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In [[mathematics]],  the '''Butcher group''', named after the New Zealand mathematician [[John C. Butcher]] by {{harvtxt|Hairer|Wanner|1974}}, is an infinite-dimensional [[group (mathematics)|group]] first introduced in [[numerical analysis]] to study solutions of non-linear [[ordinary differential equation]]s by the [[Runge–Kutta method]]. It arose from an algebraic formalism involving [[rooted tree]]s that provides [[formal power series]] solutions of the differential equation modeling the flow of a [[vector field]]. It was {{harvtxt|Cayley|1857}}, prompted by the work of [[James Joseph Sylvester|Sylvester]] on change of variables in [[differential calculus]], who first noted that the [[Faà di Bruno's formula|derivatives of a composition of functions]] can be conveniently expressed in terms of rooted trees and their combinatorics.  
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{{harvtxt|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 [[Alain Connes|Connes]]' work with [[Henri Moscovici|Moscovici]] on local [[index theorem]]s. This Hopf algebra, often called the ''Connes-Kreimer algebra'', is essentially equivalent to the Butcher group, since its dual can be identified with the [[universal enveloping algebra]] of the [[Lie algebra]] of the Butcher group.<ref>{{harvnb|Brouder|2004}}</ref> As they commented:
{{cquote|We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results.}}
==Differentials and rooted trees==
[[File:Caylrich-first-trees.png|thumb|250px|right|Rooted trees with two, three and four nodes, from Cayley's original article]]
A rooted tree is a [[graph theory|graph]] with a distinguished node, called the ''root'', in which every other node is connected to the root by a unique path.  If the root of a tree '''t''' is removed and the nodes connected to the original node by a single bond are taken as new roots, the tree '''t''' breaks up into rooted trees '''t'''<sub>1</sub>, '''t'''<sub>2</sub>, ... Reversing this process a new tree '''t''' = ['''t'''<sub>1</sub>, '''t'''<sub>2</sub>, ...] can be constructed by joining the roots of the trees to a new common root. The number of nodes in a tree is denoted by |'''t'''|. A ''heap-ordering'' of a rooted tree '''t''' is an allocation of the numbers 1 through |'''t'''| to the nodes so that the numbers increase on any path going away from the root. Two heap orderings are ''equivalent'', if there is an [[automorphism]] of rooted trees mapping one of them on the other. The number of [[equivalence class]]es of heap-orderings on a particular tree is denoted by α('''t''') and can be computed using the Butcher's formula:<ref name="Butcher2008">{{harvnb|Butcher|2008}}</ref><ref>{{harvnb|Brouder|2000}}</ref>
:<math>\displaystyle \alpha(t)= {|t|!\over t! |S_t|},</math>
where ''S''<sub>'''t'''</sub> denotes the [[symmetry group]] of '''t''' and the tree factorial is defined recursively by
:<math>[t_1,\dots,t_n]! = |[t_1,\dots,t_n]| \cdot t_1! \cdots t_n!</math>
with the tree factorial of an isolated root defined to be 1
:<math>\bullet ! =1.</math>
The ordinary differential equation for the flow of a [[vector field]] on an open subset ''U'' of '''R'''<sup>N</sup> can be written
:<math>\displaystyle {dx(s)\over ds} = f(x(s)),\,\, x(0)=x_0, </math>
where ''x''(''s'') takes values in ''U'', ''f'' is a smooth function from ''U'' to  '''R'''<sup>N</sup> and ''x''<sub>0</sub> is the starting point of the flow at time ''s'' = 0.
