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Heller, Michaël; Odrzygóźdź, Zdzisław; Pysiak, Leszek; Sasin, Wiesław

doi:10.1023/a:1024429613783

Cited by 6 publications

(12 citation statements)

References 13 publications

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“…Singularities emerge from the noncommutative regime together with the macroscopic spacetime. This result remains in agreement with our previous works on classical singularities with the help of noncommutative methods Sasin, 1995b, 1999;Heller et al, 2003).…”

Section: Perspectivessupporting

confidence: 93%

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Conceptual Unification of Gravity and Quanta

2007

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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.

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Section: Perspectivessupporting

confidence: 93%

“…Of course, for commutative algebras all such derivations vanish. It is important to notice that the mapping Φ(a) = ad(a), for every a ∈ A, establishes the isomorphism between the algebra A and the space Inn(A) as Z-moduli (Heller et al, 2004).…”

Section: Differential Algebramentioning

confidence: 99%

Conceptual Unification of Gravity and Quanta

2007

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“…for details see (Heller et al, 2004a)]. Every a ∈ A generates a random operator r a = (π p (a)) p∈E .…”

Section: Algebra Of Random Operatorsmentioning

confidence: 99%

“…In (Heller et al 2004a) we have tested the program by constructing a simplified (but still mathematically interesting) model and computing many of its details, and in (Heller et al 2004b) we have discussed its observables with a special emphasis on the position and momentum observables. In the present work, we study its dynamic and probabilistic aspects.…”

Section: Introductionmentioning

confidence: 99%

“…We thus have the bundle (Ẽ, π M , M =Ẽ/G), and we can think of it as of the frame bundle, withG the Lorentz group, over spacetime M. To simplify our construction, we choose a finite subgroup G ofG and a cross section S : M →Ẽ of the above bundle (it need not be continuous). Then we define E = x∈M S(x)G. The free action of G (to the right) on E, defines the transformation groupoid structure on the Cartesian product Γ = E × G [for details see (Heller et al 2004a)]. The choice of the cross section S : M →Ẽ can be regarded as the choice of a gauge for our model.…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative Dynamics of Random Operators

2005

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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.

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Fundamental Problems in the Unification of Physics

2011

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We discuss the following problems, plaguing the present search for the "final theory": (1) How to find a mathematical structure rich enough to be suitably approximated by the mathematical structures of general relativity and quantum mechanics? (2) How to reconcile nonlocal phenomena of quantum mechanics with time honored causality and reality postulates? (3) Does the collapse of the wave function contain some hints concerning the future quantum gravity theory? (4) It seems that the final theory cannot avoid the problem of dynamics, and consequently the problem of time. What kind of time, if this theory is supposed to be background free? (5) Will the dynamics of the "final theory" be probabilistic? Quantum probability exhibits some essential differences as compared with classical probability; are they but variations of some more general probabilistic measure theory? (6) Do we need a radically new interpretation of quantum mechanics, or rather an entirely new theory of which the present quantum mechanics is an approximation? (7) If the final theory is to be background free, it should provide a mechanism of space-time generation. Should we try to explain not only the generation of space-time, but also the generation of its material content? (8) As far as the existence of the initial singularity is concerned, one usually expects either "yes" or "not" answers from the final theory. However, if the mathematical structure of the future theory is supposed to be truly more general that the mathematical structures of M. Heller correspondence address: ul. Powstańców Warszawy 13/94, Found Phys (2011) 41: 905-918 the present general relativity and quantum mechanics, is a "third answer" possible?Could this third answer be related to the probabilistic character of the final theory? We discuss these questions in the framework of a working model unifying gravity and quanta. The analysis reveals unexpected aspects of these rather wildly discussed issues.

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Cited by 6 publications

References 13 publications

Conceptual Unification of Gravity and Quanta

Conceptual Unification of Gravity and Quanta

Noncommutative Dynamics of Random Operators

Fundamental Problems in the Unification of Physics

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