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Motivated by cosmological Hartle-Hawking and microcanonical density matrixprescriptions for the quantum state of the Universe we developSchwinger-Keldysh in-in formalism for generic nonequilibrium dynamical systemswith the initial density matrix. We build the generating functional of in-inGreen's functions and expectation values for a generic density matrix of theGaussian type and show that the requirement of particle interpretation selectsa distinguished set of positive/negative frequency basis functions of the waveoperator of the theory, which is determined by the density matrix parameters.Then we consider a special case of the density matrix determined by theEuclidean path integral of the theory, which in the cosmological context can beconsidered as a generalization of the no-boundary pure state to the case of themicrocanonical ensemble, and show that in view of a special reflection symmetryits Wightman Green's functions satisfy Kubo-Martin-Schwinger periodicityconditions which hold despite the nonequilibrium nature of the physical setup.Rich analyticity structure in the complex plane of the time variable revealsthe combined Euclidean-Lorentzian evolution of the theory, which depending onthe properties of the initial density matrix can be interpreted as a decay of aclassically forbidden quantum state.

Nonequilibrium Schwinger-Keldysh formalism for density matrix states: analytic properties and implications in cosmology