Quantum field theory on a cosmological, quantum space-time

In loop quantum cosmology, Friedmann-LeMa\^{}tre-Robertson-Walker space-times arise as well-defined approximations to specific 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 Leibniz, 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 Friedmann-LeMa\^{}tre-Robertson-Walker models arises as a well-defined reduction of this more fundamental theory.