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Noncommutative version of Borcherds’ approach toquantum field theory

Abstract : Richard Borcherds proposed an elegant geometric version of renormalized perturbative quantumfield theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle overa smooth manifold. However, this framework looses its geometric meaning when Borcherdsintroduces a (graded) commutative normal product. We present a fully geometric version ofBorcherds’ 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 containsall the terms of the perturbative expansion and we quantize it to recover the expectation values ofstandard quantum field theory when the Hopf algebra fiber is (graded) cocommutative. This constructionenables to the second quantize any theory described by a cocommutative Hopf algebrabundle.
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https://hal.archives-ouvertes.fr/hal-01222770
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Submitted on : Friday, October 30, 2015 - 3:04:04 PM
Last modification on : Saturday, September 24, 2022 - 3:36:05 PM

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  • HAL Id : hal-01222770, version 1

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Christian Brouder, Nguyen Viet Dang, Alessandra Frabetti. Noncommutative version of Borcherds’ approach toquantum field theory. PoS - Proceedings of Science, SISSA, 2015. ⟨hal-01222770⟩

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