  In loop quantum cosmology, Friedmann-LeMaitre-Robertson-Walker (FLRW)
space-times arise as well-defined approximations to specific \emph{quantum}
geometries. We initiate the development of a quantum theory of test scalar
fields on these quantum geometries. Emphasis is on the new conceptual
ingredients required in the transition from classical space-time backgrounds to
quantum space-times. These include a `relational time' a la Leibnitz, the
emergence of the Hamiltonian operator of the test field from the quantum
constraint equation, and ramifications of the quantum fluctuations of the
background geometry on the resulting dynamics. The familiar quantum field
theory on classical FLRW models arises as a well-defined reduction of this more
fundamental theory.
