**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)