Is loop quantum gravity based on discretisations?

Two spin networks, which can be interpreted as Hilbert space elements
describing truncations (or discretisations) of general relativity,
are summed. What is the physics of such states? 

This post is related to last week’s about criticism of loop quantum gravity and a comment I found in Sabine Hossenfelder’s blog. Saying that loop quantum gravity is based on discretisations quickly leads one to doubt that Lorentz invariance may be a property of LQG, as happened in the comment. So it seems worth to clear up this issue and precisely say in what context discretisations appear, in what context they don’t, and what this means for the physics that are described by LQG.

The canonical formalism of LQG is most suited to discuss this question, since one has a direct relation to general relativity as the theory which is quantised to obtain LQG. In particular, one really quantises continuum general relativity and the Hilbert space that one obtains thus has to be interpreted as the continuum Hilbert space. However, the elementary excitations on this Hilbert space, comparable to particles on a Fock space, are discrete quanta of geometry. They are the quantum operators corresponding to certain holonomies and fluxes that one can compute in the classical theory. A finite number of such holonomies however only captures a limited amount of the continuum geometry that is roughly equivalent to a discretisation of GR. Therefore, the (simplest) quantum states that one usually discusses in LQG, which also form a basis of the Hilbert space, have a certain interpretation as discrete geometries. This however doesn’t mean that there cannot exist any continuum states in this Hilbert space, since we can of course take arbitrary linear combination of basis elements, and thus, morally speaking, arbitrary superpositions of lattices. Gaining control over such states and understanding their physics is currently the main open problem in LQG. A computation in a simplified toy model explicitly showed that such quantum geometries can be consistent with local Lorentz invariance.

In conclusion, the Hilbert space of LQG contains elements which can be interpreted as discrete quantum geometries. In different approaches to LQG, these states are given a more or less fundamental status which might suggest that one is really dealing with a discretisation. However, one always needs to keep in mind that one can arbitrarily superpose such states and that the state of quantum geometry that correctly describes our universe at all scales should be searched for in a suitable continuum limit.

(As a short remark, this does not mean that states containing only a few quanta of geometry cannot be useful when one is considering certain truncations of the theory, such as cosmology. However, one then has to interpret those states in a coarse grained setting and properly understand the renormalisation group flow which relates the dynamics of continuum states to those of coarse grained few quanta states.)