Skip to main content
SpringerLink
Log in

The quantum geometric phase as a transformation invariant

  • Quantum Mechanics
  • Published:
Pramana Aims and scope Submit manuscript

Abstract

The kinematic approach to the theory of the geometric phase is outlined. This phase is shown to be the simplest invariant under natural groups of transformations on curves in Hilbert space. The connection to the Bargmann invariant is brought out, and the case of group representations described.

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. M V Berry,Proc. R. Soc. (London) A392, 45 (1984)

    ADS  MathSciNet  Google Scholar 

  2. S M Rytov,Dokl. Akad. Nauk. (USSR) 28, 263 (1938)

    Google Scholar 

  3. V V Vladimirskii,Dokl. Akad. Nauk (USSR) 31, 31 (1941)

    Google Scholar 

  4. S Pancharatnam,Proc. Indian Acad. Sci. A44, 247 (1956)

    MathSciNet  Google Scholar 

  5. See, for instance, A Shapere and F Wilczek,Geometric phases in physics (World Scientific Publishing Co. Pte. Ltd., Singapore, 1989)

    MATH  Google Scholar 

  6. N Mukunda and R Simon,Ann. Phys. (N.Y.) 228, 205, 269 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. H Weyl,The theory of groups and quantum mechanics (Dover Publications Inc., 1931) p. 4, 20

  8. See any standard text on quantum mechanics such as, for instance, L I Schiff,Quantum Mechanics (McGraw-Hill Book Company Inc., New York, 2nd edition, 1955) p. 213

    MATH  Google Scholar 

  9. Y Aharonov and J Anandan,Phys. Rev. Lett. 58, 1593 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  10. J Samuel and R Bhandari,Phys. Rev. Lett. 60, 2339 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  11. See the first of refs. [4] above

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. V Bargmann,J. Math. Phys. 5, 862 (1964)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. E P Wigner,Group theory and its application to the quantum mechanics of atomic spectra (Academic Press, New York, 1959) appendix to Ch. 20

    MATH  Google Scholar 

  14. R Simon and N Mukunda,Phys. Rev. Lett. 70, 880 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. See the second of refs. [4] above

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. See, for instance, S Helgason,Differential geometry and symmetric spaces (Academic Press, New York, 1962)

    MATH  Google Scholar 

  17. B Schutz,Geometrical methods of mathematical physics (Cambridge University Press, London, 1980)

    MATH  Google Scholar 

  18. C J Isham,Modern differential geometry for physicists (World Scientific Publishing Co. Pte. Ltd., Singapore, 1989)

    MATH  Google Scholar 

  19. G Khanna, S Mukhopadhyay, R Simon and N Mukunda,Ann. Phys. (N.Y.) 253, 55 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Theoretical Studies and Department of Physics, Indian Institute of Science, 560 012, Bangalore, India

    N Mukunda

  2. Honorary Professor, Jawaharlal Nehru Centre for Advanced Scientific Research, 560 064, Jakkur, Bangalore, India

    N Mukunda

Authors
  1. N Mukunda
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mukunda, N. The quantum geometric phase as a transformation invariant. Pramana - J Phys 49, 33–40 (1997). https://doi.org/10.1007/BF02856336

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02856336

Keywords

PACS No.

Search

Navigation