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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Seiler, Schwinger Functions for the Yukawa model in two dimensions with space time cut off, Comm. Math. Phys., 42, 2 (1975).
O. McBryan, Volume dependence of Schwinger Functions in the Yukawa 2 Quantum Field Theory, Rockefeller University preprint.
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.
O. McBryan, Convergence of the Vacuum Energy Density, ⌽-bounds and Existence of Wightman Functions for the Yukawap model, Rockefeller University preprint.
J. Magnen, R. Seneor, The Wightman Axioms for the Weakly Coupled Yukawa Model in two Dimensions, E. Polytechnique, Palaiseau, preprint.
J. Glimm, A. Jaffe, Positivity of the Hamiltonian, Forts, der Phys., 21, 327–376 (1973).
J. Glimm, A. Jaffe, T. Spencer, The Cluster Expansion, in Constructive Quantum Field Theory, Lecture notes in Physics, no. 25, Springer, Berlin (1973).
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)
J. Magnen, R. Seneor, The Infinite Volume Limit of the \( \varphi _3^4 \) model, to appear in Ann. Inst. H. Poincaré.
E. Nelson, Probability Theory and Euclidean Field Theory, in Constructive Qunatum Field Theory, Lecture Notes in Physics, No. 25, Springer, Berlin (1973).
J. Dimock, J. Glimm, Measures on the Schwartz distribution space and Application to P(⌽)2 field theories, to appear in Adv. Math.
O. McBryan, Finite mass renormalizations in the Yukawa2 Quantum Field Theory, Comm. Math. Phys. to appear.
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].
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1976 Springer-Verlag
About this paper
Cite this paper
Seneor, R. (1976). The Cluster Expansion for Y2 . In: Streit, L. (eds) Quantum Dynamics: Models and Mathematics. Acta Physica Austriaca, vol 16/1976. Springer, Vienna. https://doi.org/10.1007/978-3-7091-8473-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-7091-8473-8_7
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-8475-2
Online ISBN: 978-3-7091-8473-8
eBook Packages: Springer Book Archive