Renormalization Group: From a general concept to numbers

2019 ◽  
pp. 69-80
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Theory ◽  
Renormalization Group ◽  
General Concept ◽  
Quantum Field ◽  
Fixed Dimension ◽  
Perturbative Renormalization ◽  
Borel Transformation

Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.

scholarly journals Perturbative Renormalization by Flow Equations

2003 ◽  
Vol 15 (05) ◽  
pp. 491-558 ◽  
Author(s):  
Volkhard F. Müller
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Theory ◽  
Renormalization Group ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Quantum Field ◽  
Flow Equations ◽  
Yang Mills ◽  
Perturbative Renormalization

In this article a self-contained exposition of proving perturbative renormalizability of a quantum field theory based on an adaption of Wilson's differential renormalization group equation to perturbation theory is given. The topics treated include the spontaneously broken SU(2) Yang–Mills theory. Although mainly a coherent but selective review, the article contains also some simplifications and extensions with respect to the literature.

Field theory: Perturbative expansion and summation methods

2019 ◽  
pp. 451-470
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Conformal Mapping ◽  
Critical Exponents ◽  
Perturbative Expansion ◽  
Divergent Series ◽  
Quantum Field ◽  
Summation Methods ◽  
Borel Transformation

Chapter 23 examines perturbative expansion and summation methods in field theory. In quantum field theory, all perturbative expansions are divergent series in the mathematical sense. This leads to a difficulty when the expansion parameter is not small. In the case of Borel summable series, using the knowledge of the large order behaviour, a number of summation techniques have been developed to derive convergent sequences from divergent series. Some methods apply directly on the series like Padé approximants or order–dependent mapping (the ODM method). Others involve first a Borel transformation, like the Padé–Borel method. The method of Borel transformation, suitably modified, followed by a conformal mapping, has been applied to renormalization group (RG) functions of the phi4 3 field theory and has led to precise values of critical exponents.

Renormalization

2021 ◽  
pp. 171-222
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Dimensional Regularization ◽  
General Concept ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Subtraction Procedure ◽  
Renormalization Group Equations

This chapter describes in detail the general concept of renormalization. It starts with a discussion of the regularization of Feynman diagrams. After that, the subtraction procedure is explained in detail, followed by an introduction to the notion of a superficial degree of divergence of the diagram. On this basis, the models of quantum field theory are classified as renormalizable or non-renormalizable theories. The main arbitrariness of the subtraction procedure is fixed by imposing renormalization conditions. Special sections of this chapter are devoted to renormalization in dimensional regularization and renormalization group equations.

Zeta-function regularization of quantum field theory

1990 ◽  
Vol 68 (7-8) ◽  
pp. 620-629 ◽  
Author(s):  
A. Y. Shiekh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Zeta Function ◽  
Analytic Continuation ◽  
Quantum Field ◽  
Four Dimensions ◽  
Group Functions

Analytic continuation leads to the finite renormalization of a quantum field theory. This is illustrated in a determination of the two loop renormalization group functions for [Formula: see text] in four dimensions.

scholarly journals The KAM theorem and renormalization group

2009 ◽  
Vol 29 (2) ◽  
pp. 419-431 ◽  
Author(s):  
E. DE SIMONE ◽  
A. KUPIAINEN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Elementary Proof ◽  
Renormalization Group Equation ◽  
Quadratic Nonlinearity ◽  
Picard Iteration ◽  
Group Equation ◽  
Quantum Field ◽  
Kam Theorem

AbstractWe give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a certain PDE with quadratic nonlinearity, the so-called Polchinski renormalization group equation studied in quantum field theory.

Renormalization Group

2020 ◽  
pp. 289-318
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Critical Phenomena ◽  
Degrees Of Freedom ◽  
Gaussian Model ◽  
Coupling Constants ◽  
Quantum Field ◽  
Important Notion ◽  
Effective Hamiltonians

Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is led to the important notion of relevant, irrelevant and marginal operators and then to the universality of the critical phenomena. Furthermore, the chapter also covers (as regards the RG) transformation laws, effective Hamiltonians, the Gaussian model, the Ising model, operators of quantum field theory, universal ratios, critical exponents and β‎-functions.

scholarly journals CAUSAL PERTURBATION THEORY IN TERMS OF RETARDED PRODUCTS, AND A PROOF OF THE ACTION WARD IDENTITY

2004 ◽  
Vol 16 (10) ◽  
pp. 1291-1348 ◽  
Author(s):  
MICHAEL DÜTSCH ◽  
KLAUS FREDENHAGEN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Theory ◽  
Ward Identity ◽  
Quantum Field ◽  
Algebraic Quantum Field Theory ◽  
Algebraic Quantum ◽  
Local Construction ◽  
Free Parameters ◽  
Classical Fields

In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann [42]. In our formalism the entries of the retarded products are local functionals of the off-shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's "Action Ward Identity" [45]. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local framework independently of the infrared behavior, as was first established by Hollands and Wald [32]. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.

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