Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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(p,

U) denote connected component of W

(p)

containing p, and defin e W

(p,

U) similarly. We may

choose C

coordinates (x, y) so that in some small

neighborhood

U of p, the point p corresponds to

(0, 0), the set W

(p,

U) corresponds to (y = 0), and

the set W

(p,

U) corresponds to (x = 0). We assume

that

U is small enough that f in

U is closely

approximated by its derivative Df

(0, 0)

. Hence, f

nearly contracts vertical directions and expands

horizontal directions in

Take compact arcs I

W

(p) and I

W

(p)

both containing the points p and q as in Figure 5.

Let D be a curvilinear rectangle which is a slight

thickening of I

. The forward iterates f

(D) will stay

near I

for a while and then start to approach I

If we choose D appropriately, we can arrange for

some high iterate f

(D) to be a slight thickening

of I

as in Figure 6. This looks geometrically like the

horseshoe map. Let A

be the connected component

of the intersection D

(D) containing p, and let

be the connected component of the intersection

(D) containing q. These sets (which are

shaded in Figure 6) play the role of the rectangles

and R

, respectively, in the horseshoe construc-

tion. We use the set A

for U in Theorem 1.

The He´ non Family

To give explicit formulas for the horseshoe map

above is somewhat tedious, and it is of interest to

note that similar properties occur in maps with

simple formulas. Indeed, such properties occur quite

often in a well-known family of maps known as the

‘‘He´non family.’’ As we have mentioned, the map in

Figure 1 provides an example.

One may simply define a He´non map as a

diffeomorphism H = (H

(x, y), H

(x, y)) with inverse

G(x, y) = (G

(x, y), G

(x, y)) such that all the maps

(x, y), G

(x, y) are polynomials of degree at most

two. It is known (see, e.g., Friedland and Milnor

(1989)) that such maps H have constant Jacobian

determinant, and, up to affine conjugacy, may be

represented in the form H = H

a, b

(x, y) = (a  x



by, x)witha, b constants and b 6¼0. This makes

sense when all the terms are real or complex. In the

real case, we speak of the real He´non family and,

in the complex case, we speak of the complex

He´non family.

The real He´non family was first presented by the

physicist M He´non in 1976 as perhaps the simplest

nonlinear diffeomorphism of the plane exhibiting a

so-called ‘‘strange attractor.’’ These mappings in the

real and complex cases have been the focus of much

attention. Our interest here is that, at least for

certain parameters a, b, they provide concrete

globally defined maps whose dynamics are analo-

gous to that of the horseshoe diffeomorphism. In

fact, Devaney and Nitecki (1979) pro ved (in the real

case) that for fixed b 6¼0, there is a constant a

> 0

such that if a > a

, then the set B

a, b

of bounded

orbits of H

a, b

is a compact zero-dimensional set and

the pair (H

a, b

, B

a, b

) is topologically conjugate to

(, 

). In addition, it can be shown that the

invariant set B

a, b

is a single hyperbolic h-closure.

Analogous results are true for the complex He´non

family and proofs were originally given in the thesis

of Ralph Oberste–Vorth (unpublished) under the

supervision of John Hubbard at Cornell University.

More recent proofs are in Newhouse (2004) and

Hruska (2004). Many interesting results have been

obtained for the complex He´non map by Bedford

and Smillie and Sibony and Fornaess (see the

references in Hruska (2004).

Homoclinic Points in Systems with

Positive Topological Entropy

There is an invariant of topological conjugacy which is

known as the topological entropy. In a certain sense,

this gives a quantitative measurement of the amount of

complicated or chaotic motion in the system.

Figure 5 The curves I

 W

(p) and I

 W

(p).

(D)

f(D)

Figure 6 The curvilinear rectangle D and its N th iterate f

(D)

are geometrically like the horseshoe map.

