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Polynomial Coefficients of Dyson SchwingerEquations

Bachelor thesisBachelor of science, Physics

Niklas [email protected] - TU Berlin

Supervisors:Prof. Dr. Dirk Kreimer - HU BerlinProf. Dr. Andreas Knorr - TU Berlin

HU/TU Berlin, 24.10.2012

AbstractCoeffients of the solution of the Dyson-Schwinger equations are translated from the Hopf

algebra of rooted trees to the Hopf algebra of words. Therein they are expressed as shuffles ofLie brackets of Lyndon words. This is done for up to four nested primitives and up to threedifferent ones. Furthermore the necessary algebraic structures are presented in an accessible

way.

Poly. Coef. of DSE N. Affolter 24.10.2012, HU & TU Berlin

Contents

1 Introduction 1

2 Hopf algebras 22.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Graduation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Primitive elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Lie algebras 53.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Graduation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Lie algebras from Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Universal enveloping algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Lie algebras from pre-Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 Free Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 The Hopf algebra of decorated rooted trees HR 84.1 Motivation for trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Decorated rooted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 The commutative product m(·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Grafting operator Bd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 Coproduct ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.6 Antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.7 Lie bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Hopf algebras of words 135.1 Concatenation algebra HC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Shuffle algebra HS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Mapping φ from VW to HR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Higher order primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Dyson-Schwinger equations 17

7 Polynomial Coefficients 217.1 Counting trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 Trees written in words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Words in shuffles of Lie brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4 Coefficients in shuffles of Lie brackets . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Conclusion 27

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1 Introduction

One common way to characterize a quantum field theory is to present its Lagrangian. Another,more visual way is to write down its Dyson-Schwinger equations in terms of graphs. And whilequantum field theories are very popular due to them being the framework of the StandardModel of particle physics, there are still many and somewhat serious problems when trying toformulate them in a mathematical sensible way. The main issue is the process of renormal-ization, to which the introduction of Hopf algebras by Kreimer brought new understanding(see [1]). Another good review of the ongoing research in that matter was done by Kuruschand Kreimer [5]. This is also strongly linked to the trouble of getting to a non-perturbativequantum field theory. This is also where the study of the behaviour of the Dyson-Schwingerequations becomes interesting.

The goal of this thesis is to calculate low order coefficients (up to nesting-order four) of thesolution of single Dyson-Schwinger equations with up to three different primitives of the sametype. And to reexpress these coefficients as shuffle of Lie brackets in the Lyndon basis. Inthis form it is hoped to pave the way for further research in order to gain a better under-standing of non-linear Dyson-Schwinger equations, especially regarding quantum field theories.With respect to the calculations and the problem in general it is intended that all prerequisiteare explained and summarized in an easily accessible way. Therefore section two and threeintroduce the necessary aspects of Hopf respectively Lie algebras which form the frameworkfor all to follow. Sections four focuses on the Hopf algebra of rooted trees HR in which theDyson-Schwinger equations are first expressed. This will be followed by a description of theshuffle Hopf algebra into which the solutions of the Dyson-Schwinger equations will be renderedby calculation. Having explained the necessary structures the Dyson-Schwinger equations areintroduced themselves in section six. Section seven will present some of the important in-termediate results (including some which may be of value for calculations of higher orders)as well as combine these to present the already mentioned coefficients of the solutions of theDyson-Schwinger equations.

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2 Hopf algebras

2.1 Definition

Hopf algebras will be the foundation of all calculations done in this thesis, yet they will notbe the center of studies themselves. Therefore it is important to get a solid understanding ofthem, while not all mathematical aspects are elaborated here. Several Hopf algebras will bepresented in the following sections, where the maps will be presented in more detail.Formally defined a Hopf algebra is a co-/associative bialgebra H over a field K with a K-linearmap (antipode) S such that diagram 1 commutes. The notion of bialgebra and the necessarymathematical background can be read in Erik Panzer’s master thesis [11].Let us look at the mappings occuring in a Hopf algebra and some important axioms:

H

H⊗H

H⊗H

K

H⊗H

H⊗H

H

∆

∆

ε η

S ⊗ id

id⊗ S

m

m

Figure 1: Commuting diagram of the antipode.

m : H⊗H → H (inner product) (2.1.1)∆ : H → H⊗H (coproduct) (2.1.2)

η : K→ H (unit map) (2.1.3)ε : H → K (counit map) (2.1.4)S : H → H (antipode) (2.1.5)

m ◦ (m⊗ id) = m ◦ (id⊗m) (associativity) (2.1.6)m ◦ (η ⊗ id) = id = m ◦ (id⊗ η) (neutral map) (2.1.7)

m ◦ (id⊗ S) ◦∆ = η ◦ ε = m ◦ (S ⊗ id) ◦∆ (inverse map) (2.1.8)

This is very familiar to everyone who knows the structure of a group. Immediately we canidentify m with the product of the group and S with the inverse. We have to keep in mind though

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that Hopf algebras live on vector spaces, while groups live on sets. Therefore it is important tonote, that all Hopf algebra maps satisfy K-linearity. The unit and counit maps arise in groupaxioms in a similar matter when one defines a group solely by morphisms (η : ∗ → G, ε : G→ ∗).However the really interesting additional structure lies in the coproduct ∆, it replaces thediagonal map (g 7→ g ⊗ g,∀g ∈ G) in the axioms of a group. In graduated Hopf algebrasit describes a way to assign to every element a sum of its components. Also note that thecoproduct is an algebra homomorphism, while the antipode is an antihomomorphisms:

S(h1h2) = S(h2)S(h1),∀h1, h2 ∈ H ←→ (g1g2)−1 = g−12 g−1

1 ,∀g1, g2 ∈ G (2.1.9)

As the antipode is an important aspect of a Hopf algebra, it is interesting to note that for everyconnected filtered bialgebras it is possible to construct an antipode and therefore to achievea hopf algebra structure. This and many other properties of Hopf algebras are derived inDominique Manchons extended lecture notes [8].

2.2 Graduation

All the Hopf algebras in this thesis will be N0-graded. This means the underlying vector spaceV can be written as the direct sum of disjoint subvectorspaces V k. Every element of one of thissubspaces V k is said to be of grade k.

