A Unique Continuation Result for Klein-Gordon Bisolutions on a 2-dimensional Cylinder

TL;DR

This paper establishes a new unique continuation property for Klein-Gordon bisolutions on a 2D cylinder, showing that local vanishing implies global invariance, with implications for quantum field theory.

Contribution

It introduces a novel unique continuation theorem for Klein-Gordon bisolutions on a 2D cylinder, utilizing Beurling's interpolation methods.

Findings

Bisolutions vanishing near a large surface are globally invariant.

The proof employs Beurling's interpolation theory.

Application to quantum field theory on 2D cylinders is forthcoming.

Abstract

We prove a novel unique continuation result for weak bisolutions to the massive Klein-Gordon equation on a 2-dimensional cylinder M. Namely, if such a bisolution vanishes in a neighbourhood of a `sufficiently large' portion of a 2-dimensional surface lying parallel to the diagonal in the product manifold of M with itself, then it is (globally) translationally invariant. The proof makes use of methods drawn from Beurling's theory of interpolation. An application of our result to quantum field theory on 2-dimensional cylinder spacetimes will appear elsewhere.