What goes in what spaces in LQG? I read again the first chapter of Rovelli 2005 quantum gravity book, and I am still puzzled by the different spaces for each concept.
It seems spin foams inhabit Hilbert spaces.
But i read elsewhere the quanta of area and volume of Planck size inhabit the space time manifold, therefore not Hilbert espaces.
And from the same source, I read that networks existed linking these areas and volumes, therefore in spacetime manifold, therefore not Hilbert spaces.
Can a specialist with vulgarisation skills can enlight me with a good vision of what goes into what kind of spaces?
Thanks a lot in advance!
Florent 
 A: Well, LQG is a quantum theory of gravity, so it describes the geometry of spacetime using the language of Hilbert spaces, pretty much by design.
Spin networks are abstract graphs colored with irreducible representations and intertwining tensors of the group $SU(2)$. There are a countable number of different spin networks, and so we have as usual the linear vector space of finite-norm superpositions of different spin networks. That space is called the kinematical Hilbert space of LQG, and Rovelli denotes it by $\mathcal{K}$ I think.
Each spin network looks like a cartoony representation of a classical spatial slice. Nodes represent "lumps" of space, it is possible to interpret them as "elementary tetrahedra" forming space. Links represent connections between the lumps and the colors (irreducible representations) of links are related to area of the connections. In the simplicial interpretation these are the common triangles shared by the neighboring tetrahedra.
However, as all intuitive pictures, this one is naive and inaccurate:


*

*One might wonder what happened to time, why do the spin networks describe a spatial slice and not the whole spacetime? This is reminiscent to how in Quantum Mechanics, the wavefunctions are functions of the spatial position at time zero $x(0)$ and not of the entire trajectory $x(t)$. Similarly to how these functions of $x(0)$ give rise to the full trajectory after taking the equation of motion into account, in LQG spin networks are actually secretly states of space-time and not only one spatial slice. The exact reason why it is so is much more involved than in ordinary quantum mechanics. In fact, to establish this is one of the conceptual challenges to quantum gravity known under the name of "problem of time".

*Actually, individual spin networks are like energy levels of the harmonic oscillator - they are ultra quantum and have no classical counterparts. The quantum states that describe realistic spacetimes are always superpositions of spin networks, just like a classical motion of the harmonic oscillator is a superposition of many different eigenfunctions of the quantized oscillator.

*The real Hilbert space $\mathcal{H}$ is actually a subspace of $\mathcal{K}$. There are two basic definitions of $\mathcal{H}$, the equivalence of which is an open question. The first definition is well-formed mathematically, but it proves hard, if not impossible, to establish that it leads to a well-behaved physical theory. The second definition is heuristic and uses mathematical objects (spinfoam state sums) the existence of which hasn't been proven. However, it leads to promising physical results, including the recovery of classical General Relativity in the limit $\hbar \rightarrow 0$.


I hope this helps, let me know in the comments if you want me to add more detail on either of these points.
