Noncommutative version of Borcherds' approach to quantum field theory

TL;DR

This paper develops a noncommutative geometric framework for perturbative quantum field theory, extending Borcherds' approach by replacing the commutative normal product with a noncommutative tensor product, enabling second quantization of theories.

Contribution

It introduces a fully geometric noncommutative version of Borcherds' quantization, constructing a noncommutative Hopf algebra for perturbative expansions.

Findings

Constructed a noncommutative many-body Hopf algebra.

Reproduces standard quantum field theory expectation values.

Enables second quantization of theories with cocommutative Hopf algebra bundles.

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.