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American Journal of Physics

Galilean transformation and the path integral propagator for a crossed electric and magnetic field

Carlos Farina de Souza, ALVARO DE SOUZA DUTRA

Published Apr 1, 1989• 7 citations
Publisher

Abstract

A Galilean transformation is made in the path integral formalism to obtain the Feynman propagator of a charged particle in the presence of a crossed uniform electromagnetic field in terms of the propagator of a charged particle in the presence of only a magnetic one. Also noted is how the result makes it possible to relate the initial propagator, with E and B, to that of the free particle.

DOI
10.1119/1.16041

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Canadian Journal of Physics•Sep 1, 1994•2 citations
A Galilean transformation is made in the path integral formalism to obtain the Feynman propagator of a charged particle in the presence of a crossed uniform electromagnetic field in terms of the propagator of a charged particle in the presence of only a magnetic one. Also noted is how the result makes it possible to relate the initial propagator, with E and B, to that of the free particle.
In the present paper the author evaluates the path integral of a charged anisotropic Harmonic Oscillator (HO) in crossed electric and magnetic fields by two alternative methods. Both methods enable a rather formal calculation and circumvent some mathematical delicate issues such as the occurrence of an infinite Normalization constant and ambiguities with path integral calculations when magnetic fields are present. The \emph{1st1^{\text{st}}1st} method uses complex Fourier series and a regularization scheme via the Riemann-ζζζ-function. The \emph{2nd2^{\text{nd}}2nd} method evaluates the path integral by transforming the Lagrangian to a uniformly rotating system. The latter method uses the fact that the Lorentz- and Coriolis force have the same functional form. Both forces cancel each other within the rotating system given that it rotates with Larmor frequency ωLω_{L}ωL​. This fact simplifies considerably the calculation of the path integral.