Quantengravitation - Fortschritte und neue Entwicklungen

[tomS]

In Korfu fand eine hochkarätig besetzte Tagung zur Quantengravitation statt:

http://www.physics.ntua.gr/corfu2009/qg.html

Prof. Abhay Ashtekar
Loop Quantum Gravity
Abstract: This set of lectures will provide an introduction to loop quantum gravity through the simpler setting of loop quantum cosmology. The goal will be to provide a concise summary of the conceptual framework, salient results and open issues. The time limitation will not permit me to give detailed proofs and technical details for which I will provide a guide to literature.

Prof. John Baez
Categorification in Fundamental Physics
Abstract: Categorification is the process of replacing set-based mathematics with analogous mathematics based on categories or n-categories. In physics, categorification enters naturally as we pass from the mechanics of particles to higher-dimensional field theories. For example, higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we must categorify familiar notions from gauge theory and consider connections on "principal 2-bundles" with a given "structure 2-group". One of the simplest 2-groups is the shifted version of U(1). U(1) gerbes are really principal 2-bundles with this structure 2-group, and the B field in string theory can be seen as a connection on this sort of 2-bundle. The relation between U(1) bundles and symplectic manifolds, so important in the geometric quantization, extends to a relation between U(1) gerbes and "2-plectic manifolds", which arise naturally as phase spaces for 2-dimensional field theories, such as the theory of a classical string. More interesting 2-groups include the "string 2-group" associated to a compact simple Lie group G. This is built using the central extension of the loop group of G. A closely related 3-group plays an important role in Chern-Simons theory, and it appears that n-groups for higher n are important in the study of higher-dimensional membranes.

Prof. John Barrett
Spin networks and quantum gravity
Abstract: The series of lectures will be devoted to explaining techniques of spin networks and outlining their use in models of quantum space-time and quantum gravity. The lectures will start with the classical SU(2) spin networks, explaining the diagrammatical techniques and the construction of the Ponzano-Regge model of 3d quantum gravity. Then the q-deformation of spin networks and the Turaev-Viro model are constructed, together with an explanation of the completion to a topological quantum field theory. Next, observables are introduced in these models, and some related models of quantum space-time are also mentioned. Finally, there will be an introduction to some four-dimensional models, both the topological ones, and, briefly, an outline of four dimensional gravity models.

Prof. Vincent Rivasseau
Renormalization in Fundamental Physics
Abstract: Renormalization was first invented to cure the short distance singularities in quantum field theory. Simultaneously constructive field theory developed combinatoric tools to also attack the neglected divergence of perturbation theory. It was later understood that the renormalization group is the correct tool to track the change of physical phenomena under change of observation scale. Then it was realized that the correct notion of scale is not always naively related to short or long distance phenomena, but rather to the spectrum of the propagator. This allowed in the recent years to understand how to renormalize noncommutative field theory, and to attack with a fresh look and new hopes the problem of renormalizing quantum gravity.

Prof. Carlo Rovelli
Covariant loop quantum gravity and its low-energy limit
Abstract and content: I present a new look on Loop Quantum Gravity, aimed at giving a better grasp on its dynamics and its low-energy limit. Following the highly succesfull model of QCD, general relativity is quantized by discretizing it on a finite lattice, quantizing, and then studying the continuous limit of expectation values. The quantization can be completed, and two remarkable theorems follow. The first gives the equivalence with the kinematics of canonical Loop Quantum Gravity. This amounts to an independent re-derivation of all well known Loop Quantum gravity kinematical results. The second the equivalence of the theory with the Feynman expansion of an auxiliary field theory. Observable quantities in the discretized theory can be identifies with general relativity n-point functions in appropriate regimes. The continuous limit turns out to be subtly different than that of QCD, for reasons that can be traced to the general covariance of the theory. I discuss this limit, the scaling properties of the theory, and I pose the problem of a renormalization group analysis of its large distance behavior.

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J. Baez schreibt auf seiner Webseite (auszugsweise):

I worked on spin foams for about 5 years. I love them because they offer the hope of building spacetime from abstract algebra - higher category theory, in fact. But I gave up because a lot of puzzle pieces just didn't seem to be fitting together. Back then, the best candidate for a spin foam model of gravity was the Barrett-Crane model. But there were three big problems:
A) The Barrett-Crane model used spin networks of a different kind from the usual ones in loop quantum gravity. Instead of spin networks with edges labelled by unitary representations of SU(2) (the double cover of the rotation group), it used unitary representations of SL(2,C) (the double cover of the Lorentz group). This is because it's all about spacetime, while loop quantum gravity focuses on space. And instead of using spin networks with vertices labelled by arbitrary intertwiners, it only used a special intertwiner called the "Barrett-Crane intertwiner".
B) While loop quantum gravity in its modern formulation includes the Immirzi parameter - a dimensionless constant that sets the scale of area quantization - the Barrett-Crane model did not. If the currently accepted calculations are right, we need to choose a special and rather peculiar value of the Immirzi parameter if we want loop quantum gravity to get the right answer for the entropy of black holes. So, along with problem A), this makes it even harder to connect the Barrett-Crane model to loop quantum gravity.
C) At first people hoped for various clues linking the Barrett-Crane model to general relativity. For example, we hoped that the asymptotic value of the amplitude for a large 4-simplex in the Barrett-Crane model was nicely related to the action for general relativity. But this turned out to be false: in the Barrett-Crane model, the amplitude for a large 4-simplex is dominated by certain degenerate geometries where the 4-simplex is squashed down to 3 dimensions. ...
Carlo Rovelli raised our hopes again in a more sophisticated way: he tried to compute the propagator for gravitons starting from the Barrett-Crane model. For a beautiful and physically very sensible reason, the degenerate geometries don't dominate this calculation. Rovelli got some promising results for certain components of the graviton propagator, and left a student to work out the rest of the components... but it didn't work!

