Conceptual Unification of Gravity and Quanta

Abstract

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.

The Geometry of Noncommutative Spacetimes

“…Moreover, there exist various competing models of noncommutative spacetimes, differing from the presented one both on the conceptual and the mathematical side (cf. for instance [5,58,59,[74][75][76][77][78][79]). However, the operational approach enhanced by noncommutative geometry à la Connes offers a unique testing ground to hunt for specific physical effects via the almost-commutative models.…”

Fundamental Problems in the Unification of Physics

“…In [19,32], we checked for consistency the finite version of the model and considered some of its observational consequences; in particular, model's probabilistic properties were discussed in [20]. Mathematical aspects of the model were dealt with in [21], and its physical content in [22]. Details of the Friedman cosmological model were computed in [11,24].…”

“…The generalized differential geometry is constructed in terms of the algebra A (and a subset of its derivations) out of which the standard space-time geometry can be recovered. In [22] we have shown that the usual space-time geometry is obtained by suitably averaging corresponding algebraic quantities. The quantum sector is given by the (regular) representation π p : A → B(H p ) of the algebra A in a Hilbert space H p ; B(H p ) denotes here bounded operators on H p , and H p = L 2 ( p ) where p is a suitable fiber of , p ∈ E (for details see [21,22]).…”

“…In [22] we have shown that the usual space-time geometry is obtained by suitably averaging corresponding algebraic quantities. The quantum sector is given by the (regular) representation π p : A → B(H p ) of the algebra A in a Hilbert space H p ; B(H p ) denotes here bounded operators on H p , and H p = L 2 ( p ) where p is a suitable fiber of , p ∈ E (for details see [21,22]). The fact that there exists the isomorphism between A and π(A), π = p∈E π p , allows us to translate the differential geometry constructed in terms of the algebra A (and its derivations) into a geometry defined in terms of bounded operators on the Hilbert bundle H = (H p ) p∈E .…”

A noncommutative Friedman cosmological model

“…where r = TrR, and R is the adjoint Ricci operator (all these quantities are defined in terms of the algebra A and its submodules V 1 and V 2 , for details see [2]). However, we modify Einstein's equation itself.…”