2001/04/02 by M. Heller, Michaël Heller, W. Sasin +6
Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/0104003
LaTex, 38 pages
arxiv created 2001/04/02 · openalex publication_date 2001/04/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We further develop a model unifying general relativity with quantum mechanics proposed in our earlier papers (J. Math. Phys. 38, 5840 (1998); 41, 5168 (2000)). The model is based on a noncommutative algebra A defined on a groupoid Γ= E × G where E is the total space of a fibre bundle over space-time and G a Lie group acting on E. In this paper, the algebra A is defined in such a way that the model works also if G is a noncompact group. Differential algebra based on derivations of this algebra is elaborated which allows us to construct a "noncommutative general relativity". The left regular representation of the algebra A in a Hilbert space leads to the quantum sector of our model. Its position and momentum representations are discussed in some detail. It is shown that the model has correct correspondence with the standard theories: with general relativity, by restricting the algebra A to a subset of its center; with quantum mechanics, by changing from the groupoid Γ to its algebroid; with classical mechanics, by changing from the groupoid Γ to its tangent groupoid. We also construct a noncommutative Fock space based on the proposed model.