Classical and Quantum Implications of the Causality Structure of Two-Dimensional Spacetimes with Degenerate Metrics

TL;DR

This paper explores how the causality structure of two-dimensional spacetimes with degenerate metrics affects wave solutions and quantum field theory, revealing novel features and obstructions to unitarity.

Contribution

It introduces a new analysis of causality in degenerate 2D spacetimes, showing how solutions can be determined despite the absence of a traditional Lorentzian Cauchy surface.

Findings

Unique global solutions can be determined from certain Cauchy data.

Obstructions prevent the construction of a bounded operator for asymptotic states.

Implications for the existence of a unitary quantum field theory are discussed.

Abstract

The causality structure of two-dimensional manifolds with degenerate metrics is analysed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain it is possible to uniquely determine a global solution (if it exists), satisfying well defined matching conditions at the degeneracy curve, from Cauchy data on certain spacelike curves in the Lorentzian region. In general, however, no global solution satisfying such matching conditions will be consistent with this data. Attention is drawn to a number of obstructions that arise prohibiting the construction of a bounded operator connecting asymptotic single particle states. The implications of these results for the existence of a unitary quantum field theory are discussed.