Communications in Mathematical Physics

, Volume 53, Issue 1, pp 19–30

The cluster expansion for potentials with exponential fall-off

  • David Brydges
  • Paul Federbush
Article

Abstract

Continuing the work of a previous paper, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by potentials that fall off exponentially at large distance. The HamiltonianH0+V need be stable in the extended sense thatH0+4V+BN≧0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. A natural system included in the present treatment is that of matter with ther−1 potential replaced bye−ar/r. The Hamiltonian is stable, but the system would collapse in the absence of the exclusion principle—the potential is unstable. Therefore this system cannot be handled by the classic work of Ginibre, which requires stable potentials.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brydges, D., Federbush, P.: The cluster expansion in statistical mechanics. Commun. math. Phys.49, 233–246 (1976)Google Scholar
  2. 2.
    Dimock, J., Glimm, J.: Adv. Math.12, 58 (1974)Google Scholar
  3. 3.
    Dyson, F.J.: J. Math. Phys.8, 1538 (1967)Google Scholar
  4. 4.
    Dyson, F.J., Lenard, A.: J. Math. Phys.8, 423 (1967); J. Math. Phys.9, 698 (1968)Google Scholar
  5. 5.
    Federbush, P.: J. Math. Phys.16, 347 (1975); J. Math. Phys.16, 706 (1975)Google Scholar
  6. 6.
    Federbush, P.: J. Math. Phys.17, 200 (1976); J. Math. Phys.17, 204 (1976)Google Scholar
  7. 7.
    Ginibre, J.: Some applications of functional integration in statistical mechanics. In: Statistical mechanics and quantum field theory, Les Houches 1970 (ed. C. Dewitt, R. Stora). New York: Gordon and Breach 1971Google Scholar
  8. 8.
    Glimm, J., Jaffe, A., Spencer, T.: The cluster expansion. In: Constructive quantum field theory, The 1973 “Ettore Majorana” International School of Mathematical Physics. (ed. G. Velo, A. Wightman). Berlin-Heidelberg-New York: Springer 1973Google Scholar
  9. 9.
    Glimm, J., Jaffe, A., Spencer, T.: Ann. Math.100, 585 (1974)Google Scholar
  10. 10.
    Lieb, E.H., Thirring, W.E.: Phys. Rev. Lett.35, 687 (1975)Google Scholar
  11. 11.
    Ruelle, D.:Statistical Mechanics. New York: W. A. Benjamin, Inc. (1969)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • David Brydges
    • 1
  • Paul Federbush
    • 1
  1. 1.Department of MathematicsThe University of MichiganAnn ArborUSA

Personalised recommendations