Skip to main content
SpringerLink
Log in

Radon transform of the Wheeler-De Witt equation and tomography of quantum states of the universe

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

The notion of standard positive probability distribution function (tomogram) which describes the quantum state of the universe alternatively to the wave function or to the density matrix is introduced. Connection of the tomographic probability distribution with the Wigner function of the universe and with the star-product (deformation) quantization procedure is established.

Using the Radon transform, the Wheeler-De Witt generic equation for the probability function is written in tomographic form. Some examples of the Wheeler-DeWitt equation in the minisuperspace are elaborated explicitly for homogeneous isotropic cosmological models. Some interpretational aspects of the probability description of the quantum state are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Radon, J.: Ber. Verh. Sächs. Acad. 69, 269 (1917)

    Google Scholar 

  2. von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)

    Google Scholar 

  3. Landau, L.D., Physik, Z.: 45 430 (1927)

  4. Wigner, E.: Phys. Rev. 40, 749 (1932)

    Article  MATH  ADS  Google Scholar 

  5. Vogel, K., Risken, H.: Phys. Rev. A 40, 2847 (1989)

    Article  ADS  Google Scholar 

  6. Smithey, D.T., Beck, M., Raymer, M. G., Faridani A.: Phys. Rev. Lett. 70, 1244 (1993)

    Article  ADS  Google Scholar 

  7. Bertrand, J., Bertrand, D.: Found. Phys. 17, 397 (1987)

    ADS  MathSciNet  Google Scholar 

  8. Mancini, S., Manko, V.I., Tombesi, P.: Quant. Semiclass. Opt. 7, 615 (1995)

    ADS  Google Scholar 

  9. Mancini, S., Manko, V.I., Tombesi, P.: Phys. Lett. A 213, 1, (1996)Manko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: J. Russ. Laser Res. 20, 421 (1999) Manko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: Phys. Lett. A 273, 31 (2000) Manko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: J. Phys. A: Math. Gen. 35, 7137 (2002)

  10. Manko, O.V., Manko, V.I., Marmo, G.: Phys. Scr. 62, 446 (2000) Manko, O.V., Manko, V.I., Marmo, G.: J. Phys. A 35, 699 (2002)

    Google Scholar 

  11. Moyal, J.E.: Proc. Cambridge Philos. Soc. 45, 99 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hawking, S.W.: Nucl. Phys. B 239, 257 (1984)

    ADS  MathSciNet  Google Scholar 

  13. DeWitt, B.S.: Phys. Rev. 160, 1113 (1983) Wheeler, J.A.: In Battelle Rencontres. In DeWitt, C., Wheeler, J.A. (eds.) Benjamin, New York (1968)

  14. Hartle, J.B., Hawking, S.W.: Phys. Rev. D 28, 2960 (1983)

    ADS  MathSciNet  Google Scholar 

  15. Hawking, S.W.: Intersection between Elementary Particle Physics and Cosmology. In Piran, T., Weinberg, S. (eds.) World Scientific, Singapore (1986)

  16. Page, D.N.: Phys. Rev. D 34, 2267 (1986)

    ADS  MathSciNet  Google Scholar 

  17. Halliwell, J.J.: Phys. Rev. D 38, 2468 (1988)

    ADS  MathSciNet  Google Scholar 

  18. Rubakov, V.A.: Quantum cosmology [arXiv:gr-qc/9910025]

  19. Parentani, R.: Phys. Rev. D 56, 4618 (1997)

    ADS  MathSciNet  Google Scholar 

  20. Antonsen, F.: Deformation quantisation of Gravity [gr-qc/9712012]

  21. Kodama, H.: Quantum cosmology in terms of the wigner function, KUCP-0014. Presented at 5th Marcel Grossmann Mtg., Perth, Western Australia, 8–12 Aug. 1988

  22. Manko, V.I., Mendes, R.V.: Phys. Lett. A 263, 53 (1999)

    ADS  MathSciNet  Google Scholar 

  23. Ashtekar, A.: Lectures on Non-perturbative Canonical Gravity. World Scientific, Singapore (1991) Ashtekar, A.: New Perspectives in Canonical Gravity. Bibliopolis, Napoli (1988)

  24. Baez, J.C.: Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations. In: Crane, L., Yetter, D. (eds.) Proceedings of the Conference on Quantum Topology, pp. 213–223. World Scientific, Singapore, 1994

  25. Manko, V.I., Rosa, L., Vitale, P.: Phys. Lett B 439, 328 (1998)

    ADS  MathSciNet  Google Scholar 

  26. Bajen, F., Flato, M., Fronsdal, M., Lichnerowicz, C., Sternheimer, D.: Lett. Math. Phys. 1, 521 (1975)

    ADS  MathSciNet  Google Scholar 

  27. Manko, M.A.: J. Russ. Laser Res. 21, 421 (2000)

    Google Scholar 

  28. Manko, O.V., Manko, V.I.: J. Russ. Laser Res. 18, 407 (1997)

    Google Scholar 

  29. Manko, M.A.: J. Russ. Laser Res. 22, 168 (2001)

    Google Scholar 

  30. Louko, J.: Class. Quantum Grav. 4, 581 (1987)

    ADS  MathSciNet  Google Scholar 

  31. Manko, V.I., Shchukin, E.V.: J. Russ. Laser Res. 22, 545 (2001)

    Google Scholar 

  32. Gousheh, S.S., Sepangi, H.R.: Phys. Lett. A 272, 304 (2000)

    MATH  ADS  MathSciNet  Google Scholar 

  33. Basini, G., Capozziello, S., Longo, G.: Astron. Nach. 324, 275 (2003) Basini, G., Capozziello, S., Longo, G.: La Rivista del N. Cim. 25, N.11 (2002)

  34. Manko, V.I., Mendes, R.V.: Physica D 145, 330 (2000)

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio N’via Cinthia, 45-80126, Napoli

    V. I. Man’ko, G. Marmo & C. Stornaiolo

  2. Dipartimento di Scienze Fisiche, Università “Federico II” di Napoli, Complesso Universitario di Monte S. Angelo Edificio N’via Cinthia, 45-80126, Napoli

    V. I. Man’ko, G. Marmo & C. Stornaiolo

Authors
  1. V. I. Man’ko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Marmo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Stornaiolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Stornaiolo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Man’ko, V.I., Marmo, G. & Stornaiolo, C. Radon transform of the Wheeler-De Witt equation and tomography of quantum states of the universe. Gen Relativ Gravit 37, 99–114 (2005). https://doi.org/10.1007/s10714-005-0005-3

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-005-0005-3

Keywords

Search

Navigation