Number theory as the ultimate physical theory

  • Igor V. Volovich
Research Articles


At the Planck scale doubt is cast on the usual notion of space-time and one cannot think about elementary particles. Thus, the fundamental entities of which we consider our Universe to be composed cannot be particles, fields or strings. In this paper the numbers are considered as the fundamental entities. We discuss the construction of the corresponding physical theory. A hypothesis on the quantum fluctuations of the number field is advanced for discussion. If these fluctuations actually take place then instead of the usual quantum mechanics over the complex number field a new quantum mechanics over an arbitrary field must be developed. Moreover, it is tempting to speculate that a principle of invariance of the fundamental physical laws under a change of the number field does hold. The fluctuations of the number field could appear on the Planck length, in particular in the gravitational collapse or near the cosmological singularity. These fluctuations can lead to the appearance of domains with non-Archimedean p-adic or finite geometry. We present a short review of the p-adic mathematics necessary, in this context.

p-adic mathematical physics quantum mechanics and field theory string theory 


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  1. 1.
    M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory (CUP, Cambridge, UK 1987); I. Ya. Arefeva and I. V. Volovich, Usp. Fiz. Nauk 146, 655 (1985).MATHGoogle Scholar
  2. 2.
    C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco 1973); J. A. Wheeler, in Quantum Theory and Gravitation, ed. A. R. Marlow (Academic Press, 1980).Google Scholar
  3. 3.
    I. V. Volovich, “p-Adic string”, Class. Quan. Grav. (1987), to be published; Teor. Mat. Fiz. 71, 337 (1987).MathSciNetGoogle Scholar
  4. 4.
    V. S. Vladimirov and I. V. Volovich, Teor. Mat. Fiz. 59, 3 (1983).MathSciNetGoogle Scholar
  5. 5.
    J. A. Wheeler, Ann. Phys. 2, 604 (1957); T. Regge, Nuovo Cimento 7, 215 (1958); A. Peres and N. Rosen, Phys. Rev. 118, 335 (1960); B. De Witt, “The quantization of geometry,” in Gravitation: an Introduction to Current Research, ed. L. Witten (John Wiley and Sons, New York and London 1962); M. A. Markov, Progr. Theor. Phys. Suppl. 85 (1965); H.-J. Treder, in Relativity, Quanta and Cosmology, ed. F. de Finis (New York, 1979).MATHCrossRefGoogle Scholar
  6. 6.
    G.’ t Hooft, Nucl. Phys. B 256, 727 (1985); “Gravitational Collapse and Quantum Mechanics”, Lectures given at the 5th Adriatic Meeting on Particle Physics, Dubrovnik, June 16–28, 1986.CrossRefGoogle Scholar
  7. 7.
    N. Bohr and L. Rosenfeld, Kgl. Danske Videnskab Selskab Mat.-fis. Medd. 12, 1 (1933); Phys. Rev. 78, 794 (1950).Google Scholar
  8. 8.
    H. Poincare, La Science et I’Hypothese (Flammarion, Paris, 1923).Google Scholar
  9. 9.
    H. Weyl, Philosophy of Mathematics and Natural Science (Princeton Univ. Press, 1949).Google Scholar
  10. 10.
    D. Hilbert, Grundlagen der Geometrie (Leipzig, 1930).Google Scholar
  11. 11.
    A Passion for Physics. Essays in honour of Geoffrey Chew, including an interview with Chew, eds. C. de Tar, J. Finkelstein and Chung-1 Tan (World Scientific, 1985).Google Scholar
  12. 12.
    I. Ya. Arefeva and V. Korepin, Pis’ma Zh. Eksp. Teor. Fiz. 20, 680 (1974).Google Scholar
  13. 13.
    R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The Analytic S Matrix (Cambridge Univ. Press, 1966); G. F. Chew, The Analytic S Matrix (Benjamin, New York, 1966); N. N. Bogolyubov, A. A. Logunov and I. Todorov, Introduction to Axiomatic Quantum Field Theory (Reading, MA, Benjamin, 1975).Google Scholar
  14. 14.
    M. Jacob, ed. Dual Theory, Phys. Rep. Reprint 1 (North Holland, Amsterdam, 1974).Google Scholar
  15. 15.
    J. A. Wheeler, Geometrodynamics (Academic Press, New York and London 1962); S. W. Hawking, Nucl. Phys. B 144, 349 (1978).MATHGoogle Scholar
  16. 16.
    T. D. Lee, Phys. Lett. B 122, 217 (1983).CrossRefGoogle Scholar
  17. 17.
    M. B. Green, QMC Preprints:QMC-87-10, QMC-87-11 (1987).Google Scholar
  18. 18.
    H. J. de Vega and N. Sanchez, CERN Preprint TH. 4681 (1987); N. Sanchez, CERN Preprint TH. 4733 (1987).Google Scholar
