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.Comment: 29 pages; v2: typos corrected, references adde
