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Combinatorics of graphs and trees in perturbative quantum field theory

Very roughly, perturbative quantum field theory is about assigning amplitudes to Feynman graphs via the Feynman rules. Unfortunately, these amplitudes are divergent integrals, for graphs with loops. Renormalisation is about eliminating these infinities. One method is the BPHZ renormalisation, which amounts to recursively subtracting counterterms for all 'divergent' subgraphs. This is a complicated overcounting/undercounting procedure, that it took some 15 years to sort out, thanks in particular to the work of Bogoliubov, Parasiuk, Hepp and Zimmermann, from 1955 to 1969.

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.

A main point of the tree paper is that trees are special cases of polynomial endofunctors, and that a useful notion of decorated tree results: trees can be decorated by other polynomial endofunctors. If P is a polynomial endofunctor, then the set of (isoclasses of) P-trees is the set of operations of the free monad on P.

This is exploited in the paper

where it is shown that the Connes-Kreimer comultiplication lifts from the level of algebra to the level of sets, that the comultiplication functor is polynomial, and that its representing sets and maps are the same as found in the polynomial representation of the free monad construction.

These constructions can be generalised from sets to groupoids, which is important for applications. Some details of this can be found in

where we show how recent work by van Suijlekom resulting in Faà di Bruno formulae in the Hopf algebra of graphs has an analogue in the bialgebra of trees — provided the correct operadic trees are considered. Here is a longer abstract, of a talk given recently by Imma:
   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 Fał 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 Fał 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 Fał 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 except that that paper deals with nestings of trees instead of nestings of graphs, and iterated the construction in order to construct opetopes in all dimensions.
Last updated: 2012-08-09 by Joachim Kock.