Foundations of Physics

, Volume 45, Issue 3, pp 295–332 | Cite as

Quantum Mechanics and the Principle of Least Radix Economy

Article

Abstract

A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is then accomplished and it is used to derive the Schrödinger and Dirac equations and the breaking of the commutativity of spacetime geometry. The formulation provides an explanation of how determinism and random statistical behavior coexist in spacetime and a framework is developed that allows dynamical processes to be formulated in terms of chains of digits. These methods lead to a new (pre-geometrical) foundation for Lorentz transformations and special relativity. The Parker-Rhodes combinatorial hierarchy is encompassed within our approach and this leads to an estimate of the interaction strength of the electromagnetic and gravitational forces that agrees with the experimental values to an error of less than one thousandth. Finally, it is shown how the principle of least-radix economy naturally gives rise to Boltzmann’s principle of classical statistical thermodynamics. A new expression for a general (path-dependent) nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics.

Keywords

Quantum mechanics Variational methods Entropy  Discrete physics 

References

  1. 1.
    Wolfram, S.: A New Kind of Science. Wolfram Media Inc., Champaign (2002)MATHGoogle Scholar
  2. 2.
    Fredkin, E.: Digital mechanics. Physica D 45, 254–270 (1990)CrossRefADSMATHMathSciNetGoogle Scholar
  3. 3.
    McCauley, J.L.: Chaos, Dynamics and Fractals: An Algorithmic Approach to Deterministic Chaos. Cambridge University Press, New York (1993)Google Scholar
  4. 4.
    A. M. Turing: Systems of Logic Based on Ordinals. Ph.D. Thesis, Princeton University, (1939)Google Scholar
  5. 5.
    t’Hooft, G.: Quantum gravity as a dissipative deterministic system. Class. Quantam Gravity 16, 3263–3279 (1999)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    Bastin, T., Kilmister, C.W.: Combinatorial Physics. World Scientific, Singapore (1995)MATHGoogle Scholar
  7. 7.
    Parker-Rhodes, A.F.: The Theory of Indistinguishables. D. Reidel, Dordrecht (Holland) (1981)CrossRefMATHGoogle Scholar
  8. 8.
    Noyes, H.P.: Bit-String Physics. World Scientific, Singapore (2001)MATHGoogle Scholar
  9. 9.
    Kauffman, L.H.: Non-commutative worlds. New J. Phys. 6, 173 (2004)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Kauffman, L.H., Noyes, H.P.: Discrete physics and the Dirac equation. Phys. Lett. A 218, 139–146 (1996)CrossRefADSGoogle Scholar
  11. 11.
    McCauley, J.L.: Chaotic dynamical systems as automata. Z. Naturforsch. A 42, 547–555 (1987)MathSciNetGoogle Scholar
  12. 12.
    McCauley, J.L., Palmore, J.I.: Computable chaotic orbits. Phys. Lett. A 115, 433–436 (1986)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Garcia-Morales, V.: Nonlocal and Global Dynamics of Cellular Automata: A Theoretical Computer Arithmetic for Real Maps, arXiv:1312.6534 [math-ph] (2013)
  14. 14.
    Garcia-Morales, V.: Universal map for cellular automata. Phys. Lett. A 376, 2645–2657 (2012)CrossRefADSMATHMathSciNetGoogle Scholar
  15. 15.
    Garcia-Morales, V.: Symmetry analysis of cellular automata. Phys. Lett. A 377, 276–285 (2013)CrossRefADSMATHMathSciNetGoogle Scholar
  16. 16.
    Garcia-Morales, V.: Origin of complexity and conditional predictability in cellular automata. Phys. Rev. E 88, 042814 (2013)CrossRefADSGoogle Scholar
  17. 17.
    Garcia-Morales, V.: Universal Map for Substitution Systems, arXiv:1309.5254 [math-ph] (2013)
  18. 18.
    Wigner, E.P.: The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math. 13, 1–14 (1960)CrossRefADSMATHGoogle Scholar
  19. 19.
    Knuth, D.E.: The Art of Computer Programming. Seminumerical Algorithms, vol. II, 3rd edn. Addison Wesley, Reading (1998)Google Scholar
  20. 20.
    Hurst, S.L.: Multiple-valued logic: its status and its future. IEEE Trans. Comput. C33, 1160–1179 (1984)CrossRefGoogle Scholar
  21. 21.
    Schroeder, M.: Number Theory in Science and Communication. Springer, Berlin (2009)MATHGoogle Scholar
  22. 22.
    Dirac, P.A.M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1988)Google Scholar
  23. 23.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1996)MATHGoogle Scholar
  24. 24.
    Titchmarsh, E. C., Heath-Brown, D. R.: The Theory of the Riemann Zeta-function. Oxford University Press, Oxford, UK (1986), Eq. 2.1.7Google Scholar
  25. 25.
    Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, London (1972)MATHGoogle Scholar
