Algebraic QFT and Quantum Gravity

[selfAdjoint]

Kea posted this on another thread:

On a slightly different note (nothing to do with the SM): I actually went to a very interesting NCG talk today by Paolo Bertozzini (maybe I'll blog about it) which made a couple of things a little bit clearer to me. Paolo works on a kind of categorification of the basic (spectral triple / manifold) duality, and thinks of this Tomita-Takesaki stuff that they're keen on (http://arxiv.org/abs/math-ph/0511034) as providing a C* version of the Cosmic Galois Group somehow. But he ends up doing bundles instead of manifolds and then he says they might be like gerbes or stacks ... and he wants to put it all into a more categorical language.

The Tomita-Takesaki results are and exciting breakthrough in AQFT, by now getting to be pretty well understood. The beginnings of it are in Haag's book Local Quantum Physics.

Another line of work in AQFT that approaches the idea of matter in QG is represented by the work of Klaus Fredenhagen and his colleagues. A recent example is gr-qc/0603079, Towards a Background Independent Formulation of Perturbative Quantum Gravity, by Romeo Brunetti and Klaus Fredenhagen.
A brief quotation will give the flavor.

We adopt the point of view [3] of algebraic quantum field theory and identify physical systems with *-algebras with unit (if possible, C*-algebras) and subsystems with subalgebras sharing the same unit. In quantum field theory the subsystems
can be associated to spacetime regions. Every such region may be considered as a spacetime in its own right, in particular it may be embedded into different spacetimes. It is crucial that the algebra of the region does not depend on the way it is embedded into a larger spacetime. For instance, in a Schwartzschild spacetime the physics outside the horizon should not depend on a possible extension to a
Kruskal spacetime.

 

[Kea]

Hi selfAdjoint

Naturally I'm more a fan of the Higher Category approach, but the impression I got yesterday is that the emerging dictionary between Kontsevich, operads etc. on the one hand, and NCG (or AQFT) on the other is beginning to look substantial. That doesn't make me enthusiastic to go away and learn Tomita-Takesaki. On the contrary: it only makes me more convinced that the right way to tell phenomenologists and experimentalists how to calculate stuff is the easier way...with operads. Now on that side, admittedly, the geometry of manifolds is not yet so clear. But we deliberately set out from a different starting point, for physical reasons. Connes and Marcolli want the Riemann hypothesis. We just care about Yang-Mills and mass generation.

 

[Entropia]

This is the content of the principle of local covariance.

In the context of quantum gravity, the idea of using algebraic quantum field theory (AQFT) to study the relationship between matter and space-time is a promising approach. AQFT provides a rigorous mathematical framework for studying quantum field theory and its connection to space-time. By identifying physical systems with *-algebras, it allows for the study of subsystems and their relationship to different spacetimes.

The Tomita-Takesaki results and the work of Klaus Fredenhagen and his colleagues are both important contributions to this field. They provide a deeper understanding of how matter and space-time interact and how this relationship can be described in a more categorical language.

Paolo Bertozzini's work on categorification of the basic duality is also an interesting development in this area. By using bundles instead of manifolds, it opens up the possibility of exploring the connection between quantum field theory and quantum gravity in a new way. The idea that these bundles may be related to gerbes or stacks adds to the potential for further understanding and insights.

Overall, the combination of AQFT and quantum gravity has the potential to shed light on some of the most fundamental questions in physics, such as the nature of matter and space-time. The exciting breakthroughs in this field show that we are making progress towards a better understanding of these complex concepts.