are discussed. 1. In the present paper, we consider some methodological questions of relativistic quantum mechanics associated with quantization of the classical Hamtltonian of a relativistic charged particle with rest mass m in external fields: electromagnetic Au(r, t) (p = 0, 1, 2, 3) and scalar V(r, t):
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.
This paper is a review of path integral representations for semigroups {exp(-t H_r/ħ)}{t≥0} where H_r's are relativistic quantum Hamiltonians. We consider three different cases: in the first one Hr is a relativistic Schrödinger operator, in the second is the Hamiltonian associated to Klein-Gordon equation and in the third is that coming from the Dirac one. The paths are trajectories of diffusion processes and, for the Dirac case, they differ from previous constructions based on jump Markov processes. All formulas describe one relativistic particle in an external static electromagnetic field therefore they are relativistic versions of the Feymann-Kac representation for Schrödinger semigroups.