Local covariant quantum field theory over spectral geometries

Details Local covariant quantum field theory over spectral geometries Local covariant quantum field theory over spectral geometries

2004-05-11

arxiv.org/abs/gr-qc/0405057v1

Class.Quant.Grav. 21 (2004) 5299-5316

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

21 pages, 2 figures

Scientific paper

10.1088/0264-9381/21/23/001

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.

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