Rovelli recently gave a thumbnail sketch of LQG. Around 10 slides illustrated by drawings. You can look at the original slides and listen to the audio here:
http://relativity.phys.lsu.edu/ilqgs/panel050509.pdf
http://relativity.phys.lsu.edu/ilqgs/panel050509.mp3
It is a 3-way discussion and Rovelli is the second in line, so when you open the slides PDF you must scroll down about 10 slides first to come to Rovelli's section, likewise with the Audio mp3 you have to fast forward to about one third of the way, and just get the middle third of the recording. He will tell you in the audio which slide, when to switch slides and so forth. The other presentations, by Ashtekar and Freidel, are also good. But here I am focusing just on this very short 20 minute sequence by Rovelli.
==quote Rovelli's slides, text only==
Some questions
1. Is quantum space made out of loops and spin-networks or tetrahedra and 4-simplices?
2. Is flat space formed by many small tetrahedra with low-spin, or by few large tetrahedra with with high-spin?
3. Is low-energy physics given by quantum gravity on a single 4-simplex? Or by an infinite triangulation limit?
4. How do we study the the continuum limit from Planck scale discretness to the macroscopic continuum?
5. A spinfoam model is like a new version of quantum Regge calculus. So, why it should work better than quantum Regge calculus?
Claim:
these questions are ill posed.
***
1. Is quantum space made out of loops and spin-networks or tetrahedra and 4-simplices?
The meaningful question in quantum theory is not how something is, but how it responds to a measurement.
There is no space “between” quanta of space, and it makes no sense to ask what is the geometry between one quantum and another, or inside a quantum, or what is the “geometry of quantum”. It is like asking for the “shape of a photon”. Or “What do I measure if I measure the energy in the space occupied by half a photon?”
***
2. Is flat space formed by few tetrahedra with high spin, or by many tetrahedra (or loops) with low spin?
How many particles are there in the Fock vacuum?
How many particles are invoved in a two-particle interaction?
Quantum theory gives the probability for measurement outcomes: it does not describe “what is between measurements”.
“In between pictures” are just descriptions of the ways I decide to do calculations. They are different for different measurements, and at different orders in perturbation theory.
[my comment. Rov sometimes like Feyn has flashes of unusual common sense. they illuminate.]
***
The good question, I think, is:
What can we compute that makes sense?
and
How can we compute it?
***
The problem of quantum gravity is–in a sense–two problems:
1. What is the right (background independent) theory?
2. How do extract physics from a background independent QFT?
Difficulties:
1. In standard QFT distances and time intervals gives locations in spacetime where the field is measured. In quantum gravity distances and time interval are quantum measurements of the gravitational field.
2. I think it is interesting to compute scattering amplitudes. These depend on a background: they describe interactions of excitations in a flat space context. How do we tell a background independent theory that there is a background?
***
The only solution I know:
1. Boundary formalism
2. Vertex expansion
3. Large spin expansion
***
1. Boundary formalism
– Scattering amplitudes depend on the measured geometry around the scattering region.
– It is the boundary state that tells the theory about the background.
– Not different than in standard QFT:
W (x, x′) = ⟨0|φ(x)φ(x′)|0⟩ (1)
= ⟨0| e
iHt φ(⃗x)e
−iHt e
iHt′ φ(⃗x′ )e
−iHt′ |0⟩ (2)
= ⟨φ(⃗x)0
t | e
−iH(t′−t) |φ(⃗x′)0
t′ ⟩. (3)
W (x, x′) = ⟨e
−iH(t′−t) | φ(⃗x) φ(⃗x′) |0
t ⊗ 0
t′ ⟩
Hin ⊗Hout (4)
In quantum gravity →
W (x, y, Ψ
boundary ) = ⟨W | φ(⃗x′) φ(⃗x′) |Ψ
boundary ⟩
Hboundary
This quantity is a 4d diffeomorpism invariant and well-defined. It reduces to standard 2-point function in the flat space theory. Locations of x and x′ are well defined with respect to the boundary state.
***
2. Vertex expansion
1. There is no way in physics you can compute without a suitable approximation scheme.
2. QFT expansion = truncation to a finite number of degrees of freedom.
QED: a finite order in perturbation theory has a finite number of particles.
Lattice QCD: finite lattice with # of cells determined by (size of the phenomenon L )/(minimal relevant wavelength λ).
3. Is there a truncation to a finite number of d.of f. in gravity, which is physically good in some regimes?
4. Yes! Truncate GR to a finite triangulation of spacetime (vertex expansion). # of simplices determined by (size of the phenomenon L)/(minimal relevant wavelength λ).
5. It is background independent, in the same sense in which Regge calculus is.
6. Where is it good? Many instances: Cosmology! Long wavelength at fixed distance. Large distance expansion of the propagator...
7. Precise characterization of the regime of validity (on the boundary state)
require us to compute higher orders and compare.
***
2. Large spin limit
• In quantum gravity there is a built in scale, l
Planck . Unlikely quantum Regge calculus.
• The vertex expansion is not a large distance L ≪ l
Planck approximation.
• At fixed order in the vertex expansion, large distance with respect to the Planck scale means high spins; because Area ∼ j .
• High spins = high quantum numbers = semiclassical limit. Therefore the theory must go to GR (truncated on a fixed lattice) for high spins, at fixed triangulation.
***
Summary
1. “Loopy, polymer, triangulated” spaces are helps for intuition, not descriptions of reality. No incompatibility between them.
2. In quantum gravity, flat space is neither many small Planck scale things not few big large-spin 4 simplices. It is a process with a transition amplitudes. We can represent it with different pictures, according to the measurements we are considering, the calculation scheme, and the approximation scheme.
3. We must compute diff-invariant amplitudes, including when dealing with excitations over a flat space. The only way of doing so that I know is to code the background into the boundary space. (Boundary formalism.)
4. We need an approximation scheme. For scattering amplitudes, we can truncate degrees of freedom to a finite number, very much like is done in computing in QED and QCD. (Vertex expansion.)
5. Regime of validity of the vertex expansion: processes whose size L is not much larger than the minimal relevant wavelength λ. Includes the large distance behavior of the scattering amplitudes in coordinate space.
6. At given ratio λ/L, the Large-spin Limit captures processes at scales larger than the Planck length. It gives the semiclassical limit.
→ This does not mean that flat space is “made out of large 4-simplices”!
→ It means that we describe measurements performed at scales larger that the
Planck scale, at low order.
)
I am loosing hope. Also, won't you comment about 
