Compactness condition on boundary formalism for loop quantum gravity

Question

I'm going over Rovelli's  Covariant Loop Quantum Gravity notes http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf and in section 2.4.2 "Boundary formalism" he briefly develops how to think about quantum fields in the boundary formalism. However, I noted that he restricts his discussions to "finite" bulks of spacetime and their boundaries. He also mentions this restriction in section 3.32 and even talks about the topology of such region. However I can't seem to understand exactly why this restriction is imposed. It seems the boundary datum for fields shouldn't in general be restricted to a compact subset, instead one specifies values at infinity, so how can you have a full theory when you only consider these finite portions?

Prof. Legolasov · Accepted Answer

There are two points here: (1) can we use compact regions to do physics and (2) are those enough or do we need non-compact regions as well in the full theory.
Re (1), the answer is most certainly yes. Conceptually nothing is stopping us from doing the same in ordinary QFT, in fact, in CFT (Conformal Field Theory) this is a well known technique known as radial quantization. What's fundamentally different in GR is that the resulting space of states doesn't depend on the shape of the chosen compact region at all, due to diffeomorphism invariance. This is why it makes sense to talk about the Hilbert space of states for the boundary formalism without specifying the region and its boundary.
Re (2), this is a philosophical question that I don't want to get into. For practical purposes, experimental setups and areas of interest are localized in space-time, so those can be surrounded by a compact region and described quantum-mechanically as states of the gravitational / other fields at the boundary of that region.

Compactness condition on boundary formalism for loop quantum gravity

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