Local covariant quantum field theory over spectral geometries

TL;DR

This paper develops a framework combining noncommutative geometry with covariant quantum field theory, defining quantum fields over spectral geometries and ensuring covariance across geometries, with implications for quantum gravity.

Contribution

It introduces globally hyperbolic spectral geometries and a covariant functorial quantum field theory framework over these geometries, extending the concept to noncommutative spacetimes.

Findings

Defines globally hyperbolic spectral triples for noncommutative spacetimes

Constructs a covariant functor assigning quantum fields to spectral geometries

Suggests a dynamical selection of geometries in quantum gravity

Abstract

A framework which combines ideas from Connes' noncommutative geometry, or spectral geometry, with recent ideas on generally covariant quantum field theory, is proposed in the present work. A certain type of spectral geometries modelling (possibly noncommutative) globally hyperbolic spacetimes is introduced in terms of so-called globally hyperbolic spectral triples. The concept is further generalized to a category of globally hyperbolic spectral geometries whose morphisms describe the generalization of isometric embeddings. Then a local generally covariant quantum field theory is introduced as a covariant functor between such a category of globally hyperbolic spectral geometries and the category of involutive algebras (or *-algebras). Thus, a local covariant quantum field theory over spectral geometries assigns quantum fields not just to a single noncommutative geometry (or noncommutative spacetime), but simultaneously to ``all'' spectral geometries, while respecting the covariance principle demanding that quantum field theories over isomorphic spectral geometries should also be isomorphic. It is suggested that in a quantum theory of gravity a particular class of globally hyperbolic spectral geometries is selected through a dynamical coupling of geometry and matter compatible with the covariance principle.