A Stationary Composition Law for Schwinger-Keldysh Effective Actions

Abstract

Let $W_{\mathcal{O}}[J]$ and $W_{\mathcal{C}}[J]$ denote the connected generating functionals of an open system and its closed system counterpart, and define the environment-induced contribution $W_{\mathrm{IF}}[J]$ by the difference: $W_{\mathrm{IF}}[J]\equiv W_{\mathcal{O}}[J]-W_{\mathcal{C}}[J].$ The corresponding Legendre transforms, ${\Gamma }_{\mathcal{O}}$, ${\Gamma }_{\mathcal{C}}$, and ${\Gamma }_{\mathrm{IF}}$, do not obey an analogous additive relation but satisfy a stationary composition law given by: \[ Γ_{\cal O}[Φ] = \operatorname*{Stat}_Ψ \left\{ Γ_{\cal C}[Ψ] + Γ_{\rm IF}[Φ-Ψ] \right\} \] where $\mathrm{Stat}$ denotes evaluation at a solution of the stationarity condition with respect to $\Psi$. This variational composition applies to both local and nonlocal Schwinger-Keldysh effective actions in nonequilibrium quantum field theory. We illustrate its linear and nonlinear realizations through quadratic theories and general time independent effective actions, respectively.

22 pages