RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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55. Leszek M. Sokołowski
Metric gravity theories and cosmology: I. Physical interpretation and viability.
Class. Quantum Grav., vol. 24, pp. 3391-3411 (2007).
[abstract] [preprint]

Abstract:
We critically review some concepts underlying current applications of gravity theories with Lagrangians depending on the full Riemann tensor to cosmology. We argue that it is impossible to reconstruct the underlying Lagrangian from the observational data: the Robertson-Walker spacetime is so simple and "flexible" that any cosmic evolution may be fitted by infinite number of Lagrangians. Confrontation of a solution with the astronomical data is obstructed by the existence of many frames of dynamical variables and the fact that initial data for the gravitational triplet depend on which frame is minimally coupled to ordinary matter. Prior to any application it is necessary to establish physical contents and viability of a given gravity theory. A theory may be viable only if it has a stable ground state. We provide a method of checking the stability and show in eleven examples that it works effectively.

56. Leszek M. Sokołowski
Metric gravity theories and cosmology: II. Stability of a ground state in f(R) theories.
Class. Quantum Grav., vol. 24, pp. 3713-3734 (2007).
[abstract] [preprint]

Abstract:
A fundamental criterion of viability of any gravity theory is existence of a stable ground-state solution being either Minkowski, dS or AdS space. Stability of the ground state is independent of which frame is physical. In general, a given theory has multiple ground states and splits into independent physical sectors. All metric gravity theories with the Lagrangian being a function of Ricci tensor are dynamically equivalent to Einstein gravity with a source and this allows us to study the stability problem using methods developed in GR. We apply these methods to f(R) theories. As is shown in 13 cases of Lagrangians the stability criterion works simply and effectively whenever the curvature of the ground state is determined. An infinite number of gravity theories have a stable ground state and further viability criteria are necessary.

57. 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.

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

Abstract:
Abstract

59. 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.

60. 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.

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