RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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25. B. Iochum, T. Masson, T. Schücker, A. Sitarz
Kappa-deformation and Spectral Triples
Acta Phys. Polon. Supp., vol. 4, p. 305 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
The aim of the paper is to answer the following question: does kappa -deformation fit into the framework of noncommutative geometry in the sense of spectral triples? Using a compactification of time, we get a discrete version of kappa -Minkowski deformation via C*-algebras of groups. The dynamical system of the underlying groups (including some Baumslag–Solitar groups) is used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation.

26. Michael Heller, Leszek Pysiak and Wiesław Sasin
Fundamental Problems in the Unification of Physics
Foundations of Physics, vol. 41, pp. 905-918 (2011).
[abstract] [preprint] [journal] [download]

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.

27. 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).
[abstract] [preprint] [journal]

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.

28. P. Olczykowski, A. Sitarz
On spectral action over Bieberbach manifolds
Acta Phys. Pol., B , vol. 42, p. 1189 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
We compute the leading terms of the spectral action for orientable three dimensional Bieberbach manifolds using two different methods: the Poisson summation formula and the perturbative expansion. Assuming that the cut-off function is not necessarily symmetric we find that the scale invariant part of the perturbative expansion might only differ from the spectral action of the flat three-torus by the value of the eta invariant.

29. Piotr T. Chru¶ciel, Michał Eckstein, Luc Nguyen and Sebastian J. Szybka
Existence of singularities in two-Kerr black holes
Class. Quantum Grav., vol. 28, p. 245017 (2011).
[abstract] [preprint] [journal]

Abstract:
We show that the angular momentum—area inequality 8π|J| ≤ A for weakly stable minimal surfaces would apply to I+-regular many-Kerr solutions, if any existed. Hence, we remove the undesirable hypothesis in the Hennig–Neugebauer proof of non-existence of well-behaved two-component solutions.
*supported by the grant from The John Templeton Foundation

30. Piotr T. Chru¶ciel, Sebastian J. Szybka
Stable causality of the Pomeransky-Senkov black holes
Adv. Theor. Math. Phys., vol. 15, pp. 175-178 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
We show stable causality of the Pomeransky-Senkov black rings.

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