Jump to ContentJump to Main Navigation
Quantum Field Theory and Critical Phenomena$
Users without a subscription are not able to see the full content.

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

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 24 January 2021

General Non-Linear Models In Two Dimensions

General Non-Linear Models In Two Dimensions

Quantum Field Theory and Critical Phenomena


Oxford University Press

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

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .