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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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General Renormalization Group. The Critical Theory Near Dimension Four

General Renormalization Group. The Critical Theory Near Dimension Four

Quantum Field Theory and Critical Phenomena


Oxford University Press

The RG theory, as applied to Critical Phenomena, has been developed by Kadano, Wilson, Wegner, and many others. This chapter first describes the basic renormalization group ideas in a somewhat abstract and intuitive framework. The formulation will lack precision and the arguments will be largely heuristic. The importance of fixed points in hamiltonian space will be stressed. The special role of gaussian models and their universal properties will be related to the existence of fixed point, the gaussian fixed point. It is shown that the RG equations which appear as a consequence of the necessity of renormalization of local field theories are directly connected with the abstract RG equations introduced in Section 25.1. Universality in the theory of critical phenomena is thus directly related to the property that local field theories are insensitive to the short distance structure, and physics can, therefore, be described by renormalized correlation functions. Conversely, in the statistical sense, QFTs are always close to criticality and their existence, beyond perturbation theory, relies, from the abstract RG point of view, on the presence of IR fixed points in hamiltonian space.

Keywords:   renormalization group, gaussian fixed point, local field theories, hamiltonian space

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