{{harvtxt|Cayley|1857}} gave a method to compute the higher order derivatives ''x''<sup>(''m'')</sup>(''s'') in terms of rooted trees. His formula can be conveniently expressed using the ''elementary differentials'' introduced by Butcher. These are defined inductively by
:<math> \delta_\bullet^i= f^i, \,\,\, \delta^i_{[t_1,\dots,t_n]} = \sum_{j_1,\dots,j_n=1}^N (\delta^{j_1}_{t_1} \cdots \delta^{j_n}_{t_n})\partial_{j_1} \cdots \partial_{j_n} f^i.</math>
With this notation
:<math> {d^m x\over ds^m} = \sum_{|t|=m} \alpha(t) \delta_t,</math>
giving the power series expansion
:<math>\displaystyle x(s) = x_0 + \sum_{t}  {s^{|t|}\over |t|!} \alpha(t) \delta_t(0).</math>
As an example when ''N'' = 1, so that ''x'' and ''f'' are real-valued functions of a single real variable, the formula yields
:<math> x^{(4)} = f^{\prime\prime\prime}f^3  + 3 f^{\prime\prime}f^{\prime} f^2 + f^{\prime}f^{\prime\prime} f^2 +(f^\prime)^3 f,</math>
where the four terms correspond to the four rooted trees from left to right in Figure 3 above.
In a single variable this formula is the same as [[Faà di Bruno's formula]] of 1855; however in several variables it has to be written more carefully in the form
:<math> x^{(4)} = f^{\prime\prime\prime}(f,f,f)  + 3f^{\prime\prime}(f,f^\prime(f))  +  f^\prime(f^{\prime\prime}(f,f))            +f^\prime(f^\prime(f^\prime(f))),</math>
where the tree structure is crucial.
==Definition using Hopf algebra of rooted trees==
The [[Hopf algebra]] '''H''' of rooted trees was defined by {{harvtxt|Connes|Kreimer|1998}} in connection with [[Dirk Kreimer|Kreimer]]'s previous work on [[renormalization]] in [[quantum field theory]]. It was later discovered that the Hopf algebra was the dual of a Hopf algebra defined earlier by {{harvtxt|Grossman|Larsen|1989}} in a different context. The characters of '''H''', i.e. the homomorphisms of the underlying commutative algebra into '''R''', form a group, called the '''Butcher group'''. It corresponds to the [[formal group]] structure discovered in [[numerical analysis]] by {{harvtxt|Butcher|1972}}.
The '''Hopf algebra of rooted trees''' '''H''' is defined to be the [[polynomial ring]] in the variables '''t''', where '''t''' runs through rooted trees.
*Its [[comultiplication]] <math> \Delta:H\rightarrow H \otimes H</math> is defined by
:<math>\Delta(t) = t\otimes I + I \otimes t +\sum_{s\subset t} s\otimes [t\backslash s],</math>
where the sum is over all proper rooted subtrees '''s''' of '''t'''; <math>[t\backslash s]</math> is the monomial given by the product the variables '''t'''<sub>i</sub> formed by the rooted trees that arise on erasing all the nodes of '''s''' and connected links from '''t'''. The number of such trees is denoted by ''n''('''t'''\'''s''').
*Its [[counit]] is the homomorphism ε of '''H''' into '''R''' sending each variable '''t''' to zero.
*Its [[antipode (algebra)|antipode]] ''S'' can be defined recursively by the formula
:<math> S(t) = -t - \sum_{s \subset t}(-1)^{n(t\backslash s)}S([t\backslash s])s, \,\,\, S(\bullet)= -\bullet.</math>
The '''Butcher group''' is defined to be the set of algebra homomorphisms φ of '''H''' into '''R''' with group structure
:<math>\varphi_1 \star \varphi_2 (t)= (\varphi_1\otimes \varphi_2)\Delta(t).</math>
The inverse in the Butcher group is given by
:<math>\varphi^{-1}(t)=\varphi(St)</math>
and the identity by the counit ε.
==Butcher series and Runge&ndash;Kutta method==
The non-linear ordinary differential equation
:<math> {dx(s)\over ds} = f(x(s)),\,\,\, x(0)=x_0,</math>
can be solved approximately by the [[Runge-Kutta method]]. This iterative scheme requires an ''m'' x ''m'' matrix
:<math>A=(a_{ij})</math>
and a vector
:<math>b=(b_i)</math>
with ''m'' components.