It seems all these problems have been solved now. There's a new model sometimes called the EPRL model, after Engle, Pereira, Rovelli, and Livine, although other people were involved as well - I'll list some papers later.

The basic idea of the EPRL model is to start with the Holst Lagrangian for general relativity. In 1995, Soren Holst came up with a nice Lagrangian for gravity:

3) Soren Holst, Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D53 (1996), 5966-5969.

It looks like this:

tr(e ^ e ^ *F) + (1/γ) tr(e ^ e ^ F)

I'll explain this in detail later, because there was a student who twice asked about the math behind this Lagrangian, and Rovelli and I brushed the question off by saying "it's just like Palatini Lagrangian". I feel guilty, so someone find that student and tell him to read my explanation below.

But that gets a bit technical, so for now let me say: "it's just like the Palatini Lagrangian". Namely, the first term is the usual Palatini Lagrangian for gravity. The second term involves the Immirzi parameter, γ. The second term doesn't affect the classical equations of motion, because its variation is a total derivative. But it does affect the quantum theory!

If we triangulate spacetime and carry out a spin foam quantization of this theory - which is a bit like doing lattice gauge theory - we can show (in a rough-and-ready physicist's way) that the partition function of the quantum theory is computed as a sum over spin foams where the spin foams are labelled by certain special representations of SL(2,C).

Physicists don't learn the unitary representations of the Lorentz group in school the way they do for the Poincare group. But the unitary representations of the Lorentz group - or its double cover SL(2,C) - are very nice. Except for the trivial representation they're all infinite-dimensional, which is a bit scary at first... but there's a bunch called the "principal series" indexed by a spin j = 0,1/2,1,3/2,... and a nonnegative real number I'll call k. Very roughly speaking the spin j has to do with rotations, while k is an analogous quantity related to boosts. If you want more details, the only online explanation I can find is this:

4) Wikipedia, Representation theory of the Lorentz group, http://en.wikipedia.org/wiki/Representa ... entz_group

It may be better to read some of the many books cited there.

Anyway, the special representations of SL(2,C) that show up in the EPRL model are those with

k = γ j

This is beautiful because there's one for each spin. So, the category of these representations and their direct sums is equivalent to the category of finite-dimensional unitary representations of SU(2)!

This is how the EPRL model gets around problem A) listed above. Spin networks in this new model are nicely compatible with spin networks in loop quantum gravity, because you can think of their edges either as labelled by special representations of SL(2,C), or as labelled by arbitrary representations of SU(2). The first is the "spacetime" or Lagrangian viewpoint, the second is the "space" or Hamiltonian viewpoint.

This is also the key to how the EPRL model gets around problem B). The Immirzi parameter is built into the model in a very natural way. As a result, the quantization of area and volume in this model is compatible with that in loop quantum gravity.

I don't think I'll describe the rest of the model, which consists of a formula for computing the amplitude for a 4-simplex with edges labelled by spins. But it's this formula that solves problem C). The EPRL model gets the graviton propagator right!

Of course there are even bigger tests still ahead for this spin foam model. We need to see if it reduces to general relativity in the classical limit. In other words, we need to get Einstein's equations out of it. And we need to see if it reduces to the usual perturbative theory of quantum gravity in some other limit. In other words, we need to compute, not just graviton propagators (which describe the probability of a lone graviton zipping from here to there on the background of Minkowski spacetime), but graviton scattering amplitudes (which describe the probability of various outcomes when two or more gravitons collide).

Both these tasks are both computationally and conceptually difficult. In other words, it's not just hard to do the calculations: it's hard to know what calculations to do! When I said "in some limit" and "in some other limit", I know what limits these are in a physical sense, but not how to describe them using spin foams. Actually we seem closer to understanding graviton scattering amplitudes, thanks to the work of Rovelli. But it seems miraculous and strange that we can compute graviton propagators (much less scattering amplitudes) using very simple spin foams, as Rovelli and his collaborators have done. Every time I meet him, I ask Rovelli what's going on here: how we can describe the behavior of a graviton in terms of just a few 4-simplices of spacetime.

So, the road is still long, steep, and fraught with danger. But three problems that had everyone completely stumped have now been solved in one elegant blow.

...

Ich werde die Vorträge (sobald sie elektronisch verfügbar sind) sowie die Kommentare von J. Baez hier weiter erläutern und diskutieren. Klingt jedenfalls sehr vielversprechend!