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 $\hatΩ$ to any classical equations of motion, Lagrangian or not. The generating functional of Green's functions is defined by the equation $\hatΩZ (J) = 0$ that is reduced to the standard Schwinger-Dyson equation whenever the classical field equations are Lagrangian. The corresponding probability amplitude $Ψ$ of a field $ϕ$ is defined by the same equation $\hatΩΨ(ϕ) = 0$ although in another representation. When the classical dynamics are Lagrangian, the solution for $Ψ(ϕ)$ is reduced to the Feynman amplitude $e^{\frac{i}{\hbar}S}$, while in the non-Lagrangian case this amplitude can be a more general distribution.