In-in 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,An $iϵ$ tilt of the early-time contour projects onto the interacting vacuum.

The interaction Hamiltonian generates interaction-picture time evolution. To leading order for many cubic field interactions it is the negative of the cubic interaction Lagrangian, although derivative interactions require the canonical Legendre transform to justify this relation.