Physics, Combinatorics and Hopf Algebras

A number of problems in theoretical physics share a common nucleus of combinatoric nature. It is argued here that Hopf algebraic concepts and techiques can be particularly efficient in dealing with such problems. As a first example, a brief review is given of the recent work of Connes, Kreimer and collaborators on the algebraic structure of the process of renormalization in quantum field theory. Then the concept of $k$-primitive elements is introduced – these are particular linear combinations of products of Feynman diagrams – and it is shown, in the context of a toy-model, that they significantly reduce the computational cost of renormalization. As a second example, Sorkin’s proposal for a family of generalizations of quantum mechanics, indexed by an integer $k>2$, is reviewed (classical mechanics corresponds to $k=1$, while quantum mechanics to $k=2$). It is then shown that the quantum measures of order $k$ proposed by Sorkin can also be described as $k$-primitive elements of the Hopf algebra of functions on an appropriate infinite dimensional abelian group.