A stochastic derivation of the geodesic rule

TL;DR

This paper derives the geodesic rule for global defects by modeling the Goldstone field's evolution as a random walk on the vacuum manifold, using a Fokker-Planck equation and heat kernel analysis.

Contribution

It provides a stochastic derivation of the geodesic rule, connecting random field fluctuations to geometric evolution on the vacuum manifold.

Findings

Derivation of a Fokker-Planck equation for the Goldstone field

Identification of the heat kernel as the fundamental solution

Establishment of the geodesic rule from asymptotic behavior

Abstract

We argue that the geodesic rule, for global defects, is a consequence of the randomness of the values of the Goldstone field $\phi$ in each causally connected volume. As these volumes collide and coalescence, $\phi$ evolves by performing a random walk on the vacuum manifold $\mathcal{M}$. We derive a Fokker-Planck equation that describes the continuum limit of this process. Its fundamental solution is the heat kernel on $\mathcal{M}$, whose leading asymptotic behavior establishes the geodesic rule.