Minisuperspace Quantisation via Conditional Symmetries

  • Adamantia ZampeliEmail author
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


We review the canonical quantisation of minisuperspace models by promoting to operators the constraints as well as the additional symmetries of the metric of the configuration space of variables. We describe the classical and quantum formulation of the theory and give an application of this approach to the FLRW spacetime coupled to a massless scalar field.


Minisuperspace FLRW universe Bohmian interpretation 



I would like to thank the organisers of the 1st Domoschool for their kind hospitality and the high level of lectures they provided during the school. This work was supported by the grant GAČR 14-37086G.


  1. 1.
    T. Christodoulakis, N. Dimakis, P.A. Terzis, Lie point and variational symmetries in minisuperspace Einstein gravity. J. Phys. A 47, 095202 (2014). [1304.4359]ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    H. Stephani, M. MacCallum, Differential Equations: Their Solution Using Symmetries (Cambridge University Press, Cambridge, 1989)zbMATHGoogle Scholar
  3. 3.
    K. Kuchar, Conditional symmetries in parametrized field theories. J. Math. Phys. 23, 1647–1661 (1982)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Christodoulakis, N. Dimakis, P.A. Terzis, G. Doulis, T. Grammenos, E. Melas, et al., Conditional symmetries and the canonical quantization of constrained minisuperspace actions: the Schwarzschild case. J. Geom. Phys. 71, 127–138 (2013). [1208.0462]ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    P.A. Terzis, N. Dimakis, T. Christodoulakis, A. Paliathanasis, M. Tsamparlis, Variational contact symmetries of constraint Lagrangians. arXiv:1503.00932Google Scholar
  6. 6.
    P.A.M. Dirac, Lectures on Quantum Mechanics. Belfer Graduate School of Science. Monographs (Belfer Graduate School of Science, New York, 1964)Google Scholar
  7. 7.
    T. Christodoulakis, J. Zanelli, Operator ordering in quantum mechanics and quantum gravity. Nuovo Cim. B 93, 1–21 (1986)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    T. Christodoulakis, J. Zanelli, Consistent algebra for the constraints of quantum gravity. Nuovo Cim. B 93, 22–35 (1986)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables. 1. Phys. Rev. 85, 166–179 (1952)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables. 2. Phys. Rev. 85, 180–193 (1952)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Bohm, B. Hiley, Measurement understood through the quantum potential approach. Found. Phys. 14, 255–274 (1984)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    T. Christodoulakis, E. Korfiatis, G. Papadopoulos, Automorphism inducing diffeomorphisms and invariant characterization of Bianchi type geometries. Commun. Math. Phys. 226, 377–391 (2002). [gr-qc/0107050]Google Scholar
  13. 13.
    T. Christodoulakis, T. Gakis, G. Papadopoulos, Conditional symmetries and the quantization of Bianchi type I vacuum cosmologies with and without cosmological constant. Class. Quant. Gravit. 19, 1013–1026 (2002). [gr-qc/0106065]ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Theoretical Physics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

Personalised recommendations