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WORLD SCIENTIFIC eBooks

The Path Integral Formalism

Published Sep 1, 2012• 0 citations
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Abstract

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):

DOI
10.1142/9789814397759_0021

Related papers

Relativistic quantum mechanics and path integral for Klein-Gordon equation

Gian Fabrizio De Angelis, Maurizio Serva

The subject of this paper is a path integral representations for the semigroup {exp(-t H_1)}{t≥0} generated by the quantum Hamiltonian H_1 of a relativistic spinless particle in an external electromagnetic field. The result is compared with the "Feynman-Kac" formula which holds for relativistic Schroedinger operators.

•Jan 1, 1995•0 citations

Path Integrals over Velocities in Quantum Mechanics

Dmitri M Gitman, Sh. M. Shvàrtsman

Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals over velocities of space-time coordinates. In the representations the integration over velocities is not restricted by any boundary conditions; matrices, which have to be inverted in course of doing Gaussian integrals, do not contain any derivatives in time, and spinor and isospinor structures of the propagators are given explicitly. One can define universal Gaussian and quasi-Gaussian integrals over velocities and rules of handling them. Such a technique allows one effectively calculate propagators in external fields. Thus, Klein-Gordon propagator is found in a constant homogeneous electromagnetic field and its combination with a plane wave field.

arXiv:hep-th/9310074•Oct 13, 1993•0 citations

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

M. Merad, Tahar Boudjedaa, L. Chétouani

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.

TURKISH JOURNAL OF PHYSICS•Apr 4, 2001•1 citation

Path integrals in relativistic quantum mechanics

Gian Fabrizio De Angelis, Maurizio Serva

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.

•Jan 1, 1994•0 citations

Covariant Relativistic Quantum Mechanics Analysis of a Linearly Accelerated Scalar Particle

Karol McDonald

A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.

•Jan 1, 2009•0 citations

Path integral treatment for relativistic particles in external fields and quantized plane wave

Baya Bentag, Tahar Boudjedaa, L. Chétouani

Abstract The dynamics of relativistic particles of spin 0 and 1/2, interacting with an external electromagnetic field and a quantized plane wave, is studied using the path integral framework. We take advantage of the existing properties of the interaction to introduce a delta functional in order to calculate Green's functions. This simply reduces the problem to two coupled oscillators. The energy spectrum and wave functions are calculated for the spin 0 case and the analogy with Jaynes‐Cummings model is made to derive the energy spectrum for the spin 1/2 case.

Annalen der Physik•Feb 1, 2000•0 citations
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):
Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals over velocities of space-time coordinates. In the representations the integration over velocities is not restricted by any boundary conditions; matrices, which have to be inverted in course of doing Gaussian integrals, do not contain any derivatives in time, and spinor and isospinor structures of the propagators are given explicitly. One can define universal Gaussian and quasi-Gaussian integrals over velocities and rules of handling them. Such a technique allows one effectively calculate propagators in external fields. Thus, Klein-Gordon propagator is found in a constant homogeneous electromagnetic field and its combination with a plane wave field.
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.
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.
A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.
Abstract The dynamics of relativistic particles of spin 0 and 1/2, interacting with an external electromagnetic field and a quantized plane wave, is studied using the path integral framework. We take advantage of the existing properties of the interaction to introduce a delta functional in order to calculate Green's functions. This simply reduces the problem to two coupled oscillators. The energy spectrum and wave functions are calculated for the spin 0 case and the analogy with Jaynes‐Cummings model is made to derive the energy spectrum for the spin 1/2 case.