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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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From Langevin Equation To Supersymmetry

From Langevin Equation To Supersymmetry

Quantum Field Theory and Critical Phenomena


Oxford University Press

A particular class of Langevin equations is of special interest: equations with gaussian noise, in which the drift force derives from a static action and is thus conservative (the corresponding stochastic processes satisfy the detailed balance condition). This chapter shows that the dynamic action then has a second Grassmann symmetry which, combined with the first one, provides the simplest example of supersymmetry: quantum mechanics supersymmetry. It discusses a few consequences of the supersymmetry like WT identities, renormalization properties, and equilibrium distribution. It describes some subtleties of the formalism in the case of two-dimensional models defined on various manifolds. In the two last sections, the chapter briefly discusses supersymmetric quantum field theories, since supersymmetry has been proposed as a principle to solve the so-called hierarchy problem in particle physics by relating the masses of scalar particles (like Higgs fields) to those of fermions which can be protected against ‘large’ mass renormalization by chiral symmetry.

Keywords:   Langevin equations, gaussian noise, Grassmann symmetry, quantum mechanics supersymmetry

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