Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model |
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Authors: | Michael Heller Zdzislaw Odrzygóźdź Leszek Pysiak Wieslaw Sasin |
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Institution: | (1) Vatican Observatory, V-00120 Vatican City State;(2) ul. Powsta ców Warszawy 13/94, 33-110 Tarnów, Poland;(3) Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland |
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Abstract: | 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 on a space E. We define the algebra
of smooth complex valued functions on , 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 algebra
, and its correspondence with the standard quantum mechanics is established. |
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Keywords: | General relativity quantum mechanics unification theory noncommutative geometry groupoid |
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