The procedure leads to some of the best tested physical theories, but from a mathematical viewpoint it was quite mysterious and appeard rather ad hoc, until Kreimer in 1997 discovered that the combinatorics is governed by a Hopf algebra of rooted trees, encoding nestings of Feynman diagrams. The over-under counting is expressed by a sort of antipode, in analogy with many situations in combinatorics (like for example the inclusion/exclusion principle).
The subsequent work of Connes and Kreimer (1998-) established deep connections to noncommutative geometry, number theory, Lie theory and combinatorics, stimulating a lot of further activity by many mathematicians and physicists.
While there is now a lot of rigourous mathematical theory about all this, it could be that the basic combinatorics has not yet found its most natural form: there are still many aspects that are justified by applications to physics rather than by intrinsic principles.
I am working on this, trying to understand the constructions from a categorical viewpoint. The overall idea is that the key point of the whole theory is nesting and substitution, and that the theory of operads is a good framework for this. In fact, rather than operads in the classical sense, I prefer to work with polynomial monads and polynomial functors. This can be seen as a coordinatised version of the theory of operads.
The following two papers lay out a convenient categorical formalism for trees and graphs, respectively, which turn out to be useful in the combinatorics of quantum field theory.
This is exploited in the paper
Abstract: We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs.
Abstract: The Connes-Kreimer Hopf algebra of trees (or of Feynman graphs) encodes the combinatorics of the BPHZ renormalisation procedure in pQFT. The comultiplication of a tree returns all the ways of "cutting" the tree. However, the individual trees (or graphs) do not have direct physical interpretation; rather certain infinite sums, the so-called Green functions, carry the physical meaning. Van Suijlekom recently discovered that the Green functions satisfy a version of the classical Faˆ di Bruno formula for substitution of power series. In this talk we will show how the theory of groupoids can be used to give a very conceptual proof of the Faˆ di Bruno formulae for Green functions in the bialgebra of trees. In this framework a Green function is (the cardinality of) a groupoid and the Faˆ di Bruno formula is shown to be essentially an equivalence of groupoids.The interest in Green functions of trees comes from a result in which will eventually be a sort of cornerstone of all these developments: this paper is in part a survey of the formalisms developed in the above papers, and in part a demonstration of their usefullness. In particular it is shown that with an appropriate choice of polynomial endofunctor P (consisting of 1PI graphs and residues), there is a functor π from P-trees to graphs which together with the forgetful functor from P-trees to trees (and from trees to combinatorial trees a la Kreimer) yields Kreimer's assignment of trees to graphs, and that pullback along π is a bialgebra homomorphism, and that it preserves the Green function. Hence the Hopf algebra of graphs is a subbialgebra of the bialgebra of trees, and both contain a subbialgebra isomorphic to the Faà di Bruno bialgebra.
I have talked about this at several conferences, but it is a bit tricky to write down nicely, and it may take some time before I finish... The most recent talk had this abstract:
In quantum field theory, trees serve to express nestings of Feynman graphs. Important aspects of the combinatorics of renormalisation are captured nicely by the Connes-Kreimer Hopf algebra of rooted trees. However, the isolated graphs do not have a physical meaning: the meaning is carried rather by the Green functions, which are sums of graphs weighted by their symmetry factors. Now the trees of Connes and Kreimer do not capture anything about symmetries of graphs. I will explain how this can be 'fixed' by using instead operadic trees, or more precisely P-trees for certain polynomial endofunctors P defined over groupoids.The key point in this is an equivalence between groupoids of graphs equipped with a nesting and P-trees (defined as above). This equivalence is completely analogous to the main theorem of the paper