Report on the Quantum Gravity School: the lectures (1)
In the Spin Foam lectures, Friedel defined the basic structure of spin foam models, introduced the diagrammatic notation used for them, and discussed 3D quantum gravity extensively as an example, both without and with matter. It was only by the end of the last lecture that he reached the connections of the latter case with an effective non-commutative field theory with deformed Poincare invariance, which is probably the most exciting ofshoot of this research. Livine discussed four-dimensional models, which arise from writing gravity as a constrained BF theory. He centered on the Barrett-Crane model, and sketched the calculation of the 10j symbol, showing that from it the Regge discretization of gravity arises naturally but with extra “bad” terms. He them talked about the calculation made by Rovelli and his collaborators of the graviton propagator from spin foams, which eluded this problem by introducing a boundary semiclassical state with a phase that, when the calculation is carried out, cancels exactly the “bad” terms of the 10j symbol.
I may add some personal remarks here. When I first heard about the graviton propagator calculation I was excited, but I was unsure as to whether that phase had not been but “by hand” to ensure a nice result which otherwise the theory refused to provide. That is often the problem when you don’t actually work in a field (and you lack the experience and smartness required to check claims by yourself if you don’t work in it): you read a paper and if it gives you this kind of suspicion, you have no way of deciding whom to trust. This is one reason why it is so great to travel to conferences and get an opportunity to discuss face to face with people who don’t mind answering your questions even if they are naïve. In this case it was a dinner table conversation with Eugenio Bianchi that reassured me that the calculation was sound. The phase of the boundary state is not arbitrary or chosen by hand: it is the correct phase to use for the state to be really semiclassical, being peaked not only in the “configuration variables” that specify the classical boundary geometry but also in the “momentum variables” conjugate to them. In fact, in the LQG lectures Thiemann constructed a precise mathematical definition of “semiclassical states” and, according to Bianchi, the state that Rovelli, he and the rest used was the same kind of state, independently found by physical ansatz instead of rigourous mathematical definition. (All this was in fact actually explained in their paper; but reading it after a personal explanation is so much clearer!)
Going back to the lectures, the most intensive ones were Thiemann’s twelve lectures on canonical Loop Quantum Gravity. I certainly learnt a lot from them, and not only abouth LQG per se but also about more general things like constrained systems and how a quantum algebra is constructed from a classical one. The Master Constraint program for dealing with the Hamiltonian (which replaces the constraint at each point of space by the integral over all space of the square of the constraint) was introduced, and after some discussions of the subtleties and ambiguities to be solved, Thiemann enunciated (without proving –that would have probably another 12 lectures!) an important theorem: A particular “quantum Master Constraint operator” was defined in such a way that satisfies all desired properties, including finiteness, dipheomorphism invariance, and most importantly, good semicalssical behaviour. By this Thiemann meant that given a classical field configuration (a pair A, E of Ashtekar variables) one can find a semiclassical state (built as a superposition of spin networks which peaks on it) that makes the expectation value of the MC operator agree with the classical value of the MC to within any desired accuracy. (It is important for this that the classical state and the semiclassical one that approaches it are not physical states, in which the constraint vanishes).
After this, Thiemann also defined in a formal way the scalar product in the physical Hilbert space, which suggestively can be seen as the “time integral” of a transition amplitude. He commented on the possibility that spin foam models might benefit from using the master constraint instead of ordinary constraints, something that has not been tried (and I have no idea how it could be tried). He also described a way to compute expectation values of the volume operator to arbitrary precission in semicalssical states, and mentioned (unfortunately briefly) his more recently developed "Algebraic Quantum Gravity". After the lectures ended I imprudently tried to ask him (in personal conversation) a question concerning “the unsolved problem of the classical limit of LQG” and cut me by saying that the problem was already solved! What emerged from our follwing conversation (which on my side had a rather bewildered face) was this: the “problem of the classical limit” consists in showing that there exist in LQG physical states that, looked at large scales, resemble a nice classical manifold on which GR holds; the semiclassical behaviour of the Master Constraint operator described above implies, by some subtle argument involving projection which I could not follow, that there will be physical states with this property; therefore, the problem is solved. It does not matter at all that most combinations of spin networks, or even most physical states, will not give anything resembling a classical state at large scales. This is the same that happens in ordinary quantum mechanics: coherent states, with nice semiclassical properties, are a very particular kind of state, and most combinations of large number of individual quantum objects do not look at all like anything classical. It is enough that there be nice semiclassical states, not that they be generic.
Of course, this does not mean that one knows how to write a physical state approaching Minkowski (or Schwarzschild or any other classical solution). One doesn’t even know how to write any physical state! But I think that Thiemann’s argument does show, somehow, that this problem is merely technical and that there is no fundamental problem in LQG with respect to the existence of the classical limit. I am interested in hearing my readers’ perspectives on this.
This post is already rather long, and I still have much to say about Martin Reuter’s lectures on Asymptotic Safety, so I think it is a good idea to stop here for the moment. Expect the next post within a couple of days.