Doppler shift in semi-Riemannian signature and the non-uniqueness of the Krein space of spinors

TL;DR

This paper investigates how different spacetime decompositions in semi-Riemannian spin manifolds lead to non-equivalent spinor space norms, affecting the structure of the associated Krein spaces and implications for noncommutative geometry.

Contribution

It provides a necessary and sufficient condition for norm equivalence based on a generalized Doppler shift, highlighting non-uniqueness issues in the Krein space of spinors.

Findings

Different spacetime splittings yield non-equivalent norms on spinor sections.

A generalized Doppler shift characterizes when two splittings define equivalent norms.

Implications for the noncommutative geometry framework are discussed.

Abstract

We give examples illustrating the fact that the different space/time splittings of the tangent bundle of a semi-Riemannian spin manifold give rise to non-equivalent norms on the space of compactly supported sections of the spinor bundle, and as a result, to different completions. We give a necessary and sufficient condition for two space/time splittings to define equivalent norms in terms of a generalized Doppler shift between maximal negative definite subspaces. We explore some consequences for the Noncommutative Geometry program.