  26. 26.
    Lanczos, C.: Linear Differential Operators. Van Nostrand, London (1961)MATHGoogle Scholar
  27. 27.
    Heisenberg, W.: The Physical Principles of the Quantum Theory. Dover, New York (1949)Google Scholar
  28. 28.
    Niven, I.: Irrational Numbers. The Mathematical Association of America (Carus Mathematical Monographs No. 11). The Mathematical Association of America, Washington (1956)Google Scholar
  29. 29.
    Bohm, D.: Quantum Theory. Dover, New York (1979)Google Scholar
  30. 30.
    Tanner, G., Richter, K., Rost, J.M.: The theory of two-electron atoms: between ground state and complete fragmentation. Rev. Mod. Phys. 72, 497 (2000)CrossRefADSGoogle Scholar
  31. 31.
    Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, New York (1990)CrossRefMATHGoogle Scholar
  32. 32.
    Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (1959)MATHGoogle Scholar
  33. 33.
    Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover, New York (1950)Google Scholar
  34. 34.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover, New York (1961)MATHGoogle Scholar
  35. 35.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and all That. W. A. Benjamin, New York (1964)MATHGoogle Scholar
  36. 36.
    Halmos, P.R.: Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York (1951)MATHGoogle Scholar
  37. 37.
    t’Hooft, G.: The Cellular Automaton Interpretation of Quantum Mechanics. arXiv:1405.1548v2 (2014)
  38. 38.
    Bruckner, B.S., Bruckner, J.B., Thomson, A.M.: Real Analysis. Prentice Hall, Upper Saddle River (1996)Google Scholar
  39. 39.
    Tinkham, M.: Group Theory and Quantum Mechanics. Dover, New York (1964)MATHGoogle Scholar
  40. 40.
    Feynman, R.P.: Space–time approach to non-relativistic quantum mechanics. Rev. Modern Phys. 20, 367–387 (1948)CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys. 44, 326–352 (1927)CrossRefADSMATHGoogle Scholar
  42. 42.
    Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)CrossRefADSGoogle Scholar
  43. 43.
    Connes, A.: Noncommutative Geometry. Academic Press, London (1994)Google Scholar
  44. 44.
    Chevalley, C.: Sur certains groupes simples. Tohoku Math. J. 2(7), 14–66 (1955)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Carter, R.W.: Simple Groups of Lie Type. Wiley, New York (1989)MATHGoogle Scholar
  46. 46.
    Schwinger, J.: Unitary operator bases. Proc. Natl. Acad. Sci. USA 46, 570–579 (1960)CrossRefADSMATHMathSciNetGoogle Scholar
  47. 47.
    Kac, M.: Statistical Independence in Probability, Analysis and Number Theory. The Mathematical Association of America (Carus Mathematical Monographs No. 12). The Mathematical Association of America, Washington (1959)Google Scholar
  48. 48.
    Garcia-Morales, V., Pellicer, J.: Microcanonical foundation of nonextensivity and generalized thermostatistics based on the fractality of the phase space. Physica A 361, 161–172 (2006)CrossRefADSGoogle Scholar
  49. 49.
    Garcia-Morales, V., Pellicer, J., Manzanares, J.A.: Thermodynamics based on the principle of least abbreviated action: entropy production in a network of coupled oscillators. Ann. Phys. (New York) 323, 1844–1858 (2008)CrossRefADSMATHMathSciNetGoogle Scholar
  50. 50.
    de Broglie, L., La, L.: Thermodynamique de la particule isolé. Gauthier-Villars, Paris (1964)Google Scholar
  51. 51.
    de Broglie, L.: The reinterpretation of wave mechanics. Found. Phys. 1, 5–15 (1970)CrossRefADSGoogle Scholar
  52. 52.
    Koshy, T.: Triangular Arrays with Applications. Oxford University Press, New York (2011)Google Scholar
  53. 53.
    Snygg, J.: A New Approach to Differential Geometry Using Clifford’s Geometric Algebra. Springer, New York (2012)CrossRefMATHGoogle Scholar
  54. 54.
    Haag, R.: Local Quantum Physics: Fields, Particles and Algebras. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  55. 55.
    Pathria, R.K., Beale, P.D.: Statistical Mechanics. Elsevier, Amsterdam (2011)MATHGoogle Scholar
  56. 56.
    Gross, D.H.E.: Microcanonical Thermodynamics. World Scientific, Singapore (2001)Google Scholar
  57. 57.
    Ruelle, D.: Thermodynamic Formalism, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  58. 58.
    Berdichevsky, V.L.: Thermodynamics of Chaos and Order. Addison Wesley Longman, Essex (1997)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute for Advanced StudyTechnische Universität MünchenGarchingGermany
  2. 2.Nonequilibrium Chemical Physics - Physics DepartmentTechnische Universität MünchenGarchingGermany

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