In-in formalism
= In-in formalism
{wiki=Schwinger–Keldysh_formalism}
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,
$$
\langle Q(\tau)\rangle
=\langle Q_I(\tau)\rangle
-i\int_{-\infty}^{\tau}d\tau'
\langle[Q_I(\tau),H_I(\tau')]\rangle.
$$
An $i\epsilon$ tilt of the early-time contour projects onto the interacting vacuum.