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Ultraviolet problems in field theory and multiscale expansions

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Abstract

An introductory survey is given of ultraviolet problems in Euclidean quantum field theory which are heuristically interpreted either with the aid of the classical renormalization theory or with the aid of Wilson's renormalization group strategy. A unification of each of these approaches with the method of multiscale cluster expansions is necessary for strict proofs.

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Literature cited

  1. P. M. Blekher and D. Surgailis, “Self-similar random fields,” Itogi Nauki i Tekhnika, Teor. Veroyatn. Mat. Statist. Teor. Kibernet.,20, 3–51 (1983).

    Google Scholar 

  2. J. Glimm and A. Jaffe, “Positivity of the ϕ3 4 Hamiltonian,” Fortschr. Phys.,21, 327–376 (1973).

    Google Scholar 

  3. J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York-Berlin (1981).

    Google Scholar 

  4. O. I. Zav'yalov, Renormalized Feynman Diagrams [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  5. E. Seiler, Gauge Theories [Russian translation], Mir, Moscow (1985).

    Google Scholar 

  6. V. A. Malyshev, Introduction to Euclidean Quantum Theory of Fields [in Russian], Moscow State Univ. (1985).

  7. V. A. Malyshev and R. A. Minlos, Gibbs Random Fields. Method of Cluster Expansion [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  8. B. Simon, TheP(ϕ)2 Euclidean (Quantum) Field Theory, Princeton Univ. Press (1974).

  9. Ya. G. Sinai, Theory of Phase Transitions [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  10. M. Aizenman, “The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory,” Commun. Math. Phys.,97, No. 1–2, 91–100 (1985).

    Google Scholar 

  11. M. Aizenman, “Geometric analysis of Ф4 fields and Ising models. Parts I and II,” Commun. Math. Phys.,86, No. 1, 1–48 (1982).

    Google Scholar 

  12. R. R. Akhmitzjanov, V. A. Malyshev, and E. N. Petrova, “Cluster expansion for unbounded noninfinite potential,” in: J. Fritz, A. Jaffe, and D. Szasz (eds.), Statistical Physics and Dynamical Systems. Rigorous Results, Birkhäuser (1985), pp. 221–235.

  13. C. Aragao de Carvalho, C. Caracciolo, and J. Fröhlich, Polymers and\(g/\overrightarrow {\varphi /^4 } \) Theory in Four Dimensions, Preprint, IHES (1982).

  14. G. Auberson and G. Menessier, “The reciprocal of a Borel summable function is Borel Summable,” Commun. Math. Phys.,100, No. 3, 439–446 (1985).

    Google Scholar 

  15. T. Balaban, Ultraviolet Stability for a Model of Interacting Scalar and Vector Fields. I. A Lower Bound, Preprint HUTMP (1982).

  16. T. Balaban, Ultraviolet Stability for a Model of Interacting Scalar and Vector Fields. II. An Upper Bound, Preprint HUTMP (1982).

  17. T. Balaban, Ultraviolet Stability for a Model of Interacting Scalar and Vector Fields. III. Preprint HUTMP (1982).

  18. T. Balaban, “(Higgs)2,3 quantum fields in a finite volume. I. A lower bound,” Commun. Math. Phys.,85, 603–626 (1982).

    Google Scholar 

  19. T. Balaban, “(Higgs)2,3 quantum fields in a finite volume. II. An upper bound,” Commun. Math. Phys.,86, No. 4, 555–594 (1982).

    Google Scholar 

  20. T. Balaban, “(Higgs)2,3 quantum fields in a finite volume. III. Renormalization,” Commun. Math. Phys.,88, 411–445 (1983).

    Google Scholar 

  21. T. Balaban, (Proca)2,3 Quantum Fields in a Finite Volume. III. Renormalization, Preprint HUTMP (1982).

  22. T. Balaban, “Regularity and decay of lattice Green's functions,” Commun. Math. Phys.,89, No. 4, 571–597 (1983).

    Google Scholar 

  23. T. Balaban, “Ultraviolet stability in field theory. The ϕ3 4 model,” in: Scaling and Self-Similarity in Physics, J. Fröhlich (ed.), Birkhauser (1983), pp. 297–319.

  24. T. Balaban, “Propagators and renormalization transformation for lattice gauge theories. I,” Commun. Math. Phys.,95, No. 1, 17–40 (1984).

