Schwinger-dyson Equation for Non-lagrangian Field Theory

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