1985

Author: Giovanni Gallavotti

Title: Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods Reviews of Modern Physics, 57, 471--562, 1985.

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 sin--Gordon fields in two dimensions and for other super--renormalizable scalar fields. Renormalization in four dimensions (Hepp's theorem and De Calan--Rivasseau $n!$ bound) is presented and applications are made to the Coulomb gases in two dimensions and tot he convergence of the planar graph expansion in four dimensional field theories (t'Hooft--Rivasseau theorem).

Contents I. Introduction
II. Functional integral representation of the Hamiltonian of a quantum field
III.The free field and its multiscale decompositions
IV.Perturbation theory and ultraviolet stability
V.Effective potentials: the algorithm for their constructions
VI.A graphical expression for the effective potential
VII.Renormalization and renormalizability to second order
VIII.Counterterms, Effective interaction and renormalization in a graphical representation (arbitrary order)
IX.Resummations: form factors and beta function
X.Schwinger functions and effective potentials
XI.The cosine interaction model in two dimensions perturbation theory and multipole expansion
XII. Ultraviolet stability for the cosine interaction and renormalizability for $\a^2$ up to $8\p$.
XIII.Beyond perturbation theory in the cosine interaction case: asymptotic freedom and scale invariance
XIV. Large deviations: their control and the complete construction of the cosine field beyond $\a^2=4\p$
XV.The cosine field and the screening phenomena in the two-dimensional Coulomb gas and in related statistical mechanical systems
XVII.Renormalization to second order of the $\f^4$ field
XVIII. Renormalization and ultraviolet stabilit to any order for $\f^4$ fields
XIX.''$n!$ bounds'' on the effective potential
XX.An application: planar graphs and convergence problems. A heuristic approach
XXI.Constructing $\f^4$ fields in $2$ and $3$ dimensions
XXII. Comments on resummations. Triviality and non triviality. Some apologies.
Acknowledgments
Appendix A: Covariance of the free fields: hints
Appendix B: Hint for (2.1)
Appendix C: Wick monomials and their integrals
Appendix D: Proof of (16.14)
Appendix E: Proof of (19.8)
Appendix F: Estimate of the number of Feynman> graphs compatible with a tree
Appendix G: Applications to the hierarchical model
References.
(Archive version 1.0, 16 Nov 2005)