Construction of (λɸ4-σɸ2-μɸ)3 Quantum Field Models

  • Y. M. Park
Conference paper
Part of the Few-Body Systems book series (FEWBODY, volume 15/1976)


This lecture is intended to introduce the audience to some constructions of theories for the boson field models in three dimensional space-time, which exhibit ultraviolet divergences. Constructive quantum field theory has developed rapidly in the past few years. The polynomial interactions in two dimensional space-time (the P (ɸ)2 models) are the best behaved models and its detailed structure is now well-known [2, 12, 16, 18, 19, 23, 24, 28, 30, 36, 37, 40]. Most of you may have already been exposed in the construction of the P(ɸ)2 model elsewhere (at least, you will have a change again in Prof. Challifour’s lecture at this school) and so I will not 42 discuss that subject. The (λɸ4-σɸ2-μɸ) interactions in three dimensional space-time (the (λɸ4-σɸ2-μɸ)3 models), which we are considering here, are the next well behaved, models. These differ from P(ɸ)2 by having ultraviolet divergences and by requiring ultraviolet mass wave functions as well as vacuum energy renormalizations [14, 15].


Field Model Lattice Approximation Cluster Expansion Uniform Bound Ultraviolet Divergence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Albeverio and R. Høegh-Krohn, The Wightman Axioms and the Mass Gap for Strong Interaction of Exponential Type in Two-Dimensional Space-Time, J. Funct. Anal. 16, 39 (1974).MATHCrossRefGoogle Scholar
  2. 2.
    J. P. Eckman, Representations of the C.C.R. in the (ɸ4)3 Model; Independent of Space Cutoff, Comm. Math. Phys. 25, 1 (1972).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    J. P. Eckman, H. Epstein and J. Fröhlich, Asymptotic Perturbation Expansion for the S-Matrix and Definition of Time-Ordered Functions in Relativistic Field Models, University of Geneve, Preprint (1975).Google Scholar
  4. 4.
    J. P. Eckman and K. Osterwalder, On the Uniqueness of the Hamiltonian and of the Representation of the C.C.R. for the Quartic Boson Interaction in Three Dimensions, Helv. Phys. Acta 44, 884 (1971).Google Scholar
  5. 5.
    J. Fabrey, Exponential Representations of Canonical Commutation Relations, Comm. Math. Phys. 1 (1970).Google Scholar
  6. 6.
    J. Fabrey, Weyl Systems for the (ɸ4) 3 Model, J. Math. Phys. 37. 93 (1974).MathSciNetGoogle Scholar
  7. 7.
    J. Feldman, The λɸ43 Field Theory in a Finite Volume, Comm. Math. Phys. 37, 93 (1974).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    J. Feldman, The Nonperturbative Renormalization of (λɸ4)3, M.I.T. preprint (1975).Google Scholar
  9. 9.
    J. Feldman and K. Osterwalder, The Wightman Axioms and the Mass Gap for Weakly Coupled (ɸ4)3 Quantum Field Theories, Harvard University, Preprint, February (1975).Google Scholar
  10. 10.
    J. Feldman and K. Osterwalder, The Construction of Quantum Field Models, Proceedings of the International Colloquium on Mathematical Methods of Quantum Field Theory, Marseille (1975).Google Scholar
  11. 11.
    C. Fortuin, P. Kastelyn and J. Ginibre, Correlation Inequalities on Some partially Ordered Sets, Comm. Math. Phys. 22, 89 (1971).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    J. Fröhlich, New Super-Selection Sections (‘Soliton- States’) in Two Dimensional Bose Quantum Field Models, Princeton University, Preprint (1975).Google Scholar
  13. 13.
    J. Fröhlich, Existence and Analyticity in the Bare parameters of the\( \left( {\lambda \left( {\overrightarrow {\Phi .} \vec \Phi } \right)^2 - \sigma \Phi _1^2 - \mu \Phi _1 } \right)\) - Quantum Field Models, I, Princeton University, Preprint (1975).Google Scholar
  14. 14.
    J. Glimm, Boson Fields with the:ɸ4: Interaction in the Three Dimensions, Comm. Math. Phys. 10, 1 (1968).MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    J. Glimm and A. Jaffe, Positivity of the ɸ4 3 Hamiltonian, Fortschritte der Physik, 21, 327 (1973).MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    J. Glimm and A. Jaffe, On the Approach to the Critical Point, Harvard University, Preprint (1974).Google Scholar
