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.

scholarly journals SCHEME INDEPENDENCE AS AN INHERENT REDUNDANCY IN QUANTUM FIELD THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 2071-2074 ◽  
Author(s):  
JOSÉ I. LATORRE ◽  
TIM R. MORRIS
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Quantum Field ◽  
Renormalization Group Equations ◽  
Field Transformation ◽  
Exact Renormalization Group ◽  
Infinite Set

The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relation. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions.

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.

ANALYSIS OF A RENORMALIZATION GROUP EQUATION USING THE PROLONGATION METHOD

1999 ◽  
Vol 14 (36) ◽  
pp. 2507-2516
Author(s):  
PAUL BRACKEN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Vector Field ◽  
Differential Equations ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Quantum Field ◽  
First Order ◽  
First Order Differential Equations

The renormalization group equation in quantum field theory is reviewed. The prolongation method is applied to a simplified version of the resulting equation. A system of first order differential equations results which determine the coefficients in the prolonged vector field. Finally, we show how this system can be solved in several cases to determine solutions of these equations.

scholarly journals THE OBSERVABLE-STATE MODEL AND NONRENORMALIZABLE THEORIES

2013 ◽  
Vol 28 (07) ◽  
pp. 1350016 ◽  
Author(s):  
JUAN SEBASTIÁN ARDENGHI ◽  
ALFREDO JUAN ◽  
MARIO CASTAGNINO
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group Equation ◽  
Energy Scale ◽  
State Model ◽  
Group Equation ◽  
Quantum Field ◽  
Coupling Constant ◽  
Point Correlation ◽  
Observable State

The aim of this work is to apply the observable-state model for the quantum field theory of a ϕn self-interaction. We show how to obtain finite values for the two-point and n-point correlation functions without introducing counterterms in the Lagrangian. Also, we show how to obtain the renormalization group equation for the mass and the coupling constant. Finally, we find the dependence of the coupling constant with the energy scale and we discuss the validity of the observable-state model in terms of the projection procedure.

scholarly journals EFFECTIVE AVERAGE ACTION IN STATISTICAL PHYSICS AND QUANTUM FIELD THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 1951-1982 ◽  
Author(s):  
CHRISTOF WETTERICH
Keyword(s):  
Field Theory ◽  
Statistical Physics ◽  
Thermal Fluctuations ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Quantum Field ◽  
Four Dimensions ◽  
Average Action ◽  
Exact Renormalization Group ◽  
Unified Description

An exact renormalization group equation describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. It interpolates between the microphysical laws and the complex macroscopic phenomena. We present a simple unified description of critical phenomena for O(N)-symmetric scalar models in two, three or four dimensions, including essential scaling for the Kosterlitz-Thouless transition.

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.

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