Spectral Geometry and Causality

TL;DR

This paper demonstrates that spectral data can encode the structure of globally hyperbolic Lorentzian manifolds, revealing causal relationships and offering a promising approach for quantum gravity theories.

Contribution

It shows that spectral data can fully characterize Lorentzian manifolds and introduces causal relationships into spectral geometry, advancing the spectral approach to quantum gravity.

Findings

Spectral data can encode Lorentzian manifold structures.

Causal relationships emerge naturally in spectral geometry.

Spectral approach offers an efficient description of spacetime.

Abstract

For a physical interpretation of a theory of quantum gravity, it is necessary to recover classical spacetime, at least approximately. However, quantum gravity may eventually provide classical spacetimes by giving spectral data similar to those appearing in noncommutative geometry, rather than by giving directly a spacetime manifold. It is shown that a globally hyperbolic Lorentzian manifold can be given by spectral data. A new phenomenon in the context of spectral geometry is observed: causal relationships. The employment of the causal relationships of spectral data is shown to lead to a highly efficient description of Lorentzian manifolds, indicating the possible usefulness of this approach. Connections to free quantum field theory are discussed for both motivation and physical interpretation. It is conjectured that the necessary spectral data can be generically obtained from an effective field theory having the fundamental structures of generalized quantum mechanics: a decoherence functional and a choice of histories.