Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model

Details Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model

2004-10-02

arxiv.org/abs/gr-qc/0410010v1

Gen.Rel.Grav. 37 (2005) 541-555

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

20 pages. LaTex format

Scientific paper

10.1007/s10714-005-0041-z

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in {\it Gen. Rel. Grav.} (2004) {\bf 36}, 111-126. Generalized symmetries of the model are defined by a groupoid $\Gamma $ given by the action of a finite group on a space $E$. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on $\Gamma $. We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, "coparallelly" to the well known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on $\Gamma $, the model in its "quantum sector" is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its "gravity sector" such coderivations do exist. They are investigated.

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