  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.
