Renormalization and Hopf algebras

In quantum electrodynamics perturbation analysis leads to infinities for the mass and the charge of the electron due to self-energy Feynman loop diagrams. Those infinities can be isolated and cured by a renormalization process. However renormalization is much more widespread than one may think. Sidney Coleman suggested a simple experiment one can do at home to see it in action. Take a ping-pong ball and immerse it water. Compute the Archimedean force and see that it should produce an acceleration close to an amazing 11 g. At that acceleration the ping pong ball should shoot out of the water and hit the ceiling, but of course this does not happen. What happens is that the acceleration is in fact close to 2 g because the mass of the ball that should be used is an "effective mass" due to the surrounding fluid. 

Now in quantum field theory there are many renormalization techniques possible. Connes and Kreimer have had a truly outstanding and amazing result relating the combinatorial Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization technique with Hopf algebras and the Birkhoff factorization problem. I cannot do justice to this topic in a simple post, and I encourage the interested reader to study Connes' book: "Noncommutative Geometry, Quantum Fields and Motives" for the details. I will only present the big picture here.

The first concept needed is what Connes calls: one-particle irreducible Feynman graphs (1PI) \( \Gamma\):

• \( \Gamma \) is not a tree
• \( \Gamma \) cannot be disconnected by cutting a single edge

Then given a graph \( \Gamma \) it may have a 1PI subgraph \( \gamma \) and one can construct a contracted graph called \( \Gamma / \gamma\) which is obtained by collapsing each \( \gamma_i \) - 1PI of \( \Gamma\) to a vertex. 

Now the coproduct of the Hopf algebra of the Feynman graphs is given by:

\( \Delta (\Gamma) = \Gamma \otimes 1 + 1\otimes \Gamma + \sum_{\gamma} \gamma \otimes \Gamma / \gamma \)

Image from http://arxiv.org/abs/hep-th/9912092

The coalgebra so obtained is graded by the loop numbers, and so it is in fact a Hopf algebra. 

Now onto the Birkhoff decomposition. Start with the Riemann sphere and consider a closed loop \( \gamma \) on it. The loop cuts the sphere into two separate regions + and -. You are given a complex value function \( \phi \) along the loop and now we want to find two complex functions \( \phi_{+} \) defined on the + region and \( \phi_{-} \) defined on the - region such that \( \phi = {\phi_-}^{-1}  {\phi_+}\). In passing we note that this decomposition problem is intimately related with soliton theory and inverse scattering. 

Now for the main result. First the BPHZ procedure: given a Feynman diagram \( \Gamma \) one performs the Bogoliubov-Parasiuk preparation: replace the unrenormalized value \(U(\Gamma)\) by:

\( R(\Gamma) = U(\Gamma ) + \sum_{\gamma} C(\gamma) U(\Gamma / \gamma)\)

where the counterterms \(C(\gamma)\) are defined recursively as:

\(C(\Gamma) = -T(R(\Gamma))\) where \(T\) is the projection onto the pole part of the Laurent series in the variable z of dimensional regularization.

Now if we have a renormalizable theory and its associated Hopf algebra of Feynman diagrams defined above the Birkhoff factorization is given by:

\( \phi_- (X) = -T (\phi (X) + \sum\phi_- ( X^{'}) \phi ({X}^{''})) \) 

and

\(\phi_+ (X) = \phi (X) + \phi_- (X) + \sum \phi_- ({X}^{'}) \phi ({X}^{''} )\)

where the coproduct is:

\( \Delta (X) = X\otimes 1 + 1\otimes X + \sum {X}^{'} \otimes {X}^{''}\)

I hope now thing become clear: we start with dimensional regularization which extends the dimension of space into complex numbers (in fact into the Riemann sphere). The unrenormalized values corresponds to a function \(\phi\) on a closed loop in the Riemann sphere and we use the Hopf algebra of Feynman diagrams to perform the Birkhoff factorization which obtains the renormalized values \(\phi_+ \).

BPHZ is the same as Birkhoff factorization:

\(\phi = U\) - unrenormalized value
\(\phi_- = C \) -counterterms
\(\phi_+ = R \) - renormalized value

The Hopf algebra coproduct simply codifies the BPHZ procedure. Whow!!!! What a mathematical tour-de-force linking such unrelated mathematical areas. I don't know about you but I am really impressed with Connes and Kreimer result. Bravo!