Physics of Particles and Nuclei

, Volume 44, Issue 1, pp 47–91 | Cite as

Classical and quantum discrete dynamical systems

  • V. V. Kornyak


We study deterministic and quantum dynamics from a constructive “finite” point of view, since the introduction of the continuum or other actual infinities in physics poses severe conceptual and technical difficulties, and while all of these concepts are not really needed in physics, which is in fact an empirical science. Particular attention is paid to the symmetry properties of discrete systems. For a consistent description of the symmetries of dynamical systems at different time instants and the symmetries of various parts of such systems, we introduce discrete analogs of gauge connections. These gauge structures are particularly important to describe the quantum behavior. The symmetries govern the fundamental properties of the behavior of dynamical systems. In particular, we can show that the moving soliton-like structures are inevitable in a deterministic (classical) dynamical system, whose symmetry group breaks the set of states into a finite number of orbits of the group. We demonstrate that the quantum behavior is a natural consequence of symmetries of dynamical systems. This behavior is a result of the fundamental inability to trace the identity of indistinguish-able objects during their evolution. Information is only available on invariant statements and values related with such objects. Using general mathematical arguments, any quantum dynamics can be shown to reduce to a sequence of permutations. The quantum interferences occur in the invariant subspaces of permutation representations of the symmetry groups of dynamical systems. The observables can be expressed in terms of permutation invariants. We also show that in order to describe quantum phenomena it is sufficient to use cyclotomic fields—the minimal extensions of natural numbers suitable for quantum mechanics, instead of a non-constructive number system—the field of complex numbers. The finite groups of symmetries play the central role in this review. In physics there is an additional reason for such groups to be of interest. Numerous experiments and observations in particle physics point to the importance of finite groups of relatively low orders in a number of fundamental processes. The origin of these groups has no explanation within presently recognized theories, such as the Standard Model.