V =⊕k≥0

V k (2.2.1)

The gradings may be the number of loops for graphs, the number of vertices or the sum ofvertice weights for trees, or the number of letters for words. All the mappings will respect thisgrading in the following sense:

m(V k ⊗ V l) ⊂ V k+l (2.2.2)

∆V n ⊂⊕k+l=n

V k ⊗ V l (2.2.3)

S(V n) ⊂ V n (2.2.4)

2.3 Primitive elements

An element h ∈ H is considered to be primitive if its coproduct has this form:

h ∈ Prim(H)⇔ ∆ (h) = 1⊗ h+ h⊗ 1 (2.3.1)

All elements of grade 1 are primitive, because the coproduct respects grading and alwaysinvolves the term 1⊗ h+ h⊗ 1 [8]. There may be primitives of higher grading, some of whichwe will construct using the convolution product.

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2.4 Convolution

The space of endomorphisms of H is naturally equipped with the concatenation operation,which renders it an associative algebra. There is another product which will prove useful: Forf, g ∈ End(H) define the convolution ∗:

∗ : End(H)× End(H)→ End(H), f ∗ g 7→ m(f ⊗ g)∆ (2.4.1)

The convolution equips the endomorphisms of a Hopf algebra with a group structure, the unitmap is given by e = η ◦ ε and the inverse is the antipode. Associativity results from theassociativity of the product and coproduct. We will use the convolution to construct primitiveelements in the Hopf algebra of rooted trees.

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3 Lie algebras

3.1 Definition

Lie algebras are well known to all mathematicians and physicists. Still a good textbook resourcewas written by Fuchs [7]. Here only in short their definition: A Lie algebra L is a K-vectorspacetogether with a bilinear mapping [·, ·] : L×L → L called Lie bracket, which fulfills [l, l] = 0 ∀l ∈L (is alternating) as well as the Jacobi identity:

[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 ∀a, b, c ∈ L (3.1.1)

Antisymmetry of the Lie bracket follows directly by:

0 = [a+ b, a+ b] = 0 + [a, b] + [b, a] + 0 (3.1.2)

3.2 Graduation

Analogous to Hopf algebras, Lie algebras can be graduated. This will also be the case for allLie algebras occuring in the following. The Lie bracket will also respect the grading:

[V k, V l] ⊂ V k+l (3.2.1)

3.3 Lie algebras from Hopf algebras

The most common Lie algebras known in physics are defined in some algebra A via the com-mutator [·, ·]−, which takes some product . and antisymmetrizes it:

[a, b]− : [·, ·]− : A× A→ A, (a, b) 7→ a.b− b.a (3.3.1)

With this commutator the alternating condition is obviously fulfilled. If in addition this productis associative, that is A is an associative algebra, then the Jacobi identity holds as well:

[a, [b, c]−]− + [b, [c, a]−]− + [c, [a, b]−]−

= a.b.c− a.c.b− b.c.a+ c.b.a

+ b.c.a− b.a.c− c.a.b+ a.c.b

+ c.a.b− c.b.a− a.b.c+ b.a.c = 0 (3.3.2)

The underlined summands add to zero, but note that if A were not associative a.(b.c)− (a.b).cmight not be zero. The commutator therefore equips every associative algebra - including Hopf

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algebras - with the structure of a Lie algebra.It is interesting to see, that with this commutator the prime elements of every Hopf algebraform a Lie algebra as well. To show this we have to demonstrate that the coproduct will mapcommutated elements into the prime elements again:

∆ ([p, q]−) = ∆ (p.q − q.p)= (p⊗ 1 + 1⊗ p).(q ⊗ 1 + 1⊗ q)− (q ⊗ 1 + 1⊗ q).(p⊗ 1 + 1⊗ p)= [p, q]− ⊗ 1 + 1⊗ [p, q]− (3.3.3)

3.4 Universal enveloping algebras

Universal enveloping algebras are a way of finding the smallest associative algebra U(L) whosecommutator corresponds to the Lie bracket in L. U(L) is constructed by taking the tensor spaceT (L) of the vectorspace L and then factoring out the difference between the commutator andrespective Lie bracket of all elements:

U(L) = T (L)/(I), I = 〈{l1 ⊗ l2 − l2 ⊗ l1 − [l1, l2], l1, l2 ∈ L}〉 (3.4.1)

The ensuing enveloping algebra carries a natural Hopf algebra structure in the manner describedlater on the concatenating algebra 5.1. Furthermore the Milnor-Moore theorem [9] states thatif the Hopf algebra H is connected, graded and of finite type then it is isomorphic to theenveloping of its prime Lie algebra U(Prim(H)).

3.5 Lie algebras from pre-Lie algebras

There is also a structure named pre-Lie algebra, which is non associative but where the com-mutator still gives rise to a Lie algebra. The condition for the mapping / to be pre-Lie is:

(a / b) / c− a / (b / c)− (a / c) / b+ a / (c / b) = 0 (3.5.1)

If we now calculate the Jacobi identity again:

[a, [b, c]−]− + [b, [c, a]−]− + [c, [a, b]−]−

= a / (b / c)− a / (c / b)− (b / c) / a+ (c / b) / a

+ b / (c / a)− b / (a / c)− (c / a) / b+ (a / c) / b

+ c / (a / b)− c / (b / a)− (a / b) / c+ (b / a) / c = 0 (3.5.2)

Underlined is one occurence of the pre-Lie condition. In subsection 4.7 the appending operationon rooted trees will be introduced, which will give an example of a pre-Lie algebra.

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3.6 Free Lie algebras

A good resource to read extensively about free Lie algebras is Reutenauers textbook “Free LieAlgebras” [14]. In a free Lie algebra there is defined a generating set. No relations beyond thedefinitions of a Lie algebra are assumed. This leads to the search for a basis of the generatedvectorspace and the so called Hall bases [14]. The question is to find some rules as to whichnested commutators should be considered as in the basis. For example in the case of a gener-ating set with three elements we can express all nested commutators up to depth 2 by linearcombinations of:

B = {a, b, c, [a, b], [b, c], [c, a], [a, [b, c]], [b, [a, c]]} (3.6.1)

Because of the Jacobi identity, we may omit one of the double commutators. While this iseasy to see, it becomes much more complex when looking at higher depths. As the aim of thisthesis is to express some tree polynomials in terms of shuffles of Lie brackets, it is important tohave defined a specific basis to the Lie algebra. The basis will be formed by the Lyndon wordswhich will therefore be introduced here. Let all the elements of the generating set be labelledby letters with an ordering (e.g. a < b < c, ...). Call this set G. A word can be split intotwo parts (e.g. abaa into a and baa). Now a word is a Lyndon word if and only if for everysplitting, the left word is lexicographical smaller then the right one. Some examples:

abac : a < bac, ab < ac, aba < c⇒ Lyndon (3.6.2)bababa : b < ababa, ba < baba, bab = bab⇒ not Lyndon (3.6.3)aaa : a < aa, aa > a⇒ not Lyndon (3.6.4)