    Google Scholar 

  25. T. Balaban, “Propagators and renormalization transformation for lattice gauge theories. II,” Commun. Math. Phys.,96, No. 2, 223–250 (1984).

    Google Scholar 

  26. T. Balaban, “Recent results in constructing gauge fields,” Phys. A.,124, 79–90 (1984).

    Google Scholar 

  27. T. Balaban, “Propagators for lattice gauge theories in a background field,” Commun. Math. Phys.,99, No. 3, 389–434 (1985).

    Google Scholar 

  28. T. Balaban, “Spaces of regular gauge field configurations on a lattice and gauge fixing conditions,” Commun. Math. Phys.,99, No. 1, 75–102 (1985).

    Google Scholar 

  29. T. Balaban, Renormalization Group Methods in Non-Abelian Gauge Theories, Preprint HUTMP (1983).

  30. T. Balaban, “Averaging operators for lattice gauge theories,” Commun. Math. Phys.,98, No. 1, 17–51 (1985).

    Google Scholar 

  31. T. Balaban, Ultraviolet Stability of Three-dimensional Lattice Pure Gauge Theories, Preprint, Harvard Univ. (1985).

  32. T. Balaban, J. Imbrie, and A. Jaffe, “Renormalization of the Higgs model: minimizers, propagators and the stability of mean field theory,” Commun. Math. Phys.,97, No. 1–2, 299–329 (1985).

    Google Scholar 

  33. T. Balaban, J. Imbrie, and A. Jaffe, Exact Renormalization Group for Gauge Theories, Preprint, Harvard Univ. (1983).

  34. T. Balaban, J. Imbrie, and D. Brydges, “The mass gap for Higgs models on a unit lattice,” Ann. Phys.,158, No. 2, 281–319 (1984).

    Google Scholar 

  35. G. A. Battle, III, “Pseudoscalar interaction of coupled quantum-mechanical oscillators with independent Fermi systems,” J. Math. Phys.,23, No. 5, 760–779 (1982).

    Google Scholar 

  36. G. A. Battle, III, “Non-Gaussian a-positivity of ϕd 2n field theories,” J. Funct. Anal.,51, No. 3, 312–325 (1983).

    Google Scholar 

  37. G. A. Battle, III, Non-Gaussian α-Positivity of ϕd 2n Field Theories, Preprint, Texas Univ. (1985).

  38. G. A. Battle, III, “A new combinatoric estimate for cluster expansions,” Commun. Math. Phys.,94, No. 1, 133–139 (1984).

    Google Scholar 

  39. G. A. Battle, III, An Elementary Expansion of the Yakawa2 Field Theory Without Vacuum Energy Counterterms. Part I, Preprint, Texas Univ. (1985).

  40. G. A. Battle, III, An Elementary Expansion of the Yakawa2 Field Theory Without Vacuum Energy Counterterms. Part II, Preprint, Texas Univ. (1985).

  41. G. A. Battle, III, An Intrinsic Cluster Expansion for Statistical Mechanics, Preprint, Texas Univ. (1985).

  42. G. A. Battle, III, A Technical Note on a Fermi Cluster Expansion, Preprint, Texas Univ. (1985).

  43. G. A. Battle, III, Application of the Tataru-Vinicius Identity to an Enumeration Problem, Preprint, Texas Univ. (1985).

  44. G. A. Battle, III and P. Federbush, “A phase cell cluster expansion for Euclidean field theories,” Ann. Phys.,142, No. 1, 95–139 (1982).

    Google Scholar 

  45. G. A. Battle, III and P. Federbush, “A phase cell cluster expansions for a hierarchical ϕ3 4 model,” Commun. Math. Phys.,88, No. 2, 263–293 (1983).

    Google Scholar 

  46. G. A. Battle, III and P. Federbush, “A note on cluster expansions, tree graph identities, extra 1/N! factors!!!” Lett. Math. Phys.,8, No. 1, 55–57 (1984).

    Google Scholar 

  47. G. A. Battle, III, P. Federbush, and R. W. Robinson, Tree Graphs and Quasibounded Spin Systems, Preprint (1984).

  48. G. Benfatto, An Iterated Mayer Expansion for the Yakawa Gas, Preprint, Univ. di Roma, dip. di Matematica (1984).

  49. G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicolo, E. Olivieri, E. Presutti, and E. Scacciatelli, “Ultraviolet stability in Euclidean scalar field theories,” Commun. Math. Phys.,71, No. 2, 95–130 (1980).

