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Many Degrees of Freedom in Field Theory pp 1–50Cite as

An Introduction to Some Topics in Constructive Quantum Field Theory

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Part of the book series: NATO Advanced Study Institutes Series ((ASIB,volume 30))

Summary

We give an elementary introduction to the spirit and some of the simpler techniques of constructive quantum field theory. In the first part of these notes we briefly review the general framework of relativistic quantum field theory, its Euclidean description and Euclidean field theory. In the second part we combine Euclidean with Lagrangean field theory and, as a result, formulate a concrete program for the construction of relativistic quantum field models. In the third part we exemplify this program by considering in some detail the famous λФ4-theory in two space-time dimensions and commenting on the construction of λФ4 in three dimensions. Finally, in part four, we discribe some recent results concerning the degeneracy of the physical vacuum (phase transitions), the spontaneous breaking of internal symmetries and the critical point in λФ4 in two and three dimensions.

Supported by ZiF, University of Bielefeld, 48 Bielefeld Germany, and in part by the U.S. National Science Foundation under grant MPS 75 — 11864; A.P. Sloan Foundation Fellow

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References

  1. Proceedings of the conference on Quantum Dynamics: Models and Mathematics, Bielefeld 1975; to appear.

    Google Scholar 

  2. J. Bisognano and E. Wichmann, J. Math. Phys. 16, 985, (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. H. J. Borchers and J. Yngvason, “Necessary and Sufficient Conditions for Integral Representations of Wightman Functionals at Schwinger Points”, preprint 1975; see also Borchers’ contribution to [M].

    Google Scholar 

  4. J. Bros, H. Epstein and V. Glaser, Helv. Phys. Acta 45, 149, (1972).

    Google Scholar 

  5. C. Burnap, “Isolated One Particle States in Boson Quantum Field Theory Models”, Harvard University preprint 1975. (see also J.Glimm and A.Jaffe, Phys. Rev. DII, 2816, (1975) and ref. [Sp 1]).

    Google Scholar 

  6. Proceedings of the summer school in theoretical physics, Cargèse, Corsica, 1976; to appear.

    Google Scholar 

  7. S. Coleman, “Classical Lumps and Their Quantum Descendants”, to appear in the proceedings of the 1975 International School on Subnuclear Physics “Ettore Majorana”, (Erice).

    Google Scholar 

  8. J. Dimock, Commun. math. Phys. 35, 347, (1974); see also his contribution to [E].

    Article  MathSciNet  ADS  Google Scholar 

  9. J. Dimock, “The P(Ф)2 Green’s Functions: Asymptotic Perturbation Expansion”, Buffalo preprint 1975, to appear.

    Google Scholar 

  10. J. Dimock and J.-P. Eckmann, “Spectral Properties and Bound State Scattering for Weakly Coupled λP(Ф)2 Models”, ZiF-University of Bielefeld, preprint 1976, to be published.

    Google Scholar 

  11. J. Dimock and J. Glimm, Adv. Math. 12, 58, (1974).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Doplicher, R. Haag and J. Roberts, Commun. math. Phys. 23, 199, (1971) and 35, 49,(1 974).

    Article  MathSciNet  ADS  Google Scholar 

  13. W.Driessler and J.Fröhlich, “The Reconstruction of Local Observable Algebras from the Euclidean Green’s Functions of a Relativistic Quantum Field Theory”, ZiF, Univ. Bielefeld, Preprint 1976, to be published.

    Google Scholar 

  14. J.-P.Eckmann, H.Epstein and J.Fröhlich, “Asymptotic Perturbation Expansion for the S-Matrix and ”, Ann. Inst. H. Poincaré, 25, 1, (1976).

    Google Scholar 

  15. J.-P. Eckmann, J. Magnen and R. Sénéor, Commun. math. Phys. 39, 251, (1975).

    Article  ADS  Google Scholar 

  16. G. Velo and A. Wightman (eds.), “Constructive Quantum Field Theory”, Lecture Notes in Physics 25, (1973), Springer, Berlin-Heidelberg-New York. (Proceeding of “Erice, 1973”).

