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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.
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