Abstract
Mathematical core of quantum mechanics is the theory of unitary representations of symmetries of physical systems. We argue that quantum behavior is a natural result of extraction of “observable” information about systems containing “unobservable” elements in their descriptions. Since our aim is physics where the choice between finite and infinite descriptions can not have any empirical consequences, we consider the problem in the finite background. Besides, there are many indications from observations — from the lepton mixing data, for example — that finite groups underly phenomena in particle physics at the deep level. The “finite” approach allows to reduce any quantum dynamics to the simple permutation dynamics and, thus, to express quantum observables in terms of permutation invariants of symmetry groups and their integer characteristics such as sizes of conjugate classes, sizes of group orbits, class coefficients, and dimensions of representations. Our study has been accompanied by computations with finite groups, their representations and invariants. We have used both our C implementation of algorithms for working with groups and computer algebra system GAP.
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References
Kornyak, V.V.: Quantization in discrete dynamical systems. J. Math. Sci. 168(3), 390–397 (2010)
Kornyak, V.V.: Structural and symmetry analysis of discrete dynamical systems. In: Cellular Automata, pp. 1–45. Nova Science Publishers Inc., New York (2010), http://arxiv.org/abs/1006.1754
Hall Jr., M.: The Theory of Groups. Macmillan, New York (1959)
Kirillov, A.A.: Elements of the Theory of Representations. Springer, Heidelberg (1976)
Nakamura, K., et al.: (Particle Data Group): The review of particle physics. J. Phys. G 37, 075021, 1–1422 (2010)
Smirnov, A.Y.: Discrete Symmetries and Models of Flavor Mixing, p. 14 (2011); arXiv:1103.3461
Harrison, P.F., Perkins, D.H., Scott, W.G.: Tri-bimaximal mixing and the neutrino oscillation data. Phys. Lett. B 530, 167 (2002); arXiv: hep-ph/0202074
Harrison, P.F., Scott, W.G.: Permutation symmetry, Tri-bimaximal neutrino mixing and the S3 group characters. Phys. Lett. B 557, 76 (2003); arXiv: hep-ph/0302025
Ishimori, H., Kobayashi, T., Ohki, H., Okada, H., Shimizu, Y., Tanimoto, M.: Non-abelian discrete symmetries in particle physics. Prog. Theor. Phys. Suppl. 183, 1–173 (2010); arXiv:1003.3552
Ludl, P.O.: Systematic Analysis of Finite Family Symmetry Groups and Their Application to the Lepton Sector, arXiv:0907.5587
Blum, A., Hagedorn, C.: The Cabibbo Angle in a Supersymmetric D14 Model. Nucl. Phys. B 821, 327–353 (2009)
Altarelli, G., Feruglio, F.: Discrete flavor symmetries and models of neutrino mixing. Rev. Mod. Phys. 82(3), 2701–2729 (2010)
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Kornyak, V.V. (2011). Computations in Finite Groups and Quantum Physics. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_21
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DOI: https://doi.org/10.1007/978-3-642-23568-9_21
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