The Functional Measure for the In-In Path Integral

Understand

The in-in path integral of a scalar field propagating in a fixed background is formulated in a suitable function space.

• The free kinetic operator, whose inverse gives the propagators of the in-in perturbation theory, becomes essentially self adjoint after imposing appropriate boundary conditions.
• An explicit spectral representation is given for the scalar in the flat space and the standard propagators are rederived using this representation.
• In this way the subtle boundary path integral over the field configurations at the return time is handled straightforwardly.