The scheme defines vectors ''x''<sub>''n''</sub> by first finding a solution ''X''<sub>1</sub>, ... , ''X''<sub>''m''</sub> of
:<math> X_i= x_{n-1} + h \sum_{j=1}^m a_{ij} f(X_j)</math>
and then setting
:<math>x_n=x_{n-1} +h \sum_{j=1}^m b_j f(x_j).</math>
{{harvtxt|Butcher|1963}} showed that the solution of the corresponding ordinary differential equations
:<math> X_i(s)=x_0 + s\sum_{j=1}^m a_{ij} f(X_j(s)),\,\,\, x(s)=x_0 + s \sum_{j=1}^m b_jf(X_j(s))</math>
has the power series expansion
:<math> X_i(s) = x_0 +\sum_t {s^{|t|}\over |t|!} \alpha(t) t! \sum_{j=1}^m a_{ij} \varphi_j(t)\delta_t(0),\,\,\,\,x(s) = x_0 +
\sum_t {s^{|t|}\over |t|!} \alpha(t) t! \varphi(t)\delta_t(0), </math>
where φ<sub>''j''</sub> and φ are determined recursively by
:<math>\varphi_j(\bullet)=1.\,\,\, \varphi_i([t_1,\cdots,t_k])=\sum_{j_1,\dots,j_k} a_{ij_1}\dots a_{ij_k} \varphi_{j_1}(t_1)\dots \varphi_{j_k}(t_k)</math>
and
:<math>\varphi(t) = \sum_{j=1}^m b_j \varphi_j(t).</math>
The power series above are called '''B-series''' or '''Butcher series'''.<ref name="Butcher2008" /><ref>{{citation|title=The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas|journal=Applied Numerical Mathematics|volume=15 |year=1994|pages=341–356| first=K. R.|last= Jackson|first2=A. |last2=Kværnø|first3=S.P.|last3=Nørsett|doi=10.1016/0168-9274(94)00031-X|issue=3}} (Special issue to honor professor J. C. Butcher on his sixtieth birthday)</ref> The corresponding assignment φ is an element of the Butcher group. The homomorphism corresponding to the actual flow has
:<math> \Phi(t)={1\over t!}.</math>
Butcher showed that the Runge-Kutta method gives an ''n''th order approximation of the actual flow provided that φ and Φ agree on all trees with ''n'' nodes or less. Moreover {{harvtxt|Butcher|1972}} showed that the homomorphisms defined by the Runge-Kutta method form a dense subgroup of the Butcher group: in fact he showed that, given a homomorphism  φ', there is a Runge-Kutta homomorphism φ agreeing with φ' to order ''n''; and that if given homomorphims φ and φ' corresponding to Runge-Kutta data (''A'', ''b'') and (''A' '', ''b' ''), the product homomorphism <math>\varphi\star \varphi^\prime</math> corresponds to the data
:<math> \begin{pmatrix} A & 0\\ 0 & A^\prime\\ \end{pmatrix},\,\, (b,b^\prime).</math>
{{harvtxt|Hairer|Wanner|1974}} proved that the Butcher group acts naturally on the functions ''f''. Indeed setting
:<math>\varphi\circ f= 1 +\sum_t {s^{|t|}\over |t|!} \alpha(t) t! \varphi(t)\delta_t(0),</math>
they proved that
:<math> \varphi_1\circ (\varphi_2\circ f) = (\varphi_1\star \varphi_2)\circ f.</math>
==Lie algebra==
{{harvtxt|Connes|Kreimer|1998}} showed that associated with the Butcher group '''G''' is an infinite-dimensional Lie algebra. The existence of this Lie algebra is predicted by a theorem of {{harvtxt|Milnor|Moore|1965}}: the commutativity and natural grading on '''H''' implies that the dual '''H'''* can be identified with the [[universal enveloping algebra]] of a Lie algebra <math>\mathfrak{g}</math>. Connes and Kreimer explicitly identify <math>\mathfrak{g}</math> with a space of [[derivation (abstract algebra)|derivation]]s  θ of '''H''' into '''R''', i.e. linear maps such that
:<math>\theta(ab)=\varepsilon(a)\theta(b) + \theta(a)\varepsilon(b),</math>
the formal tangent space of '''G''' at the identity ε. This forms a Lie algebra with Lie bracket
:<math>[\theta_1,\theta_2](t)=(\theta_1 \otimes \theta_2 -\theta_2\otimes\theta_1)\Delta(t).</math>
<math>\mathfrak{g}</math> is generated by the derivations θ<sub>'''t'''</sub> defined by
:<math>\theta_t(t^\prime)=\delta_{tt^\prime}, </math>
for each rooted tree '''t'''.