676 Homoclinic Phenomena

Let f : X !X be a continuous self-map of the

compact metric space (X, d). For a positive integer

n > 0, we define an n-orbit to be a finite sequence

O(x, n) = {x, f (x), ..., f

n1

(x)}. Given a positive real

number >0, we say that two n-orbits O(x, n) and

O(y, n) are ‘‘-distinguishable’’ if there is a 0  j < n

such that d(f

x, f

y) >. Another way to look at this

is the following. Define the so-called d

-metric on X

by setting d

(x, y) = max

0j<n

d( f

x, f

y). Then, the

two n-orbits O(x, n), O(y, n) are -distinguishable if

and only if d

(x, y) >. It follows from compactness

of X and the uniform continuity of each of the

maps f

,0 j < n, that the number r(n, , f )of

-distinguishable n-orbits is finite for each given >0

and each positive integer n. We define the number

hðf Þ¼lim

!0

lim sup

n!1

log rðn;;f Þ

This means that, for some sequence of inte-

gers n

< n

< ..., the map f has roughly e

h(f )

-distinguishable n

-orbits for i large and  small.

The number h(f ) is called the topological entropy

of the map f. It may be infinite for homeomorph-

isms, but it is always finite for smooth maps on

finite-dimensional manifolds. The number h(f ) has

many nice properties. For instance, h(f

) = Nh(f )

for every positive integer N, and, if f is a homeo-

morphism, then h(f

1

) = h(f ). Further, if f and g are

topologically conjugate, then h(f ) = h(g). The so-

called ‘‘variational principle for topological

entropy’’ asserts that h(f ) is the supremum of the

measure-theoretic entropies of the invariant prob-

ability measures for f. Our interest in this invariant

here is the following theorem of Katok.

Theorem 2(Katok). Let f be a C

diffeomorphism

of a compact two-dimensional manifold M to itself

with positive topological entropy. Then, f has

transverse homoclinic points.

In fact, Katok extended this theorem (see the

supplement in Hasselblatt and Katok (1995)) to

show that, if h(f ) > 0 and >0, then there is a

compact zero-dimensional hyperbolic ba sic set  for

h such that h(f , ) > h(f )  . Thus, one can find

nice invariant topologically transitive sets for f (i.e.,

sets with dense orbits) on which the topological

entropies of restriction of f are arbitrarily close to

that of f.

This theorem has the interesting consequence that

the map f !h(f ) is lower-semicontinuous on the

space of C

diffeomorphisms of a surface. It was

proved in Newhouse (1989) (and, independently by

Yomdin (1987)) that the map f !h( f ) is upper-

semicontinuous on the space of C

diffeomorphisms

of any compact manifold. Combining these results

gives the theorem that the map f !h(f ) is contin-

uous on the space of C

diffeomorphisms on a

compact surface, and that positivity of h( f ) implies

the existence of transverse homoclinic points.

It is also worth noting that, for any continuous

self-map f : M !M on a compact manifold M, one

has the inequality h(f )  log jj where  is the

eigenvalue of largest norm of the induced map f

on the first real homology group (Manning 1975).

Putting this together with Theorem 2 gives the fact

that there are whole homotopy classes of diffeo-

morphisms on surfaces all of whose elements have

transverse homoclinic points. For instance, consider

a2 2 matrix

L ¼



with integer entries, determinant 1, and eigenvalues



, 

with 0 < j

j < 1 < j

j. Let

L : T

the induced diffeomorphism on the two-dimensional

torus T

. This is an example of what is called an

‘‘Anosov’’ diffeomorphism. In this case the number

 above is simply 

, and this holds for any

diffeomorphism f of T

which can be continuously

deformed into

L. Hence, any such f must have

transverse homoclinic points.

Homoclinic Tangencies

Let {f



,  2[0, 1]} be a parametrized family of C

diffeomorphisms of the plane with  an external

parameter. It frequently occurs that there is a

hyperbolic saddle fixed point p



for each parameter

 moving continuously with  such that, at some

value 

, a homoclinic tangency is created at a point

. This means that there are an >0, a small

neighborhood U of q

, and curves 



W







W



) such that 







= ; for 

 <<



, 







= {q

}, and 







consists of two

distinct points for 

<<þ . In most cases,

the tangency of 



and 



at q

will be of the

second order, and we will assume that occurs here.

