Projective limits of state spaces IV. Fractal label sets

Abstract

Abstract Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski (1977) to represent quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces (see Lanery (2016) [1] for a concise introduction to this formalism). 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 (2013), a projective state space was introduced for this algebra in Lanery and Thiemann (2016). However, the non-trivial structure of the holonomy–flux algebra prevents the construction of satisfactory semi-classical states (Lanery and Thiemann, 2017). 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, quantum states can then be constructed which are more regular than was possible on the original algebra. In particular, this allows the design of semi-classical states whose semi-classicality is enforced step by step, starting from collective, macroscopic degrees of freedom and going down progressively toward smaller and smaller scales.