Massless scalar fields in 1+1 dimensions and Krein spaces

TL;DR

This paper explores the Krein space formulation for massless scalar fields in 1+1 dimensions, establishing convergence criteria and analyzing the implications for the interpretation of Fourier components as probability amplitudes.

Contribution

It introduces a Krein space realization for the massless scalar field and examines the mathematical properties and physical interpretations within this framework.

Findings

Established convergence criteria for the Krein space

Completed the space of test functions with Krein topology

Found the interpretation of Fourier components as probability amplitudes is lost

Abstract

We consider the Krein realization of the Hilbert space for a massless scalar field in 1+1 dimensions. We find convergence criteria and the completion of the space of test functions ${\cal S}$ with the topology induced by the Krein scalar product. Finally, we show that the interpretation for the Fourier components as probability amplitudes for the momentum operator is lost in this case.