Finite Group Cellular Automaton Irreducible Component Invariant Subspace Primitive Root 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries and Models of Neutrino Mixing,” Rev. Mod. Phys. 82(3), 2701–2729 (2010).ADSCrossRefGoogle Scholar
  2. 2.
    G. G. Athanasiu, E. G. Floratos, and S. Nicolis, “Holomorphic Quantization on the Torus and Finite Quantum Mechanics,” J. Phys. A: Math. Gen. 29(21), 6737–6745 (1996).MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    J. C. Baez, “Spin Foam Models,” Class. Quant. Grav. 15, 1827–1858 (1998).MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    A. Blum and C. Hagedorn, “The Cabibbo Angle in a Supersymmetric D14 Model,” Nucl. Phys. B 821, 327–353 (2009).ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    P. J. Cameron, Permutation Groups (Cambridge Univ. Press, 1999).Google Scholar
  6. 6.
    J. H. Conway et al., Atlas of Finite Groups (Clarendon Press, Oxford, 1985).zbMATHGoogle Scholar
  7. 7.
    J. D. Dixon, B. Mortimer, Permutation Groups (Springer, Berlin, 1996).zbMATHCrossRefGoogle Scholar
  8. 8.
    R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals (Mir, Moscow, 1968) [in Russian].Google Scholar
  9. 9.
    E. G. Floratos and G. K. Leontaris, “Uncertainty Relation and Non-Dispersive States in Finite Quantum Mechanics,” Phys. Lett. B 412(1–2), 35–41 (1997).MathSciNetADSGoogle Scholar
  10. 10.
    E. G. Floratos and S. Nicolis, “Non-Commutative Solitons in Finite Quantum Mechanics,” Nucl. Phys. B 119, 947–949 (2003).zbMATHCrossRefGoogle Scholar
  11. 11.
    GAP—Groups, Algorithms, Programming—a System for Computational Discrete Algebra.
  12. 12.
    M. Hall, Group Theory (IL, Moscow, 1962) [in Russian].Google Scholar
  13. 13.
    P. F. Harrison, D. H. Perkins, and W. G. Scott, “Tri-Bimaximal Mixing and the Neutrino Oscillation Data,” Phys. Lett. B 530, 167 (2002) (Preprint arXiv: hep-ph/0202074).ADSCrossRefGoogle Scholar
  14. 14.
    P. F. Harrison and W. G. Scott, “Permutation Symmetry, Tri-Bimaximal Neutrino Mixing and the S3 Group Characters,” Phys. Lett. B 557, 76 (2003) (Preprint arXiv: hep-ph/0302025).MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    D. F. Holt, B. Eick, and E. A. O’Brien, Handbook of Computational Group Theory (Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2005).zbMATHCrossRefGoogle Scholar
  16. 16.
    G.’ t Hooft, “Dimensional Reduction in Quantum Gravity,” Preprint THU-93/26 (Utrecht, 1993) (Preprint arXiv: gr-qc/9310006).Google Scholar
  17. 17.
    G.’ t Hooft, “Quantum Gravity as a Dissipative Deterministic System,” Class. Quant. Grav. 16, 3263 (1999).ADSzbMATHCrossRefGoogle Scholar
  18. 18.
    G.’ t Hooft, “The Mathematical Basis for Deterministic Quantum Mechanics”, Preprint ITP-UU-06/14, Preprint SPIN-06/12 (Preprint arXiv: quant-ph/0604008).Google Scholar
  19. 19.
    H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, and M. Tanimoto, “Non-Abelian Discrete Symmetries in Particle Physics,” Prog. Theor. Phys. Suppl. 183, 1–173 (2010) (Preprint arXiv: 1003.3552).ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Th. Juffmann, A. Milic, M. Mullneritsch, et al., “Real-Time Single-Molecule Imaging of Quantum Interference,” Nature Nanotechnology (2012). doi 10.1038/nnano.2012.34Google Scholar
  21. 21.
    A. A. Kirillov, Elements of Representation Theory (Nauka, Moscow, 1972) [in Russian].Google Scholar
  22. 22.
    F. Klein, Vorlesungen über das Ikosaeder (Teubner, Leipzig, 1884), Russian translation: F. Klein, Lectures on Icosahedron and Solution to Fifth-Degree Equations (Nauka, Moscow, 1989) [in Russian].Google Scholar
  23. 23.
    Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, 1 (Nauka, Moscow, 1981) [in Russian].Google Scholar
  24. 24.
    V. V. Kornyak, “On Compatibility of Discrete Relations”, Lect. Notes Comp. Sci. 3718, 272–284 (2005). MathSciNetCrossRefGoogle Scholar
  25. 25.
    V. V. Kornyak, “Discrete Relations on Abstract Simplicial Complexes”, Prog. Comp. Soft. 32(2), 84–89 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    V. V. Kornyak, “Cellular Automata with Symmetric Local Rules”, Lect. Notes Comp. Sci. 4194, 240–250 (2006). MathSciNetCrossRefGoogle Scholar
  27. 27.
    V. V. Kornyak, “Symmetries of Discrete Dynamical Systems,” Teor. Fiz. (Izd. Samara State Univ., Samara) 8, 158–169 (2007).Google Scholar
  28. 28.
    V. V. Kornyak, “Discrete Symmetries and Dynamics of Cellular Automata,” in Computer Algebra Systems in Teaching and Research (Wydawnictwo Akademii Podladkiej, Siedlce, 2007), pp. 192–201.Google Scholar
  29. 29.
    V. V. Kornyak, “Symmetró Analysis of Discrete Dynamical Systems,” in Computer Algebra and Differential Equations, Acta Academiae Aboensis, Ser. B 67(2), 154–166 (2007).MathSciNetGoogle Scholar
  30. 30.
    V. V. Kornyak, “Symmetric Cellular Automata”, Prog. Comp. Soft. 33(2), 87–93 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    V. V. Kornyak, “Symmetries and Dynamics of Discrete Systems”, Lect. Notes Comp. Sci. 4770, 236–251 (2007). CrossRefGoogle Scholar
  32. 32.
    V. V. Kornyak, “Simulation of Nanostructures by Discrete Dynamical Systems,” in Nuclear Physics and Nanotechnologies: Nuclear Physics Aspects of Formation, Study, and Application of Nanostructures, Ed. by A.N. Sisakyan (JINR, Dubna, 2008), pp. 320–338 [in Russian].Google Scholar
  33. 33.
    V. V. Kornyak, “Gauge Invariance in Discrete Models,” in Computer Algebra Systems in Teaching and Research (Wydawnictwo Akademii Podladkiej, Siedlce, 2009), pp. 39–48.Google Scholar
