7. 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.

8. M. Heller, L. Pysiak, W. Sasin
Inner Geometry of Random Operators
Demonstratio Mathematica, vol. 39, pp. 971-978 (2006).

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Abstract

9. L. Pysiak, M. Heller, Z. Odrzygóźdź, W. Sasin
Observables in a Noncommutative Approach to the Unification of Quanta and Gravity: A Finite Model
General Relativity and Gravitation, vol. 37, pp. 541-555 (2005).

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10. Michael Heller, Leszek Pysiak and Wiesłw Sasin
Noncommutative Dynamics of Random Operators
International Journal of Theoretical Physics, vol. 44, pp. 619-628 (2005).
[abstract] [preprint] [journal] [download]

Abstract:
We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А 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 ∊ А 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 А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, 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 (А, 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.

11. Michael Heller, Leszek Pysiak, and Wiesław Sasin
Noncommutative unification of general relativity and quantum mechanics
J. Math. Phys., vol. 46, p. 122501 (2005).
[abstract] [preprint] [journal]

Abstract:
We present a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry is developed in terms of a noncommutative algebra A which is defined on a transformation groupoid Γ given by the action of a noncompact group G on the total space E of a principal fiber bundle over space-time M. The case is important since to obtain physical effects predicted by the model we should assume that 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.

12. M. Heller, Z. Odrzygóźdź, L. Pysiak, W. Sasin
Quantum Groupoids of the Final Type and Quantization on Orbit Spaces
Demonstratio Mathematica , vol. 37, pp. 671-678 (2004).

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