    Google Scholar 

  50. G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicolo, E. Olivieri, E. Presutti, and E. Scacciatelli, “Some probabilistic techniques in field theory,” Commun. Math. Phys.,59, No. 2, 143–166 (1978).

    Google Scholar 

  51. G. Benfatto, G. Gallavotti, and F. Nicolo, “Elliptic equations and Gaussian processes,” J. Funct. Anal.,36, No. 3, 343–400 (1980).

    Google Scholar 

  52. G. Benfatto, G. Gallavotti, and F. Nicolo, “On the massive sine-Gordon equation in the first few regions of collapse,” Commun. Math. Phys.,83, No. 3, 387–410 (1982).

    Google Scholar 

  53. A. Bovier and G. Felder, “Skeleton inequalities and the asymptotic nature of perturbation theory for ϕ4-theories in two and three dimensions,” Commun. Math. Phys.,93, No. 2, 259–275 (1984).

    Google Scholar 

  54. S. Breen, “Leading large order asymptotics for (ϕ4)2 perturbation theory,” Commun. Math. Phys.,92, No. 2, 179–194 (1983).

    Google Scholar 

  55. D. Brydges, “Field theories and Symanzik's polymer representation,” in: Gauge Theor. Fundam. Interact. and Rigor. Results. Lect. Int. Summer Sch. Theor. Phys., Poiana Brasov, 1981, Boston (1982), pp. 311–337.

  56. D. Brydges and P. Federbush, “A new form of the Mayer expansion in classical statistical mechanics,” J. Math. Phys.,19, No. 10, 2064–2067 (1978).

    Google Scholar 

  57. D. Brydges, J. Fröhlich, and A. D. Sokal, “A new proof of the existence and nontriviality of the continuum ϕ2 4 and ϕ3 4 quantum field theories,” Commun. Math. Phys.,91, No. 2, 141–186 (1983).

    Google Scholar 

  58. D. Brydges, J. Fröhlich, and E. Seiler, “On the construction of quantized gauge field. I. General results,” Ann. Phys.,121, No. 1–2, 227–284 (1979).

    Google Scholar 

  59. D. Brydges, J. Fröhlich, and E. Seiler, “On the construction of quantized gauge field. II. Construction of the lattice approximation,” Commun. Math. Phys.,71, 159–205 (1980).

    Google Scholar 

  60. D. Brydges, J. Fröhlich, and E. Seiler, “On the construction of quantized gauge field. III. The two-dimensional Abelian Higgs model without cutoffs,” Commun. Math. Phys.,79, No. 3, 353–399 (1981).

    Google Scholar 

  61. C. de Calan, D. Petritis, and V. Rivasseau, “Local existence of the Borel transform in Euclidean massless ϕ4 4,” Commun. Math. Phys.,101, No. 4, 559–577 (1985).

    Google Scholar 

  62. C. de Calan and V. Rivasseau, “Renormalization in the complete Mellin representation of Feynman amplitudes,” Commun. Math. Phys.,78, 531–544 (1981).

    Google Scholar 

  63. C. de Calan and V. Rivasseau, “Local existence of the Borel transform in Euclidean ϕ4 4,” Commun. Math. Phys.,82, No. 1, 69–100 (1981).

    Google Scholar 

  64. C. de Calan and V. Rivasseau, “A comment on the local existence of the Borel transform in Euclidean ϕ4 4,” Commun. Math. Phys.,91, No. 2, 265 (1983).

    Google Scholar 

  65. C. de Calan and V. Rivasseau, “The perturbation series ϕ3 4 field theory is divergent,” Commun. Math. Phys.,83, No. 1, 77–82 (1982).

    Google Scholar 

  66. C. Cammarota, “Decay of correlations for infinite range interactions in unbounded spin systems,” Commun. Math. Phys.,85, No. 4, 517–528 (1982).

    Google Scholar 

  67. Ph. Combe, R. Hoegh-Krohn, S. Rodriguez, M. Sirugue, and M. Sirugue-Collin, “Zero mass, 2-dimensional real time sine-Gordon model without u.v. cutoffs,” Ann. Inst. H. Poincaré,37, No. 2, 115–127 (1982).

