Quantization of Big Bang in Crypto-Hermitian Heisenberg Picture

  • Miloslav Znojil
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)


A background-independent quantization of Universe near its Big Bang singularity is considered. Several conceptual issues are addressed in Heisenberg picture. (1) The observable spatial-geometry non-covariant characteristics of an empty-space expanding Universe are sampled by (quantized) distances \(Q=Q(t)\) between space-attached observers. (2) In Q(t) one of the Kato’s exceptional-point times \(t=\tau _{(EP)}\) is postulated real-valued. At such an instant the widely accepted “Big Bounce” regularization of the Big Bang singularity gets replaced by the full-fledged quantum degeneracy. Operators \(Q(\tau _{(EP)})\) acquire a non-diagonalizable Jordan-block structure. (3) During our “Eon” (i.e., at all \(t>\tau _{(EP)}\)) the observability status of operators Q(t) is guaranteed by their self-adjoint nature with respect to an ad hoc Hilbert-space metric \(\varTheta (t) \ne I\). (4) In adiabatic approximation the passage of the Universe through its \(t=\tau _{(EP)}\) singularity is interpreted as a quantum phase transition between the preceding and the present Eon.


Hilbert Space Quantum Phase Transition Unitary Evolution Exceptional Point Heisenberg Picture 
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.


  1. 1.
    C.L. Bennett, D. Larson et al., Astrophys. J. Suppl. Ser. 208 (2013). UNSP 20Google Scholar
  2. 2.
    V. Mukhanov, Physical Foundations of Cosmology (CUP, Cambridge, 2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    C. Rovelli, Quantum Gravity (CUP, Cambridge, 2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    M. Znojil, Non-self-adjoint operators in quantum physics: ideas, people, and trends, in [21], pp. 7–58Google Scholar
  5. 5.
    F.J. Dyson, Phys. Rev. 102, 1217 (1956)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    F.G. Scholtz, H.B. Geyer, F.J.W. Hahne, Ann. Phys. (NY) 213, 74 (1992)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); C.M. Bender, D.C. Brody, H.F. Jones, Phys. Rev. Lett. 89, 270401 (2002); Phys. Rev. Lett. 92, 119902 (2004) (erratum)Google Scholar
  8. 8.
    C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    A. Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. 7, 1191 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.V. Smilga, J. Phys. A: Math. Theor. 41, 244026 (2008)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Znojil, SIGMA 5, 001 (2009). arXiv:0901.0700
  12. 12.
    M. Znojil, Phys. Lett. A 379, 2013 (2015)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Znojil, Phys. Rev. D 78, 085003 (2008)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Mostafazadeh, private communicationGoogle Scholar
  15. 15.
    W. Piechocki, APC seminar “Solving the general cosmological singularity problem”. Paris, 15 Nov 2012Google Scholar
  16. 16.
    P. Malkiewicz, W. Piechocki, Class. Quant. Gravity 27, 225018 (2010)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Ashtekar, A. Corichi, P. Singh, Phys. Rev. D 77, 024046 (2008)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    M.H. Stone, Ann. Math. 33, 643 (1932)CrossRefGoogle Scholar
  19. 19.
    A. Ashtekar, T. Pawlowski, P. Singh, Phys. Rev. D 74, 084003 (2006)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966)CrossRefzbMATHGoogle Scholar
  21. 21.
    F. Bagarello, J.-P. Gazeau, F.H. Szafraniec, M. Znojil (eds.), Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects (Wiley, Hoboken, 2015)zbMATHGoogle Scholar
  22. 22.
    J.-P. Antoine, C. Trapani, “Metric operators, generalized Hermiticity and lattices of Hilbert spaces,,, in [21], pp. 345–402Google Scholar
  23. 23.
    M. Znojil, SIGMA 4, 001 (2008). arXiv:0710.4432v3
  24. 24.
    T. Thiemann, Modern Canonical Quantum General Relativity (CUP, Cambridge, 2007)CrossRefzbMATHGoogle Scholar
  25. 25.
    M. Znojil, J. Phys. A: Math. Theor. 40, 4863 (2007); M. Znojil, J. Phys. A: Math. Theor. 40, 13131 (2007)Google Scholar
  26. 26.
    R. Penrose, Found. Phys. 44, 873 (2014)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    M. Znojil, J.-D. Wu, Int. J. Theor. Phys. 52, 2152 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    D.I. Borisov, F. Ruzicka, M. Znojil, Int. J. Theor. Phys. 54, 4293 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    S. Albeverio, S. Kuzhel, “PT-symmetric operators in quantum mechanics: Krein spaces methods”, in [21], pp. 293–344Google Scholar
  31. 31.
    M. Znojil, H.B. Geyer, Fort. d. Physik—Prog. Phys. 61, 111 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    M. Znojil, Ann. Phys. (NY) 361, 226 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Nuclear Physics Institute ASCRŘežCzech Republic

Personalised recommendations