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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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Symmetries And Renormalization

Symmetries And Renormalization

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter deals with linear continuous symmetries corresponding to compact Lie groups because they imply interesting formal properties; consequences of discrete symmetries can also be studied but with somewhat different methods. It also deals only with infinitesimal group transformations and, therefore, topological properties of groups will play no role. The chapter proceeds as follows. It first introduces a regularization which preserves the symmetry. It then proves identities — generally called Ward–Takahashi (WT) identities — consequences of the symmetry of the action and satisfied by the generating functional of 1PI correlation functions. These identities imply relations between the divergences of correlation functions and thus between the counter-terms which render the theory finite. From these relations, the generic form of the counter-terms is derived. Such an analysis is based on a loop expansion of perturbation theory. Finally, the chapter reads off the properties of the renormalized action.

Keywords:   linear continuous symmetries, compact Lie groups, Ward–Takahashi identities

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