    MATH  Google Scholar 

  17. J. Fröhlich, Ann. Phys. 97, 1, (1976).

    Article  ADS  Google Scholar 

  18. J. Fröhlich, Helv. Phys. Acta 47, 265, (1974), and “Schwinger Functions and Their Generating Functionals, II … ” to appear in Adv. Math.

    MathSciNet  MATH  Google Scholar 

  19. J. Fröhlich, “Phase Transitions, Goldstone Bosons and Topological Superselection Rules” in “Current Problems in Elementary Particle and Mathematical Physics”, P. Urban (ed.), Springer, Wien-New York, 1976.

    Google Scholar 

  20. J. Fröhlich, Commun. math. Phys. 47, 269, (1976).

    Article  ADS  Google Scholar 

  21. J. Fröhlich, contribution to ref. [M], and “Existence and Analyticity in the Bare Parameters of the \(\lambda {(\vec \phi \cdot \vec \phi )^2}\)-Quantum Field Model”, (report, 1975; so far unpublished).

    Google Scholar 

  22. J. Fröhlich and E. Seiler, “The Massive Thirring Schwinger Model (QED2): Convergence of Perturbation Theory and Particle Structure”, to appear in Helv. Phys. Acta, 1976.

    Google Scholar 

  23. J. Fröhlich and B. Simon, “Pure States for General P(Ф)2 Theories: Construction, Regularity and Variational Equally”, to appear in Ann. Math.

    Google Scholar 

  24. J. Fröhlich, B. Simon and T. Spencer, “Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking”, Commun. math. Phys. 50, 79, (1976), (see also Phys. Rev. Letters 36, 804, (1976)).

    Article  ADS  Google Scholar 

  25. V. Glaser, Commun. math. Phys. 37, 257, (1974).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. J. Glimm, Commun. math. Phys. 8, 12, (1968).

    Article  ADS  MATH  Google Scholar 

  27. J. Glimm and A. Jaffe, in “Statistical Mechanics and Quantum Field Theory”, Les Houches 1970, C. de Witt and R. Stora (eds.), Gordon and Breach, New York, 1971.

    Google Scholar 

  28. J. Glimm and A. Jaffe, in “Mathematics of Contemporary Physics”, R. Streater (ed.), Academic Press, New York, 1972.

    Google Scholar 

  29. J. Glimm and A. Jaffe, Fortschritte d. Physik 21, 327, (1973).

    Article  MathSciNet  ADS  Google Scholar 

  30. J. Glimm and A. Jaffe, Acta Math. 125, 203, (1970).

    Article  MathSciNet  Google Scholar 

  31. J. Glimm and A. Jaffe, J. Math. Phys. 13, 1568, (1972).

    Article  MathSciNet  ADS  Google Scholar 

  32. J. Glimm and A. Jaffe, Ann.Inst. H. Poincaré, 22, 109, (1975); see also Ann. Inst. H. Poincaré 22, 97, (1975).

    MathSciNet  Google Scholar 

  33. J. Glimm and A. Jaffe, “Critical Exponents and Elementary Particles”, Rockefeller university, preprint 1976, to be published.

    Google Scholar 

  34. J. Glimm and A. Jaffe, see [GJ4] and Commun. Math. Phys. 22, 253, (1971); also [GJ2].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. J. Glimm, A. Jaffe and T. Spencer, see their contribution to [E], Part I.

    Google Scholar 

  36. J. Glimm, A.Jaffe and T. Spencer, see their contribution to [E], Part II; (“Cluster Expansion”).

    Google Scholar 

  37. J. Glimm, A. Jaffe and T. Spencer, Ann. Math. 100, 585, (1974).

    Article  MathSciNet  Google Scholar 

  38. J. Glimm, A. Jaffe and T. Spencer, Commun. math. Phys. 45, 203, (1975) and “A Convergent Expansion about Mean Field Theory”, (Part I and II), to appear in Ann. Phys.