To a given generating set G (an alphabet), call the set of all Lyndon words W . Then thecorresponding Lie brackets L(w) to a Lyndon word w are calculated recursivly as follows: Ifw ∈ G then L(w) = w. If not find the factorization w = uv so that u and v are Lyndon wordsand v has maximal length. Then L(w) = [L(u), L(v)]. Some examples:

L(aaba) = [L(aab), a] = [[a, [a, b]], a] (3.6.5)L(abac) = [L(ab), L(ac)] = [[a, b], [a, c]] (3.6.6)L(abc) = [a, [b, c]] (3.6.7)

The Lyndon basis to a given set of letters G is now simply the Lie bracketing of all Lyndonwords formed by these letters. For example the Lyndon basis for the alphabet G = {a, b}involving up to four letters is:

BL = {a, b, [a, b], [a, [a, b]], [[a, b], b], [a, [a, [a, b]]], [[a, [a, b]], b], [[[a, b], b], b]} (3.6.8)

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4 The Hopf algebra of decorated rooted trees HR

4.1 Motivation for trees

Many of the interesting structures (and problems) in quantum field theories arise in the processof renormalization. This comes from the fact that the loop integrals associated with Feynmandiagrams do not converge in general (see any textbook on QFT [12]). A big step to tacklethese divergencies was the introduction of the forest formula. Its very name is inspired by thetree-like structure when tracing subdivergencies of Feynman diagrams:

↔�↔� and ↔� (4.1.1)

The tree associated to a graph catches its nesting-structure and factors out its - for manycalculations - superfluous differentiations on where subdivergencies are inserted. For the latterfact there will be need for some compensation in counting, as we will see later in section 6 onDyson-Schwinger equations.

4.2 Decorated rooted trees

There are two ways to construct the object tree: One is to consider an undirected graph (N,E)defined by the sets of nodes N and edges E. If we choose a node and call it root and demandthat the graph has no loops (i.e. simply connected) we have defined a rooted tree. The set ofall rooted trees will be called T from now on. If we furthermore define the set of decorationsD, and assign to every node an element of D, the rooted trees become decorated rooted trees.We will also use the weight of a decoration which we identify with the weight of a node, and

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the weight of a tree which is simply the sum of the weights of its nodes. The number of nodesof a tree will be denoted by |t|:

wD : D → N (4.2.1)

wT : T → N, t 7→∑n∈N(t)

wD(n) (4.2.2)

| · | : T → N, t 7→ |t| = |N(t)| (4.2.3)

Some examples:

D = { , } , wD( ) = 1, wD( ) = 2 (4.2.4)

{t ∈ T , |t| ≤ 2} ={, , , , ,

}(4.2.5)

{t ∈ T , wT (t) = 3} =

{, , ,

}(4.2.6)

We call the nodes adjacent to the root its children, and the root the parent of its children anddraw the root above its children. We recursively repeat this for the drawing process therebyobtaining for every node except the root a node which we call its parent. Also one has to beaware of the fact that the nature of our two-dimensional notation forces us to always denoteplanar representatives of rooted trees. One could be mislead into assuming an ordering of thechildren of a node. But we have defined the children of a node solely by the set of edges andtherefore all orderings are equal. For example:

= (4.2.7)

The K-vector space VR is the vector space on which the Hopf algebra of decorated rooted treesHR lives. K is considered to be any field of characteristic 0, but we will do all calculations in Q.The basis of VR is generated by T ∪ {I} via the free commutative product, which will also bethe algebra product m in HR. As a consequence VR is infinite dimensional (as is its generatingset T ).

4.3 The commutative product m(·, ·)

In the Hopf algebra of decorated rooted trees HR the product m is the free commutativeproduct. Such products of trees are called forests (which then are just one or more disjointtrees). For example:

m(

,)

= = m(

,)

(4.3.1)

In essence this multiplication is the same as the union operation in multiset theory. Multisetsare sets in which elements may appear multiple times [3]. The union is commutative and

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associative, and so is m. Also define the empty tree I without nodes to be the neutral elementof the multiplication:

m (t, I) = t = m (I, t) ,∀t ∈ HR (4.3.2)

4.4 Grafting operator Bd

The action of a grafting operator Bd on a forest t ∈ HR is defined by adding a new root withdecoration d ∈ D, to which all the roots of t are connected. For example:

B ( ) = (4.4.1)

This definition is extended by linearity:

Bd(k1t1 + k2t2) = k1Bd(t1) + k2B

d(t2),∀t1, t2 ∈ HR, k1, k2 ∈ K (4.4.2)

4.5 Coproduct ∆

The coproduct of a tree is a linear combination of tensor products of possible stems and crownsof the tree. To get the coefficients right we introduce admissable cuts and the complete cut.An admissable cut is a set of edges to be severed, these edges will neither appear in the stemnor in the crown. In order to be admissable, every severed edge has to be connected to the rootby unsevered edges. The stem is the sets of those nodes and edges which are still connectedto the root. The nodes which got an edge severed and are not in the stem are the roots of thetrees which form a forest called the crown. The complete cut is no cut in the just defined sense,but delivers an empty stem and the crown is the tree (complete cut: t 7→ I ⊗ t). The emptycut is just the cut where the set of severed edges is empty. Call Ac(t) the set of all possibleadmissable cuts and the complete cut, SC and CC the stem respectively the crown of a cut:

∆ (t) =

{I⊗ I if t = I∑

C∈A(t) CC ⊗ SC else(4.5.1)

Some examples:

∆

( )= ⊗ I + 2 ⊗ + ⊗ + ⊗ + I⊗ (4.5.2)

∆( )

= ⊗ I + ⊗ + ⊗ + ⊗ + I⊗ (4.5.3)

∆ ( ) = ∆ ( ) ∆ ( ) = ⊗ I + ⊗ + ⊗ + I⊗ (4.5.4)

The coproduct can also be defined recursively by demanding (see [11] section 2.3.1):

∆ ◦Bd = Bd ⊗ I +(id⊗Bd

)◦∆ (4.5.5)

The coproduct definition is also extended linearly.