The geometry is as in Figure 7.

λ < λ

λ = λ

λ > λ

Figure 7 Creation of a homoclinic tangency.

Homoclinic Phenomena 677

The creation of homoclinic tangencies is part of

the general subject of ‘‘homoclinic bifurcations.’’ A

recent survey of this subject is in the book by

Bonatti et al. (2005). Typical results are the

following. If p = p



is a saddle fixed point whose

derivative is area-decreasing (i.e., jDet(Df (p))j < 1),

then there are infinitely many parameters  near 

for which each transverse homoclini c point of p



is a

limit of periodic sinks (asymptotically stable peri-

odic orbits) (Newhouse 1979, Robinson 1983). In

addition, so-called strange attractors and SRB

measures appear (Mora and Viana 1993).

Finally, we mention that recently it has been

shown that, generically in the C

topology for r  2,

homoclinic closures associated to a homoclinic

tangency (in dimension 2) have maximal Hausdorff

dimension (Theorem 1.6 in Downarowicz and

Newhouse (2005)).

See also: Chaos and Attractors; Fractal Dimensions in

Dynamics; Generic Properties of Dynamical Systems;

Hyperbolic Dynamical Systems; Lyapunov Exponents

and Strange Attractors; Saddle Point Problems;

Singularity and Bifurcation Theory; Solitons and Other

Extended Field Configurations.

Further Reading

Birkhoff GD (1960) Nouvelles recherches sur les syste` mes

dynamiques. In: George David Birkhoff, Collected Mathema-

tical Papers, Vol. II, pp. 620–631. New York: American

Mathematical Society.

Bonatti C, Diaz L, and Viana M (2005) Dynamics Beyond

Uniform Hyperbolicity, Encyclopedia of Mathematical

Sciences, Subseries: Mathematical Physics III, vol. 102,

Berlin–Heidelberg–New York: Springer.

Devaney R and Nitecki Z (1979) Shift automorphism in the

henon mapping. Communications in Mathematical Physics 67:

137–148.

Downarowicz T and Newhouse S (2005) Symbolic extensions in

smooth dynamical systems. Inventiones Mathematicae 160(3):

453–499.

Friedland S and Milnor J (1989) Dynamical properties of plane

polynomial automorphisms. Ergodic Theory and Dynamical

Systems. 9: 67–99.

Guckenheimer J and Holmes P (1983) Nonlinear Oscillations,

Dynamical Systems, and Bifurcation of Vector Fields, Applied

Mathematical Sciences, vol. 42, New York: Springer.

Hasselblatt B and Katok A (1995) Introduction to the Modern

Theory of Dynamical Systems, Encyclopedia of Mathematics

and Its Applications, vol. 54, Cambridge: Cambridge

University Press.

Hruska SL (2004) A numerical method for proving hyperbolicity

in complex he´non mappings (http://arxiv.org).

Manning A (1975) Topological entropy and the first homology

group. In: Manning A (ed.) Dynamical Systems – Warwick

1974, Lecture Notes in Math. vol. 468, pp. 567–573. New

York: Springer.

Moser J (1973) Stable and Random Motions in Dynamical

Systems, Annals of Mathematical Studies, Number 77.

Princeton: Princeton University Press.

Mora L and Viana M (1993) Abundance of strange attractors.

Acta Mathematica 171: 1–71.

Newhouse S (1979) The abundance of wild hyperbolic sets and

non-smooth stable sets for diffeomorphisms. Publications

Mathe´matiques de l’I.H.E.S. 50: 101–151.

