Physics, Combinatorics and Hopf Algebras

TL;DR

This paper explores how Hopf algebra techniques can address combinatorial problems in physics, demonstrating their efficiency in renormalization and generalizations of quantum mechanics through algebraic structures.

Contribution

It introduces the concept of $k$-primitive elements in Hopf algebras and applies them to simplify renormalization and to describe generalized quantum measures.

Findings

$k$-primitive elements reduce computational cost in renormalization

Hopf algebra framework describes generalized quantum measures

Demonstrates efficiency of algebraic methods in theoretical physics

Abstract

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.