==Renormalization==
{{harvtxt|Connes|Kreimer|1998}} provided a general context for using [[Hopf algebra]]ic methods to give a simple mathematical formulation of [[renormalization]] in [[quantum field theory]]. Renormalization was interpreted as [[Riemann–Hilbert problem|Birkhoff factorization]] of loops in the character group of the associated Hopf algebra.  The models considered by {{harvtxt|Kreimer|1999}} had Hopf algebra '''H''' and character group '''G''', the Butcher group. {{harvtxt|Brouder|2000}} has given an account of this renormalization process in terms of Runge-Kutta data.
In this simplified setting, a ''renormalizable model'' has two pieces of input data:<ref>{{harvnb|Kreimer|2007}}</ref>
   
   
* a set of ''Feynman rules'' given by an algebra homomorphism Φ of '''H''' into the algebra ''V'' of [[Laurent series]] in ''z'' with poles of finite order;
* a ''renormalization scheme'' given by a linear operator ''R''  on ''V'' such that ''R'' satisfies the [[Rota-Baxter algebra|Rota-Baxter identity]]
::<math>R(fg) + R(f)R(g) = R(fR(g)) + R(R(f)g)\,</math>
:and the image of ''R'' – ''id'' lies in the algebra ''V''<sub>+</sub> of [[power series]] in ''z''.
Note that ''R'' satisfies the Rota-Baxter identity if and only if ''id'' –  ''R'' does. An important example is the ''[[minimal subtraction scheme]]''
:<math>\displaystyle R(\sum_{n} a_n z^n )= \sum_{n< 0} a_n z^n.</math>
In addition there is a projection ''P'' of '''H''' onto the [[augmentation ideal]] ker ε given by
:<math>\displaystyle P(x) = x -\varepsilon(x)1.</math>
To define the renormalized Feynman rules, note that the antipode ''S'' satisfies
:<math> m\circ (S\otimes {\rm id}) \Delta (x) =\varepsilon(x)1</math>
so that
:<math>S = - m\circ (S\otimes P)\Delta,</math>
The ''renormalized Feynman rules'' are given by a homomorphism <math>\Phi_S^R</math> of '''H''' into ''V'' obtained by twisting the homomorphism Φ • S. The homomorphism <math>\Phi_S^R</math> is uniquely specified by
:<math>\Phi_S^R = -m(S\otimes \Phi_S^R\circ P)\Delta.</math>
Because of the precise form of Δ, this gives a recursive formula for <math>\Phi_S^R</math>.
For the minimal subtraction scheme, this process can be interpreted in terms of Birkhoff factorization in the complex Butcher group. Φ can be regarded as a map γ of the unit circle into the complexification '''G'''<sub>'''C'''</sub> of '''G''' (maps into '''C''' instead of '''R'''). As such it has a Birkhoff factorization
:<math> \displaystyle \gamma(z)=\gamma_-(z)^{-1} \gamma_+(z),</math>
where  γ<sub>+</sub> is [[holomorphic]] on the interior of the closed unit disk and γ<sub>–</sub> is holomorphic on its complement in the [[Riemann sphere]] '''C''' <math>\cup\{\infty\}</math> with γ<sub>–</sub>(∞) = 1. The loop γ<sub>+</sub> corresponds to the renormalized homomorphism. The evaluation at ''z'' =  0 of γ<sub>+</sub> or the renormalized homomorphism gives the ''dimensionally regularized'' values for each rooted tree.
In example, the Feynman rules depend on additional parameter μ, a "unit of mass". {{harvtxt|Connes|Kreimer|2001}} showed that
:<math>\partial_\mu \gamma_{\mu-} =0,</math>
so that γ<sub>μ–</sub> is independent of μ.
