Noncommutative version of Borcherds approach to quantum field theoryReport as inadecuate

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* Corresponding author 1 IMPMC - Institut de minéralogie, de physique des matériaux et de cosmochimie 2 LPP - Laboratoire Paul Painlevé 3 ICJ - Institut Camille Jordan Villeurbanne

Abstract : Richard Borcherds proposed an elegant geometric version of renormalized perturbative quantum field theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle over a smooth manifold. However, this framework looses its geometric meaning when Borcherds introduces a graded commutative normal product. We present a fully geometric version of Borcherds- quantization where the external tensor product plays the role of the normal product. We construct a noncommutative many-body Hopf algebra and a module over it which contains all the terms of the perturbative expansion and we quantize it to recover the expectation values of standard quantum field theory when the Hopf algebra fiber is graded cocommutative. This construction enables to the second quantize any theory described by a cocommutative Hopf algebra bundle.

Keywords : quantum field theory Hopf algebra quantization

Author: Christian Brouder - Nguyen Viet Dang - Alessandra Frabetti -



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