  Quantum Field Theory on Curved Spacetime (QFT on CS) is a well established
theoretical framework which intuitively should be a an extremely effective
description of the quantum nature of matter when propagating on a given
background spacetime. If one wants to take care of backreaction effects, then a
theory of quantum gravity is needed. It is now widely believed that such a
theory should be formulated in a non-perturbative and therefore background
independent fashion. Hence, it is a priori a puzzle how a background dependent
QFT on CS should emerge as a semiclassical limit out of a background
independent quantum gravity theory. In this article we point out that the
Born-Oppenheimer decomposition (BOD) of the Hilbert space is ideally suited in
order to establish such a link, provided that the Hilbert space representation
of the gravitational field algebra satisfies an important condition. If the
condition is satisfied, then the framework of QFT on CS can be, in a certain
sense, embedded into a theory of quantum gravity. The unique representation of
the holonomy-flux algebra underlying Loop Quantum Gravity (LQG) violates that
condition. While it is conceivable that the condition on the representation can
be relaxed, for convenience in this article we consider a new classical
gravitational field algebra and a Hilbert space representation of its
restriction to an algebraic graph for which the condition is satisfied. An
important question that remains and for which we have only partial answers is
how to construct eigenstates of the full gravity-matter Hamiltonian whose BOD
is confined to a small neighbourhood of a physically interesting vacuum
spacetime.
