X Critical Exponents and Renormalization in the φ4 Scaling Limit

  • J. Glimm
  • A. Jaffe


For dimensions d ≤ 3, the φ4 scaling limit defines a nonrenormalizable field theory. The standard relations between critical exponents and renormalization are presented. Arguments supporting the existence of the scaling limit are based on correlation inequalities and the numerical values of Ising model exponents, 2η < ηE for d=2,3.


Ising Model Critical Exponent Point Function Scaling Limit Distance Scale 
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.
    G. BAKER, Self-interacting boson quantum field theory and the thermodynamic limit in d dimensions, J. Math. Phys. 16, 1324–1346 (1975).ADSCrossRefGoogle Scholar
  2. 2.
    E. BAROUCH, B. MCCOY, C. TRAY and T. T. WU, The spin-spin correlation functions for the two dimensional Ising-model Exact theory in the scaling limit, to appear.Google Scholar
  3. 3.
    J. FELDMAN, On the absence of bound states in φ4 the e quantum field model without symmetry breaking, Cand. J. Phys. 52, 1583–1587 (1974).ADSGoogle Scholar
  4. 4.
    J. FELDMAN and K. OSTERWALDER, The Wightman axioms and the mass gap for weakly coupled quantum field theories, to appear.Google Scholar
  5. 5.
    M. FISHER, Rigorous inequalities for critical point correlation exponents, Phys. Rev. 180, 594–600 (1969).CrossRefGoogle Scholar
  6. 6.
    J. Fröhlich, Existence and analyticity in the bare parameters of the quantum field models, I. Manuscript.Google Scholar
  7. 7.
    J. GLIMM and A. JAFFE, The (λø4)2 quantum field theory with-out cutoffs III. The physical vacuum, Acta Math. 125, 203–261 (1970).MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. GLIMM and A. JAFFE The (λø4)2 quantum field theory without cutoffs IV. Perturbation of the Hamiltonian, J. Math. Phys. 13, 1558–1584 (1972).MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    J. GLIMM and A. JAFFE, quantum field theory in the single phase region: Differentiability of the mass and bounds on critical exponents, Phys. Rev. 10, 536–539 (1974).Google Scholar
  10. 10.
    J. GLIMM and A. JAFFE, Two and three body equations in quantum field models, Commun. Math. Phys. 44, 293–320 (1974).MathSciNetADSGoogle Scholar
  11. 11.
    J. GLIMM and A. JAFFE, A remark on the existence of. Phys. Rev. Lett. 33, 440–442 (1974).MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    J. GLIMM and A. JAFFE, Three particle structure of interactions and the scaling limit, Phys. Rev. 11, 2816–2827 (1975).MathSciNetGoogle Scholar
  13. 13.
    J. GLIMM, A. JAFFE, and T. SPENCER, The particle structure of the weakly coupled P(0) models and other applications of high temperature expansions. In: Constructive quantum field theory, G. Velo and A. Wightman (eds.) Springer Verlag, Berlin, 1973. 166Google Scholar
  14. 14.
    J. GLIMM, A. JAFFE, and T. SPENCER, Existence of phase transitions for quantum fields, Commun. Math. Phys. To appear.Google Scholar
  15. 15.
    J. GLIMM, A. JAFFE, and T. SPENCER, A cluster expansion for the quantum field theory in the two phase region. In preparation.Google Scholar
  16. 16.
    F. GUERRA, L. ROSEN, and B. SIMON, In: Constructive quantum field theory, G. Velo and A. Wightman (eds.) Springer Verlag, Berlin, 1973.Google Scholar
  17. 17.
    D. ISAACSON, The critical behavior of the autoharmonic oscillator, NYU Thesis.Google Scholar
  18. 18.
    B. JOSEPHSON, Inequality for the specific heat, I Derivation, II Applications, Proc. Phil. Soc. 92, 269–284 (1967).Google Scholar
  19. 19.
    D. MARCHESIN, Work in progress.Google Scholar
  20. 20.
    J. MAGNEN and R. SENEOR, The infinite volume limit of the 4 model, Ann. Inst. H. Poincaré, to appear.Google Scholar
  21. 21.
    K. OSTERWALDER and R. SCHRADER, Axioms for Euclidean Green’s functions, Commun. Math. Phys. 31, 83–112 (1974).MathSciNetADSGoogle Scholar
  22. 22.
    K. OSTERWALDER and R. SCHRADER, Axioms for Euclidean Green’s functions II, Commun. Math. Phys. 42, 281–305 (1975).MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Y. PARK, Uniform bounds of the pressure of the X4): 3 model. Preprint.Google Scholar
  24. 24.
    J. ROSEN, Mass renormalization for adz Euclidean lattice field theory.Google Scholar
  25. 25.
    J. ROSEN, Private communication.Google Scholar
  26. 26.
    E. SEILER and B. SIMON, Nelson’s symmetry and all that in the Yukawa and e field theories. Preprint.Google Scholar
  27. 27.
    T. SPENCER, The absence of even bound states in Commun. Math. Phys. 39, 77–79 (1974).ADSCrossRefGoogle Scholar
  28. 28.
    E. BREZIN, J. C. LEGUILLORE and J. ZINN-JUSTIN, Field theoretical approach to critical phenomena. In: Phase transitions and critical phenomena, Vol. VI., Ed. by Domb and Green, Academic Press, New York, to appear.Google Scholar

Copyright information

© Birkhäuser Boston Inc. 1985

Authors and Affiliations

  • J. Glimm
    • 1
  • A. Jaffe
    • 2
  1. 1.Rockefeller UniversityNew YorkUSA
  2. 2.Harvard UniversityCambridgeUSA

Personalised recommendations