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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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The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry

The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter considers models possessing global symmetries non-linearly realized on the fields. This implies, in particular, that under an in infinitesimal group transformation, the variation of the field is a non-linear function of the field itself. Since such models have non-trivial geometric properties, the chapter first extensively discusses the simplest example, the non-linear σ-model, a model with an O(N) symmetry, the field being an N-vector of fixed length. A simple analysis reveals that in the non-linear σ-model, in the tree approximation, the O(N) symmetry is always spontaneously broken, unlike what happens in a φ4-like theory with the same symmetry: the action describes the interactions of N - 1 massless fields, the Goldstone modes. Power counting shows that the model is renormalizable in two dimensions. Therefore, the field is dimensionless and creates a problem already mentioned in Section 9.3: although the degree of divergence of Feynman diagrams is bounded, an infinite number of counter-terms is generated because all correlation functions are divergent. The chapter shows that due to the special geometric properties of the model, the coefficients of all counter-terms can be calculated as a function of two of them so that the renormalized theory depends only on a finite number of parameters.

Keywords:   global symmetries, non-linear function, power counting, Goldstone modes

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