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4.6 Antipode

Call A(t) the admissable cuts of a tree t (without the complete cut), then the antipode isdefined recursively by:

S(t) =

{I if t = I−∑

C∈A(t) m(SC , S(CC)) else(4.6.1)

Note that the empty cut is included, contributing a term -t. The Antipode too is extended bylinearity and the algebra homomorphism property. Some examples:

S ( ) = − (4.6.2)

S( )

= − − S( ) = − + (4.6.3)

S

( )= − − 2 S( )− S( )− S

( )= − + 2 − −

(− − 2 S( )− S( )

)= − + 2 + + + (4.6.4)

4.7 Lie bracket

As HR is commutative, the Lie algebra structure induced by the commutator 3.3 is abelianand therefore not very interesting. However there is a second Lie algebra structure: Define the“appending” operation VR × VR → VR, (f1, f2) → f1 / f2, as all possible forests arising whenappending every root of f2 to a node of f1. Some examples:

/ = (4.7.1)

/ = + + (4.7.2)

/ = + 2 + (4.7.3)

/ = 2 (4.7.4)

This operation is not associative, the associator is non vanishing:

( / ) / − / ( / ) = / − / = + − = 6= 0 (4.7.5)

But / fulfills the pre-Lie condition 3.5.1:

(f1 / f2) / f3 − f1 / (f2 / f3) = (f1 / f3) / f2 − f1 / (f3 / f2) (4.7.6)

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This comes quite easily, when trying to reexpress (f1 / f2) / f3:

(f1 / f2) / f3 = f1 / (f2 / f3) + (f1 / f3) / f2 − f1 / (f3 / f2) (4.7.7)

Appending f3 to f1 / f2 is the same as appending f3 directly to f2 and then appending this tof1 (term 1), plus appending f3 first to f1 followed by appending f2 to this (term 2). But nowwe have to subtract where we actually appended f2 to f3 (term 3). An example:

( / ) / = + (4.7.8)

/ ( / ) + ( / ) / − / ( / ) =(/)

+(/)−(/)

(4.7.9)

= + + − = + (4.7.10)

Note that if the left tree t1 has one node the appending operation coincides with the graftingoperator Bt1 :

t1 / f2 = Bt1(f2) ∀t1 ∈ T , f2 ∈ VR (4.7.11)

Now define the Lie bracket as the commutator:

[·, ·] : HR ×HR → HR, (f1, f2) 7→ f2 / f1 − f1 / f2 (4.7.12)

The Lie bracket is by definition antisymmetric. As the appending operation is not associative,the Jacobi identity is non-trivial. But a simple evaluation using the pre-Lie property will indeedyield the Jacobi identity (see section 3.5). The Lie bracket is defined to be K-linear as usual.Examples: [

,]

= + − (4.7.13)

[ + , ] = 2 + 2 − − (4.7.14)

With the so defined bracket HR is equipped with a second Lie algebra structure.

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5 Hopf algebras of words

On both Hopf algebras presented in this section can be read in great detail in Reutenauer [14].The basis for theK-vectorspace VW which underlies both following Hopf algebras is generated bya set of letters G and the free associative noncommutative product m, also called concatenationproduct.

5.1 Concatenation algebra HC

The inner product in the Hopf algebra HC is the same product as the one generating thevectorspace VW , for example:

m(l1l2, l3) = l1l2l3 (5.1.1)

The neutral element is the empty word e. The associated coproduct δ is defined for letters as

δ(l) = l ⊗ I + I⊗ l (5.1.2)

which generalizes to words by the homomorphism property. For example:

δ(aab) = δ(a)δ(a)δ(b) = (aa⊗ I + 2a⊗ a+ I⊗ aa)(b⊗ I + I⊗ b)= aab⊗ I + 2ab⊗ a+ b⊗ aa+ aa⊗ b+ 2a⊗ ab+ I⊗ aab (5.1.3)

The antipode as well has a very simple structure:

S(l1l2...ln) = (−1)nlnln−1...l1,∀l ∈ G (5.1.4)

It is also possible to define a Lie bracket via the usual commutator:

[·, ·] : HC ×HC → HC , (w1, w2) 7→ w2w1 − w1w2 (5.1.5)

The concatenation is associative and consequently a Lie algebra structure emerges, as outlinedin section 3.3.

5.2 Shuffle algebra HS

It is also possible to define the shuffle product �, which shuffles the letters of two words whilepreserving the ordering of letters of each word. It can be defined recursively:

l1w1� l2w2 = l1(w1� l2w2) + l2(l1w1� w2) · (5.2.1)e� w = w = w� e (5.2.2)

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Two examples:

a� b = a(e� b) + b(a� e) = ab+ ba

ab� ac = a(be� ac) + a(ab� ce) = a(bac+ abc+ abc+ cab)

= abac+ 2aabc+ acab (5.2.3)

The shuffle algebra on words can also be extended to a Hopf algebra HS, by introducing thedecomposition coproduct δ�:

δ�(w) =∑uv=w

u⊗ v (5.2.4)

The antipode of this Hopf algebra is the same as in the concatenation algebra 5.1.4. Moredetails can be read in Reutenauer [14] chapter 1.5.As with all algebras the question of finding a basis is of interest. Reutenauer [14] cites a paperof Perrin and Viennot (1981) stating that over Q the Lie brackets of the Lyndon words forma basis for the shuffle algebra. As the paper is unpublished a short argument is provided hereas well. Radford [13] showed that the Lyndon words (not their bracketing) generate by theshuffle a basis for the shuffle algebra. Obviously there are as many bracketings of Lyndon wordsas there are Lyndon words. We first show that the bracketings of Lyndon words of specifiedlength are linearly independent. Then we will show that a bracketed Lyndon word can not beexpressed as some sum of shuffles of bracketings of less length.First observe the following triangular property of the Lyndon bracketings:

∀l ∈ BL : l ≤ L(l) (5.2.5)

Where the ordering is the lexicographical one and applies to every summand on each side. Theproof by induction on the length of the words (assuming the properties explained in 3.6):

∀l ∈ G : l ≤ L(l) = l (5.2.6)∀l ∈ BL : L(l) = [L(u), L(v)] = L(u)L(v)︸︷︷︸

≥uv

−L(v)L(u)︸︷︷︸≥vu

≥ uv = l (5.2.7)

Where we used the induction hypothesis and the Lyndon property. For every multiset wechoose out of G, we can then take the greatest Lyndon bracketing and observe that the nextlesser one will introduce a new word. Therefore they are linearly independent.To show that bracketings of Lyndon words cannot be expressed as sum of shuffles of Lyndonbracketings of less length, we borrow the binary operation called right residual B from [10],where a detailed explanation can be found. It can be described as removing the right wordfrom the beginning of the left word. If this is not possible it gives zero. Some examples:

abcB ab = c, acbB ab = 0 (5.2.8)

14


The right residual of bracketings is a differentiation for the shuffle product, see also [10]. Nowassume a bracketing of a Lyndon word could be written as a sum of shuffles of bracketings ofless length:

L(l) =∑|ki|<|l|

L(ki)� ...� L(kn) (5.2.9)

Take the right residual of L(l) on both sides:

2|l| = 0 (5.2.10)

The right side equals zero because of the differentiation property, letting the right residual ofL(l) act on words of lesser length, which will always give zero. We arrive at a contradictionand therefore have proven it is not possible to write a bracketing of a Lyndon word by shufflesof bracketings of less length.We can repeat these two steps for Lyndon words of all lengths and have therefore deductedfrom the Lyndon words being a basis that also the bracketings of Lyndon words form a basisof the shuffle algebra of words. In shuffles of this basis the results of the calculations of thisthesis will be presented in section 7.4.

