Action and Vertices in the Worldine Formalism

TL;DR

This paper explores the worldline formalism to analyze local interactions and vertex factors in both flat and curved spacetimes, revealing how gravitational fields influence momentum conservation and proposing a semi-classical quantum gravity modification.

Contribution

It introduces a covariant Fourier transform in the worldline formalism and studies the effects of gravitational fields on vertex factors and momentum conservation.

Findings

Vertex factors in curved spacetime match flat spacetime results.

Loop momenta are determined solely by external momenta in gravitational fields.

A Lorentz invariant de-localization modifies the vertex without affecting momentum conservation.

Abstract

Utilizing the worldline formalism we study the effects of demanding local interactions on the corresponding vertex factor. We begin by reviewing the familiar case of a relativistic particle in Minkowksi space, showing that localization gives rise to the standard conservation of momentum at each vertex. A generalization to curved geometry is then studied and a notion of covariant Fourier transform is introduced to aid in the analysis. The vertex factor is found to coincide with the one derived for flat spacetime. Next, we apply this formalism to a loop immersed in a gravitational field, demonstrating that the loop momenta is determined entirely by the external momenta. Finally, we postulate that the semi--classical effects of quantum gravity on the Feynman path integral can be accounted for by a modification to the vertex factor which de-localizes the vertex. We study one particular Lorentz invariant de-localization which, remarkably, has no effect on conservation of vertex momenta.