  This article, as the first of three, aims at establishing the
(time-dependent) Born-Oppenheimer approximation, in the sense of space
adiabatic perturbation theory, for quantum systems constructed by techniques of
the loop quantum gravity framework, especially the canonical formulation of the
latter. The analysis presented here fits into a rather general framework, and
offers a solution to the problem of applying the usual Born-Oppenheimer ansatz
for molecular (or structurally analogous) systems to more general quantum
systems (e.g. spin-orbit models) by means of space adiabatic perturbation
theory. The proposed solution is applied to a simple, finite dimensional model
of interacting spin systems, which serves as a non-trivial, minimal model of
the aforesaid problem. Furthermore, it is explained how the content of this
article, and its companion, affect the possible extraction of quantum field
theory on curved spacetime from loop quantum gravity (including matter fields).