5.3 Mapping φ from VW to HR

If the letters G generating VW consist of the decorations D of HR there is a mapping φ fromVW to HR. Define it on words by:

φ : W → HR, w = l1...ln 7→ (l1 / (...(ln−1 / ln))), li ∈ G (5.3.1)

Extend it to a definition on all of VW by K-linearity. Φ maps solely onto the trees inHR withoutsidebranches and without forests. It is straightforward to see that this mapping is injective andwhen limited to the trees without sidebranches it is a bijection.

5.4 Higher order primitives

With the knowledge of Φ and the shuffle product, we introduce a possibility to generate primi-tives in HR beyond trees with one node. Start with a tuple of letters u = (l1, ..., ln) ∈ HW andcalculate their full shuffle l1� ...� ln, set its coefficient to one and map u to HW . So far:

(l1, ..., ln) 7→ t′ = Nφ(l1� ...� ln) (5.4.1)

15


Where N is a factor setting the summands coefficients to one. Now define the derivation D,which maps a tree t to |t|t and use the convolution product to define:

P ′(t′) :=1

|t′|(S ∗D)(t′) (5.4.2)

P (u) :=N

n(S ∗D)φ(l1� ...� ln) (5.4.3)

Due to the shuffle the ordering of the li in the tupel u does not matter. The generated sums offorests are primitives in HR and the process is called polarization [2].

∆ (P (u)) = P (u)⊗ I + I⊗ P (u) (5.4.4)

Examples:

P (a, b) =1

2(S ∗D)φ(a� b) =

1

2m(S ⊗D)∆

(+)

=1

2m(S ⊗D)

((+)⊗ I + ⊗ + ⊗ + I⊗

(+))

=1

2

(− − + 2 + 2

)= + − (5.4.5)

P (P (a, b), a) = a / P (a, b) + P (a, b) / a− P (a, b)

= + − + + + + − − − − +

= + 2 + + − − 2 − + (5.4.6)

The Lie brackets (4.7) in the Hopf algebra of rooted trees also provide new primitives, due tothe fact that the evaluation of the term ( ⊗ − ⊗ ) vanishes for primitives and , see [4].We will not need those in the calculations, but it is this relation which allows us to apply theMilnor-Moore theorem to HR (section 3.4).

16


6 Dyson-Schwinger equations

If we consider quantum electrodynamics and look at the vertex function while neglecting loopcorrections to the propagators, we get an equation expressed in Feynman graphs (6.0.7). Itdescribes the fact that if we want to do calculations to higher orders of perturbations theory,we have to consider nested vertex functions as well. In addition at higher orders new vertexdiagrams appear which cannot be expressed by nesting one loop vertex functions into each other.Lets take a look at the first orders of only the vertex diagrams in quantum electrodynamics:

� =� + α� + α2

� +O(α3)

(6.0.7)Where the blobs represent the nesting of the vertex graph in itself and α counts the loop order.We can rewrite the nesting structure in terms of trees of HR as:

T = 1 + αB (T 3) + α2B (T 5) +O(α3)

(6.0.8)

where represents the primitive one-loop vertex and the primitive two-loop vertex graph. Thesolution to this self-consistency equation can be written as a power series in α with coefficientsin HF : Γ =

∑∞i=0 α

iγi. Because it will prove useful to calculate the coefficients, we also do asubstitution T = 1 + αX which in this example gives:

X = B ((1 + αX)3) + αB ((1 + αX)5) +O(α2)

(6.0.9)

One can proceed similarly with only propagators (2-point functions), but has to keep in mindthat the propagators can repeat themselves on internal lines. This could be expressed bysumming up all positive integer powers of T . In that case we would add up an infinite amountof appending the 1 though, which is why we choose to sum up (T − 1)k including zero. This isjust the geometric series which is the power series of 1

1−(T−1)= T−1. In Φ3-theory this would

lead to the following equation:T = 1− αB (T−2) (6.0.10)

Which by the substitution T = 1− αX would produce

X = B (1− αX)−2 (6.0.11)

17


Now Foissy [6] has discovered in his work, that there exist classes of Dyson-Schwinger equationsin trees, so that their coefficients generate a Hopf subalgebra. We will choose those whichcorrespond to the propagators of quantum field theories and calculate the first orders of theirsolutions.The α coefficient keeps track of the loop order of the corresponding graphs:

X =∞∑i=0

ci+1αi (6.0.12)

While the trees themselves have coefficients in N. These can be calculated as a product of somevalues of ther vertices:

ci =∑t∈Ti

f(t)t (6.0.13)

f : T → K, t 7→∏v∈t

v (6.0.14)

From Foissy we know that the equations generating Hopf subalgebras in general look like(J ⊂ N):

X =∑j∈J

Bj

((1− µX)−

λjµ

+1), µ 6= 0 (6.0.15)

or:X =

∑j∈J

Bj

(eλjX

)(6.0.16)

We set λ = ±1, µ ∈ {0, 1} and replace X → αX, Bj → αjBvj :

X =∑j∈J

αj−1Bvj

((1± αX)±j+1

)(6.0.17)

X =∑j∈J

αj−1Bvj

(e±jαX

)(6.0.18)

Where both signs in the exponential generate the same Hopf subalgebra, as can be seen by:

α→ −α, e+jαX → e−jαX , X =∞∑i=0

ciαi → X =

∞∑i=0

(−1)iciαi (6.0.19)

For |J | = 1 This leads in all cases to

X = αj−1∑k=0

dkαkB(Xk) (6.0.20)

The dk ∈ K determine the factor a vertex with k children contributes to the coefficient of atree:

v = dkA(v) (6.0.21)

18


Where A(v) is a multinomial coefficient, counting the number of different permutations ofv’s subtrees when embedded into the plane. This multinomial coefficient comes directly fromXk. In the exponential case the dks follow directly from the formal series/definition of theexponential:

Bj(ejαX) =

∞∑k=0

αkjk

k!Bj(X

k)⇒ dk =jk

k!(6.0.22)

In the case of positive powers we get binomial coefficients

Bj−1

((1 + αX)j

)=

j∑k=0

αk(j

k

)Bj−1(Xk)⇒ dk =

(j

k

)=

j!