    Google Scholar 

  68. P. Federbush, “A mass zero cluster expansion. Part 1. The expansion,” Commun. Math. Phys.,81, No. 3, 327–340 (1981).

    Google Scholar 

  69. P. Federbush, “A mass zero cluster expansion. Part 2. Convergence,” Commun. Math. Phys.,81, No. 3, 341–360 (1981).

    Google Scholar 

  70. P. Federbush, A Phase Cell Approach to Yang-Mills Theory. O. Introductory Exposition, Preprint, Univ. of Michigan (1984).

  71. P. Federbush and T. Kennedy, Surface Effects in Debye Screening, Preprint (1985).

  72. J. Feldman “The λϕ3 4 field theory in a finite volume,” Commun. Math. Phys.,37, No. 2, 93–120 (1974).

    Google Scholar 

  73. J. Feldman, Ph.D. Thesis, Harvard Univ. (1974).

  74. J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, “Bounds on completely Euclidean Feynman graphs,” Commun. Math. Phys.,98, No. 2, 273–288 (1985).

    Google Scholar 

  75. J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, Infrared Red ϕ4 4. Lectures at XLIII Session of Les Houches Summer School on Critical Phenomena, Random Systems, Gauge Theories (1934).

  76. J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, Large Order Bounds for the Perturbative ϕ4 4 Theory. Lectures at XLIII Session of Les Houches Summer School on Critical Phenomena, Random Systems, Gauge Theories, Preprint (1984).

  77. J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, “Bounds on renormalized Feynman graphs,” Commun. Math. Phys.,100, No. 1, 23–55 (1985).

    Google Scholar 

  78. J. Feldman and K. Osterwalder, “The Wightman axioms and the mass gap for weakly coupled (ϕ4)3 quantum field theories,” Ann. Phys.,97, 80–135 (1976).

    Google Scholar 

  79. J. Fröhlich and J. Imbrie, Improved Perturbation Expansion for Disordered Systems: Beating Griffith Singularities, Preprint HUTMP (1985).

  80. G. Gallavotti, “Some aspects of the renormalization problems in statistical mechanics and field theory,” Atti. Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I,15, No. 2, 23–59 (1978).

    Google Scholar 

  81. G. Gallavotti, “On the ultraviolet stability in statistical mechanics and field theory,” Ann. Math. Pura Appl.,120, 1–23 (1979).

    Google Scholar 

  82. G. Gallavotti, Renormalization Theory and Ultraviolet Stability for Scalar Fields via Renormalization Group Methods, Preprint Univ. di Roma, dip. di Matematica (1984).

  83. G. Gallavotti and F. Nicolo, “Renormalization theory in four-dimensional scalar fields (I),” Commun. Math. Phys.,100, No. 4, 545–590 (1985).

    Google Scholar 

  84. G. Gallavotti and F. Nicolo, “Renormalization theory in four-dimensional scalar fields (II),” Commun. Math. Phys.,101, No. 2, 247–282 (1985).

    Google Scholar 

  85. G. Gallavotti and V. Rivasseau, “Φ4 field theory in dimension 4: a modern introduction to its unsolved problems,” Ann. Inst. H. Poincaré,40, No. 2, 185–210 (1984).

    Google Scholar 

  86. K. Gawedzki, Block Spin Renormalization, Preprint Institute des Hautes Etudes des Scientifiques, France, May (1984).

  87. K. Gawedzki and A. Kupiainen, “Renormalization group study of a critical lattice model. I. Convergence to the line of fixed points,” Commun. Math. Phys.,82, No. 3, 407–433 (1981).

    Google Scholar 

  88. K. Gawedzki and A. Kupiainen, “Renormalization group study of a critical lattice model. II. The correlation functions,” Commun. Math. Phys.,83, No. 4, 469–492 (1982).

    Google Scholar 

  89. K. Gawedzki and A. Kupiainen, “Renormalization group for a critical lattice model. Effective interactions beyond the perturbation expansion or bounded spins approximation,” Commun. Math. Phys.,88, No. 1, 77–94 (1983).

    Google Scholar 

  90. K. Gawedzki and A. Kupiainen, “A rigorous block spin approach to massless lattice theories,” Commun. Math. Phys.,77, No. 1, 31–64 (1980).

    Google Scholar 

  91. K. Gawedzki and A. Kupiainen, “Triviality of ϕ4 4 and all that in hierarchical model approximation,” J. Statist. Phys.,29, No. 4, 683–698 (1982).

