Quantum Field Theory and Statistical Mechanics pp 383-418 | Cite as

# A Tutorial Course in Constructive Field Theory

## Abstract

Prior to 1970, the major focus of constructive field theory was the mathematical framework required to establish the existence of quantum fields [1, 2]. Since 1970, the emphasis has gradually shifted, first toward verifying physical properties of the known models, and more recently toward bringing constructive field theory closer to the mainstream of physics [3]. In fact, by 1973 it had become more or less clear that the mathematical framework developed to give the first examples in d = 2, 3 space-time dimensions would be adequate to study d = 4. However, it was also clear that tosolve the ultraviolet problem in d = 4, it would be necessary to incorporate into mathematical physics a deeper physical understanding of the questions being studied. In particular ideas of scaling and of critical behavior may be useful to select and analyze a suitable nontrivial critical point as the first step in dealing with the d = 4 ultraviolet problem. An infinite scaling transformation connects the problem of removing the ultraviolet cutoff with the problem of existence of scaling behavior at the critical point.

## Keywords

Ising Model Scaling Limit Cluster Expansion Single Phase Region Asymptotic Completeness## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J. Glimm and A. Jaffe, Quantum Field Models, in Statistical Mechanics, C. DeWitt and R. Stora (eds.) 1970 Les Houches Lectures, Gordon and Breach Science Publishers, New York, 1971.Google Scholar
- [2]J. Glimm and A. Jaffe, Boson Quantum Field Theory, in Mathematics of Contemporary Physics, R. Streater (ed.), London Mathematical Society, Academic Press, 1972.Google Scholar
- [3]J. Glimm, A. Jaffe and T. Spencer, The Particle Structure of the Weakly Coupled P(ϕ)
_{2}Model and Other Applications of High Temperature Expansions, Part I. Physics of Quantum Field Models, in Constructive Quantum Field Theory, G. Velo and A. S. Wightman (eds.), 1973 Erice Lectures, Springer Lecture Notes in Physics, Vol. 25, Springer-Verlag, 1973.Google Scholar - [4]J. Glimm and A. Jaffe, Critical Problems in Quantum Fields, presented at the International Colloquium on Mathematical Methods of Quantum Field Theory, Marseille, June 1975.Google Scholar
- [5]J. Glimm and A. Jaffe, Critical Exponents and Renormalization in the ϕ
^{4}Scaling Limit, to appear in Quantum Dynamics: Models and Mathematics, L. Streit (ed.), Springer.Google Scholar - [6]J. Glimm and A. Jaffe, Functional Integral Methods in Quantum Field Theory, proceedings of the 1976 Cargèse Summer Institute.Google Scholar
- [7]T. Spencer, The Decay of the Bethe-Salpeter Kernel in P(ϕ)
_{2}Quantum Field Models, Commun. Math. Phys. 44, 143–164 (1975).ADSCrossRefGoogle Scholar - [8]T. Spencer and F. Zirilli, Scattering States and Bound States in λP(ϕ)
_{2}, Commun. Math. Phys. 49, 1–16 (1976).MathSciNetADSCrossRefGoogle Scholar - [9]J. Glimm and A. Jaffe, 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.Google Scholar
- [10]R. Schrader, A Constructive Approach to ϕ
_{4}^{4}. I. Commun. Math. Phys. 49, 131–153 (1976)MathSciNetADSCrossRefGoogle Scholar - R. Schrader, A Constructive Approach to ϕ
_{4}^{4}. II. preprint. 49, 131–153 (1976)MathSciNetGoogle Scholar - R. Schrader, A Constructive Approach to ϕ
_{4}^{4}. III. Commun. Math. Phys. 50, 97–102 (1976).MathSciNetADSCrossRefGoogle Scholar - [11]J. Rosen, The Ising Model Limit of ϕ
^{4}Lattice Fields, preprint, 1976.Google Scholar - [12]J. Glimm and A. Jaffe, The ϕ
_{2}^{4}Quantum Field Model in the Single Phase Region: Differentiability of the Mass and Bounds on Critical Exponents, Phys. Rev. D10, 536–539 (1974).ADSGoogle Scholar - [13]O. McBryan and J. Rosen, Existence of the Critical Point in ϕ
^{4}Field Theory, preprint 1976.Google Scholar - [14]J. Glimm, A. Jaffe and T. Spencer, The Particle Structure of the Weakly Coupled P(ϕ)
_{2}Model and Other Applications of High T mperature Expansions, Part II. The Cluster Expansion, in Constructive Quantum Field Theory, G. Velo and A. S. Wightman (eds.), 1973 Erice Lectures, Springer Lecture Notes in Physics, Vol. 25, Springer-Verlag, 1973.Google Scholar - [15]J. Glimm, A. Jaffe and T. Spencer, The Wightman Axioms and Particle Structure in the P(ϕ)