  34. 34.
    V. V. Kornyak, “Discrete Dynamical Systems with Symmetries: Computer Analysis,” Prog. Comp. Soft. 34(2), 84–94 (2008). MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    V. V. Kornyak, “Symmetries, Gauge Invariance, and Quantization in Discrete Models,” Num. Meth. Prog. Adv. Comp. 10, 415–421 (2009). Google Scholar
  36. 36.
    V. V. Kornyak, “Discrete Symmetry Analysis of Lattice Systems,” Phys. Part. Nuc. Let. 6(7), 554–558 (2009). CrossRefGoogle Scholar
  37. 37.
    V. V. Kornyak, “Discrete Dynamics: Gauge Invariance and Quantization,” Lect. Notes Comp. Sci. 5743, 180–194 (2009).ADSCrossRefGoogle Scholar
  38. 38.
    V. V. Kornyak, “Computer Algebra Study of Structural and Symmetry Properties of Discrete Dynamical Systems,” in Modern Optics and Photonics (World Scientific, 2010), pp. 197–221.Google Scholar
  39. 39.
    V. V. Kornyak, “Quantization in Discrete Dynamical Systems,” J. Math. Sci. 168(3), 390–397 (2010). MathSciNetCrossRefGoogle Scholar
  40. 40.
    V. V. Kornyak, “Constructive Study of Quantum Behavior,” in Mathematical Modeling in Physics, Civil Engineering, Economics and Finance (Wydawnictwo Collegium Mazovia Innowacyjna Szkola Wyzsza, Siedlce, 2011), pp. 100–114.Google Scholar
  41. 41.
    V. V. Kornyak, “Structural and Symmetry Analysis of Discrete Dynamical Systems,” in Cellular Automata (Nova Science Publishers, Inc, New York, 2011), pp. 119–163.Google Scholar
  42. 42.
    V. V. Kornyak, “Finite Quantum Models: Constructive Approach to Description of Quantum Behavior,” Zap. Nauch. Sem. POMI RAN 387, 122–144 (2011).MathSciNetGoogle Scholar
  43. 43.
    V. V. Kornyak, “Computation in Finite Groups and Quantum Physics,” Lect. Notes Comp. Sci. 6885, 263–279 (2011). CrossRefGoogle Scholar
  44. 44.
    V. V. Kornyak, “Finite Quantum Models: Constructive Approach to Description of Quantum Behavior,” J. Math. Sci. 179(6), 702–713 (2011). MathSciNetCrossRefGoogle Scholar
  45. 45.
    V. V. Kornyak, “Modeling of Finite Quantum Systems,” Lect. Notes Comp. Sci. 7125, 79–93 (2012). CrossRefGoogle Scholar
  46. 46.
    V. V. Kornyak, “Permutation Interpretation of Quantum Mechanics,” J. Phys. Conf. Ser. 343(1), 1–20 (2012). Google Scholar
  47. 47.
    R. Lidl and G. Niederreiter, Finite Fields Vols. 1–2 (Mir, Moscow, 1988) [in Russian].zbMATHGoogle Scholar
  48. 48.
    B. D. McKay, “Practical Graph Isomorphism,” Congressus Numerantium 30, 45–87 (1981). MathSciNetGoogle Scholar
  49. 49.
    P. O. Ludl, “Systematic Analysis of Finite Family Symmetry Groups and Their Application to the Lepton Sector” (Preprint arXiv: 0907.5587, 2009).Google Scholar
  50. 50.
    W. Magnus, “Residually Finite Groups,” Bull. Amer. Math. Soc. 75(2), 305–316 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    A. I. Mal’tsev, “On Isomorphic Representation of Infinite Groups by Matrices,” Mat. Sb. 8(50), 405–422 (1940).zbMATHGoogle Scholar
  52. 52.
    F. Markopoulou, “The Internal Description of a Causal Set: What the Universe Looks Like from the Inside,” Comm. Math. Phys. 211, 559–583 (2000).MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. 53.
    O. Nairz, M. Arndt, and A. Zeilinger, “Quantum Interference Experiments with Large Molecules,” Am. J. Phys 71(4), 319–325 (2003).ADSCrossRefGoogle Scholar
  54. 54.
    K. Nakamura, et al. (Particle Data Group), “The Review of Particle Physics,” J. Phys. G 37, 075021, 1–1422 (2010).CrossRefGoogle Scholar
  55. 55.
    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).zbMATHGoogle Scholar
  56. 56.
    R. Oeckl, Discrete Gauge Theory (From Lattices to TQPT) (Imperial College Press, London, 2005).CrossRefGoogle Scholar
  57. 57.
    H. Poincaré, La valeur de la Science (Flammarion, Paris, 1904) Russian translation: A. Poincaré, About science. Ed. by L.S. Pontryagin (Nauka, Moscow, 1990) [in Russian].Google Scholar
  58. 58.
    V. V. Prasolov and A. B. Sosinsky, Knots, Links, Braids and 3-Manifolds (MTsNMO, Moscow, 1997) [in Russian].zbMATHGoogle Scholar
  59. 59.
    J. J. Rotman, An Introduction to the Theory of Groups (Springer, Berlin, 1995).zbMATHCrossRefGoogle Scholar
  60. 60.
    N. Seiberg, “Emergent Spacetime,” (Preprint arXiv:hep-th/0601234).Google Scholar
  61. 61.
    I. R. Shafarevich, Basic Notions of Algebra (RKhD, Izhevsk, 2001).Google Scholar
  62. 62.
    A. Yu. Smirnov, “Discrete Symmetries and Models of Flavor Mixing”, (Preprint arXiv: 1103.3461).Google Scholar
  63. 63.
    E. Spanier, Algebraic Topology (Mir, Moscow, 1971) [in Russian].zbMATHGoogle Scholar
  64. 64.
    R. Zulanke and P. Vintgen, Differential Geometry and Bindles (Mir, Moscow, 1975) [in Russian].Google Scholar
  65. 65.
    H. Weyl, Group Theory and Quantum Mechanics (Nauka, Moscow, 1986) [in Russian].Google Scholar
  66. 66.
    H. Weyl, Ars Combinatoria. Appendix B in Philosophy of Mathematics and Natural Science (Princeton Univ. Press, 1949). Russian translation in Applied Combinatorial Mathematics (Mir, Moscow, 1968).Google Scholar
  67. 67.
    H. Wielandt, Finite Permutation Groups (Acad. Press, New York and London, 1964).zbMATHGoogle Scholar
  68. 68.
    S. Wolfram, A New Kind of Science (Wolfram Media, Inc, 2002).Google Scholar
  69. 69.

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • V. V. Kornyak
    • 1
  1. 1.Joint Institute for Nuclear ResearchDubnaRussia

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