A quantum field algebra

Abstract

The Laplace Hopf algebra created by Rota and coll. is generalized to provide an algebraic tool for combinatorial problems of quantum field theory. This framework encompasses commutation relations, normal products, time-ordered products and renormalisation. It considers the operator product and the time-ordered product as deformations of the normal product. In particular, it gives an algebraic meaning to Wick's theorem and it extends the concept of Laplace pairing to prove that the renormalised time-ordered product is an associative deformation of the normal product involving an infinite number of parameters. The parameters themselves form a group: the renormalisation group, which acts on the product instead of on the algebra.