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TURKISH JOURNAL OF PHYSICS

Path Integral Treatment for Spinless Relativistic Equation in the Two Component Theory

M. Merad, Tahar Boudjedaa, L. Chétouani

Published Apr 4, 2001• 1 citation
OpenAlex

Abstract

In this paper we have set up a path integral formalism for Feshbach-Villars equation by using the fermionic Schwinger model for Pauli matrices which describe an isocharge symmetry. This choice is made in analogy with spin model and the coherent state representation is then used. We have also given a general method of treating the problem of vanishing scalar potential by reducing it the to non-relativistic case and then, via Foldy-Wouthuysen canonical transformation, an explicit solution is constructed. The free case and constant magnetic field interaction are explicitly exposed. In each cases the propagators are evaluated and the energy spectrum and the corresponding wave functions are deduced. P.A.C.S.03.65 Ca.Formalism P.A.C.S.03.65 Db.Functional analytical methods P.A.C.S.03.65 Pm.Relativistic wave equations P.A.C.S.03.65 Ge.Solutions of wave equations: bound states.

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The Path Integral Formalism

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In this paper we have set up a path integral formalism for Feshbach-Villars equation by using the fermionic Schwinger model for Pauli matrices which describe an isocharge symmetry. This choice is made in analogy with spin model and the coherent state representation is then used. We have also given a general method of treating the problem of vanishing scalar potential by reducing it the to non-relativistic case and then, via Foldy-Wouthuysen canonical transformation, an explicit solution is constructed. The free case and constant magnetic field interaction are explicitly exposed. In each cases the propagators are evaluated and the energy spectrum and the corresponding wave functions are deduced. P.A.C.S.03.65 Ca.Formalism P.A.C.S.03.65 Db.Functional analytical methods P.A.C.S.03.65 Pm.Relativistic wave equations P.A.C.S.03.65 Ge.Solutions of wave equations: bound states.
In this paper, we propose to construct a path integral formalism for a spinless particle in the Feshbach-Villars formalism. The Pauli matrices describing the isocharge symmetry are replaced by the bosonic Schwinger model and the coherent state representation is then used. The radial free case is determined via the Foldy-Wouthuysen canonical transformation and the important case of the Coulomb potential is investigated. The Green functions are evaluated and the energy spectrum and corresponding wave functions are exactly deduced. PACS. 03.65.Db – Functional analytical methods. PACS. 03.65.Pm – Relativistic wave equations. PACS. 03.65.Ge – Solutions of wave equations: bound states. I.
Exact Green’s function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in [Formula: see text]) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green’s function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease.
Exact solutions of the neutral spinning particle in interaction with rotating magnetic field and a scalar potential using path integral formalism have been found. The propagator is first written in the standard Feynman form by replacing the spin by a unit vector aligned along the polar and azimuthal directions (θ φ). Then we use the phase-space and rotations in the space of spin coherent states to simplify the calculations. Thus the exact energy spectrum with a corresponding wave functions are deduced.
We consider the relativistic quantum scattering of spinless particles in one-spatial dimension using the Feshbach-Villars formalism. We construct the general form of the scattering matrix, for symmetric and non-symmetric potentials, based on the symmetry properties of the Feshbach-Villars equation. Then, since in one dimension there are only two partial waves associated with even and odd parities, we show, in a simple and comprehensive way, how to describe this transmission-reflection problem using partial-wave decomposition. As an illustration, we also discuss the special case of scattering by a symmetric square-well potential.
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):