Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences

K.G. Zloshchastiev1

Abstract

In the framework of a generic generally covariant quantum theory, we introduce a logarithmic correction to the quantum wave equation. We demonstrate the emergence of evolution time from the group of automorphisms of the von Neumann algebra governed by this nonlinear correction. It turns out that such a parametrization of time is essentially energy-dependent and becomes global only asymptotically, as the energies become very small as compared to the effective quantum gravity scale. We show how the logarithmic nonlinearity deforms the vacuum wave dispersion relations and explains certain features of the astrophysical data coming from the recent observations of high-energy cosmic rays. In general, the estimates imply that, ceteris paribus, particles with higher energy propagate slower than those with lower energy, therefore, for a high-energy particle the mean free path, lifetime in a high-energy state and thus the travel distance from the source can be significantly larger than one would expect from the conventional theory. In addition, we discuss the possibility and conditions of transluminal phenomena in the physical vacuum such as Cherenkov-type shock waves.

References

  1. S. Weinberg, Annals Phys. 194, 336 (1989).
  2. N. Gisin, Helv. Phys. Acta 62, 363 (1989); Phys. Lett. A 143, 1 (1990).
  3. J. Polchinski, Phys. Rev. Lett. 66, 397 (1991).
  4. M. Czachor, Phys. Rev. A 57, 4122 (1998); Phys. Lett. A 225, 1 (1997); M. Czachor and H.-D. Doebner, Phys. Lett. A 301, 139 (2002).
  5. H. D. Doebner and G. A. Goldin, Phys. Rev. A 54, 3764 (1996).
  6. A. Ashtekar and T. A. Schilling, arXiv: gr-qc/9706069.
  7. P. M. Pearle, Phys. Rev. D 13, 857 (1976); A. N. Grigorenko, J. Phys. A: Math. Gen. 28, 1459 (1995); T. P. Singh, J. Phys. Conf. Ser. 174, 012024 (2009).
  8. I. Bialynicki-Birula and J. Mycielski, Annals Phys. 100, 62 (1976).
  9. C. J. Murphy, "C*-Algebras and operator theory," Academic Press, Boston, 1990.
  10. M. Kastner and I. Snyman, private communications.
  11. E. T. Jaynes, Phys. Rev. 108, 171 (1957) [Sec. 8].
  12. A. Wehrl, Rep. Math. Phys. 16, 353 (1979).
  13. A. Connes and C. Rovelli, Class. Quant. Grav. 11, 2899 (1994).
  14. C. Rovelli, Class. Quant. Grav. 10, 1549 (1993).
  15. M. Heller and W. Sasin, Phys. Lett. A 250, 48 (1998).
  16. J. R. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Lett. B 293, 37 (1992).
  17. L. J. Garay, Phys. Rev. Lett. 80, 2508 (1998).
  18. G. Amelino-Camelia et al, Nature 393, 763 (1998); J. R. Ellis et al, Astrophys. J. 535, 139 (2000).
  19. J. I. Latorre, P. Pascual and R. Tarrach, Nucl. Phys. B 437, 60 (1995).
  20. G. S. Asanov, Finsler Geometry, Relativity And Gauge Theories (Dordrecht, Netherlands: Reidel, 1985), 370 p.; F. Girelli, S. Liberati and L. Sindoni, Phys. Rev. D 75, 064015 (2007).
  21. N. D. Birrell and P. C. W. Davies, Quantum Fields In Curved Space (Cambridge, UK: Univ. Pr., 1982), 340 p.
  22. A. A. Abdo et al. [Fermi LAT/GBM Collaborations], Science 323, 1688 (2009).
  23. T. Kifune, Astrophys. J. 518, L21 (1999).
  24. R. J. Protheroe and H. Meyer, Phys. Lett. B 493, 1 (2000); J. Albert et al. [MAGIC Collaboration and Other Contributors Collaboration], Phys. Lett. B 668, 253 (2008).
  25. J. Lukierski et al, Phys. Lett. B 264, 331 (1991); Phys. Lett. B 293, 344 (1992); Annals Phys. 243, 90 (1995).
  26. G. Amelino-Camelia, Phys. Lett. B 392, 283 (1997).
  27. E. Tkalya, private communication.
  28. Here we use the original definition of the Cauchy's formula, in other versions the square of the refractive index is often omitted, due to the smallness of the refraction constant; also, the latter is rescaled by a factor of two.
  29. B. M. Bolotovskii and V. L. Ginzburg, Usp. Fiz. Nauk 106, 577 (1972).
  30. E. F. Beall, Phys. Rev. D 1, 961 (1970).
  31. S. R. Coleman and S. L. Glashow, Phys. Lett. B 405, 249 (1997);
  32. T. Jacobson, S. Liberati and D. Mattingly, Phys. Rev. D 67, 124011 (2003).
  33. R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004); Phys. Rev. D 70, 125010 (2004) [Erratum-ibid. D 70, 129906 (2004)].
  34. P. Castorina, A. Iorio and D. Zappala, Europhys. Lett. 69, 912 (2005).
  35. C. Kaufhold and F. R. Klinkhamer, Nucl. Phys. B 734, 1 (2006); Phys. Rev. D 76, 025024 (2007).
  36. B. Altschul, Phys. Rev. Lett. 98, 041603 (2007); Phys. Rev. D 75, 105003 (2007); Nucl. Phys. B 796, 262 (2008).
  37. R. Mignani and E. Recami, Lett. Nuovo Cim. 7, 388 (1973).
  38. M. Thulasidas, Int. J. Mod. Phys. D 16, 983 (2007).
  39. G. Amelino-Camelia, Phys. Lett. B 528, 181 (2002).
  40. J. Chang et al., Nature 456, 362 (2008).
  41. F. W. Stecker and S. L. Glashow, Astropart. Phys. 16, 97 (2001); F. W. Stecker and S. T. Scully, New J. Phys. 11, 085003 (2009).
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