Superoperator Representation of Nonlinear Response; Unifying Quantum Field and Mode Coupling Theories

TL;DR

This paper introduces a superoperator approach in Liouville space for computing nonlinear response functions, unifying quantum field theory and classical mode coupling, simplifying calculations, and avoiding complex diagrammatic techniques.

Contribution

It presents a superoperator formalism that unifies quantum and classical response theories, enabling simpler and more direct calculations of nonlinear response functions.

Findings

Superoperator method maintains natural time ordering.

Wick's theorem for superoperators facilitates factorization.

Classical responses derived without stability matrices.

Abstract

Computing response functions by following the time evolution of superoperators in Liouville space (whose vectors are ordinary Hilbert space operators) offers an attractive alternative to the diagrammatic perturbative expansion of many-body equilibrium and nonequilibrium Green functions. The bookkeeping of time ordering is naturally maintained in real (physical) time, allowing the formulation of Wick's theorem for superoperators, giving a factorization of higher order response functions in terms of two fundamental Green's functions. Backward propagations and the analytic continuations using artificial times (Keldysh loops and Matsubara contours) are avoided. A generating functional for nonlinear response functions unifies quantum field theory and the classical mode coupling formalism of nonlinear hydrodynamics and may be used for semiclassical expansions. Classical response functions may be obtained without the explicit computation of stability matrices.