Two-point functions in (loop) quantum cosmology

TL;DR

This paper explores the mathematical structure of two-point functions in quantum cosmology, especially within loop quantum cosmology, clarifying their definitions, relations, and physical implications for single and multiverse models.

Contribution

It provides a comprehensive analysis of two-point functions, their derivations, and the connections between different formalisms in quantum cosmology, including loop quantum cosmology.

Findings

Clarified the relation between group averaging and deparametrized definitions.

Derived vertex expansions and composition laws for two-point functions.

Discussed implications for single universe and multiverse quantum theories.

Abstract

The path-integral formulation of quantum cosmology with a massless scalar field as a sum-over-histories of volume transitions is discussed, with particular but non-exclusive reference to loop quantum cosmology. Exploiting the analogy with the relativistic particle, we give a complete overview of the possible two-point functions, pointing out the choices involved in their definitions, deriving their vertex expansions and the composition laws they satisfy. We clarify the origin and relations of different quantities previously defined in the literature, in particular the tie between definitions using a group averaging procedure and those in a deparametrized framework. Finally, we draw some conclusions about the physics of a single quantum universe (where there exist superselection rules on positive- and negative-frequency sectors and different choices of inner product are physically equivalent) and multiverse field theories where the role of these sectors and the inner product are reinterpreted.