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Noncommutative unification of general relativity and quantum mechanics. (English summary) J. Math. Phys. 46 (2005), no. 12, 122501, 15 pp. The paper is addressed to a unified mathematical description of general relativity and quantum mechanics. The presented setup is based on ideas of noncommutative geometry in terms of groupoids. More specifically, the authors study the geometry of a noncommutative algebra that is defined on a transformation groupoid. The latter is assumed to be given by the action of a non-compact group on the total space of a given principal G-bundle over space-time. Within this setting the authors show how to obtain a specific generalization of Einstein’s field equation as an eigenvalue equation for the Ricci operator. Moreover, in the quantum sector the relevant operators turn out to be random. The authors also study the “commutative limit ” and show how to recover ordinary gravity and quantum mechanics. The interesting feature here is that this limit is related to the measurement of some observable

Year: 2013

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