Advertisement

The Cluster Expansion for Y2

  • R. Seneor
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 16/1976)

Abstract

For long time, the results concerning the Euclidean Yukawa quantum field theory (Y2) in two dimension were very far from the ones obtained in P(⌽)2 quantum field theories. This difference was essentially due to the difficulties one has in describing Euclidean Fermi fields. However since the definition of this model solely in term of bose fields (i.e. with the fermions “integrated out”) given by E. Seiler [1] considerable progress has been made. Upper bounds depending exponentially on the interaction volume as in P(⌽)2 have been obtained by O. McBryan [2] and E. Seiler and B. Simon [3]. The comparison with P(⌽)2 theories is even more complete since McBryan [4] has obtained the proof of ⌽-bounds, and by the way, of the existence of Wightman functions. The next step to complete the analogy with P(⌽)2 theories was to prove the convergence of a cluster expansion. This is the result obtained in collaboration with J. Magnen [5] that I will report here.

Keywords

Pseudo Differential Operator Cluster Expansion Fermion Propagator Wightman Function Relativistic Quantum Field Theory 
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]
    E. Seiler, Schwinger Functions for the Yukawa model in two dimensions with space time cut off, Comm. Math. Phys., 42, 2 (1975).MathSciNetCrossRefGoogle Scholar
  2. [2]
    O. McBryan, Volume dependence of Schwinger Functions in the Yukawa 2 Quantum Field Theory, Rockefeller University preprint.Google Scholar
  3. [3]
    E. Seiler, B. Simon, Bounds in the Yukawa2 Quantum Field Theory: Upper Bound on the Pressure, Hamiltonian Bound and Linear Lower Bound, Princeton University preprint.Google Scholar
  4. [4]
    O. McBryan, Convergence of the Vacuum Energy Density, ⌽-bounds and Existence of Wightman Functions for the Yukawap model, Rockefeller University preprint.Google Scholar
  5. [5]
    J. Magnen, R. Seneor, The Wightman Axioms for the Weakly Coupled Yukawa Model in two Dimensions, E. Polytechnique, Palaiseau, preprint.Google Scholar
  6. [6]
    J. Glimm, A. Jaffe, Positivity of the Hamiltonian, Forts, der Phys., 21, 327–376 (1973).MathSciNetCrossRefGoogle Scholar
  7. [7]
    J. Glimm, A. Jaffe, T. Spencer, The Cluster Expansion, in Constructive Quantum Field Theory, Lecture notes in Physics, no. 25, Springer, Berlin (1973).Google Scholar
  8. [8]
    J. P. Eckmann, J. Magnen, R. Seneor, Decay Properties and Borel summability for the Schwinger Functions in P(⌽)2 theories, Comm. Math. Phys., 39, 4 (1975)MathSciNetCrossRefGoogle Scholar
  9. [9]
    J. Magnen, R. Seneor, The Infinite Volume Limit of the \( \varphi _3^4 \) model, to appear in Ann. Inst. H. Poincaré.Google Scholar
  10. [10]
    E. Nelson, Probability Theory and Euclidean Field Theory, in Constructive Qunatum Field Theory, Lecture Notes in Physics, No. 25, Springer, Berlin (1973).Google Scholar
  11. [11]
    J. Dimock, J. Glimm, Measures on the Schwartz distribution space and Application to P(⌽)2 field theories, to appear in Adv. Math.Google Scholar
  12. [12]
    O. McBryan, Finite mass renormalizations in the Yukawa2 Quantum Field Theory, Comm. Math. Phys. to appear.Google Scholar
  13. [13]
    E. Seiler, B. Simon, On finite mass renormalizations in the two dimensional Yukawa model, Princeton University preprint. For a more complete list of references, see [2], [3] or [5].Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • R. Seneor
    • 1
  1. 1.Centre de Physique ThéoriqueE. PolytechniquePalaiseauFrance

Personalised recommendations