  17. 17.
    J. Glimm, A. Jaffe, A Remark on the Existence of ɸ 4 4, Phys. Rev. Letter, 33, 440 (1974)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    J. Glimm, A. Jaffe and T. Spencer, The Particle Structure of the Weakly Coupled P (ɸ)2 Model and Other Applications of High Temperature Expansions, Contribution to [40].Google Scholar
  19. 19.
    J. Glimm, A. Jaffe and T. Spencer, Phase Transitions for the ɸ4 2 Quantum Field Theory, Comm. Math. Phys. 45, 203 (1975).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    R. Griffiths, Correlation in Ising Ferromagnetics, I, II, III, J. Math. Phys. 8, 478–483, 484–489 (1967).Google Scholar
  21. 21.
    R. Griffiths, C. Hurst and S. Sherman, Concavity of Magnatization of an Ising Ferromagnet in a positive External Field, J. Math. Phys. 11, 790 (1970).MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    F. Guerra, External Field Dependence of Magnetization and Long Range Order in Quantum Field Theory, I.A.S., Preprint, December (1975).Google Scholar
  23. 23.
    F. Guerra, L. Rosen and B. Simon, The P (ɸ)2 Euclidean Quantum Field Theory as Classical Statistical Mechanics, Ann. of Math. 101, 111 (1975).MathSciNetCrossRefGoogle Scholar
  24. 24.
    F. Guerra, L. Rosen and B. Simon, Boundary Conditions in the P (ɸ) 2 Euclidean Quantum Field Theory, Princeton University, Preprint (1975).Google Scholar
  25. 25.
    J. Lebowitz, GHS and Other Inequalities, Comm. Math. Phys. 35, 87 (1974).MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    J. Magnen and R. Seneor, Infinite Volume Limit of the ɸ 4 3 Model, Ecole Polytechnique, Preprint, February (1975).Google Scholar
  27. 27.
    J. Magnen and R. Seneor, The Wightman Axioms for the Weakly Coupled Yukawa Model in Two Dimensions, Ecole Polytechnique, Preprint (1975).Google Scholar
  28. 28.
    E. Nelson, Probability Theory and Euclidean Field Theory, Contribution to [40].Google Scholar
  29. 29.
    K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s Functions, I and II, Comm. Math. Phys. 31, 83 (1973), Comm. Math. Phys. 42, 281 (1975).Google Scholar
  30. 30.
    K. Osterwalder and R. Seneor, The Scattering Matrix is Nontrivial for Weakly Coupled P (ɸ) 2 Models, Preprint (1975).Google Scholar
  31. 31.
    Y. Park, Lattice Approximation of the (λɸ4-μɸ)3 Field Theory, J. Math. Phys. 16, 1065 (1975); The λɸ 4 3 Euclidean Quantum Field Theory in a periodic Box, J. Math. Phys. 16, 2183 (1975).Google Scholar
  32. 32.
    Y, Park, Uniform Bounds of the Pressures of the λɸ4 3Field Model, to appear in J. Math. Phys.Google Scholar
  33. 33.
    Y. Park, Uniform bounds of the Schwinger Functions in Boson Field Models, to appear in J. Math. Phys.Google Scholar
  34. 34.
    Y. Park, Convergence of Lattice Approximations and Infinite Volume Limit in the (λɸ4-σɸ2-μɸ)3 Field Theory, University of Bielefeld (ZiF), preprint, December(1975).Google Scholar
  35. 35.
    E. Seiler and B. Simon, Nelson’s Symmetry and All That in the Yukawa2 and (ɸ 4) 3 Field Theories, Princeton University, Preprint (1975).Google Scholar
  36. 36.
    B. Simon, The P (ɸ) 2 Euclidean ( Quantum) Field Theory, Princeton University Press (1974).Google Scholar
  37. 37.
    B. Simon and R. Griffiths, The (ɸ4)2 Field Theory as a classical Ising Model, Comm. Math. Phys. 33, 145 (1973).MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    R. Schrader, A Possible Constructive Approach to ɸ 4 4 Free University, Preprint (1975).Google Scholar
  39. 39.
    K. Symanzik, Euclidean Quantum Field Theory, in: Local Quantum Theory, ed. R. Jost, Academic press, New York (1969).Google Scholar
  40. 40.
    G. Velo and A. Wightman, Constructive Quantum Field Theory, Lecture Notes in Physics, Vol. 25, Springer- Verlag, Berlin (1973).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Y. M. Park
    • 1
    • 2
  1. 1.Department of Theoretical PhysicsUniversity of Bielefeld48 Bielefeld 1F.R. Germany
  2. 2.Department of MathematicsYonsei UniversitySeoulKorea

Personalised recommendations