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MP: Theoretische und Mathematische Grundlagen der Physik

MP 14: Quantenfeldtheorie: Algebraische QFT und Verwandtes (Forts.)

MP 14.2: Fachvortrag

Montag, 18. März 2002, 16:40–17:00, SR 1035/36

Connes–Kreimer renormalization and Hopf algebras beyond perturbation theory — •Bertfried Fauser — Universität Konstanz, Fachbereich Physik, Fach M678, D-78457 Konstanz

Recently A. Connes and D. Kreimer showed that the Zimmermann forest formula can be reformulated using the antipode of the Hopf algebra of rooted trees. This mechanism relays strongly on the recursive evaluation of the antipode as given by Milnor and Moore in 1965 and generally developed by W. R. Schmitt in 1987. However, a flawed lemma of Kuperberg 1991 which is used e.g. in knot theory, for Jones-polynomials, Möbius inversions etc. is used for the evaluation of these antipodes.

We will concentrate on Clifford Hopf gebras which do not belong to the class of ‘non interacting‘ Hopf algebras, where the Kuperberg lemma applies, to show the following crucial results:

1) There are no non-trivial integrals and co-integrals in Clifford Hopf gebras.

2) Clifford Hopf gebras are non-connected or ’interacting’ Hopf gebras.

3) The recursive antipode formula does not hold in these algebras.

4) Wick normal-ordering connects topological distinct instances of Hopf gebras, i.e. interacting and non-interacting ones.

5) Quantized algebras, e.g Clifford versus Grassmann, are in general non-local and non-connected.

Since spinors and thus Clifford Hopf gebras play a prominent role in QFT, our results exhibit a major drawback of the conventional and the Connes–Kreimer renormalization theory of perturbative QFT if one tries to extend it to the non-perturbative regime.