The complex Butcher group comes with a natural one-parameter group λ<sub>''w''</sub> of automorphisms, dual to that on '''H'''
:<math>\lambda_{w}(t)= w^{|t|}t</math>
for ''w'' ≠ 0 in '''C'''.
The loops γ<sub>μ</sub> and λ<sub>''w''</sub> · γ<sub>μ</sub> have the same negative part and, for ''t'' real,
:<math>\displaystyle F_t=\lim_{z=0} \gamma_-(z) \lambda_{tz}(\gamma_-(z)^{-1})</math>
defines a one-parameter subgroup of the complex Butcher group '''G'''<sub>'''C'''</sub> called the [[renormalization group| renormalization group flow]] (RG).
Its infinitesimal generator β is an element of the Lie algebra of '''G'''<sub>'''C'''</sub> and is defined by
:<math>\beta=\partial_t F_t|_{t=0}.</math>
It is called the [[beta-function]] of the model.
In any given model, there is usually a finite-dimensional space of complex coupling constants. The complex Butcher group acts by diffeomorphims on this space. 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 diagram]]s 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.
==Example==
{{harvtxt|Kreimer|2007}} has given a "toy model" involving [[dimensional regularization]] for '''H''' and the algebra ''V''. If ''c'' is a positive integer and ''q''<sub>μ</sub> = ''q'' / μ is a dimensionless constant, Feynman rules can be defined recursively by
:<math>\displaystyle \Phi([t_1,\dots, t_n])=\int {\Phi(t_1)\cdots \Phi(t_n) \over |y|^2 + q_\mu^2} (|y|^2)^{-z({c\over 2} -1)} \, d^D y,</math>
where ''z'' = 1 – ''D''/2 is the regularization parameter. These integrals can be computed explicitly in terms of the [[Gamma function]] using the formula
:<math>\displaystyle \int  {(|y|^2)^{-u}\over |y|^2 +q_\mu^2} \, d^Dy =  \pi^{D/2} (q_\mu^2)^{-z-u} {\Gamma(-u +D/2)\Gamma(1+u-D/2)\over \Gamma(D/2)}.</math>


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:<math>\displaystyle \Phi(\bullet)=\pi^{D/2}(q_\mu^2)^{-zc/2}{\Gamma(1+cz)\over cz}.</math>
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Taking the renormalization scheme ''R'' of minimal subtraction, the renormalized quantities <math>\Phi_S^R(t)</math> are [[polynomial]]s in <math>\log q_\mu^2</math> when evaluated at ''z'' = 0.
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==References==
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*{{citation|title=On the Butcher group and general multi-value methods|journal=Computing|volume= 13|year= 1974|pages=1&ndash;15|first=E. |last=Hairer|first2=G.|last2= Wanner|url=http://www.springerlink.com/content/e6r7327737lq3516/|doi=10.1007/BF02268387}}
*{{citation|last=Kreimer|first= Dirk|authorlink=Dirk Kreimer|title=On the Hopf algebra structure of perturbative quantum field theories|journal=Adv. Theor. Math. Phys.|volume=2|year=1998|pages= 303–334|arxiv=q-alg/9707029}}
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*{{citation|last=Kreimer|first= Dirk|authorlink=Dirk Kreimer|title= Factorization in Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities|
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*{{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | last2=Moore | first2=John C. | title=On the structure of Hopf algebras | jstor=1970615 | mr=0174052 | year=1965 | journal=[[Annals of Mathematics]] | series = Second Series | volume=81 | issue=2 | pages=211–264 | doi=10.2307/1970615}}


[[Category:Combinatorics]]
for the earn, a suction mouth is on the shore facet wins nevertheless not touch the floor, A lot of have commented positively on the simple fact that  [http://tinyurl.com/mdm2hs2 ghd straightener] it is a unique edition and that it life up to it with the situation getting likened to a clutch bag, so can be used for other issues other than just carrying your straighteners in.
[[Category:Numerical analysis]]
[[Category:Quantum field theory]]
[[Category:Renormalization group]]
[[Category:Hopf algebras]]

Latest revision as of 18:54, 19 July 2014

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