Solenodon Paradoxus

1. What do you mean by "compactify"? Do you propose to start with a higher-dimensional spacetime manifold? 2. Ashtekar connections and the densitized triad are a preliminary step which we do prior to doing LQG. The real LQG consists of (a) the kinematical Hilbert space of spin networks and (b) the projection operator on the physical states (given through the spinfoam vertex). 3. I still don't see how you propose to modify the existing projection operator given by the EPRL vertex to include SO(10) GUT since every time I bring this up you end up replying about something completely unrelated. I am sorry, but I feel like this conversation is going nowhere, because you obviously don't know what you are talking about. I won't reply unless I see something to support your initial claim that SO(10) GUT can be incorporated in LQG.

Dear Mordred, I am aware of the way LQG is constructed and how the SU(2) group enters this construction. I still don't see though how replacing SU(2) with SO(10) would describe anything useful and you still haven't answered how you propose to modify the spinfoam amplitude, which is, and I couldn't emphasize this any harder, dependent on the structure of unitary representations of SU(2) and SL(2,C). You can find this on page 141 of that "Covariant Quantum Gravity" textbook. LQC does not consider spinfoams (do you mean spinfoam cosmology?), it has always been a polymer quantization of symmetry-reduced cosmological models of spacetime.

Ok, that's something new, finally. Thank you. But in the paper it is not mentioned how this relates to LQG. That is the point I would like clarified.

I'm gonna try to reach out to you for the last time. I have no doubt that what you said about the double cover is correct. In fact, I couldn't agree more! I am interested in generalizing this construction to include the SO(10) GUT. Can you or can you not provide any arguments/links to support *this* claim of yours (and not any other)?

Thank you for this, but it wasn't the original question. Let me repeat the question here so that there's no misunderstanding: I am interested in your claim that "there's a full treatment of SO(10) GUT in LQG" (or did I misunderstand the claim?). Please follow up on your previous post where you proposed replacing SU(2) with SO(10), which I don't understand because it is a rough sketch with not enough details to make things clear. Specifically, I want to understand which group and how the usual SU(2) of spin networks extends to, and which group does the SL(2,C) of spinfoams extend to and how. In fact, let me ask this same question in an even more unambiguous way. Say we go with the spinfoam formulation. Then there's a mapping from SU(2) representation theory to the SL(2,C) unitary (principal series) representation theory called the upsilon-gamma map, which determines the amplitude of the spinfoam vertex. This completely defines the projection operator as a limit of spinfoam amplitudes. Now how would you modify this to include SO(10) GUT gauge&matter fields? Do you simply add another labels to the spinfoam faces (= spin network links), or do you consider a gauge-gravity unifying group (which one?). How does the spinfoam amplitude change? Please provide all these details or just a reference, because this is what "full treatment of SO(10) GUT in LQG" means in my opinion. And it happens to be a subject I am currently extremely interested in. Note that: 1. You can't extend SU(2) to SO(10) because SO(10) GUT *does not* include gravity, and you are making the gravitational SU(2) a part of it which means that you no longer model the GUT theory. You could, however, extend SU(2) to something encompassing both SU(2) and SO(10) and say that in the regime where a classical spacetime emerges this splits into a semidirect product (this is demanded by Coleman-Mandula theorem). Is this what you mean? 2. You can't extend SL(2,C) to SO(10) or its double-cover because of the same reason, and in addition, because noncompact groups (SL(2,C)) can't be subgroups of compact groups (SO(10)). Update: no, it is not true that $SO(3,1) = SU(2) \times SU(2)$. It is true for the complexified Lie algebras that $so(3,1) = d_2 = su(2) + su(2)$, but not true for the Lie groups. And I am not just being nitpicky, this is actually significant to LQG.

Could you be more specific? 1. Afaik, there's two different (equivalence hasn't been proven so far) approaches to LQG The one due to Thiemann is based on the Hamiltonian constraint operator acting on the node, and the one due to Rovelli is based on the projection operator approximated by a series of (renormalized) EPRL spinfoams. Which one are we talking about here (or does it matter at all)? 2. SO(10) GUT *does not* include gravity. In fact, there's a famous no-go theorem due to Coleman and Mandula, which forbids any but trivial unification of gravitational SO(3,1) and internal symmetries in a larger Lie group. I agree that it is hardly valid in the background independent context (that is, prior to "breaking" of background independence and emergence of classical spacetime and matter QFT on it). But this at least deserves a detailed explanation. 3. Strictly speaking, even the "SO(3,1) part of SO(10)" in your post doesn't make sense, since a noncompact Lie group SO(3,1) cannot be a subgroup of a compact SO(10).

Please kindly provide a link to the full treatment of SO(10) GUT in Loop Gravity.