The Functional Measure for the In-In Path Integral

TL;DR

This paper formulates the in-in path integral for scalar fields in a fixed background within a suitable function space, providing a spectral representation and clarifying boundary conditions, which enhances the mathematical rigor and understanding of the path integral's structure.

Contribution

It introduces a rigorous function space formulation for the in-in path integral, explicitly handles boundary conditions, and clarifies the nature of solutions like instanton-like configurations.

Findings

Spectral representation of the kinetic operator in flat space.

Explicit derivation of standard propagators from the spectral representation.

Identification of boundary conditions involving both fields and their derivatives.

Abstract

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. It turns out that not only the values of the forward (+) and the backward (-) evolving fields but also their time derivatives must be matched at the return time, which is mainly overlooked in the literature. This formulation also determines the field configurations that are included in the path integral uniquely. We show that some of the recently suggested instanton-like solutions corresponding to the stationary phases of the cosmological in-in path integrals can be rigorously identified as limits of sequences in the function space.