The functional measure for the in–in path integral

DOI10.1088/0264-9381/32/9/095008zbMATH Open1328.83199arXiv1212.3066OpenAlexW2014815513MaRDI QIDQ5254366

Publication date: 9 June 2015

Published in: Classical and Quantum Gravity (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1212.3066

zbMATH Keywords

path integral cosmology in-in formalism

Mathematics Subject Classification ID

Relativistic cosmology (83F05) Quantum field theory on curved space or space-time backgrounds (81T20) Quantization of the gravitational field (83C45) Path integrals in quantum mechanics (81S40) Exact solutions to problems in general relativity and gravitational theory (83C15)