DOI: 10.1007/jhep11(2026)032 ISSN: 1029-8479

A stationary composition law for Schwinger-Keldysh effective actions

Denis Comelli
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A
bstract

Let W 𝒪 [ J ] and W 𝒞 [ J ] denote the connected generating functionals of an open system and its closed system counterpart, and define the environment-induced contribution W IF [ J ] by the difference: W IF [ J ] ≡ W 𝒪 [ J ] − W 𝒞 [ J ] . The corresponding Legendre transforms, Γ 𝒪 , Γ 𝒞 , and Γ IF , do not obey an analogous additive relation but satisfy a stationary composition law given by:

$$ {\Gamma}_{\mathcal{O}}\left[\Phi \right]=\underset{\Psi}{\mathrm{Stat}}\left\{{\Gamma}_{\mathcal{C}}\left[\Psi \right]+{\Gamma}_{\mathrm{IF}}\left[\Phi -\Psi \right]\right\} $$ Γ O Φ = Stat Ψ Γ C Ψ + Γ IF Φ − Ψ

where Stat denotes evaluation at a solution of the stationarity condition with respect to Ψ. 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.

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