    Google Scholar 

  92. K. Gawedzki and A. Kupiainen, “Non-Gaussian fixed points of the block spin transformation hierarchical model approximation,” Commun. Math. Phys.,89, No. 2, 191–220 (1983).

    Google Scholar 

  93. K. Gawedzki and A. Kupiainen, “Massless lattice ϕ4 4 theory: rigorous control of a renormalized asymptotically free model,” Commun. Math. Phys.,99, No. 2, 197–252 (1985).

    Google Scholar 

  94. K. Gawedzki and A. Kupiainen, “Block spin renormalization group for dipole gas and (Δϕ)4,” Ann. Phys.,147, No. 1, 198–243 (1983).

    Google Scholar 

  95. K. Gawedzki and A. Kupiainen, “Lattice dipole gas and (Δϕ4) models at long distances: decay of correlations and scaling limit,” Commun. Math. Phys.,92, No. 4, 531–553 (1984).

    Google Scholar 

  96. K. Gawedzki and A. Kupiainen, “Rigorous renormalization group and asymptotic freedom,” in: J. Fröhlich (editor), Scaling and Self-Similarity in Physics, Birkhäuser (1983), pp. 227–262.

  97. K. Gawedzki and A. Kupiainen, Non-Gaussian Scaling Limits. Hierarchical Model Approximations, IHES, July (1983).

  98. K. Gawedzki and A. Kupiainen, Rigorous Renormalization Group and Large N, Preprint, Univ. Bielefeld (1983).

  99. K. Gawedzki and A. Kupiainen, “Nontrivial Continuum Limit of a ϕ4 4 Model with Negative Coupling Constant, Preprint, Harvard Univ. (1984).

  100. K. Gawedzki and A. Kupiainen, Gross-Neveu Model through Convergent Perturbation Expansions, Preprint, Univ. Helsinki (1985).

  101. K. Gawedzki and A. Kupiainen, Renormalization of a Non-Renormalizable Quantum Field Theory, Preprint, Univ. Helsinki (1985).

  102. K. Gawedzki and A. Kupiainen, Asymptotic Freedom Beyond Perturbation Theory, Preprint, Harvard Univ. (1985).

  103. K. Gawedzki, A. Kupiainen, and B. Tirozzi, “Borel summability of the perturbation series in a hierarchicalλ(Δϕ)4 model,” J. Statist. Phys.,36, No. 1–2, 145–162 (1984).

    Google Scholar 

  104. K. Gawedzki, A. Kupiainen, and B. Tirozzi, Renormalons: A Dynamical System Approach, Preprint HUTMP (1984).

  105. J. Glimm, “Yukawa coupling of quantum fields in two dimensions. I,” Commun. Math. Phys.,5, No. 5, 343–386 (1967).

    Google Scholar 

  106. J. Glimm, “Boson fields with the:ϕ 4: interaction in three dimensions,” Commun. Math. Phys.,10, No. 1, 1–47 (1968).

    Google Scholar 

  107. L. Gross, “Convergence of U(1)3 lattice gauge theory to its continuum limit,” Commun. Math. Phys.,92, No. 2, 137–162 (1983).

    Google Scholar 

  108. G. Hooft, “On the convergence of planar diagram expansion,” Commum. Math. Phys.,86, No. 4, 449–464 (1982).

    Google Scholar 

  109. G. Hooft, “Rigorous construction of planar diagram field theories in four dimensional Euclidean space,” Commum. Math. Phys.,88, No. 1, 1–26 (1983).

    Google Scholar 

  110. J. Imbrie, Renormalization Group Methods in Gauge Field Theories, Preprint, Harvard Univ. (1985).

  111. K. R. Ito, “Construction of two-dimensional quantum electrodynamics based on a Hamiltonian formalism,” Lett. Math. Phys.,2, No. 5, 357–365 (1978).

    Google Scholar 

  112. K. R. Ito. “Construction of two-dimensional quantum electrodynamics,” J. Math. Phys.,26, No. 6, 1473–1494 (1980).

    Google Scholar 

  113. K. R. Ito, “Construction of Euclidean (QED)2 via lattice gauge theory. Boundary condition and volume dependence,” Commum. Math. Phys.,83, No. 4, 537–561 (1982).

