Quantum Gravity Colloquium: the talks
This time we had four seminar-like talks, plus an open discussion session on the topic "Are the foundational problems of quantum mechanics relevant for quantum gravity?" Starting with the talks: Leron Borsten talked on the entropy of black holes in supergravity theories, and intriguing connections it has with entanglement in quantum information theory. There is no clear picture yet in which to view the black hole entropy as entanglement entropy, but there are some identities or isomorphisms between the mathematical description of both concepts that do not look as a coincidence. Yousef Ghazi-Tabatabai talked on an approach to interpreting quantum mechanics which seemed related to the ideas pushed forward by Rafael Sorkin in his plenary talk at Morelia.
Frank Hellmann (who also must be given the lion's share of the credit for organizing the colloquium) talked on "Partial Observables", an approach to defining quantum observables in generally covariant theories and, potentially, solving the problem of time. It has been long championed by Carlo Rovelli, with whom Frank worked before coming to Nottingham. The talk explained how observables that are evolving in a conventional framework can be recast as Dirac observables when the dynamics is written in a generally covariant way. These are fit to answer the question "If the system is in physical state Rho, what is the probability of seeing the correlation (x,t)?" (where x,t are the variables of the classical configuration space). The answer to this question is Tr (Rho P(x,t)), where P(x,t) is the operator that projects states into the physical state which is itself the projection, onto the physical Hilbert space, of the kinematical state corresponding to correlation (x,t). Sadly little time was left by the end of the talk for Frank to discuss the thorny case of multi-time measurements, which is the real centrepiece of his paper.
Eugenio Bianchi gave an excellent talk on Perturbative Regge Calculus and Loop Quantum Gravity. It was a version of the talk he gave at Morelia, but with the math replaced by the concepts, which was much better! I think this stuff is extremely important and I am hope to start working into it in the future, so I will summarise the talk in more detail than the previous ones. I would be extremely happy to receive comments discussing it or pointing out mistakes in my exposition.
Loop Quantum Gravity is an essentially non-perturbative theory. Any attempt to find a "semiclassical limit" and connect it to established physics is complicated by the fact that semiclassical physics is essentially perturbative; so there is the problem of even mathematically connecting the two frameworks, before a concrete calculation to see if they agree can be done. One way of doing this connection is the boundary amplitude formalism introduced by Rovelli. Take a kinematical semiclassical state, a kinematical state given by a superposition of spin networks which is a Gaussian peaked on on a classical spacetime (and choose this to be flat space). Fix this as the state on the boundary of a region, and you can compute correlations of observables, measured in the boundary, due to the dynamics in the interior. Use a spin foam model to specify the dynamics: for example, the Barrett-Crane model. You can then calculate, based on a nonperturbative theory, semiclassical correlations of your dynamical variables, which are spins jmn. You obtain results. However, you don't know if your initial theory that defines the boundary state (LQG) is correct, nor if the spin foam model you have chosen to encode the dynamics is correct, and besides that as the whole conceptual and calculational framework looks very different from things used in other areas of physics, you would really really want something to compare your results to as a check.
Enter Perturbative Regge Calculus. Regge Calculus is an approximation scheme to GR in which the curved manifold is replaced by a skeleton triangulation, with the geometry encoded in the discrete edges and vertices. As Eugenio stressed, it can also be thought of as exact (not approximated) GR but with piecewise flat metrics instead of continuous metrics. Choose a triangulation that discretizes flat space; it is described by the connectivity C and a set of edge lengths Li. Now add small perturbations to the edge length variables, and quantize these perturbations. You are now doing Quantum Perturbative Regge Calculus, which is a straightforward background-dependent, perturbative quantum theory, in which all the standard rules of the game apply. Change your variables from edge lengths to face areas, and calculate the quantum area-area correlations on the boundary of a region. Compare them with the fluctuation in spins jmn calculated from the non-perturbative theory. Are there equal?
The answer is that, for the Barrett-Crane model dynamics, all quantities compared between both calculations up to the 3-point function match exactly, provided that one identifies the spins used as variables in LQG with the areas used in Regge calculus, up to the factor 8 Pi G b, with b being the Imirizi parameter! This is an independent and nontrivial check of the famous LQG spectrum area, which was derived at a purely kinematical level. Here the dynamics is necessary to ensure the right correspondence between areas and spin variables. For example, if one uses a non graph-changing Hamiltonian to define the dynamics, the correspondence is not recovered. The calculation with the new spin foam model recently proposed by Rovelli, Pereira and Engle, which cures certain problems with the Barrett-Crane model, is still not completed and its results are eagerly awaited.
It should be stressed that, though nontrivial, this check does not amount yet to confirming that LQG has the "correct" semiclassical limit. The only perturbative theory of quantum gravity that has right to be acknowledged as "correct" (because it uses uncontroversial quantum field theory principles in an unobjectionable way) is the Effective Field Theory approach popularized by Donoghue. It is not known if quantum perturbative Regge calculus is a sort of discrete equivalent of this; understanding the connection between them would be a key step forward. But the calculation Eugenio talked about has great importance by itself, because it shows that a fully nonperturbative approach to quantum gravity, when used in conjunction with a semiclassical state, can give the same answers as a better understood perturbative approach.
As this post is getting longish, I will break it here and post in one or two days about the discussion on the relation between foundations of QM and QG. Stay tuned.