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

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 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.

arXiv comment20 LaTex pages, General Relativity and Gravitation, in press

References in corpus (9)

• Gravity coupled with matter and foundation of non-commutative geometry
• Gravity in Non-Commutative Geometry
• Quantum Groups and Noncommutative Geometry
• Quantum Space-time and Classical Gravity
• Gravity from Noncommutative Geometry
• Towards Noncommutative Quantization of Gravity
• Topology at the Planck Length
• Emergence of Time
• State Vector Reduction as a Shadow of a Noncommutative Dynamics

Cited by in corpus (7)

• Noncommutative Dynamics of Random Operators
• Noncommutative Unification of General Relativity and Quantum Mechanics
• Conceptual Unification of Gravity and Quanta
• Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model
• Canonical Noncommutativity Algebra for the Tetrad Field in General Relativity
• Anatomy of Malicious Singularities
• Geometry and General Relativity in the Groupoid Model with a Finite Structure Group