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

### Authors:

- Michael Heller,
- Zdzisław Odrzygóźdź,
- Leszek Pysiak,
- Wiesław Sasin

### 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 \$\\textbackslash\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 \$\\textbackslash\mathcal\A\\$ , and its correspondence with the standard quantum mechanics is established.

- Record ID
- WUT115026
- Author
- Journal series
- General Relativity and Gravitation, ISSN 0001-7701, 1572-9532
- Issue year
- 2004
- Vol
- 36
- No
- 1
- Pages
- 111-126
- Keywords in English
- Differential Geometry, General relativity, Groupoid, Mathematical and Computational Physics, Noncommutative geometry, Quantum mechanics, Relativity and Cosmology, Unification theory
- DOI
- DOI:10.1023/B:GERG.0000006697.80418.01 Opening in a new tab
- URL
- http://link.springer.com/article/10.1023/B%3AGERG.0000006697.80418.01 Opening in a new tab
- Score (nominal)
- 0
- Score source
- journalList
- Publication indicators
- = 15; = 13; = 22
- Citation count
- 22

- Uniform Resource Identifier
- https://repo.pw.edu.pl/info/article/WUT115026/

- URN
`urn:pw-repo:WUT115026`

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or PerishOpening in a new tab system.