Spectral geometry of spacetime

TL;DR

This paper explores how spectral data and the Hadamard condition can characterize spacetime as a smooth, globally hyperbolic manifold, integrating spectral geometry with quantum field theory principles.

Contribution

It introduces a novel approach combining spectral triples, causal relationships, and the Hadamard condition to characterize spacetime geometry.

Findings

Spectral data can characterize spacetime geometry.

The Hadamard condition serves as a smoothness criterion.

A unified framework for spectral geometry and quantum field theory.

Abstract

Spacetime, understood as a globally hyperbolic manifold, may be characterized by spectral data using a 3+1 splitting into space and time, a description of space by spectral triples and by employing causal relationships, as proposed earlier. Here, it is proposed to use the Hadamard condition of quantum field theory as a smoothness principle.