Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

Details Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

2003-11-16

arxiv.org/abs/gr-qc/0311053v1

Gen.Rel.Grav. 36 (2004) 111-126

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

20 LaTex pages, General Relativity and Gravitation, in press

Scientific paper

10.1023/B:GERG.0000006697.80418.

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group G on a space E. We define the algebra of smooth complex valued functions on the groupoid, with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the groupoid algebra, and its correspondence with the standard quantum mechanics is established.

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