Jean Zinn-Justin, Phase Transitions and Renormalisation Group

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[size=120%]Phase Transitions and Renormalisation Group (Oxford Graduate Texts)
By Jean Zinn-Justin

•    Publisher:                    Oxford University Press, USA
•    Number Of Pages:       464
•    Publication Date:         2007-08-30
•    ISBN-10 / ASIN:         0199227195
•    ISBN-13 / EAN:           9780199227198
•    Binding:                       Hardcover

Book Description:
This work tries to provide an elementary introduction to the notionsof continuum limit and universality in statistical systems with a largenumber of degrees of freedom. The existence of a continuum limitrequires the appearance of correlations at large distance, a situationthat is encountered in second order phase transitions, near thecritical temperature. In this context, we will emphasize the role ofgaussian distributions and their relations with the mean fieldapproximation and Landau's theory of critical phenomena. We will showthat quasi-gaussian or mean-field approximations cannot describecorrectly phase transitions in three space dimensions. We will assignthis difficulty to the coupling of very different physical lengthscales, even though the systems we will consider have only local, thatis, short range interactions. To analyze the unusual situation, a newconcept is required: the renormalization group, whose fixed pointsallow understanding the universality of physical properties at largedistance, beyond mean-field theory. In the continuum limit, criticalphenomena can be described by quantum field theories. In thisframework, the renormalization group is directly related to therenormalization process, that is, the necessity to cancel theinfinities that arise in straightforward formulations of the theory. Wethus discuss the renormalization group in the context of variousrelevant field theories. This leads to proofs of universality and toefficient tools for calculating universal quantities in a perturbativeframework. Finally, we construct a general functional renormalizationgroup, which can be used when perturbative methods are inadequate.

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