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.

Updates on some criticisms of loop quantum gravity

Image from MySafetySign.com

In this post, we will gather some criticism which has been expressed towards loop quantum gravity and comment on the current status of the respective issues. The points raised here are the ones most serious in my own opinion, and different lists and assessments could be expressed by other researchers. 

Approaches to quantum gravity

Again as a part of a lecture series in preparation, I compiled a list of the currently largest existing research programmes aimed at finding a quantum theory of gravity (while unfortunately omitting some smaller, yet very interesting, approaches). A much more extensive account is given in here. For an historical overview, I recommend this paper.