Newhouse S (1980) Lectures on dynamical systems. In: Coates J

and Helgason S (eds.) Dynamical Systems, CIME Lectures,

Bressanone, Italy, June 1978, Progress in Mathematics, vol. 8,

pp. 1–114. Cambridge, MA: Birkha¨user.

Newhouse S (1989) Continuity properties of entropy. Annals of

Mathematics 129: 215–235.

Newhouse S (2004) Cone-fields, domination, and hyperbolicity.

In: Brin M, Hasselblatt B, and Pesin Y (eds.) Modern

Dynamical Systems and Applications, pp. 419–432.

Cambridge: Cambridge University Press.

Nusse HE and Yorke JA (1998) Dynamics: Numerical Explora-

tions. Applied Mathematical Sciences, vol. 101, New York:

Springer.

Poincare´ H (1899) Les methodes nouvelles de la Mecanique

Celeste–Tome 3. Guathier-Villars, Original publication (New

Printing: Librairie Scientifique et Technique Albert Blanchard

9, Rue Medecin 75006, Paris, 1987).

Robinson C (1983) Homoclinic bifurcation to infinitely many

sinks. Communications in Mathematical Physics 90: 433–459.

Robinson C (1999) Dynamical Systems, Stability, Symbolic

Dynamics, and Chaos

, Studies in Advanced Mathematics,

2nd edn. New York: CRC Press.

Yomdin Y (1987) Volume growth and entropy. Israel Journal of

Mathematics 57: 285–300.

Hopf Algebra Structure of Renormalizable Quantum Field Theory

D Kreimer, IHES, Bures-sur-Yvette, France

Overview

Renormalization theory is a venerable subject put to

daily use in many branches of physics. Here, we

focus on its applications in quantum field theory,

where a standard perturbative approach is provided

through an expansion in Feynman diagrams. Whilst

the combinatorics of the Bogoliubov recursion,

solved by suitable forest formulas, has been known

for a long time, the subject regained interest on the

conceptual side with the discovery of an underlying

Hopf algebra structure behind these recursions.

Perturbative expansions in quantum field theory

are organized in terms of one-particle irreducible

(1PI) Feynman graphs. The goal is to calculate the

corresponding 1PI Green functions order by order in

the coupling constants of the theory, by applying

Feynman rules to these 1PI graphs of a

678 Hopf Algebra Structure of Renormalizable Quantum Field Theory

renormalizable theory under consi deration. This

allows one to disentangle the problem into an

algebraic part and an analytic part.

For the algebraic part, one studies Feynman graphs

as combinatorial objects which lead to the Lie and

Hopf algebras discussed below. Feynman rules then

assign analytic expressions to these graphs, with the

analytic structure of finite renormalized quantum field

theory largely dictated by the underlying algebra.

The objects of interest in qua ntum field theory are

the 1PI Green functions. They are parametrized by

the quantum numbers – masses, momenta, spin, and

such – of the particles participating in the scattering

process under consideration. We call a set of such

quantum numbers an external leg structure

r. For

example, the three terms in the Lagrangian of

massless quantum electrodynamics correspond to

r 2f ; ; g½1

Note that the Lagrangian L of massless quantum

electrodynamics is obtained accordingly as

L ¼

ð

1

ð Þþ

ð Þ

1



@ þ



A þ

½2

where

 are coordinate space Feynman ru les.

The renormalized 1PI Green function in momen-

tum space, G

({g}; {p}, {m}; ), is obtained as the

image under renormalized Feynman rules 

applied

to a series of graphs:



¼ 1 þ

k¼1

 1 þ

resðÞ¼r

jj



SymðÞ

½3

Here

r is a given such external leg structure, while c

is the finite sum of 1PI graphs having k loops,

resðÞ¼r

jj¼k



SymðÞ

½4

and 0 < g < 1 is a coupling constant. The general-

ization to the case of several couplings {g} and

masses {m} is stra ightforward. In the above, the sum

is over all 1PI graphs with the same given external

leg structure. We have denoted the map which

assigns

r to a given graph a residue, for example,

resð

Þ¼ ½5

The unrenormalized but regularized Feynman rules

 assign to a graph a function

ðÞðfgg; fpg; fmg; ; zÞ

v2

½0



ð4Þ

f incident v

e2

½1

int

Propðk

4

½6

and formally the unrenormalized Green function

ðfgg; fpg; fmg; ; zÞ

¼ ð

Þfgg; fpg; fmg; ; zðÞ½7

which is a function of a suitably chosen regulator z.

