 Solution

The early-time state is the adiabatic Bunch-Davies vacuum: modes well inside the sound horizon approach positive-frequency Minkowski modes, and the tilted early-time contour selects the interacting vacuum by damping unwanted excitations. The in-in formalism evolves this one initial state forward to the observation time and backward, computing $⟨\Omega |Q(\tau )|\Omega ⟩$. An in-out calculation instead computes a transition amplitude between prescribed initial and final states.

In the interaction picture, the free quadratic Hamiltonian evolves both $Q^I$ and the free fields inside $H_{\mathrm{int}}^I$. The background Hamiltonian determines the homogeneous solution; the cubic Hamiltonian supplies the perturbative interaction insertions. In particular, the interaction-picture fields evolve under the quadratic Hamiltonian.