Gurau et. al. posted a new paper today on arxiv (arxiv.org/0807.4122 [hep-th]), proposing a new approach to quantum field theory: marked trees.

We propose a new formalism for quantum field theory which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively {\it differential} renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse-Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a nonperturbative definition of field theory in noninteger dimension.

## 7 comments:

Interesting.

Yes, reminds me of the colored operad approach.

Well, it has to be operads (props etc) of some kind, doesn't it? Note that they do mention ribbon graphs.

Yes, the ribbon graphs arise in the noncommutative case, it seems. This would take us back to Mulase and Waldron's work.

Good, you're blogging again!

"It just classifies Feynman amplitudes differently, by breaking these amplitudes into pieces and putting these pieces into boxes labeled by trees."

This seems like what I'm doing, but in more generality. I'm in a particular application, bound states of the color force, but the idea is to classify the Feynman diagrams according to how they permute the colors. Then there's only a finite number of them and you can work non perturbational calculations with them. These are easy to solve because particle number is conserved, but I wonder if this gives the more general technique.

Another thing that comes to mind is that 't Hooft's introduction to QFT says that the difference between classical and quantum physics is that the Feynman diagrams of classical mechanics do not include loops.

The state-of-the-art in new loop calculations most likely involves those techniques from twistor string theory. See the slides by Zvi Bern on the subject: The S-Matrix Reloaded.

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