Difference between Loop Quantum Gravity (LQG) and Causal Dynamical Triangulation (CDT)

CDT is based on the idea to calculate the path integral over spacetime geometries by summing over the Regge triangulations. The discreteness of these triangulations is put in by hand, moreover, it isn't clear if this discreteness is actually physical, or is just a mathematical artifact. I.e. whether the sum over these discrete triangulations itself has discrete properties.

As far as I know, there isn't a convincing mathematical argument that shows that the CDT path integral is well defined. Existing CDT computations are done numerically on a computer, they very much resemble lattice QCD computations.

LQG is based on applying quantum mechanics to the phase space of a continuous geometrical theory (General Relativity in the frame-connection formulation) in a specific way. The discreteness arises after quantization – just like the energy of a canonical oscillator which is continuous in the classical theory becomes discrete after quantization; in LQG the geometry of space (distances, areas, volumes, etc.) becomes discrete after quantization.

More about the discreteness of spacetime in LQG in this answer: https://physics.stackexchange.com/a/521712/30833.

Graphs are the mathematical extensions of the Loops. A Loop is like a Closed structure, say a ring-like flux structure, This is a closed Structure, a flux of Space-time itself, connected to another and this sort of connection is basically a 3D Penrose Spin Network, first developed by Roger Penrose, modified by many theoretical physicists like Abhay Ashtekar, Rovelli and Smolin and many more. When we deal mathematically with a Loopy representation, we have various mathematical technicalities and the point is that you mentioned Quantum Regge calculus, this is the mathematical background of Loop Quantum Gravity and Causal Dynamical Triangulation. So, the basic thing is that a loop is a closed Structure and now, in this case, to avoid technicalities, we deal with something called a Graph, Graphs are the 3D objects, These can have links, they are connected to another, these have certain nodes and also these can have many many mathematical configurations. The point is that we can carry the same integration used in a Loopy representation to the Graphical representation and many mathematical technicalities disappear. So according to this Graphical mathematical representation, a Graph is a normal representation for describing the physics of Quantum Space-time. So, a Graph is the mathematical extension of a Loop. Now enter to CDT. Causal Dynamical Triangulation is a mathematical theory. You know that what is CDT. The point is that, in pictures, you see Graphs and those polygonal Structures as same, but they are way different from one another. In the case, so both have a same mathematical background, Quantum Regge calculus, but both use these in different ways. While those polygonal are no such kind of extension like Graphs are the extensions of Loops or so. The conclusion is that they use Quantum Regge Calculus in different ways or so.

There are many similarities indeed. Both uses a foliation (even though LQG people don't call it like that, they effectively define it) and both uses Regge Calculus as its math.

But huge difference, that LQG seems to be focusing on the local aspects of the interactions of the discrete building blocks of spacetime. As the theory is working with connections and gauges, it is taking a traditional field theoretical approach, and aims to combine gr with the standard model.

In the other hand the CDT action contains only global information, like the number of vertices and simplices in of given triangulation, multiplied with the Newton and cosmological constants (and a third one quantifieing the difference between space and time. The action is global and no dynamics is put in by hand. The result of the numerical simulations, the shape of the configurations are given by the interplay of the normalizing factor in the partition function and the entropy.

Phase diagram-wise they share similarities also. LQG has a fully connected state, which probably corresponds to the crumpled phase in CDT. (which is a phase where the time extent of the universe disappears and there is only space).

Huge difference between the to, that LQG can operate with strong couplings, meaning that there are concrete couplings between the loops and fields. In CDT there is week coupling only, meaning that everything happens via the geometry.

LQG describes a chunk of spacetime, you can perform LQG calculations on a few simplices. Actually there are attempts to put calculations to qbits. (For example here).

CDT is a background independent approach, gluing the simplices creates the spacetime itself. Due to finite size effects/ Lattice artefacts it dies not make sense to describe CDT with a few simplices. This is why during numerical simulations a few 100k (or even million) simplices are used.

A CDT universe in the C phase (de-Sitter) describes a quantum universe with good semiclassical behavior and proper volume fluctuations on the top of it in accordance with the Hartle - Hawking minisuperspace model, where the only dynamical parameter is the scale factor. Furthermore, most recent finding shows, that the initial inhomogeneities of the universe and Dark Matter has probably quantum gravitational and not particle like origin (see Cosmic Fibers from quantum gravity ).

Even though they describe a 4 dimensional spacetime, both theories include (rather results) a dimensional reduction. At close distances spacetime behaves as it was effectively 2 dimensional.

LQG is a huge field, planck sized calculations are done, also tensor networks can be allied to it, and there is a loop quantum cosmology attempting to describe the early universe. I think that Dark Matter for example would be of particlish origin there.

Also, if I'm correct the LQG description contains gravitons, while CDT is a purely geometric approach.

In LQG you can take a chunc of spacetime, and calculate the dynamics of it, evolve it in time.

In CDT the whole spacetime history is calculated with fixed time extent. Every configuration of CDT is a closed history of the evution of the 3 dimensional submanifold from start to end (represented by the foliation, start and end is typically the same because the topology of the euclidean time direction is a circle, the snake bites its tail for numerical reasons ).