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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 Non-Linear Models In Two Dimensions

General Non-Linear Models In Two Dimensions

Chapter:
(p.365) 15 GENERAL NON-LINEAR MODELS IN TWO DIMENSIONS
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0015

This chapter describes the formal properties and discusses the renormalization of a class of geometric models: models based on homogeneous spaces. Homogeneous spaces are associated with non-linear realizations of group representations and these models are natural generalizations of the non-linear σ-model considered in Chapter 14. They can be studied in different parametrizations corresponding to different choices of coordinates when these spaces are considered as Riemannian manifolds. However, in contrast with arbitrary manifolds, there exist natural ways to embed these manifolds in flat euclidean spaces, spaces in which the symmetry group acts linearly. This is the system of coordinates used in the discussion of the non-linear σ-model and again used in the first part of this chapter because the renormalization properties are simpler and the physical interpretation of correlation functions more direct. It then examines some properties of these models in a generic parametrization. The renormalization problem is solved by the introduction of a symmetry (generally called BRS symmetry) with anticommuting (Grassmann) parameters which, later, will play an essential role in the renormalization of gauge theories. The second part of the chapter studies the more specific properties of models corresponding to a special class of homogeneous spaces: symmetric spaces. The chapter ends with comments about more general models based on non-compact groups and arbitrary Riemannian manifolds.

Keywords:   homogeneous spaces, renormalization, anticommuting parameters, symmetric spaces

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