RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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55. Leszek Pysiak
Time Flow in a Noncommutative Regime
International Journal of Theoretical Physics, vol. 46, pp. 16-30 (2007).
[abstract] [journal] [download]

Abstract:
We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra M of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on M depending on the state f . The state f defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair (M, f)() is a noncommutative probabilistic space and f can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.

56. M. Heller, Z. Odrzygóźdź, L. Pysiak, W. Sasin
Anatomy of Malicious Singularities
Journal of Mathematical Physics, vol. 48, pp. 092504-092511 (2007).

Abstract:
Abstract

57. Michael Heller, Leszek Pysiak and Wiesław Sasin
Conceptual Unification of Gravity and Quanta
International Journal of Theoretical Physics, vol. 46, pp. 2494-2512 (2007).
[abstract] [preprint] [journal] [download]

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 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.

58. Michael Heller, Zdzisław Odrzygóźdź, Leszek Pysiak, and Wiesław Sasin
Anatomy of malicious singularities
J. Math. Phys., vol. 48, p. 092504 (2007).
[abstract] [preprint] [journal]

Abstract:
As well known, the b boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation ρ, defined on the Cauchy completed total space math of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class [p0] related to the singularity remains in close contact with all other equivalence classes, i.e., if p0 ∊ cl[p] for every p ∊ E. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on math, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant.

59. Leszek M. Sokołowski
General relativity, gravitational energy and spin-two field
Int. J. Geom. Meth. Mod. Phys. , vol. 4, pp. 1-23 (2007).
[abstract] [preprint]

Abstract:
(Lectures given at the 42 Karpacz Winter School of Theoretical Physics, Lądek Zdrój, Poland, 6-11 February 2006, "Current Mathematical Topics in Gravitation and Cosmology"):

In my lectures I will deal with three seemingly unrelated problems: i) to what extent is general relativity exceptional among metric gravity theories? ii) is it possible to define gravitational energy density applying field-theory approach to gravity? and iii) can a consistent theory of a gravitationally interacting spin-two field be developed at all? The connecting link to them is the concept of a fundamental spin-2 field. A linear spin-2 field encounters insurmountable inconsistencies when coupled to gravity. After discussing the inconsistencies of any coupling of the linear spin-2 field to gravity, I exhibit the origin of the fact that a gauge invariant field has the variational metric stress tensor which is gauge dependent. I give a general theorem explaining under what conditions a symmetry of a field Lagrangian becomes also the symmetry of the stress tensor. It is a conclusion of the theorem that any attempt to define gravitational energy density in the framework of a field theory of gravity must fail. Finally I make a very brief introduction to basic concepts of how a certain kind of a necessarily nonlinear spin-2 field arises in a natural way from vacuum higher derivative gravity theories. This specific spin-2 field consistently interacts gravitationally.

60. Andrzej Woszczyna
Mechanika zaburzeń skalarnych w radiacyjnym wszechświecie
Prace Komisji Astrofizyki PAU, vol. 11, p. 117 (2007).
[pdf.pl]

Abstract:
Abstract

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