Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2004-08-21
Physics
High Energy Physics
High Energy Physics - Theory
16 pages. Invited talk given at the V Workshop of the DGFM of the Mexican Physical Society, Morelia, Mexico, November 2003. Al
Scientific paper
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.
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