Schwinger-Dyson equation for non-Lagrangian field theory

TL;DR

This paper introduces a novel method for constructing quantum correlators in gauge systems without relying on a Lagrangian framework, using a BRST operator to define generating functionals and amplitudes.

Contribution

It proposes a BRST-based approach to define quantum correlators for non-Lagrangian gauge systems, extending the Schwinger-Dyson formalism beyond traditional Lagrangian theories.

Findings

Defines a BRST operator for general gauge systems

Constructs generating functionals satisfying a master equation

Provides a generalized amplitude formulation for non-Lagrangian dynamics

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 $\phi$ is defined by the same equation $\hat\Omega \Psi (\phi) = 0$ although in another representation. When the classical dynamics are Lagrangian, the solution for $\Psi (\phi)$ 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.