  Motivated by cosmological Hartle-Hawking and microcanonical density matrix
prescriptions for the quantum state of the Universe we develop
Schwinger-Keldysh in-in formalism for generic nonequilibrium dynamical systems
with the initial density matrix. We build the generating functional of in-in
Green's functions and expectation values for a generic density matrix of the
Gaussian type and show that the requirement of particle interpretation selects
a distinguished set of positive/negative frequency basis functions of the wave
operator of the theory, which is determined by the density matrix parameters.
Then we consider a special case of the density matrix determined by the
Euclidean path integral of the theory, which in the cosmological context can be
considered as a generalization of the no-boundary pure state to the case of the
microcanonical ensemble, and show that in view of a special reflection symmetry
its Wightman Green's functions satisfy Kubo-Martin-Schwinger periodicity
conditions which hold despite the nonequilibrium nature of the physical setup.
Rich analyticity structure in the complex plane of the time variable reveals
the combined Euclidean-Lorentzian evolution of the theory, which depending on
the properties of the initial density matrix can be interpreted as a decay of a
classically forbidden quantum state.
