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 Γ given by the action of a finite group on a space E. We define the algebra \(\mathcal{A}\) of smooth complex valued functions on Γ, 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 \(\mathcal{A}\), and its correspondence with the standard quantum mechanics is established.
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REFERENCES
Chamseddine, A. H., Felder, G., and Fröhlich, J. (1993). Commun. Math. Phys. 155, 205–217.
Sitarz, A. (1994). Class. Quant. Grav. 11, 2127–2134.
Connes, A. (1996). Commun. Math. Phys. 182, 155–176.
Madore, J. and Mourad, J. (1998). J. Math. Phys. 39, 424–442.
Madore, J. and Saeger, L. A. (1998). Class. Quant. Grav. 15, 811–826.
Heller, M., Sasin, W., and Lambert, D. (1997). J. Math. Phys. 38, 5840–5853.
Heller, M. and Sasin, W. (1998). Phys. Lett. A 250, 48–54.
Heller, M. and Sasin, W. (1999). Int. J. Theor. Phys. 38, 1619–1642.
Heller, M., Sasin, W., and Odrzygó?d?, Z. (2000). J. Math. Phys. 41, 5168–5179.
Paterson, A. L. (1999). Groupoids, Inverse Semigroups and Their Operator Algebras, Birkhäuser, Boston, Massachusetts.
Heller, M. and Sasin, W. (1995). J. Math. Phys. 36, 3644–3662.
Madore, J. (1999). An Introduction to Noncommutative Differential Geometry and Its Physical Applications, 2nd ed., Cambridge University Press, Cambridge, United Kingdom.
Majid, S. (1995). Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, United Kingdom.
Majid, S. (2000). J. Math. Phys. 41, 3892–3942.
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Heller, M., Odrzygóźdź, Z., Pysiak, L. et al. Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model. General Relativity and Gravitation 36, 111–126 (2004). https://doi.org/10.1023/B:GERG.0000006697.80418.01
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DOI: https://doi.org/10.1023/B:GERG.0000006697.80418.01