    Google Scholar 

  114. C. King, The U(1) Higgs Model I. The Continuum Limit, Preprint HUTMP (1984).

  115. C. King, The U(1) Higgs Model II. The Infinite Volume Limit, Preprint HUTMP (1984).

  116. G. Lang and A. Lesniewski, “Axioms for renormalization in Euclidean quantum field theory,” Commun. Math. Phys.,91, No. 4, 505–518 (1983).

    Google Scholar 

  117. A. Lesniewski, “On Callan's proof of the BPHZ theorem,” Helv. Phys. Acta,56, No. 6, 1158–1167 (1983).

    Google Scholar 

  118. G. Mack and A. Pordt, “Convergent perturbation expansions for Euclidean quantum field theory,” Commun. Math. Phys.,97, No. 1–2, 267–298 (1985).

    Google Scholar 

  119. J. Magnen and R. Sénéor, “The infinite volume limit of theϕ 43 model,” Ann. Inst. H. Poincaré,24, 95–159 (1976).

    Google Scholar 

  120. J. Magnen and R. Sénéor, “Phase space cell expansion and Borel summability for the Euclideanϕ 43 theory,” Commun. Math. Phys.,56, 237–276 (1977).

    Google Scholar 

  121. J. Magnen and R. Sénéor, “The infrared behavior of (Δϕ) 43 ,” Ann. Physics,152, No. 1, 130–202 (1982).

    Google Scholar 

  122. J. Magnen and R. Sénéor, “A note on cluster expansions,” Ann. Inst. H. Poincaré Sect. A,38, No. 1, 93–98 (1983).

    Google Scholar 

  123. V. A. Malyshev, “Uniform cluster estimates for lattice models,” Commun. Math. Phys.,64, No. 2, 131–157 (1980).

    Google Scholar 

  124. F. Nicolo, “On the massive sine-Gordon equation in the higher regions of collapse,” Commun. Math. Phys.,88, No. 4, 561–600 (1983).

    Google Scholar 

  125. Y. M. Park, “Convergence of lattice approximations and infinite volume limit in the (λϕ4-σϕ2-μϕ)3)3, field theory,” J. Math. Phys.,18, No. 23, 354–366 (1977).

    Google Scholar 

  126. J. Polchinski, “Renormalization and effective Lagrangians,” Nucl. Phys. B,231, No. 2, 269–295 (1984).

    Google Scholar 

  127. J. Potthof, “Euclideanϕ 43 theory in an electromagnetic potential,” Ann. Inst. H. Poincaré Sect. A,37, No. 2, 129–154 (1982).

    Google Scholar 

  128. V. Rivasseau, “Construction and Borel summability of planar 4-dimensional Euclidean field theory,” Commun. Math. Phys.,95, No. 4, 445–486 (1984).

    Google Scholar 

  129. S. N. M. Ruijsenaars, “The Wightman axions for the fermionic Federbush model,” Commun. Math. Phys.,87, No. 2, 181–228 (1982).

    Google Scholar 

  130. R. Sénéor, “Superrenormalizable infrared theories,” in Lecture Notes in Math., Vol. 1031 (1983), pp. 108–113.

    Google Scholar 

  131. T. Spencer, “The Lipatov argument,” Commun. Math. Phys.,74, No. 3, 273–280 (1980).

    Google Scholar 

  132. D. H. Weingarten and J. L. Challifour, “Continuum limit of QED2 on a lattice,” Ann. Physics,123, No. 1, 61–101 (1979).

    Google Scholar 

  133. D. H. Weingarten and J. L. Challifour, “Continuum limit of QED2 on a lattice, II,” Ann. Phys.,126, No. 1, 154–175 (1980).

    Google Scholar 

  134. M. J. Westwater, “On Edward's model for long polymer chains,” Commun. Phys.,72, No. 2, 131–174 (1980).

    Google Scholar 

  135. M. J. Westwater, “On Edward's model for polymer chains. II. The self-consistent approach,” Commun. Math. Phys.,79, No. 1, 53–73 (1981).

    Google Scholar 

  136. M. J. Westwater, “On Edward's model for polymer chains. III. Borel summability,” Commun. Math. Phys.,84, No. 4, 459–470 (1982).

    Google Scholar 

  137. M. J. Westwater, “Edward's model for long polymer chains,” in: Lect. Notes Phys.,153 (1982), 394–396 (1982).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 24, pp. 111–189, 1986.

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Malyshev, V.A. Ultraviolet problems in field theory and multiscale expansions. J Math Sci 42, 1811–1868 (1988). https://doi.org/10.1007/BF01095509

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