  19. 19.
    G. Veneziano, CERN Preprint TH. 4397 (1986).Google Scholar
  20. 20.
    C. J. Isham, in Quantum Gravity 2, eds. C. J. Isham, R. Penrose and D. W. Sciama (Clarendon Press, Oxford, 1981).Google Scholar
  21. 21.
    A. Casher, CERN Preprint TH. 4738 (1987).Google Scholar
  22. 22.
    D. Friedan and S. Shenker, Nucl. Phys. B 281, 509 (1987).CrossRefMathSciNetGoogle Scholar
  23. 23.
    M. J. Bewick and S. G. Rajeev, MIT Preprint CTP-1414 (1986).Google Scholar
  24. 24.
    B. Riemann, Nachrichten K. Gesellschaft Wiss. Gottingen 13, 133 (1868).Google Scholar
  25. 25.
    B. Gross and N. Koblitz, Ann. Math. 109, 569 (1979).CrossRefMathSciNetGoogle Scholar
  26. 26.
    Z. I. Borevich and I. R. Shafarevich, Number Theory (Academic Press, 1966); J.-P. Serre, A Course in Arithmetic (Springer-Verlag, 1973); S. Lang, Algebra (Addison-Wesley, 1965).Google Scholar
  27. 27.
    W. H. Schikhof, “Non-Archimedean monotone functions,” Report 7916 (Mathematisch Instituut, Nijmegen, The Netherlands, 1979).Google Scholar
  28. 28.
    R. Penrose, in Quantum Gravity 2, eds. C. J. Isham, R. Penrose and D. W. Sciama (Clarendon Press, Oxford, 1981).Google Scholar
  29. 29.
    S. W. Hawking, “Quantum Cosmology,” in 300 Years of Gravity (Cambridge Univ. Press, 1986).Google Scholar
  30. 30.
    K. Mahler, Introduction to p-Adic Numbers and Their Functions (Cambridge Univ. Press, 1973); N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions (Springer Verlag, 1984); N. Koblitz, p-Adic Analysis: a Short Course on Recent Work (Cambridge Univ. Press, 1980); W. H. Schikhof, Ultrametric Calculus (Cambridge Univ. Press, 1984); S. Lang, Cyclotomic Fields (Springer Verlag, I and II, 1980).Google Scholar
  31. 31.
    B. Dwork, Lectures on p-Adic Differential Equations (Springer Verlag, 1982).Google Scholar
  32. 32.
    B. A. Dubrovin, I. M. Krichever and S. P. Novikov, Soviet Sci. Rev. 3, 1 (1982); M. Mulase, J. Diff. Geom. 19, 403 (1984); T. Shiota, Inv. Math. 83, 333 (1986).Google Scholar
  33. 33.
    Y. Morita, J. Fac. Sci. Univ. Tokyo, Sec. lA 22, 255 (1975).MATHGoogle Scholar
  34. 34.
    C. Lovelace, Phys. Lett. B 32, 703 (1970); V. Alessandrini and D. Amati, Nuovo Cimento 4A, 793 (1971).CrossRefMathSciNetGoogle Scholar
  35. 35.
    L. Alvarez-Gaume and P. Nelson, CERN Preprint TH. 4615 (1986).Google Scholar
  36. 36.
    L. Gerritzen and M. van der Put, Schottky Groups and Mumford Curves, Lecture Notes Math. 817 (Springer, 1980).Google Scholar
  37. 37.
    Yu. Manin and V. G. Drinfeld, J. Reine Angew. Math. 262/263, 239 (1973).MathSciNetGoogle Scholar
  38. 38.
    S. Saito, Tokyo Metropolitan Univ. Preprint TMUP-HEL-8613 (1986); TMUP-HEL-8615 (1986); TMUP-HEL-8701 (1987); N. Ishibashi, Y. Matsuo and H. Ooguri, Univ. Tokyo Preprint UT-499 (1986); K. Sogo, Inst. Nucl. Study, Univ. Tokyo Preprint INS-Rep-626 (1987).Google Scholar
  39. 39.
    L. Alvarez-Gaume, C. Gomez and C. Reina, CERN Preprint TH. 4641 (1987).Google Scholar
  40. 40.
    M. Martellini and N. Sanchez, CERN Preprint TH. 4680 (1987).Google Scholar
  41. 41.
    A. Neveu and P. West, CERN Preprints TH. 4697 and 4707 (1987).Google Scholar
  42. 42.
    A. D. Linde, Rep. Prog. Phys. 47, 925 (1984); L. P. Grischuk and Ya. B. Zeldovich, Preprint Inst. Space Res. 176, Moscow (1982).CrossRefMathSciNetGoogle Scholar
  43. 43.
    A. D. Sakharov, Zh. Eks. Teor. Fiz. 87, 375 (1984); I. Ya. Aref’eva and I. V. Volovich, Phys. Lett. B 164, 287 (1985); Nucl. Phys. B 274, 619 (1986); I. Ya. Aref’eva, B. G. Dragovic and I. V. Volovich, Phys. Lett. B 177, 357 (1986); M. Pollock, Phys. Lett. B 174, 251 (1986); A. Popov, “Chiral fermions in D = 11 supergravity”, Teor. Mat. Fiz., to be published.Google Scholar
  44. 44.
    I. M. Gelfand, M. I. Graev and I. I. Pjatetski-Shapiro, Theory of Representations and Automorphic Functions (Nauka, Moscow, 1966) [in Russian].Google Scholar
  45. 45.
    P. A. M. Dirac, in Mathematical Foundations of Quantum Theory, ed. A. R. Marlou (Academic Press, 1978).Google Scholar

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© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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