= 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 $\langle\Omega|Q(\tau)|\Omega\rangle$. 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 $H_{\rm int}^I$. The background Hamiltonian determines the homogeneous solution; the cubic Hamiltonian supplies the perturbative interaction insertions. In particular, \b[the interaction-picture fields evolve under the quadratic Hamiltonian].
Back to article page
