Butcher group
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group[1] first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method.
It was Cayley (1857), prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics.
Connes & Kreimer (1999) pointed out that the Butcher group is the group of characters of the Hopf algebra of rooted trees that had arisen independently in their own work on renormalization in quantum field theory and Connes' work with Moscovici on local index theorems.
The number of equivalence classes of heap-orderings on a particular tree is denoted by α(t) and can be computed using the Butcher's formula:[3][4] where St denotes the symmetry group of t and the tree factorial is defined recursively by with the tree factorial of an isolated root defined to be 1 The ordinary differential equation for the flow of a vector field on an open subset U of RN can be written where x(s) takes values in U, f is a smooth function from U to RN and x0 is the starting point of the flow at time s = 0.
Cayley (1857) gave a method to compute the higher order derivatives x(m)(s) in terms of rooted trees.
These are defined inductively by With this notation giving the power series expansion As an example when N = 1, so that x and f are real-valued functions of a single real variable, the formula yields where the four terms correspond to the four rooted trees from left to right in Figure 3 above.
It corresponds to the formal group structure discovered in numerical analysis by Butcher (1972).
The homomorphism corresponding to the actual flow has Butcher showed that the Runge–Kutta method gives an nth order approximation of the actual flow provided that φ and Φ agree on all trees with n nodes or less.
corresponds to the data Hairer & Wanner (1974) proved that the Butcher group acts naturally on the functions f. Indeed, setting they proved that Connes & Kreimer (1998) showed that associated with the Butcher group G is an infinite-dimensional Lie algebra.
is generated by the derivations θt defined by for each rooted tree t. The infinite-dimensional Lie algebra
[1] Connes & Kreimer (1998) provided a general context for using Hopf algebraic methods to give a simple mathematical formulation of renormalization in quantum field theory.
Renormalization was interpreted as Birkhoff factorization of loops in the character group of the associated Hopf algebra.
In this simplified setting, a renormalizable model has two pieces of input data:[6] Note that R satisfies the Rota–Baxter identity if and only if id – R does.
An important example is the minimal subtraction scheme In addition there is a projection P of H onto the augmentation ideal ker ε given by To define the renormalized Feynman rules, note that the antipode S satisfies so that The renormalized Feynman rules are given by a homomorphism
For the minimal subtraction scheme, this process can be interpreted in terms of Birkhoff factorization in the complex Butcher group.
The evaluation at z = 0 of γ+ or the renormalized homomorphism gives the dimensionally regularized values for each rooted tree.
Its infinitesimal generator β is an element of the Lie algebra of GC and is defined by It is called the beta function of the model.
In particular the renormalization group defines a flow on the space of coupling constants, with the beta function giving the corresponding vector field.
More general models in quantum field theory require rooted trees to be replaced by Feynman diagrams with vertices decorated by symbols from a finite index set.
Connes and Kreimer have also defined Hopf algebras in this setting and have shown how they can be used to systematize standard computations in renormalization theory.
