Schwinger-Dyson equation for non-Lagrangian field theory

Abstract

A method is proposed of constructing quantum correlators for a general gauge system whose classical equations of motion do not necessarily follow from the least action principle. The idea of the method is in assigning a certain BRST operator $\hat{\Omega }$ to any classical equations of motion, Lagrangian or not. The generating functional of Green's functions is defined by the equation $\hat{\Omega }Z(J)=0$ that is reduced to the standard Schwinger-Dyson equation whenever the classical field equations are Lagrangian. The corresponding probability amplitude $\Psi$ of a field $\varphi$ is defined by the same equation $\hat{\Omega }\Psi (\varphi )=0$ although in another representation. When the classical dynamics are Lagrangian, the solution for $\Psi (\varphi )$ is reduced to the Feynman amplitude $e^{\frac{i}{\hslash }S}$, while in the non-Lagrangian case this amplitude can be a more general distribution.

33 pages

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