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.
We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra A on the groupoid Γ = E ×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of A, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra A, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra M of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which * Correspondence address: ul. Powstańców Warszawy 13/94, 33-110 Tarnów, Poland. E-mail: mheller@wsd.tarnow.pl 1 depends on the state ϕ. The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state ϕ satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra A, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not "feel" singularities.
We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra A on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ. We show that every a ∈ A defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra A which can be used to define a state dependent dynamics; i.e., the pair (A, ϕ), where ϕ is a state on A, is a "dynamic object". Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (A, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the * Vatican Observatory, V-00120 Vatican City State. 1 usual quantum mechanics.We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra A on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ. We show that every a ∈ A defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra A which can be used to define a state dependent dynamics; i.e., the pair (A, ϕ), where ϕ is a state on A, is a "dynamic object". Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (A, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.
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 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.
We define and investigate the concept of the groupoid representation induced by a representation of the isotropy subgroupoid.Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity theorem for such representations which is a generalization of the classical Mackey's theorem known from the theory of group representations.
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