  Instead of formulating the state space of a quantum field theory over one big
Hilbert space, it has been proposed by Kijowski [Kijowski 1977] to represent
quantum states as projective families of density matrices over a collection of
smaller, simpler Hilbert spaces. One can thus bypass the need to select a
vacuum state for the theory, and still be provided with an explicit and
constructive description of the quantum state space, at least as long as the
label set indexing the projective structure is countable. Because uncountable
label sets are much less practical in this context, we develop in the present
article a general procedure to trim an originally uncountable label set down to
countable cardinality. In particular, we investigate how to perform this
tightening of the label set in a way that preserves both the physical content
of the algebra of observables and its symmetries. This work is notably
motivated by applications to the holonomy-flux algebra underlying Loop Quantum
Gravity. Building on earlier work by Okolow [arXiv:1304.6330], a projective
state space was introduced for this algebra in [arXiv:1411.3592]. However, the
non-trivial structure of the holonomy-flux algebra prevents the construction of
satisfactory semi-classical states. Implementing the general procedure just
mentioned in the case of a one-dimensional version of this algebra, we show how
a discrete subalgebra can be extracted without destroying universality nor
diffeomorphism invariance. On this subalgebra, states can then be constructed
whose semi-classicality is enforced step by step, starting from collective,
macroscopic degrees of freedom and going down progressively toward smaller and
smaller scales.
