Advertisement

Communications in Mathematical Physics

, Volume 123, Issue 4, pp 659–676 | Cite as

p-adic quantum mechanics

  • V. S. Vladimirov
  • I. V. Volovich
Article

Abstract

An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.

Keywords

Neural Network Generalize Function Complex System Quantum Mechanic Probability Measure 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Vladimirov, V.S., Volovich, I.V.: Superanalysis, Differential calculus. Theor. Math. Phys.59, 3–27 (1984)Google Scholar
  2. 2.
    Volovich, I.V.:p-adic space-time and string theory. Theor. Math. Phys.71, 337–340 (1987)Google Scholar
  3. 3.
    Volovich, I.V.:p-adic string. Class. Quant. Grav.4, L83-L87 (1987)Google Scholar
  4. 4.
    Volovich, I.V.: Number theory as the ultimate physical theory. Preprint CERN-TH. 4981/87Google Scholar
  5. 5.
    Grossman, B.:p-adic strings, the Weyl conjectures and anomalies. Phys. Lett. B197, 101–106 (1987)Google Scholar
  6. 6.
    Freund, P.G.O., Olson, M.: Non-archimedean strings. Phys. Lett. B199, 186–190 (1987)Google Scholar
  7. 7.
    Freund, P.G.O., Witten, E.: Adelic string amplitudes. Phys. Lett. B199, 191–195 (1987)Google Scholar
  8. 8.
    Volovich, I.V.: Harmonic analysis andp-adic strings. Lett. Math. Phys.16, 61–67 (1988)Google Scholar
  9. 9.
    Aref'eva, I. Ya., Dragovič, B., Volovich, I.V.: On thep-adic summability of the anharmonic oscillator. Phys. Lett. B200, 512–514 (1988)Google Scholar
  10. 10.
    Gervais, J.-L.:p-adic analyticity and Virasoro algebras for conformal theories in more than two dimensions. Phys. Lett. B201, 306–310 (1988)Google Scholar
  11. 11.
    Frampton, P.H., Okada, Y.:p-adic stringN-point function. Phys. Lett.60, 484–486 (1988)Google Scholar
  12. 12.
    Marinari, E., Parisi, G.: On thep-adic five point function. Preprint ROM2F-87/38Google Scholar
  13. 13.
    Freund, P.G.O., Olson, M.:p-adic dynamical systems. Nucl. Phys. B297, 86–97 (1988)Google Scholar
  14. 14.
    Aref'eva, I. Ya., Dragovič, B., Volovich, I.V.: On the adelic string amplitudes. Phys. Lett. B209, 445–450 (1988)Google Scholar
  15. 15.
    Frampton, P.H., Okada, Y.: Effective scalar field theory ofp-adic string. Phys. Rev. D (to appear)Google Scholar
  16. 16.
    Grossman, B.: Adelic conformal field theory. Preprint Rockefeller University DOE/ER/40325-16Google Scholar
  17. 17.
    Meurice, Y.: The classical harmonic oscillator on Galois andp-adic fields, preprint ANL-HEP-PR-87-114Google Scholar
  18. 18.
    Borevich, Z.I., Shafarevich, I.R.: Number theory. New York: Academic Press 1966Google Scholar
  19. 19.
    Gelfand, I.M., Graev, M.I., Pjatetski-Shapiro, I.I.: Theory of representations and automorphic functions. Moscow: Nauka 1966Google Scholar
  20. 20.
    Vladimirov, V.S.: Distributions over the field ofp-adic numbers. Usp. Math. Nauk43, 17–53 (1988)Google Scholar
  21. 21.
    Koblitz, N.:p-adic numbers,p-adic analysis, and zeta-functions. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  22. 22.
    Dwork, B.: Lectures onp-adic differential equations. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  23. 23.
    Vinogradov, I.M.: Elements of Number theory. Moscow: Nauka 1965Google Scholar
  24. 24.
    Weyl, H.: The theory of groups and quantum mechanics. New York: Dover 1931Google Scholar
  25. 25.
    Karasev, M.V., Maslov, V.P.: Asymptotic and geometric quantization. Usp. Math. Nauk.39, 115–173 (1984)Google Scholar
  26. 26.
    Weil, A.: Sur certains groupes d'operateurs unitaries. Acta Math.111, 143–211 (1964)Google Scholar
  27. 27.
    Manin, Yu. I.: Non-Archimedean integration andp-adicL-functions of Jacquet and Lenglands. Usp. Math. Nauk.31, 5–54 (1976)Google Scholar
  28. 28.
    Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. New York: McGraw-Hill 1965Google Scholar
  29. 29.
    Maslov, V.P.: Operator methods. Moscow: Nauka 1973Google Scholar
  30. 30.
    Glimm, J., Jaffe, A.: Quantum physics. A functional integral point of view. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  31. 31.
    Vladimirov, V.S., Volovich, I.V.: Vacuum state inp-adic quantum mechanics. Phys. Lett. B217, 411–415 (1989)Google Scholar
  32. 32.
    Vladimirov, V.S., Volovich, I.V.:p-adic Schrödinger type equation, Preprint NBI-HE-87-77 (1988)Google Scholar
  33. 33.
    Zelenov, E.I.:p-adic quantum mechanics forp=2. Theor. Math. Phys. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • V. S. Vladimirov
    • 1
  • I. V. Volovich
    • 1
  1. 1.Steklov Mathematical InstituteMoscowUSSR

Personalised recommendations