# Ad-hoc algebraic spin network quantization

In Loop Quantum Gravity, one usually embraces the functorial point of view towards (T)QFT. Canonical and spinfoam formalisms only differ in how the projection operator is defined. Canonical formalism uses so-called Thieman's form of the Hamiltonian constraint, which is very ad-hoc, and strictly speaking is just one of many equally feasible possibilities. In spinfoam description, a different projection operator is defined through the limit of amplitudes given by state sums over 2-complexes called spinfoams.

The crucial point here is that both formalisms seem to agree on the general functorial nature of the quantum theory:

• States are associated to hypersurfaces.
• Projection operator is associated with the bulk quantum spacetime.

In fact, I've seen Rovelli and others use the term "boundary formalism" in many LQG papers. I believe it essentially states that LQG is a functorial TQFT, with some of the Atiyah-Signer axioms relaxed & modified to accomodate the quantum gravity case, but those are not important for this question.

I am aware of two different conceptual frameworks that (T)QFT can be formalized in: the functorial picture described above (which is usually used in TQFT) and the algebraic picture from AQFT.

I would like to know if there's been any progress on formulating a LQG-like theory in the algebraic picture. In the following I'll try to sketch the properties that I expect such a formulation to have:

1. It shouldn't be based on the split of spacetime or on the boundary formalism. States should be truly 4d manifestly-covariant. Basically, I expect the formalism to be a deformation quantization of the space of Einsteinean (that is, pseudo-Riemannian and satisfying Einstein's equations) 4d manifolds.
2. The relevant gauge group should be $SO(3,1) \sim SL(2,\mathbb{C})$.
3. States should be given by spin networks which live in the bulk. Spin networks describe the geometry of spacetime, not spatial slices. There's no Hamiltonian constraint anymore.
4. There is a different constraint though – the quantum version of Einstein's equations $$\varepsilon_{IJKL} e^{J} \wedge F^{KL} = 0.$$ I don't know how it can be realized on the space of $SL(2,\mathbb{C})$ spin networks, but at this point I'm only interested in if any realization exists.