Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods

Giovanni Gallavotti

doi:10.1103/RevModPhys.57.471

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Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods

Giovanni Gallavotti

Dipartimento di Matematica, II Università di Roma, Via O. Raimondo, 00173 Roma, Italy

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Rev. Mod. Phys. 57, 471 – Published 1 April, 1985

DOI: https://doi.org/10.1103/RevModPhys.57.471

Abstract

A self-contained analysis is given of the simplest quantum fields from the renormalization group point of view: multiscale decomposition, general renormalization theory, resummations of renormalized series via equations of the Callan-Symanzik type, asymptotic freedom, and proof of ultraviolet stability for sine-Gordon fields in two dimensions and for other super-renormalizable scalar fields. Renormalization in four dimensions (Hepp's theorem and the De Calan-Rivasseau $n!$ bound) is presented and applications are made to the Coulomb gases in two dimensions and to the convergence of the planar graph expansions in four-dimensional field theories (t' Hooft-Rivasseau theorem).

References (127)

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Gallavotti, G., 1979a, "On the ultraviolet stability in statistical mechanics and field theory," Ann. Mat. Pura Appl. 120, 1
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[c1] Aizenman, M., 1982, "Geometric analysis of $φ^{4}$ -fields and Ising models," Commun. Math. Phys. 86, 1

[c2] Albeverio, S., G. Gallavotti, and R. Hoegh-Khron, 1979, "Some results for the exponential interaction in two or more dimensions," Commun. Math. Phys. 70, 187

[c3] Baker, G., 1973, "Ising model with a scaling interaction," Phys. Rev. B 5, 2622

[c4] Balaban, T., 1982a, " ${(Higgs)}_{2, 3}$ quantum fields in a finite volume: I. A lower bound," Commun. Math. Phys. 85, 603

[c5] Balaban, T., 1982b, " ${(Higgs)}_{2, 3}$ quantum fields in a finite volume: II. An upper bound," Commun. Math. Phys. 86, 555

[c6] Balaban, T., 1983, " ${(Higgs)}_{2, 3}$ quantum fields in a finite volume: III. Renormalization," Commun. Math. Phys. 88, 411

[c7] Balaban, T., 1984, Ultraviolet stability in field theory: the $φ_{3}^{4}$ model, in Scaling and Self-Similarity in Physics, edited by J. Fröhlich, Progress in Physics (Birkhäuser, Boston), Vol. 7, pp. 297-322

[c8] Benfatto, G., M. Cassandro, G. Gallavotti, F. Nicolò, O. Olivieri, E. Presutti, and E. Scacciatelli, 1978, "Some probabilistic techniques in field theory," Commun. Math. Phys. 59, 143

[c9] Benfatto, G., M. Cassandro, G. Gallavotti, F. Nicolò, O. Olivieri, E. Presutti, and E. Scacciatelli, 1980, "Ultraviolet stability in Euclidean scalar field theories," Commun. Math. Phys. 71, 95

[c10] Benfatto, G., G. Gallavotti, and F. Nicolò, 1980, "Elliptic equations and Gaussian processes," J. Funct. Anal. 36, 343

[c11] Benfatto, G., G. Gallavotti, and F. Nicolò, 1982, "On the massive sine-Gordon equation in the first few regions of collapse," Commun. Math. Phys. 83, 387

[c12] Bleher, P., and Y. Sinai, 1973, "Investigation of the critical point in models of the type of Dyson's hierarchical model," Commun. Math. Phys. 33, 23

[c13] Bleher, P., and Y. Sinai, 1975, "Critical indices for Dyson's asymptotically hierarchical models," Commun. Math. Phys. 45, 247

[c14] Bogoliubov, N., and D. Shirkov, 1959, Introduction to the Theory of Quantized Fields (Interscience, New York)

[c15] Brydges, D., 1978, "A rigorous approach in Debye screening in dilute classical Coulomb systems," Commun. Math. Phys. 58, 313

[c16] Brydges, D., J. Fröhlich, and A. Sokal, 1983, "A new proof of the existence and nontriviality of the continuum $φ_{2}^{4}$ and $φ_{3}^{4}$ quantum field theories," Commun. Math. Phys. 91, 141

[c17] Callan, C., 1970, "Broken scale invariance in scalar field theory," Phys. Rev. D 2, 1541

[c18] Callan, C., 1976, Introduction to renormalization theory, in Methods in Field Theory, edited by R. Balian and J. Zinn-Justin, Ecole d'été de Physique Théorique de Les Houches, 1975 (North-Holland, Amsterdam), pp. 41-77

[c19] Cammarota, C., 1982, "Decay of correlations for infinite range interactions in unbounded spin systems," Commun. Math. Phys. 85, 517

[c20] Colella, F., and O. Lanford, 1983, in Constructive Field Theory, Vol. 25 of Lecture Notes in Physics, edited by G. Velo and A. Wightman (Springer, New York), pp. 44-70

[c21] Coleman, S., and E. Weinberg, 1973, "Radiative corrections as the origin of spontaneous symmetry breakdown," Phys. Rev. D 7, 1888

[c22] Collet, P., and J.-P. Eckmann, 1978, A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics, Vol. 74 of Lecture Notes in Physics (Springer, Berlin)

