Riemann-Zeta-Regularisation of Feynman Path Integrals

TL;DR

This paper introduces a novel method for regularizing divergent Feynman path integrals using the Riemann zeta-function, applied to a charged particle in an anisotropic harmonic oscillator with electromagnetic fields.

Contribution

It presents a new regularization technique for divergent path integrals based on zeta-function regularization, demonstrated on a specific quantum system.

Findings

Successful regularization of divergent Gaussian product integrals

Application of zeta-function regularization to quantum path integrals

Potential for broader use in quantum field theory calculations

Abstract

The Feynman Propagator of a charged particle confined to an anisotropic Harmonic Oscillator potential and moving in a crossed electromagnetic field is calculated in a conceptually new way. The calculation is based on the expansion of the path variable into a complex Fourier series. The path integral then becomes an infinite product of Gaussian integrals. This product is divergent. It turns out that we can regularize this product by using the zeta-function. It is a remarkable fact that the zeta-function is so well suited as a regularizator for divergent path integrals.