Feynman Path Integral of a charged anisotropic HO in crossed electric and magnetic fields. Alternative calculational methods

TL;DR

This paper presents two alternative methods for evaluating the Feynman path integral of a charged anisotropic harmonic oscillator in crossed electric and magnetic fields, simplifying calculations and resolving mathematical ambiguities.

Contribution

It introduces novel calculational techniques using Fourier series with zeta regularization and a rotating frame transformation to evaluate complex path integrals.

Findings

Both methods successfully evaluate the path integral.

The rotating frame method simplifies calculations by canceling Lorentz and Coriolis forces.

The approaches avoid issues like infinite normalization constants.

Abstract

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{$1^{\text{st}}$} method uses complex Fourier series and a regularization scheme via the Riemann-$\zeta$-function. The \emph{$2^{\text{nd}}$} 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 $\omega_{L}$. This fact simplifies considerably the calculation of the path integral.