Advertisement

Foundations of Physics

, Volume 30, Issue 12, pp 2113–2124 | Cite as

Bose Condensation Without Broken Symmetries

  • Andrei E. Ruckenstein
Article

Abstract

This paper considers the issue of Bose–Einstein condensation in a weakly interacting Bose gas with a fixed total number of particles. We use an old current algebra formulation of non-relativistic many body systems due to Dashen and Sharp to show that, at sufficiently low temperatures, a gas of weakly interacting Bosons displays Off-diagonal Long Range Order in the sense introduced by Penrose and Onsager. Even though this formulation is somewhat cumbersome it may demystify many of the standard results in the field for those uncomfortable with the conventional broken symmetry based approaches. All the physics presented here is well understood but as far as we know this perspective, although dating from the 60's and 70's, has not appeared in the literature. We have attempted to make the presentation as self-contained as possible in the hope that it will be accessible to the many students interested in the field.

Keywords

Long Range Weakly Interact Range Order Body System Break Symmetry 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995); K. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997).Google Scholar
  2. 2.
    For a historical introduction to Bose condensation with references to the early literature on the weakly interacting Bose gas see A. Griffin, in the Proceedings of the International School of Physics “Enrico Fermi,” Course CXL, M. Inguscio, S. Stringari, and C. Wieman, eds. (IOP Press, Amsterdam, 1999).Google Scholar
  3. 3.
    F. Dafovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999); see also, Proceedings of the International School of Physics “Enrico Fermi,” Course CXL, M. Inguscio, S. Stringari, and C. Wieman, eds. (IOP Press, Amsterdam, 1999).Google Scholar
  4. 4.
    P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.) 28, 349 (1964); N. N. Bogoliubov, in Lectures on Quantum Statistics (Gordon 6 Breach, New York, 1970); see also, D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, New York, 1975).Google Scholar
  5. 5.
    M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996); C. W. Gardiner, Phys. Rev. A 56, 1414 (1997); P. Villain, M. Lewenstein, R. Dum, Y. Castin, L. You, A. Imamoglu, and T. B. A. Kennedy, J. of Modern Optics 44, 1775 (1997); Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).Google Scholar
  6. 6.
    R. F. Dashen and D. H. Sharp, Phys. Rev. 165, 1857 (1968); see also D. H. Sharp, ibid 165, 1867 (1968); C. G. Callan, R. F. Dashen, and D. H. Sharp, ibid 165, 1883 (1968); J. Grodnik and D. H. Sharp, Phys. Rev. D 1, 1531 (1970); ibid 1, 1546 (1970). To our knowledge the field theoretic approach to Bose systems using collective fields was pioneered by N. N. Bogoliubov and D. N. Zubarev, Sov. Phys. JETP 1, 83 (1955).Google Scholar
  7. 7.
    V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983).Google Scholar
  8. 8.
    S. Sunakawa, S. Yamasaki, and T. Kebukawa, Prog. Theor. Phys. 41 919 (1969); D. D. H. Yee, Phys. Rev. 184, 196 (1969); B. Tsu-Shen Chang, J. Math. Phys. 12, 971 (1971); Phys. Rev. A 5, 1480 (1972); G. S. Grest and A. K. Rajagopal, Phys. Rev. A 10, 1395 (1974); A. K. Rajagopal and G. S. Grest, ibid 10, 1837 (1974).Google Scholar
  9. 9.
    O. Penrose, Phil. Mag. 42 1373 (1951); O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).Google Scholar
  10. 10.
    We note that, by contrast, symmetry breaking also implies ODLRO in the correlation function (1).Google Scholar
  11. 11.
    Here we will not consider fragmented condensates.Google Scholar
  12. 12.
    The operator \(\hat P = \Psi \left( {{\text{r}}} \right)\left[ {{1 \mathord{\left/ {\vphantom {1 {\hat n}}} \right. \kern-\nulldelimiterspace} {\hat n}}\left( {{\text{r}}} \right)} \right]\Psi ^\dag \left( {{\text{r}}} \right)\) is a projection operator \(\left( {\hat P^2 = \hat P} \right)\) with eigenvalues λ=0, 1. Since for any finite interaction no bosonic state can be annihilated by \(\hat P,\hat P \equiv {\hat l}\) on all states in the Hilbert space.Google Scholar
  13. 13.
    G. A. Goldin and D. H. Sharp, in Group Representations in Mathematics and Physics, Battelle Seattle 1969 Rencontres, V. Bargmann, ed. (Springer, New York, 1970); G. A. Goldin, J. Math. Phys. 12, 462 (1971); G. A. Goldin, J. Grodnik, R. T. Powers, and D. H. Sharp, J. Math. Phys. 15, 88 (1974).Google Scholar
  14. 14.
    G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996).Google Scholar
  15. 15.
    S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).Google Scholar
  16. 16.
    N. N. Bogoliubov, J. Phys. USSR 11, 23 (1947); for a recent review see, for example, A. L. Fetter, in the Proceedings of the International School of Physics “Enrico Fermi,” Course CXL, M. Inguscio, S. Stringari, and C. Wieman, eds. (IOP Press, Amsterdam 1999).Google Scholar
  17. 17.
    B. D. Josephson, Phys. Lett. 1, 251 (1962).Google Scholar
  18. 18.
    M. Ma and T.-L. Ho, J. Low Temp. Phys. 115, 61 (1999).Google Scholar
  19. 19.
    P. W. Anderson, Phys. Rev. 86 694 (1952).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Andrei E. Ruckenstein
    • 1
  1. 1.Department of Physics and AstronomyRutgers UniversityPiscataway08854-0849

Personalised recommendations