(j − k)!k!(6.0.23)

In the case of negative powers, consider j = 2 first:

B2

((1− αX)−1

)=∞∑k=0

αkB2(Xk) (6.0.24)

(j − 1)!(1− αX)−j = α1−j dj−1

dXj−1(1− αX)−1 =

∞∑k=j−1

αk−j+1 k!

(k − j + 1)!Xk−j+1

=∞∑k=0

αk(k + j − 1)!

k!Xk (6.0.25)

Bj+2 ((1− αX)n) =n∑k=0

αk(k + j − 1)!

(j − 1)!k!Bj+2(Xk)⇒ dk =

(k + j − 1)!

(j − 1)!k!(6.0.26)

Also the possibility of rewriting X:

X̃ = 1± αX ⇒ X̃ = 1±∑j∈J

αjBvj

(X̃±j+1

)(6.0.27)

With the number of children n (k above) and the fertility f (j above) this leads to the vertexcoefficients:

v+ = Af !

(f − n)!n!=

1

S

f !

(f − n)!(6.0.28)

v− = A(n+ f − 1)!

n!(f − 1)!=

1

S

(n+ f − 1)!

(f − 1)!(6.0.29)

ve = Afn

n!=

1

Sfn (6.0.30)

These coefficients count how often it is possible to nest n vertex (v+) or propagator (v−) graphsinto a graph with f vertices respectively propagators. While the exponential case can be treated

19


in the same way as vertex or propagators, its physical meaning is still under research and willnot be considered hereafter.For vertices every nesting spot can only be taken once, it follows immediately that there aref(f − 1)...(f − n + 1) possibilities for that. Propagators on the other hand can be nestedmore then once on the same propagator. For every propagator nested, there is an additionaloption present for where to nest the next propagator, therefore there are f(f + 1)...(f + n− 1)possibilities. These considerations are in accordance with what we found above for v+ andv−.

Let us look at the already mentioned equation:

X = B ((1 + αX)3) + αB ((1 + αX)5) +O(α2)

(6.0.31)

X =∞∑i=0

ci+1αi, P1 := , P2 := (6.0.32)

Considering that the fertility of is 3 and the fertility of 5 the first orders of the solutioncan be calculated iteratively to:

c1 = , c2 = + 3 , c3 = 9 + 3 + 5 + 3

c4 = 27 + 9 + 18 + + 15 + 15 + 9 + 10 + 6 + 5 (6.0.33)

A small test as to wether this could generate a Hopf subalgebra so far:

∆ (c3) = c3 ⊗ 1 + 1⊗ c3 + 15 ⊗ + 9 ⊗ + 3 ⊗ + 5 ⊗ + 3 ⊗ (6.0.34)= c3 ⊗ 1 + 1⊗ c3 + 3c2 ⊗ c1 + 5c1 ⊗ c2

∆ (c4) = c4 ⊗ 1 + 1⊗ c4 + 5 ⊗ (6.0.35)

+ 63 ⊗ + 45 ⊗ + 27 ⊗ + 30 ⊗ + 9 ⊗ + 18 ⊗ + 21 ⊗ + ⊗

+ 35 ⊗ + 15 ⊗ + 21 ⊗ + 15 ⊗ + 15 ⊗ + 9 ⊗ + 10 ⊗ + 6 ⊗

= c4 ⊗ 1 + 1⊗ c4 + 7c1 ⊗ c3 + 5c2 ⊗ c2 + 3c3 ⊗ c1 + 10c21 ⊗ c2 + 6c1c2 ⊗ c1 + c3

1 ⊗ c1

In the remainder we will focus on one-equation systems (only propagator or only vertex nesting)with one to three primitives involved.

20


7 Polynomial Coefficients

7.1 Counting trees

The first step in the calculation is to list the coeffients associated to each tree. We will use thenotation

(ab

)which evaluates to v± with fertility a and number of childrens b. The fertilities n

and n1 are associated to primitive n2 to primitive and n3 to . The coefficients calculateddepend therefore only on the fertilities and are differentiated by how many different primitivesoccur in what quantity. A solution coefficient c3 of a Dyson-Schwinger equation for the looporder 3 with 1- and 2- loop primitives would then be the sum of γ3 and γ1,1.

γ1 = (7.1.1)

γ2 = n (7.1.2)

γ1,1 = n1 + n2 (7.1.3)

γ3 = n2 +1

2

(n

2

)(7.1.4)

γ2,1 = n21 + n1n2

(+

)+

1

2

(n2

2

)+

(n1

2

)(7.1.5)

γ1,1,1 = n1n2

(+

)+ n1n3

(+

)+ n2n3

(+

)+

(n1

2

)+

(n2

2

)+

(n3

2

)(7.1.6)

γ4 = n3 +n

2

(n

2

)+ n

(n

2

)+

1

6

(n

3

)(7.1.7)

γ3,1 = n31 + n2

1n2

(+ +

)+

1

2

(n1

3

)+

1

6

(n2

3

)+ n1

(n1

2

)+n1

2

(n2

2

)+n2

2

(n1

2

)+ n1

(n1

2

)(+

)+ n2

(n1

2

)+ n1

(n2

2

)(7.1.8)

In order to avoid drawing 64 different trees, we use the notation t(a1(a2)). A bracketed termafter a decoration indicates that it is appended to the preceding node and the ai indicate thedecorations:

γ1,1,1,1 =∑

(i,j,k,l)∈Π(1,2,3,4)

(ninjnkt(ai(tj(tk(tl)))) +

1

2ni

(nj2

)t(aiaj(akal)))

+

(ni2

)njt(ni(nj(nk)nl))) +

1

6

(ni3

)t(ai(ajakal))

)(7.1.9)

21


7.2 Trees written in words

Trees without sidebranches translate trivially into words and are therefore omitted here. As areminder:

= abaa (7.2.1)

The letter a is associated to and b to . Brackets in words indicate a primitive formed bythe letters in the brackets, eg. a(ab) := aP (ab). The same holds for powers of a letter, eg.a2 := P (aa). The only words difficult to find are those for trees with more then one child atthe root. This is because:

Φ−1

( )= Φ−1 ◦Ba

( )= aΦ−1

( )(7.2.2)

The trees with three nodes:

= 2(aaa− aa2) (7.2.3)

= 2(baa− ba2) (7.2.4)

= aab+ aba− a(ab) (7.2.5)

= abc+ acb− a(bc) (7.2.6)

Here too trees with fewer different decorations are just special cases of those with more. Treeswith four nodes and only one decoration:

= 2(aaaa− aaa2) (7.2.7)

= 3(aaaa− aa3) + 2(a(aa2)− aaa2 − aa2a) (7.2.8)

= 6(aaaa+ aa3 + a(aa2)− aaa2 − aa2a) (7.2.9)

22


Trees with four nodes having one node with a different decoration:

= 2(baaa− baa2) (7.2.10)

= 2(abaa− aba2) (7.2.11)

= aaba+ aaab− aa(ba) (7.2.12)

= 3(ba3 − ba2a) + 6(baa2 − baaa) (7.2.13)

= (−3aaab− 2aaba− 2abaa− a(aab)

+ 4aa(ab) + a(ab)a+ 2aba2 + aa2b) (7.2.14)

= 3baaa− ba2a− baa2 (7.2.15)

=1

2(−3aaab− 2aaba− 2abaa− a(aab)− 2a(a2b)

+ 4aa(ab) + a(ab)a+ 4aba2 + 3aa2b) (7.2.16)

=1

2(5aaab+ 6aaba+ 8abaa+ a(aab) + 2a(a2b)

− 6aa(ab)− 3a(ab)a− 8aba2 − 3aa2b) (7.2.17)

= −aa2b+ 2aaab+ aaba− aa(ba) (7.2.18)

Trees with four different nodes:

= abcd+ abdc− ab(cd) (7.2.19)

=1

2a1

(b1b2b3) +∑

(i,j,k)∈Π(1,2,3)

(2bibjbk − bi(bjbk)−

1

2(bjbk)bi

) (7.2.20)

=1

2(ab{c, d}+ a{c, d}b+ ab(cd) + a(cd)b− a(b{c, d})− a(b(cd))) (7.2.21)

Here the primitive Lie brackets in HR appear denoted as {a, b}.

7.3 Words in shuffles of Lie brackets

Words with two letters:

aa =1

2a� a (7.3.1)

ab =1

2(a� b+ [a, b]) (7.3.2)

ba =1

2(a� b− [a, b]) (7.3.3)

23


And words with three letters:

aaa =1

6a3� (7.3.4)

aab =1

12(2a� a� b+ 2[a, [a, b]] + 3a� [a, b]) (7.3.5)

aba =1

12(2a� a� b− 4[a, [a, b]]) (7.3.6)

baa =1

12(2a� a� b+ 2[a, [a, b]]− 3a� [a, b]) (7.3.7)

abc =1

6a� b� c+

1

4a� [b, c] +

1

4c� [a, b] +

1

3[a, [b, c]] +

1

6[[a, c], b] (7.3.8)

Or if we abandon writing Lie brackets in Lyndon bracketing for a moment, we observe that wedo: A full shuffle, twice a shuffle of the outer letter with the commutator of the two remainingletters, and a commutator of the first letter with a commutator of the remainder, and repeatthe last step for the word backwards:

abc =1

12(2a� b� c+ 3a� [b, c] + 3c� [a, b] + 2[a, [b, c]] + 2[c, [b, a]]) (7.3.9)

The advantage of this formula is, that we do not lose sight of the symmetries involving thecommutators. Also that aab, aba and baa are special cases of abc is immediate. Continue withwords with four letters:

aaaa =1

24a4� (7.3.10)

aaab =1

96

(4a3�

� b+ 9a2�� [a, b] + 8a� [a, [a, b]]− 6[a, [a, [a, b]]]

)(7.3.11)

aaba =1

96

(4a3�

� b+ 3a2�� [a, b]− 8a� [a, [a, b]]− 18[a, [a, [a, b]]]

)(7.3.12)

abaa =1

96

(4a3�

� b− 3a2�� [a, b]− 8a� [a, [a, b]] + 18[a, [a, [a, b]]]

)(7.3.13)

baaa =1

96

(4a3�

� b− 9a2�� [a, b] + 8a� [a, [a, b]] + 6[a, [a, [a, b]]]

)(7.3.14)

abcd =1

20(5[a, [b, [c, d]]] + 3[[a, [b, d]], c]− 2[[a, c], [b, d]] + 2[[a, [c, d]], b]

+ 2[[a, d], [b, c]] + 1[[[a, d], c], b]) +1

24a� b� c� d

+1

12(2a� [b, [c, d]] + a� [[b, d], c] + 2d� [a, [b, c]] + d� [[a, c], b])

+1

40(3[c, d]� b� a+ 1[b, d]� a� c+ 3[b, c]� a� d

− 1[a, d]� b� c+ 1[a, c]� b� d+ 3[a, b]� c� d+ 5[a, b]� [c, d]) (7.3.15)

24


To express abcd a modified algorithm for the decomposition into the Radfod basis was utilized(see [10]).

7.4 Coefficients in shuffles of Lie brackets

Finally we combine our calculations to reexpress the tree-sum-coefficients from section 7.1.Two considerations are available to check for possible faults in the results: When all involvedfertilities equal one, only the shuffle coefficient may survive. Also, if in terms of trees γy,x−y =qγx, q ∈ Q after identifying two decorations in γy,x−y, then the same has to hold for thecoeffients in terms of shuffles and Lie brackets.

γ1 = a (7.4.1)

γ2 =n

2a� a (7.4.2)

γ1,1 =1

2((n1 + n2)a1� a2 + (n1 − n2)[a1, a2]) (7.4.3)

We observe that for a1 = a2 ⇒ γ1,1 = 2γ2 as predicted.

