The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.

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Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory

A new approach to massive integrable models is considered. It allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form-factors in theSU(2) Thirring and Sine-Gordon models.

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Free field representation for massive integrable models

We construct a continuum limit for the effective low energy Lagrangians of the Gross-Neveu model in two euclidean dimensions by showing that they are related to each other through convergent perturbation expansions. This provides a rigorous control of the ultraviolet problem in a renormalizable quantum field theory.

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Gross-neveu model through convergent perturbation expansions

We review the recent approach of Grosse and Wulkenhaar to the perturbative renormalization of non-commutative field theory and suggest a related constructive program. This paper is dedicated to J. Bros on his 70th birthday.

Non-commutative Renormalization

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ4-theory on Moyal space ensures renormalisability The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed

http://www.th.u-psud.fr/page_perso/Rivasseau/NONCOM07/wallet.pdf

Noncommutative Induced Gauge Theory

Renormalized Ward identities and Callan-Symanik equations are dcveloped for vertex functions involving composite fields in a massive Thirring model with U(n) symmetry. The existence of a critical curve of fixed points of a renormalization group acting on a space of coupling constants is proved. On this curve all nonsoft axial-vector and scaling anomalies vanish. Attraction properties of this curve are investigated. The possibility of a second critical curve for strong coupling is discussed. (auth)

P. K. Mitter

Papers

Asymptotic scale invariance in a massive Thirring model with U(n) symmetry