International Journal of Theoretical Physics

, Volume 34, Issue 12, pp 2423–2433 | Cite as

p-Adic stochastics and Dirac quantization with negative probabilities

  • Andrew Khrennikov
Article
Received:

Abstract

A new mathematical apparatus, ap-adic theory of probability, is applied to realize the hypothetical world based on negative probability distributions created by Dirac for the relativistic quantization of photons. Within thep-adic theory of probability, negative probability distributions are well defined (in the language of limits of relative frequencies, but with respect to ap-adic metric). We propose that the negative Gibbs distributions arising in relativistic quantization are described byp-adic Stochastics.

Keywords

Probability Distribution Field Theory Elementary Particle Quantum Field Theory Relative Frequency 
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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References

  1. Beltrametti, E., and Cassinelli, G. (1972). Quantum mechanics andp-adic numbers,Foundations of Physics,2, 1–7.Google Scholar
  2. Borevich, Z. I., and Schafarevich, I. R. (1966).Number Theory, Academic Press, New York.Google Scholar
  3. Dirac, P. A. M. (1942). The physical interpretation of quantum mechanics,Proceedings of the Royal Society of London A,180, 1–39.Google Scholar
  4. Frampton, P. H., and Okada, Y. (1988).p-Adic stringN-point function,Physical Review Letters B,60, 484–486.Google Scholar
  5. Freund, P. G. O., and Witten, E. (1987). Adele string amplitudes,Physics Letters B,199, 191–195.Google Scholar
  6. Khrennikov, A. Yu. (1989). Non-Archimedean white noise, inProceedings of International Conference Gaussian Random Fields, Nagoya, p. 117.Google Scholar
  7. Khrennikov, A. Yu. (1990). Mathematical methods of the non-Archimedean physics,Russian Mathematical Surveys,45, 87–125.Google Scholar
  8. Khrennikov, A. Yu. (1991). Quantum mechanics withp-adic valued wave functions,Journal of Mathematical Physics,32, 932–937.Google Scholar
  9. Khrennikov, A. Yu. (1993).p-adic theory of probability and its applications. A principle of the statistical stabilization of frequencies,Teoreticheskaia i Matematicheskaia Fizika,97(3), 348–363.Google Scholar
  10. Khrennikov, A. Yu. (1994a).p-Adic Valued Distributions in Mathematical Physics, Kluwer, Boston.Google Scholar
  11. Khrennikov, A. Yu. (1994b). Probability interpretation of thep-adic valued quantum theories,Symposium on the Foundations of Modern Physics, Helsinki, June 1994, Abstracts, p. 134.Google Scholar
  12. Kolmogoroff, A. N. (1933).Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin 1933; reprint 1973 [English translation by N. Morrison, New York, 1950].Google Scholar
  13. Mahler, K. (1973).Introduction to p-Adic Numbers and their Functions, Cambridge University Press, Cambridge.Google Scholar
  14. Vladimirov, V. S., and Volovich, I. V. (1984). Superanalysis, 1. Differential calculus,Teoreticheskaia i Matematicheskaia Fizika,59(1), 3–27.Google Scholar
  15. Vladimirov, V. S., Volovich, I. V., and Zelenov, E. I. (1994).p-Adic Numbers in Mathematical Physics, World Scientific, Singapore.Google Scholar
  16. Volovich, I. V. (1987).p-Adic string,Classical and Quantum Gravity,4, 83–87.Google Scholar
  17. Von Mises, R. (1919). Grundlagen der Wahrscheinlichkeitsrechnung,Mathematische Zeitschrift,5, 52–99.Google Scholar
  18. Von Mises, R. (1964).The Mathematical Theory of Probability and Statistics, Academic Press, London.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Andrew Khrennikov
    • 1
  1. 1.Institute for MathematicsRuhr UniversityBochumGermany

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