[c23] Dahmen, H., and G. Jona-Lasinio, 1967, "A variational formulation of quantum field theory," Nuovo Cimento A 52, 807

[c24] De Calan, C., and V. Rivasseau, 1981, "Local existence of the Borel transform in Euclidean $φ_{4}^{4}$ ," Commun. Math. Phys. 82, 69

[c25] De Calan, C., and V. Rivasseau, 1982, "The perturbation series for $φ_{3}^{4}$ is divergent," Commun. Math. Phys. 83, 77

[c26] DiCastro, C., and G. Jona-Lasinio, 1969, "On the microscopic foundations of scaling laws," Phys. Lett. 29A, 322

[c27] Dyson, F., 1949, "The radiation theories of Tomonaga, Schwinger, and Feynman," Phys. Rev. 75, 486, reprinted in Selected Papers on Quantum Electrodynamics, edited by J. Schwinger (Dover, New York, 1958), pp. 275-291

[c28] Dyson, F., 1949, "The $s$ -matrix in quantum electrodynamics," Phys. Rev. 75, 1736, reprinted in Selected Papers on Quantum Electrodynamics, edited by J. Schwinger (Dover, New York), pp. 292-311

[c29] Dyson, F., 1951a, "Heisenberg operators in quantum electrodynamics," Phys. Rev. 83, 608

[c30] Dyson, F., 1951b, "The Schroedinger equation in quantum electrodynamics," Phys. Rev. 83, 1207

[c31] Dyson, F., 1971, "An Ising ferromagnet with discontinuous long range order," Commun. Math. Phys. 21, 269

[c32] Eckmann, J.-P., and H. Epstein, 1979a, "Time ordered products and Schwinger functions," Commun. Math. Phys. 64, 95

[c33] Eckmann, J.-P., and H. Epstein, 1979b, "Borel summability of the mass and the $S$ -matrix in $φ^{4}$ models," Commun. Math. Phys. 68, 245

[c34] Federbush, P., and G. Battle, 1982, "A phase space cell cluster expansion for Euclidean field theories," Ann. Phys. (N.Y.) 142, 95

[c35] Federbush, P., and G. Battle, 1983, "A phase space cell cluster expansion for a hierarchical $φ_{3}^{4}$ model," Commun. Math. Phys. 88, 263

[c36] Feldman, J., 1974, "The $φ_{3}^{4}$ theory in finite volume," Commun. Math. Phys. 37, 93

[c37] Feldman, J., and K. Osterwalder, 1976, "The Wightman exioms and the mass gap for weakly coupled $φ_{3}^{4}$ quantum field theories," Ann. Phys. (N.Y.) 97, 80

[c38] Feynman, R., 1948, "Space-time approach to non-relativistic quantum mechanics," Rev. Mod. Phys. 20, 267, reprinted in Selected Papers on Quantum Electrodynamics, edited by J. Schwinger (Dover, New York, 1958), pp. 321-341

[c39] Fröhlich, J., 1976, in Quantum sine-Gordon equation and quantum solitons in two space-time dimensions, in Renormalization Theory, edited by G. Velo and A. Wightman (Reidel, Dordrecht), pp. 371-414

[c40] Fröhlich, J., 1982, "On the triviality of $λ φ_{d}^{4}$ theories and the approach to the critical point in ( $d \geq 4$ ) dimensions," Nucl. Phys. B 200, 281

[c41] Fröhlich, J., and T. Spencer, 1981, "The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas," Commun. Math. Phys. 81, 527

[c42] Gallavotti, G., 1976, Probabilistic aspects of critical fluctuations, in Critical Phenomena, Vol. 54 of Lecture Notes in Physics, edited by J. Brey and R. Jones, Sitges International School on Statistical Mechanics (Springer, Berlin), pp. 250-275

[c43] Gallavotti, G., 1978, "Some aspects of the renormalization problems in statistical mechanics," Mem. Accad. Lincei 15, 23

[c44] Gallavotti, G., 1979a, "On the ultraviolet stability in statistical mechanics and field theory," Ann. Mat. Pura Appl. 120, 1

[c45] Gallavotti, G., 1979b, in The stability problem in $φ^{4}$ scalar field theories, in Quantum Fields—Algebras, Processes, edited by L. Streit, Bielefeld Recontres in Physics and Mathematics (Springer, Vienna), pp. 407-440

[c46] Gallavotti, G., 1981, Elliptic operators and Gaussian processes, in Aspects statistiques et aspects physiques des processus Gaussiens, Colloq. Int. Cent. Natl. Rech. Sci. 307, 349

[c47] Gallavotti, G., 1983a, The Elements of Mechanics (Springer, New York)

[c48] Gallavotti, G., 1983b, "A criterion of integrability for perturbed nonresonant oscillators. Wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum," Commun. Math. Phys. 87, 365

[c49] Gallavotti, G., and F. Nicolò, 1984a, "Renormalization theory for four dimensional scalar fields, I, II," Istituto di Fisica, Ia Università di Roma preprint, nota interna no. 824 (Commun. Math. Phys., in press)

[c50] Gallavotti, G., and F. Nicolò, 1984b, "The screening phase transitions in the two dimensional Coulomb gas," Istituto di Fisica, Ia Università di Roma, nota interna no. 830 (J. Stat. Phys., in press)

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Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods

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