Abstract
The conjectured inequality γ(6)≦0 leads to the existence of ϕ 4 d fields and the scaling (continuum) limit ford-dimensional Ising models. Assuming γ(6)≦0 and Lorentz covariance of this construction, we show that ford≧6 these ϕ 4 d fields are free fields unless the field strength renormalizationZ −1 diverges. Let λ be the bare charge and ε the lattice spacing. Under the same assumptions (γ(6)≦0, Lorentz covariance andd≧6) we show that if λε4−d is bounded as ε→0, thenZ −1 is bounded and the limit field is free.
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References
Baker, G.: Self-interacting Boson quantum field theory and the thermodynamic limit ind dimensions. J. Math. Phys.16, 1324–1346 (1975)
Cannon, J., Jaffe, A.: Lorentz covariance of the λ(ϕ4)2 quantum field theory. Commun. math. Phys.17, 261–321 (1970)
Feldman, J.: On the absence of bound states for λϕ 42 quantum field models without symmetry breaking. Canad. J. Phys.52, 1583–1587 (1974)
Feldman, J., Osterwalder, K.: The Wightman axioms and the mass gap for weakly coupled φ 42 quantum field theories. Ann. Phys., to appear
Feldman, J., Osterwalder, K.: The construction of λ(ϕ4)3 quantum field models, presented at the International Colloquium on Mathematical Methods in Quantum Field Theory, Marseille, June 1975
Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. math. Phys.22, 89–103 (1971)
Fröhlich, J.: To appear
Fröhlich, J., Simon, B.: Pure states for generalP(ϕ)2 theories: construction, regularity, and variational equality, preprint, 1976
Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. math. Phys.50, 79–85 (1976)
Glimm, J., Jaffe, A.: A λϕ4 field theory without cutoffs I. Phys. Rev.176, 1945–1951 (1968)
Glimm, J., Jaffe, A.: The λ(ϕ4)2 quantum field theory without cutoffs. II. The field operators and the approximate Hamiltonian. Ann. Math.91, 362–401 (1970)
Glimm, J., Jaffe, A.: The λ(ϕ4)2 quantum field theory without cutoffs. III. The physical vacuum. Acta Math.125, 203–267 (1970)
Glimm, J., Jaffe, A.: Quantum field models. In: Statistical mechanics (C. De Witt, R. Stora, eds.), 1970 Les Houches Lectures. New York: Gordon and Breach Science Publishers 1971
Glimm, J., Jaffe, A.: Boson quantum field models. In: Mathematics of contemporary physics. London Mathematical Society (R. Streater ed.). New York-London: Academic Press 1972
Glimm, J., Jaffe, A.: The λϕ 42 quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian. J. Math. Phys.13, 1568–1584 (1972)
Glimm, J., Jaffe, A.: The ϕ 42 quantum field model in the single phase region: Differentiability of the mass and bounds on critical exponents. Phys. Rev. D10, 536–539 (1974)
Glimm, J., Jaffe, A.: A remark on the existence of ϕ 44 . Phys. Rev. Letters33, 440–442 (1974)
Glimm, J., Jaffe, A.: On the approach to the critical point. Ann. de l'Inst. Henri Poincaré22, 13–26 (1975)
Glimm, J., Jaffe, A.: Three particle structure of ϕ4 interactions and the infinite scaling limit. Phys. Rev. D11, 2816–2827 (1975)
Glimm, J., Jaffe, A.: Particles and bound states and progress toward unitarity and scaling. Presented at the International Symposium on Mathematical Problems in Theoretical Physics. Kyoto, January 23–29, 1975
Glimm, J., Jaffe, A.: Critical problems in quantum fields, presented at the International Colloquium on Mathematical Methods of Quantum Field Theory. Marseille, June 1975
Glimm, J., Jaffe, A.: Critical exponents and renormalization in the ϕ4 scaling limit, presented at the Conference on Quantum Dynamics: Models and Mathematics. Bielefeld, September 1975
Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and particle structure in theP(ϕ)2 quantum field model. Ann. Math.100, 585–632 (1974)
Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(ϕ)2 model and other applications of high temperature expansions, Part I: Physics of quantum field models; Part II: The cluster expansion. In: Constructive quantum field theory (G. Velo, A. S. Wightman eds.), 1973 Erice Lectures, Lecture Notes in Physics, Vol. 25. Berlin-Heidelberg-New York: Springer 1973
Glimm, J., Jaffe, A., Spencer, T.: Phase transitions for ϕ 42 quantum fields. Commun. math. Phys.45, 293–320 (1975)
Isaacson, D.: The critical behavior of the anharmonic oscillator. Commun. math. Phys., to appear
Mc Bryan, O., Rosen, J.: Existence of the critical point in ϕ4 field theory. Commun. math. Phys., to appear
Magnen, J., Sénéor, R.: The infinite volume limit of the ϕ 43 model. Ann. l'Inst. H. Poincaré, to appear
Marcheson, D.: Private communication
Rosen, J.: Mass Renormalization for the λϕ4 Euclidean lattice field theory, preprint, 1975
Rosen, J.: The Ising model limit of ϕ4 lattice fields, preprint, 1976
Schrader, R.: A possible constructive approach to ϕ 44 . Commun. math. Phys.49, 131–154 (1976)
Schrader, R.: A possible constructive approach to ϕ 44 II, preprint, 1976
Spencer, T.: The absence of even bound states for λ(ϕ4)2. Commun. math. Phys.39, 77–79 (1974)
Streater, R., Wightman, A. S.: PCT, spin and statistics and all that. New York: Benjamin 1964
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Communicated by K. Hepp
Supported in part by the National Science Foundation under Grant MPS 74-13252
Supported in part by the National Science Foundation under Grant MPS 75-21212
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Glimm, J., Jaffe, A. Particles and scaling for lattice fields and Ising models. Commun.Math. Phys. 51, 1–13 (1976). https://doi.org/10.1007/BF01609048
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DOI: https://doi.org/10.1007/BF01609048