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The in-in formalism computes expectation values at a finite time by evolving a state forward and backward along a closed time contour. At first order in the interaction Hamiltonian,

⟨ Q (τ)⟩ = ⟨ Q_{I} (τ)⟩ - i \int_{- \infty}^{τ} d τ^{'} ⟨[Q_{I} (τ), H_{I} (τ^{'})]⟩ .

i ϵ

tilt of the early-time contour projects onto the interacting vacuum.

Keldysh formalism