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Space-Time Grains: Roots of Special and Doubly Special Relativity

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Abstract

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be therefore held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.

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References

  1. Anderson, P.: Science 177, 393 (1972)

    Article  ADS  Google Scholar 

  2. Laughlin, R.B.: A Different Universe: Reinventing Physics from the Bottom Down. Basic Books, London (2005)

    Google Scholar 

  3. Licata, I., Sakaji, A.: Physics of Emergence and Organization. WS, London (2008)

    Book  MATH  Google Scholar 

  4. Beck, C.: Phys. Rev. Lett. 87, 180601 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  5. Beck, C., Cohen, E.G.D.: Physica A 322, 267 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Wilk, G., Wlodarczyk, Z.: Phys. Rev. Lett. 85, 2770 (2000)

    Article  ADS  Google Scholar 

  7. Hanel, R., Thurner, S., Gell-Mann, M.: Generalized entropies and the transformation group of superstatistics. Proc. Natl. Acad. Sci. 108, 6390 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Reynolds, A.: Phys. Rev. Lett. 91, 084503 (2003)

    Article  ADS  Google Scholar 

  9. Beck, C., Miah, S.:. arXiv:1207.4062

  10. Daniels, K.E., Beck, C., Bodenschatz, E.: Physica D 193, 208 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Wilk, G., Wlodarczyk, Z.: Eur. Phys. J. A 40, 299 (2009)

    Article  ADS  Google Scholar 

  12. Beck, C.: Eur. Phys. J. A 40, 267 (2009)

    Article  ADS  Google Scholar 

  13. Beck, C.: In: Radons, G., et al. (eds.) Anomalous Transport: Foundations and Applications. Wiley, London (2007)

    Google Scholar 

  14. Jizba, P., Kleinert, H.: Phys. Rev. E 78, 031122 (2008)

    Article  ADS  Google Scholar 

  15. Jizba, P., Kleinert, H.: Phys. Rev. D 82, 085016 (2010)

    Article  ADS  Google Scholar 

  16. Allegrini, P., Barbi, F., Grigolini, P., Paradisi, P.: Phys. Rev. E 73, 046136 (2006)

    Article  ADS  Google Scholar 

  17. Bertoin, J.: Lévy Processes. CUP, Cambridge (1996)

    MATH  Google Scholar 

  18. Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, London (1966)

    MATH  Google Scholar 

  19. Jizba, P., Scardigli, F.: Phys. Rev. D 86, 025029 (2012)

    Article  ADS  Google Scholar 

  20. Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets. WS, Singapore (2009)

    Book  MATH  Google Scholar 

  21. Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  22. Gaveau, B., Jacobson, T., Kac, M., Schulman, L.S.: Phys. Rev. Lett. 53, 419 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  23. Jacobson, T., Schulman, L.S.: J. Phys. A 17, 375 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  24. Newton, T., Wigner, E.: Rev. Mod. Phys. 21, 400 (1949)

    Article  ADS  MATH  Google Scholar 

  25. Feshbach, H., Villars, F.: Rev. Mod. Phys. 30, 24 (1958)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Johnson, E.A., MacKinnon, A.: J. Phys. Condens. Matter 5, 5859 (1993)

    Article  ADS  Google Scholar 

  27. Amelino-Camelia, G.: Int. J. Mod. Phys. D 11, 35 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Magueijo, J., Smolin, L.: Phys. Rev. Lett. 88, 190403 (2002)

    Article  ADS  Google Scholar 

  29. Amelino-Camelia, G.: Nature 418, 34 (2002)

    Article  ADS  Google Scholar 

  30. Kowalski-Glikman, J.: Introduction to Doubly Special Relativity. Lecture Notes in Physics, vol. 669. Springer, Berlin (2005)

    Google Scholar 

  31. Daszkiewicz, M., Lukierski, J., Woronowicz, M.: Phys. Rev. D 77, 105007 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  32. Magueijo, J., Smolin, L.: Phys. Rev. D 67, 044017 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  33. Leutwyler, H.: Nuovo Cimento 37, 556 (1965)

    Article  Google Scholar 

  34. Marmo, G., Mukunda, N., Sudarshan, E.C.G.: Phys. Rev. D 30, 2110 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  35. Feynman, R.P.: Phys. Rev. 80, 440 (1950)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Feynman, R.P.: Phys. Rev. 84, 108 (1951)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Kleinert, H.: Gauge Fields in Condensed Matter, Vol. I Superflow and Vortex Lines. WS, Singapore (1989)

    Book  MATH  Google Scholar 

  38. Bern, Z., Kosower, D.A.: Phys. Rev. Lett. 66, 1669 (1991)

    Article  ADS  Google Scholar 

  39. Bern, Z., Kosower, D.A.: Nucl. Phys. B 379, 451 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  40. Strassler, M.J.: Nucl. Phys. B 385, 145 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  41. Schubert, C.: Phys. Rep. 335, 73 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  42. Bastianelli, F., van Nieuwenhuizen, P.: Path Integrals and Anomalies in Curved Space. CUP, Cambridge (2005)

    Google Scholar 

  43. Vilenkin, A., Ford, L.H.: Phys. Rev. D 26, 1231 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  44. Scardigli, F., Gruber, C., Chen, P.: Phys. Rev. D 83, 063507 (2011)

    Article  ADS  Google Scholar 

  45. Tegmark, M., et al.: Phys. Rev. D 74, 123507 (2006)

    Article  ADS  Google Scholar 

  46. Chibisov, G.V.: Sov. Phys. Usp. 19, 624 (1976)

    Article  ADS  Google Scholar 

  47. Lakes, R.: Phys. Rev. Lett. 80, 1826 (1998)

    Article  ADS  Google Scholar 

  48. Rumpf, H.: Phys. Rev. D 33, 185 (1986)

    Article  MathSciNet  Google Scholar 

  49. Rumpf, H.: Prog. Theor. Phys. Suppl. 111, 63 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  50. de Aguiar, T.C., Menezes, G., Svaiter, N.F.: Class. Quantum Gravity 26, 075003 (2009)

    Article  ADS  Google Scholar 

  51. Ambjorn, J., Loll, R., Westra, W., Zohren, S.: Phys. Lett. B 680, 359 (2009)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to H. Kleinert, Z. Haba, M. Sakellariadou, and L.S. Schulman for useful feedbacks.

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Authors and Affiliations

  1. FNSPE, Czech Technical University in Prague, Břehová 7, 115 19, Praha 1, Czech Republic

    Petr Jizba

  2. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Fabio Scardigli

  3. Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan

    Fabio Scardigli

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  1. Petr Jizba
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  2. Fabio Scardigli
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Correspondence to Fabio Scardigli.

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Jizba, P., Scardigli, F. Space-Time Grains: Roots of Special and Doubly Special Relativity. Found Phys 44, 512–522 (2014). https://doi.org/10.1007/s10701-013-9758-9

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  • DOI: https://doi.org/10.1007/s10701-013-9758-9

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