1. Heller, Michael; Pysiak, Leszek; Sasin, Wiesław
Quantum effects in a noncommutative Friedman world model
Canadian Journal of Physics, vol. 90, pp. 223-228 (2012).
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Abstract:
We present a noncommutative version of the closed Friedman world model and show how its classical space–time geometry can be expressed in terms of typically quantum mathematical structures, namely in terms of an operator algebra on a family of Hilbert spaces. The operator algebra can be completed to the von Neumann algebra , but the geometry cannot be prolonged from to . This mathematical fact is a stumbling block in creating full quantum gravity theory. Two effects appearing in this model, generation of matter and probabilistic properties of singularities, are also discussed. *supported by the grant from The John Templeton Foundation

2. Michael Heller, Leszek Pysiak and Wiesław Sasin
Fundamental Problems in the Unification of Physics
Foundations of Physics, vol. 41, pp. 905-918 (2011).
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Abstract:
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 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.

3. Michael Heller, Leszek Pysiak, and Wiesław Sasin
Geometry of non-Hausdorff spaces and its significance for physics
J. Math. Phys., vol. 52, p. 043506 (2011).
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Abstract:
Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and momentum operators can be defined, and they satisfy a commutation relation which, in the suitable limiting case, reproduces the Heisenberg indeterminacy relation. It should be emphasized that this is neither an additional assumption nor an effect of a quantization process, but simply the consequence of a purely geometric analysis.

4. Michael Heller, Zdzisław Odrzygóźdź, Leszek Pysiak and Wiesław Sasin
Gravitational Aharonov-Bohm Effect
International Journal of Theoretical Physics, vol. 47, pp. 2566-2575 (2008).
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Abstract:
We study a purely gravitational Aharonov-Bohm effect. The space-time curvature is concentrated in the quasiregular singularity of a cosmic string, outside of which space-time is (locally) flat. The symmetries of this field configuration are described by the groupoid symmetries rather than by the usual group symmetries. The groupoid in question is formed by homotopy classes of piecewise smooth paths in the cosmic string region. A gravitational counterpart of the Aharonov-Bohm effect occurs if the symmetry of the system, with respect to the groupoid action, is broken down.

5. Leszek Pysiak
Time Flow in a Noncommutative Regime
International Journal of Theoretical Physics, vol. 46, pp. 16-30 (2007).
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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.

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

Abstract:
Abstract