The Scalar Field Equation in the Presence of Signature Change

TL;DR

This paper investigates the behavior of a massless scalar field on a 2D manifold with changing signature, deriving unique solutions that satisfy wave equations through junction conditions and a variational approach.

Contribution

It introduces a novel analysis of scalar fields across signature-changing manifolds, establishing unique solutions via junction conditions and a variational framework.

Findings

Propagation depends only on the total size of spacelike regions.

Solutions are unique and satisfy the wave equation everywhere.

The approach unifies junction conditions with a variational principle.

Abstract

We consider the (massless) scalar field on a 2-dimensional manifold with metric that changes signature from Lorentzian to Euclidean. Requiring a conserved momentum in the spatially homogeneous case leads to a particular choice of propagation rule. The resulting mix of positive and negative frequencies depends only on the total (conformal) size of the spacelike regions and not on the detailed form of the metric. Reformulating the problem using junction conditions, we then show that the solutions obtained above are the unique ones which satisfy the natural distributional wave equation everywhere. We also give a variational approach, obtaining the same results from a natural Lagrangian. (PACS numbers 04.20.Cv and 02.40.+m.)