    Article  MathSciNet  ADS  Google Scholar 

  39. F. Guerra, Phys. Rev. Letters 28, 1213, (1972).

    Article  ADS  Google Scholar 

  40. F. Guerra, L. Rosen and B. Simon, Commun. math. Phys. 27, 10, (1972).

    Article  MathSciNet  ADS  Google Scholar 

  41. F. Guerra, L. Rosen and B. Simon, Commun. math. Phys. 29, 233, (1973).

    Article  MathSciNet  ADS  Google Scholar 

  42. F. Guerra, L. Rosen and B. Simon, Ann. Math. 101, 111, (1975).

    Article  MathSciNet  Google Scholar 

  43. F. Guerra, L. Rosen and B. Simon, Commun. math. Phys. 41, 19, (1975).

    Article  MathSciNet  ADS  Google Scholar 

  44. F. Guerra, L. Rosen and B. Simon, “Boundary Conditions in the P(Ф)2 Euclidean Field Theory”, to appear in Ann.Inst. H. Poincaré, 1976.

    Google Scholar 

  45. K. Hepp, “Théorie de la Renormalisation”, Lecture Notes in Physics 2, Springer Verlag, Berlin-Heidelberg-New York, 1969.

    Google Scholar 

  46. R. Haag and D. Kastler, J. Math. Phys. 5, 848, (1964).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. R. Jost, “The General Theory of Quantized Fields”, Ann. Math. Soc. Publ., Providence, R.I., 1965.

    Google Scholar 

  48. J. Lebowitz and A. Martin-Löf, Commun. math. Phys. 25, 276, (1972).

    Article  ADS  Google Scholar 

  49. J. Lebowitz and O. Penrose, Commun. math. Phys. 11, 99, (1968).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. M. Lüscher, “Dynamical Charges in the Quantized, Renormalized Massive Thirring Model”, DESY preprint, (Hamburg), 1976.

    Google Scholar 

  51. J. Magnen and R. Sénéor, “L’Interaction Ф4 à trois Dimensions: Positivité de 1’Hamiltonien, Développement à Faible Constante de Couplage et Analyticité” Orsay 1976, C.N.R.S. document A.O. 12.829.

    Google Scholar 

  52. “Les Méthodes Mathématiques de la Théorie Quantique des Champs”, Editions du C.N.R.S., Paris, 1976; (proceedings of the 1975 Colloquium in Marseille).

    Google Scholar 

  53. E. Nelson, in “Mathematical Theory of Elementary Particles”, R. Goodman and I. Segal, (eds.), M.I.T. Press, Cambridge, Mass. 1966.

    Google Scholar 

  54. E. Nelson, in “Partial Differential Equations”, D. Spencer (ed.), Symp. in Pure Math. Vol. 23, A.M.S. Publ. 1973.

    Google Scholar 

  55. E. Nelson, J. Funct. Analysis 12, 97, (1973).

    Article  MATH  Google Scholar 

  56. E. Nelson, his contribution to [E].

    Google Scholar 

  57. E. Nelson, J. Funct. Analysis 11, 211, (1972).

    Article  MATH  Google Scholar 

  58. C. Newman, his contribution to [E] and [M] and refs. given there.

    Google Scholar 

  59. K. Osterwalder, his contribution to [E].

    Google Scholar 

  60. K. Osterwalder and R. Schrader, Commun, math. Phys., 31, 83, (1973), and 42, 281, (1975).

    Article  MathSciNet  ADS  Google Scholar 

  61. Y. Park, “uniform Bounds of the Schwinger Functions in Boson Field Models”, to appear in J. Math. Phys.

    Google Scholar 

  62. refs. marked with *

    Google Scholar 

  63. J. Roberts, “Local Cohomology and Superselection Structure”, C.N.R.S. Marseille preprint 1976.

    Google Scholar 

  64. D. Ruelle, Nuovo Cimento 19, 356, (1961).

    Article  MathSciNet  MATH  Google Scholar 

  65. see [EEF], [D2] and K. Osterwalder and R. Sénéor “The Scattering Matrix is Non-Trivial for Weakly Coupled P(Ф)2 Models”, to appear in Helv. Phys. Acta, 1576, also Eckmann’s and Osterwalder’s contribution to [Bi].

    Google Scholar 

  66. R. Schrader, “A Possible Constructive Approach to Ф4 4I-III”, preprints 1975–76, to be published; his contribution to [M].

    Google Scholar 

  67. E. Seiler and B. Simon, “Nelson’s Symmetry and All That in the (Yukawa) 2 and (Ф4)3 Field Theories”, to appear in Ann. Phys. 1976.

