The Cluster Expansion for Y2
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  considerable progress has been made. Upper bounds depending exponentially on the interaction volume as in P(⌽)2 have been obtained by O. McBryan  and E. Seiler and B. Simon . The comparison with P(⌽)2 theories is even more complete since McBryan  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  that I will report here.
KeywordsPseudo Differential Operator Cluster Expansion Fermion Propagator Wightman Function Relativistic Quantum Field Theory
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