Noncommutative unification of general relativity and quantum mechanics

Abstract

In Gen. Rel. Grav. (36, 111-126 (2004); in press, gr-qc/0410010) we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry was developed in terms of a noncommutative algebra A defined on a transformation groupoid Γ given by the action of a group G on a space E. 1 that also in this case the model works well. The case is important since to obtain physical effects predicted by the model we should assume thet G is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalue equation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model "collapses" to the usual quantum mechanics.

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

“…In our model this fact is preserved in its space-time sector, but in its quantum sector one looks for the derivations. This fact was also signalled in one of our previous works [6]. It was Madore who first demonstrated that in some derivation based noncommutative geometries the metric could bne unique [8, p. 75].…”

“…In a series of works [4,5,6,7] we have also proposed a model aimed at a unification of relativity and quanta which differs from other models of this type by the ample use of the groupoid concept. We consider a transformation groupoid Γ = E × G, where E is typically the frame bundle over space-time M and G a group acting on E, and the noncommutative algebra A of compactly supported complex valued functions on Γ with convolution as multiplication.…”

Conceptual Unification of Gravity and Quanta

“…The differential geometry of the groupoid Γ is based on the algebra A and its derivations. Derivations are classified into three types: horizontal V 1 , vertical V 2 and inner V 3 (Heller et al, 2005c). The pair (A, V ), where V is a subset of the module Der(A) of all derivations of the algebra A, is called a differential algebra.…”

“…We first construct the corresponding differential geometry (connection, curvature, Einstein operator), and then we postulate that the eigenvalue equation for the Einstein operator should play the role of a generalized Einstein's equation (no energy-momentum tensor is assumed). This is an important modification with respect to (Heller et al, 2005c); its best justification being the result obtained in section 5, where the components of this equation are computed for the closed Friedman world model. It turns out (by comparison with the usual Friedman model) that the (generalized) eigenvalues of the Einstein operator should be interpreted as matter sources.…”

“…Thus the pair (M, ϕ) is both a "dynamic object" and a "probabilistic object" of the model. For the full discussion of these properties one should refer to (Heller et al, 2005b(Heller et al, , 2005c; here a new element has been added: if the state ϕ satisfies the Kubo-Martin-Schwinger (KMS) condition, it can be interpreted as describing a generalized equilibrium state (Rovelli, 1993;Martinetti and Rovelli, 2003). Therefore, on the fundamental level, dynamics, probability and at least some thermodynamic properties are encoded in the same mathematical structure.…”