    Google Scholar 

  68. B. Simon, “The P(Ф)2 Euclidean (Quantum) Field Theory”, Princeton University Press, Princeton N.J., 1974.

    Google Scholar 

  69. B. Simon and R. Griffiths, Commun. math. Phys. 33, 145, (1973).

    Article  MathSciNet  ADS  Google Scholar 

  70. T. Spencer, Commun. math. Phys. 44, 143, (1975).

    Article  ADS  Google Scholar 

  71. T. Spencer and F.Zirilli, Commun. math. Phys. 49, (1976).

    Google Scholar 

  72. T. Spencer, Commun. math. Phys. 39, 63 (1974).

    Article  ADS  Google Scholar 

  73. T. Spencer, Commun. math. Phys. 39, 77, (1974).

    Article  ADS  Google Scholar 

  74. R. Streater and I. Wilde, Nuc1. Phys. B24, 561, (1970), R. Streater, Acta Physica Austriaca, Suppl. XI, 317, (1973), P. Bonnard and R. Streater, Helv. Phys. Acta 49, (1976).

    Article  Google Scholar 

  75. R. Streater and A.S. Wihgtman, “PCT, Spin and Statistics and All That ”, Benjamin, New York, 1964.

    Google Scholar 

  76. K. Symanzik, “A Modified Model of Euclidean Quantum Field Theory”, NYU Report, 1964.

    Google Scholar 

  77. K. Symanzik, J. Hath. Phys. 7, 510, (1966).

    Article  MathSciNet  ADS  Google Scholar 

  78. K. Symanzik, in “Local Quantum Theory”, R. Jost (ed.), Academic Press, New York, 1969.

    Google Scholar 

  79. A. Wightman, in “Fundamental Interactions in Physics and Astrophysics”, B.Kursunoglu et al. (eds.), Plenum Press 1973.

    Google Scholar 

  80. Hamiltonian formalism: see [GJ1] and refs. given there. Euclidean formalism: E. Seiler, Commun. math. Phys. 42, 163, (1975). E. Seiler and B. Simon, J. Math. Phys. (1976). E. Seiler and B. Simon, Commun. math. Phys. 45 99, (1975). E. Seiler and B. Simon, ref. [Ses]. O. McBryan, Commun. math. Phys. 44, 237, (1975). O. McBryan, Commun. math. Phys. 45, 279, (1975). O. McBryan, his contribution to [M]. J. Magnen and R. Sénéor, “The Wightman Axioms for the Weakly Coupled Yukawa Model in Two Dimensions”, to appear in Commun. math. Phys., 1976. A. Cooper and L. Rosen, “The Weakly Coupled (Yukawa)2 Field Theory; Cluster Expansion and Wightman Axioms”, Princeton university, preprint 1976. A semi-Euclidean formalism has been proposed and applied to Yukawa models by P. Federbush and D. Brydges, (J. Math. Phys., (1975)).

    Google Scholar 

  81. [GJ3] [MS] [F5] [Pa] [SeS], and J. Feldman, Commun. math. Phys. 37, 93, (1974). J. Feldman and K. Osterwalder, Ann. Phys. 97, 80, (1976). J. Feldman and K. Osterwalder, their contribution to | M |. J. Magnen and R. Sénéor, Ann.Inst.H. Poincaré 24, 95, (1976) Y. Park, J. Math. Phys. 16, 1065, (1975); in “Current Problems in Elementary Particle and Mathematical Physics”, P. Urban, (ed.), Springer-Verlag, Wien-New York, 1976.

    Article  ADS  Google Scholar 

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  1. Department of Mathematics, Princeton University, Princeton, N.J., 08540, USA

    Jürg Fröhlich

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  1. Jürg Fröhlich
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  1. University of Bielefeld, Federal Republic of Germany

    L. Streit

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© 1978 Plenum Press, New York

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Fröhlich, J. (1978). An Introduction to Some Topics in Constructive Quantum Field Theory. In: Streit, L. (eds) Many Degrees of Freedom in Field Theory. NATO Advanced Study Institutes Series, vol 30. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-8924-2_1

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  • DOI: https://doi.org/10.1007/978-1-4615-8924-2_1

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