γ3 =1

6a3�

(n2 +

(n

2

))− 1

2

(n

2

)(a� a2 + [a, a2]) (7.4.4)

γ2,1 =1

6a2�

1 � a2

(n2

1 + 2n1n2 + 2

(n1

2

)+

(n2

2

))(7.4.5)

+1

6[a1, [a1, a2]]

(n2

1 − n1n2 −(n1

2

)+

(n2

2

))+

1

4a1� [a1, a2]

(n2

1 − n1n2 +

(n1

2

)−(n2

2

))− 1

2

(n2

2

)(a2� a2

1 + [a2, a21])− 1

2

(n1

2

)(a2� (a1a2) + [a1, (a1a2)])

This time we have to keep in mind that after identifying a1 with a2 the primitive (a1a2) becomestwo times a2. Incorporating this we see that γ2,1 = 3γ3 again in accordance with prediction.The

∑cycl signify that the triplet (i, j, k) runs through the cyclic permutations of (1, 2, 3)

25


γ1,1,1 =1

3

3∑i=1

((ni2

)+

3∑j>i

ninj

)a1� a2� a3

+1

6

∑cycl

(ni(nj − nk)−

(nj2

)+

(nk2

))[ai, [aj, ak]]

+1

4

∑cycl

(ni(nj − nk) +

(nj2

)−(nk2

))ai� [aj, ak]

− 1

2

∑cycl

(ni2

)(ai� (ajak) + [ai, (ajak)]) (7.4.6)

As there are two non vanishing cyclic permutations this again fulfills γ1,1,1 = 2γ2,1 if we identifya3 with a1. Finally the coefficient with four nodes of one decoration:

γ4 =1

24a4�

(n3 + 4n

(n

2

)+

(n

3

))− 1

12a2�� a2

(10n

(n

2

)− 3

(n

3

))(7.4.7)

+1

6[a, [a, a2]]

(n

(n

2

)+

(n

3

))− 1

4a� [a, a2]

(3n

(n

2

)+

(n

3

))+

1

2

(a� (aa2) + [a, (aa2)]

)(n

(n

2

)+

(n

3

))− 1

2

(a� a3 + [a, a3]

)(3n

(n

2

)−(n

3

))There are some intermediate results presented for the calculation of the γ1,1,1,1 coefficient. Butdue to the rapidly increasing complexity of the involved terms it has not been calculated inthis thesis.

26


8 Conclusion

Except for γ4 all the results and intermediate results were calculated seperately for combinationsof equal and different decorations. This allowed to check the results for selfconsistency, acircumstance increasing the reliability of the results significantly, as the computations are quitetedious and lengthy. Additionally some necessary precalculations were done for γ1,1,1,1, whichmay be of use in further work. For higher order calculations it will propably prove unavoidableto write and use some computer algorithms to prevent errors from occuring.A next possible step would be the search for patterns in the results of different orders, mayberevealing a direct formula to calculate the rational factors of the shuffles. For this the coefficientswith all different decorations are the most interesting, as they are the most general casesimmediately yielding the other coefficients. However when calculating higher order coefficients(greater then three), the conversion from trees into words stops to be one to one even if excludingLie bracket primitives. This will make the results even more complex, but at the same timethe freedom of choice gained may be valuable in order to simplify the terms.The coefficients in their current form also do not include Dyson-Schwinger equation systemswhich incorporate propagators as well as vertices (or different propagators/vertices). Takingthose into account would not change the nature of the calculations, but the v± factors whichcount trees (7.1) would need to distinguish their children, adding to the complexity of theresults.It may also be that it will be most useful to focus on one quantum field theory, and makeuse of additional simplifications coming from explicit evaluation of Feynman graphs. This mayaccording to Professor Kreimer be well one of the next steps his group is going to undertake.

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References

[1] C Bergbauer and Dirk Kreimer. Hopf algebras in renormalization theory: Locality anddyson-schwinger equations from hochschild cohomology. IRMA Lect.Math.Theor.Phys.,10:133–164, 2006.

[2] Isabella Bierenbaum, Dirk Kreimer, and Stefan Weinzierl. The next-to-ladder approxima-tion for linear dyson-schwinger equations. Physics Letters B, 646:129–133, 2007.

[3] Wayne D Blizard. Multiset theory. Notre Dame Journal of Formal Logic, 30:36–66, 1988.

[4] Alain Connes and Dirk Kreimer. Renormalization in quantum field theory and the riemann-hilbert problem ii: the β-function, diffeomorphisms and the renormalization group. Com-mun.Math.Phys., 216:215–241, 2001.

[5] Kurusch Ebrahimi-Fard and Dirk Kreimer. The hopf algebra approach to feynman diagramcalculations. Journal of Physics A, 2005.

[6] Loïc Foissy. Faà di bruno subalgebras of the hopf algebra of planar trees from combinatorialdyson-schwinger equations. to appear in Advances in Mathematics, 2007.

[7] Jürgen Fuchs and Christoph Schweigert. Symmetries, lie algebras and representations.Cambridge Univ. Press, 1997.

[8] Dominique Manchon. Hopf algebras, from basics to applications to renormalization. Re-vised and updated version, may 2006.

[9] J. W. Milnor and J.C Moore. On the structure of hopf algebras. Ann. of Mathematics,2:211–264, 1965.

[10] Minh Hoang Ngoc and Michel Petitot. Lyndon words, polylogarithms and the riemann ζfunction. Discrete Maths, 217:273–292, 2000.

[11] Erik Panzer. Hopf-algebraic renormalization of kreimer’s toy model. master thesis, 2012.

[12] Michael E. Peskin and Daniel V. Schroeder. An introduction to quantum field theory.Westview Pr., 2007.

[13] D.E. Radford. A natural ring basis for the shuffle algebra and an application to groupschemes. Journal of Algebra, 58:432–454, 1979.

[14] Christophe Reutenauer. Free Lie Algebras, volume 7 of London Mathematical SocietyMonographs New Series. Oxford Sciene Publications, 1993.

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Acknowledgements

Hereby I would like to thank Professor Kreimer for supervising my thesis even though I am nota Humboldt university student and for the various confusing and enlightening talks we had.I also give my thanks to Professor Knorr for supervising from the TU Berlin side making thisthesis possible.

Zusammenfassung

In dieser Bachelorarbeit werden einzelne Dyson-Schwinger Gleichungen betrachtet die Vertexbeziehungsweise Propagator Verschachtelungen in Quantenfeldtheorien beschreiben. Es wer-den Lösungskoeffizienten bis zur vierten Ordnung in der Hopf Algebra der dekorierten Wurzel-bäume ausgerechnet. Diese werden im Anschluss in die Shuffle Hopf Algebra in der Basisder Lyndon-Lie-Klammern umgerechnet. Die Zwischen-/Resultate werden präsentiert und aufSelbstkonsistenz geprüft. Zusätzlich werden die vielfältigen algebraischen Begriffe und Struk-turen im Vorfeld erläutert, in einer Weise die einen einfachen und schnellen Zugang zur Materieermöglichen soll.

Version

An addition was made to the argumentation on why the bracketings of Lyndon words generatea basis of the shuffle algebra over words. Apart from that this is the bachelor thesis as it washanded in on the 25th of October 2012.

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