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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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St And Brs Symmetries, Stochastic Field Equations

St And Brs Symmetries, Stochastic Field Equations

Quantum Field Theory and Critical Phenomena


Oxford University Press

Section 15.3 introduced a transformation depending on anticommuting parameters, to prove the geometric stability of homogeneous spaces under renormalization. There is a set of topics, stochastic field equations, gauge theories, in which similar transformations are met. These new problems have one common feature: they all involve a constraint equation to which, by a set of formal transformations, is associated a quantum action. This action has an anticommuting type symmetry which has no geometric origin but is merely a consequence of these transformations. This chapter first discusses this mathematical structure from a rather formal point of view, using a notation adapted to a finite number of degrees of freedom. It explains the appearance of Slavnov–Taylor symmetry, which is a conventional non-linear symmetry, in the integral representation of constraint equations. It then shows how it leads to a symmetry with anticommuting parameters first discovered in quantized gauge theories by Becchi, Rouet and Stora and, therefore, called BRS symmetry. Its generator has a vanishing square, and generalizes exterior differentiation. This symmetry is remarkably stable against a number of algebraic deformations and this explains its role in the context of stochastic equations. In some cases it can be expressed in compact form by introducing Grassmann coordinates. The chapter shows how BRS symmetry can encode the compatibility conditions of a system of linear first order differential equations. It exhibits the special form BRS symmetry takes when the constraint equations apply to group manifolds.

Keywords:   Slavnov–Taylor symmetry, Grassmann coordinates, stochastic field equations, group manifolds

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