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Modular theory, non-commutative geometry and quantum gravity. (English) Zbl 1221.46068

Summary: This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes’s non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics, and to the detailed discussion of the main foundational issues raised by the proposal.

MSC:

46L87 Noncommutative differential geometry
18F99 Categories in geometry and topology
58B34 Noncommutative geometry (à la Connes)
81R60 Noncommutative geometry in quantum theory
81T05 Axiomatic quantum field theory; operator algebras
83C65 Methods of noncommutative geometry in general relativity
46L60 Applications of selfadjoint operator algebras to physics
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