A Stationary Composition Law for Schwinger-Keldysh Effective Actions

Abstract

Let $W_{\cal O}[J]$ and $W_{\cal C}[J]$ denote the connected generating functionals of an open system and its closed system counterpart, and define the environment-induced contribution $W_{\rm IF}[J]$ by the difference: $ W_{\rm IF}[J]\equiv W_{\cal O}[J]-W_{\cal C}[J].$ The corresponding Legendre transforms, $\Gamma_{\cal O}$, $\Gamma_{\cal C}$, and $\Gamma_{\rm IF}$, do not obey an analogous additive relation but satisfy a stationary composition law given by: \[ \Gamma_{\cal O}[\Phi] = \operatorname*{Stat}_{\Psi} \left\{ \Gamma_{\cal C}[\Psi] + \Gamma_{\rm IF}[\Phi-\Psi] \right\} \] where ${\rm Stat}$ 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.