_{2}Quantum Field Model, Ann. Math. 100, 585–632 (1974).MathSciNetCrossRefGoogle Scholar - [16]J. Fröhlich, B. Simon and T. Spencer, Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking, Commun. Math. Phys. 50, 79–95 (1976).ADSCrossRefGoogle Scholar
- [17]F. Guerra, L. Rosen, B. Simon, The P(ϕ)
_{2}Euclidean Quantum Field Theory as Classical Statistical Mechanics, Ann. Math. 101, 111–259 (1975).MathSciNetCrossRefGoogle Scholar - [18]J. Glimm and A. Jaffe, A Remark on the Existence of ϕ
_{4}^{4}, Phys. Rev. Lett. 33, 440–442 (197 4).MathSciNetADSCrossRefGoogle Scholar - [19]C. Newman, preprint.Google Scholar
- [20]J. Glimm and A. Jaffe, Particles and Scaling for Lattice Fields and Ising Models; Commun. Math. Phys. 51, 1–14 (1976).MathSciNetADSCrossRefGoogle Scholar
- [21]J. Glimm and A. Jaffe, Critical Exponents and Elementary Particles, Commun. Math. Phys., to appear.Google Scholar
- [22]R. Schrader, preprint.Google Scholar
- [23]J. Feldman, On the Absence of Bound States in the λϕ
_{2}^{4}Quantum Field Model without Symmetry Breaking, Canad. J. Phys. 52, 1583–1587 (1974).ADSGoogle Scholar - [24]T. Spencer, The Absence of Even Bound States for λ(ϕ
^{4})_{2}, Commun. Math. Phys. 39, 77–79 (1974).ADSCrossRefGoogle Scholar - [25]J. Glimm and A. Jaffe, Three Particle Structure of ϕ4 Interactions and the Scaling Limit, Phys. Rev. Dll, 2816–2827 (1975).MathSciNetADSGoogle Scholar
- [26]J. Glimm, A. Jaffe and T. Spencer, A Convergent Expansion about Mean Field Theory, Part I. The Expansion, Ann. Phys. 101, 610–630 (1976).MathSciNetADSCrossRefGoogle Scholar
- [27]J. -P. Eckmann, J. Magnen, and R. Sénéor, Decay Properties and Borel Summability for the Schwinger Functions in P(ϕ)
_{2}Theories, Commun. Math. Phys. 39, 251–271 (1975).ADSCrossRefGoogle Scholar - [28]J. Dimock, The P(ϕ)
_{2}Green’ s Functions: Asymptotic Perturbation Expansion, Helv. Phys. Acta 49, 199–216 (1976).MathSciNetGoogle Scholar - [29]K. Osterwalder and R. Sénéor, The Scattering Matrix is Non-Trivial for Weakly Coupled P(ϕ)
_{2}Models, Helv. Phys. Acta 49, 525–534 (1976).MathSciNetGoogle Scholar - [30]J. -P. Eckmann, H. Epstein and J. Fröhlich, Asymptotic Perturbation Expansion for the S-Matrix and the Definition of Time-Ordered Functions in Relativistic Quantum Field Models, Ann. de 1’ Inst. H. Poincaré 25, 1–34 (1976).Google Scholar
- [31]J. Dimock and J. -P. Eckmann, Spectral Properties and Bound State Scattering for Weakly Coupled P(ϕ)
_{2}Models, to appear in Annals of Physics.Google Scholar - [32]Y. Park, Lattice Approximation of the (λϕ
^{4}- µϕ)_{3}Field Theory in a Finite Volume, J. Math. Phys. 16, 1065–1075 (1975).ADSCrossRefGoogle Scholar - [33]G. Sylvester, Continuous-Spin Ising Ferromagnets, Ph. D. thesis, M. I. T., 1976; J. Stat. Phys.,to appear.Google Scholar
- [34]J. Lebowitz, GHS and Other Inequalities, Commun. Math. Phys. 35, 87–92 (1974).MathSciNetADSCrossRefGoogle Scholar
- [35]J. Glimm and A. Jaffe, Absolute Bounds on Vertices and Couplings, Ann. de l’ Inst. H. Poincaré 22, 1–11 (1975).Google Scholar
- [36]J. Rosen, Mass Renormalization for Lattice λϕ
_{2}^{4}Fields, preprint, 1976.Google Scholar - [37]D. Marchesin, private communication.Google Scholar
- [38]T. Spencer, The Mass Gap for the P(ϕ)
_{2}Quantum Field Model with a Strong External Field, Commun. Math. Phys. 39, 63–76 (1974).ADSCrossRefGoogle Scholar - [39]J. Glimm, A. Jaffe and T. Spencer, A Convergent Expansion about Mean Field Theory, Part II. Convergence of the Expansion, Annals of Phys. 101, 631–669 (1976).MathSciNetADSCrossRefGoogle Scholar
- [40]J. Magnen and R. Sénéor, The Infinite Volume Limit of the ϕ
_{3}^{4}Model, Ann. de l’ Inst. H. Poincaré 24, 95–159 (1976).Google Scholar - [41]J. Feldman and K. Osterwalder, The Wightman Axioms and the Mass Gap for Weakly Coupled (ϕ
^{4})_{3}Quantum Field Theories, Annals of Phys. 97, 80–135 (1976).MathSciNetADSCrossRefGoogle Scholar - [42]J. Magnen and R. Sénéor, Wightman Axioms for the Weakly Coupled Yukawa Model in Two Dimensions, to appear in Commun. Math. Phys.Google Scholar
- [43]A. Cooper and L. Rosen, The Weakly Coupled Yukawa
_{2}Field Theory: Cluster Expansion and Wightman Axioms, preprint.Google Scholar