The n-Category Café

the phenomenon

In the study of perturbative quantum field theory one is concerned with functions – called amplitudes – that take a collection of graphs – called Feynman diagrams – to Laurent polynomials in a complex variable $z$ – called the (dimensional) regularization parameter – $$Amplitude : CertainGraphs \to LaurentPolynomials$$ and wishes to extract a “meaningful” finite component when evaluated at vanishing regularization parameter $z = 0$.

A prescription – called renormalization scheme – for adding to a given amplitude in a certain recursive fashion further terms – called counterterms – such that the resulting modified amplitude – called the renormalized amplitude – is finite at $z=0$ was once given by physicists and is called the BPHZ-procedure.

This procedure justifies itself mainly through the remarkable fact that the numbers obtained from it match certain numbers measured in particle accelerators to fantastic accuracy.

its combinatorial Hopf-algebraic interpretation

The combinatorial Hopf algebraic approach to perturbative quantum field theory starts with the observation that the BPHZ-procedure can be understood

- by noticing that there is secretly a natural group structure on the collection of amplitudes;

- which is induced from the fact that there is secretly a natural Hopf algebra structure on the collection of graphs;

- and with respect to which the BPHZ-procedure is simply the Birkhoff decomposition of group valued functions on the circle into a divergent and a finite part.

the Connes-Kreimer theorem

A Birkhoff decomposition of a loop $\phi : S^1 \to G$ in a complex group $G$ is a continuation of the loop to

- a holomorphic function $\phi_+$ on the standard disk inside the circle

- a holomorphic function $\phi_-$ on the complement of this disk in the projective complex plane

- such that on the unit circle the original loop is reproduced as

$$\phi = \phi_+ \cdot (\phi_-)^{-1} \,,$$ with the product and the inverse on the right taken in the group $G$. Notice that by the assumption of holomorphicity $\phi_+(0)$ is a well defined element of $G$.

Now: CK-Theorem:

a) If $G$ is the group of characters on any graded connected commutative Hopf algebra $H$ – $G = Hom(H,\mathbb{C})$ – then the Birkhoff decomposition always exists and is given by the formula $$\phi_- : (X \in H) \mapsto Counit(X) - PolePartOf( Product(\phi_- \otimes \phi) \circ (1 \otimes (1 - Counit)) \circ Coproduct (X) ) \,.$$

b) There is naturally the structure of a Hopf algebra, $H = Graphs$, on the graphs considered in quantum field theory. As an algebra this is the free commutative algebra on the “1-particle irreducible graphs”. Hence QFT amplitudes can be regarded as characters on this Hopf algebra.

c) The BPHZ renormalization-procedure for amplitudes is nothing but a) applied to the special case b).

Proof: A. Connes, D. Kreimer, Renormalization in quantum field theory I.

the Hopf algebra perspective on QFT

This result first of all makes Hopf algebra an organizational principle for (re-)expressing familiar operations in quantum field theory.

Computing the renormalization $\phi_+$ of an amplitude $\phi$ amounts to using the above formula to compute the counterterm $\phi_-$ and then evaluating the right hand side of $$\underbrace{\phi_+}_{renormalized amplitude} = \underbrace{\phi}_{amplitude} \underbrace{\cdot}_{convolution product} \underbrace{\phi_-}_{counterterm} \,,$$ where the product is the group product on characters, hence the convolution product of characters.

Every elegant reformulation has in it the potential of going beyond mere reformulation by allowing to see structures invisible in a less natural formulation. For instance Dirk Kreimer claims that the Hopf algebra language allows him to see patterns in perturbative quantum gravity previously missed.

gauge theory and BV-BRST with Hopf algebra

I learned most of what I know about this from Walter von Suijlekom, who I stayed with at the program at the Hausdorff Institute in Bonn over last summer.

Walter is thinking about the Hopf-algebraic formulation of BRST-BV methods in nonabelian gauge theory and gave a couple of talks to us about his work in Bonn.

In his nicely readable Renormalization of gauge fields using Hopf algebra he reviews the central idea: the BRST formulation of Yang-Mills theory manifests itself at the level of the resulting bare i.e. unnormalized amplitudes in certain relations satisfied by these, the Slavnov-Taylor identies.

Renormalization of gauge theories is consistent only if these relations are still respected by renormalized amplitudes, too. We can reformulate this in terms of Hopf algebra now:

the relations between amplitudes to be preserved under renormalization must define a Hopf ideal in the Hopf algebra of graphs. Walter proves this to be the case for Slavnov-Taylor in his theorem 9 on p. 12.

As a payoff, he obtains a very transparent way to prove the generalization of Dyson’s formula to nonabelian gauge theory, which expresses renormalized Green’s functions in terms of unrenormalized Green’s functions “at bare coupling”. This is his corollary 12 on p. 13.

In the context of BRST-BV quantization these statements are subsumed, he says, by the structure encoded in the Hopf ideal which corresponds to imposing the BV-master equation. This is in his latest:

W. van Suijlekom: Representing Feynman graphs on BV-algebras.