Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 067, 47 pages      arXiv:1007.4094      https://doi.org/10.3842/SIGMA.2010.067
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Modular Theory, Non-Commutative Geometry and Quantum Gravity

Paolo Bertozzini a, Roberto Conti b and Wicharn Lewkeeratiyutkul c
a) Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand
b) Dipartimento di Scienze, Università di Chieti-Pescara ''G. D'Annunzio'', Viale Pindaro 42, I-65127 Pescara, Italy
c) Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Received March 30, 2010, in final form July 26, 2010; Published online August 19, 2010

Abstract
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 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 a 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.

Key words: modular theory; non-commutative geometry; spectral triple; category theory; quantum physics; space-time.

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