Renormalization and Computation 3

Posted in Uncategorized by Mike Stay on 2009 October 10

This is the third in a series of posts on Yuri Manin’s recent pair of papers applying Hopf algebra renormalization to the Halting problem. Last time I talked about the way people usually do renormalization; this time I’ll talk about the recent work by Kreimer, Connes, and others in exposing the underlying Hopf algebra in this process.

A Hopf algebra is

  • An R-module for a commutative rig R, which means you can add vectors and multiply them by a scalar.
  • An algebra, which means you can take two vectors and multiply them. This operation is associative; there’s also a unit vector that satisfies left- and right-unit laws.
  • A bialgebra, which means there’s also a coassociative comultiplication and counit, and the structures all work together. When the tensor product is the cartesian product, the comultiplication duplicates the vector and the counit is the constant map to 1 in the base field. Even when the tensor product isn’t the cartesian product, it can still be useful to think of it that way.
  • A bialgebra with an involution, called the antipode.

A group is very like a Hopf algebra; in fact, a group object in the category of vector spaces and linear maps is a cocommutative Hopf algebra. You can multiply group elements and there’s a multiplicative unit; you can duplicate and delete them in equations; and you can invert them.

It turns out that Feynman diagrams form a Hopf algebra if you poke yourself in one eye and squint with the other. First, a cut C of an oriented graph g (i.e directed graph with no parallel edges) picks an upper set g^C and a lower set g_C of vertices such that

  • given an oriented wheel in the graph, its vertices either all belong to the upper set or all belong to the lower set, and
  • any edge connecting a vertex v in the upper set to a vertex w in the lower set must be directed from v to w.

Now, given a set of Feynman diagrams, consider all formal linear combinations of graph cuts. This is a vector space because you can add these things pointwise and multiply them by a scalar. We can make it into a bialgebra by defining multiplication to be a linear map

\displaystyle m(g \otimes g') = g \coprod g'

with unit

\displaystyle \begin{array}{rl}e:I & \to H \\ 1 & \mapsto 0, \end{array}

and comultiplication to be a linear map

\displaystyle \Delta(g) = \sum_C g^C \otimes g_C,

where C ranges over all cuts of g, with counit

\displaystyle \begin{array}{rl}\epsilon:H & \to I \\ \sum_g a_g g& \mapsto a_0. \end{array}

It’s graded: just count the number of vertices. And we can turn it into a Hopf algebra by defining the antipode S:H \to H to be a linear map such that

\displaystyle \begin{array}{rl}S(1) & = 1 \\  S(g) & \displaystyle = -g-\sum_C S(g^C) \coprod g_C \\ & \displaystyle = -g-\sum_C g^C \coprod S(g_C).\end{array}

Each algebra homomorphism (not necessarily preserving the Hopf algebra structure) from H to an algebra \mathcal{A} defines a way to assign a (generalized) probability amplitude to each diagram. The set \mbox{hom}(H, \mathcal{A}) of such homomorphisms becomes a group when we note that the functor \mbox{hom}(-, \mathcal{A}) is contravariant, so the comultiplication in H gets mapped to a multiplication.

Next: given a complex group G (that is, a group that’s also a complex manifold so the multiplication and inverse are complex-analytic functions), a Birkhoff decomposition of a loop \phi:S^1 \to G is an analytic 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

    \displaystyle \phi = \phi_{+} \phi_{-}^{-1},

    where the product and the inverse on the right are taken in the group G. Notice that \phi_+(0) is a well defined element of G.

Take G = \mbox{hom}(H, \mathbb{C}). Now imagine our regularization parameter is a complex number \Lambda and we have some map \phi:\mathbb{C} \to G that’s singular at \Lambda = 0. Then the Connes-Kreimer theorem says that the Birkhoff decomposition always exists and gives an explicit formula. Hopf algebra renormalization is simply rearranging the terms in the Birkhoff decomposition:

\displaystyle \phi_{+} = \phi \star \phi_{-}^{-1},

where \star is the convolution product.

As I understand this, \phi is isomorphic to \tilde{\phi}:H \to \mbox{hom}(\mathbb{C},\mathbb{C}). Given a linear combination of graphs, \tilde{\phi} gives you back a Laurent polynomial in \Lambda which you can split into terms with negative exponents (the polar part) and those with positive exponents (the renormalized part).


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