Note that in [6] the four-dimensional Dirac-

distribution guarantees momentum conservation at

each vertex and restricts the number of four-

dimensional integrations to the number of indepen-

dent cycles in the graph. It is assumed that the

reader is familiar with the readily established fact

that these integrals suffer from UV singularities,

which render the integration over the momenta in

internal cycles ill-defined. We also remind the reader

that the problem persists in coordinate space, where

one confronts the continuation of products of

distributions to regions of coinciding support. We

restrict ourselves here to a discussion of the situation

in momentum space and refer the reader to the

literature for the situation in coordinate space.

Ignoring problems of convergence in the sum over

all graphs, the problem of renormalization is to

make sense of these functions term by term: We

have to determine invertible series Z

({g}, z) in the

couplings g such that the modified Lagrangian

L ¼

ðfgg; zÞ

ðrÞ½8

produces a perturbation series in graphs that allows

for the removal of the regulator z.

This amounts to a transition from unrenorma-

lized to renormalized Feynman rules  !

.Letus

first describe how this transition is achieved using

the Lie and Hopf algebra struc ture of the per turba-

tive expansion, which is described in detail below:



Decide on the free fields and local interactions of

the theory, appropriately specifying quantum

numbers (spin, mass, flavor, color, and such) of

fields, restricting interactions so as to obtain a

renormalizable theory.



Consider the set of all 1PI graphs with edges

corresponding to free-field propagators. Define

vertices for local interactions. This allows one to

construct a pre-Lie algebra of graph insertions.

Antisymmetrize this pre-Lie product to get a Lie

algebra L of graph insertions and define the Hopf

algebra H which is dual to the enveloping algebra

U(L) of this Lie algebra.



Realize that the coproduct and antipode of this

Hopf algebra give rise to the forest formula,

which generates local counter-terms upon intro-

ducing a Rota–Baxter map, a renormalization

scheme in physicists’ parlance.

Hopf Algebra Structure of Renormalizable Quantum Field Theory 679



Use the Hochschild cohomology of this Hopf

algebra to show that one can absorb singularities

in local counter-terms.



Determine the corepresentations of this Hopf

algebra to identify the sub-Hopf algebras corre-

sponding to time-ordered products in physical

fields. This is most easily achie ved by rewriting

the Dyson–Schwinger equations using Hochschild

1-cocycles.

The last point exhibits close connections, in parti-

cular, between the structure of gauge theories and

the corepresentation theory of their perturbative

Hopf algebras which we discuss below in brief.

This program can be carried out in coordinate

space as well as momentum space renormalization.

It has given a firm mat hematical background to the

process of renormalization, justifying the practice of

quantum field theory. The notion of locality has

achieved a precise formulation in terms of the

Hochschild cohomology of the perturbation expan-

sion. In momentum space, this approach emphasizes

the connections to number theory, which emerge

when one investigates the role of the Hopf algebra

primitives, which in turn furnish the Hochschild

1-cocycles underlying locality.

The next sections describe the above setup in

some detail.

Lie and Hopf Algebras of Graphs

All algebras are supposed to be over some field K of

characteristic zero, associative and unital, and

similarly for coalgebras . The unit (and, by abuse of

notation, also the unit map) will be denoted by I,

the counit map by



e. All algebra homomorphisms

are supposed to be unital. A bialgebra

(A =

i = 0

, m, I, ,



e) is called graded connected

if A

 A

iþj

and (A

) 

jþk = i

 A

, and if

(I) = I  I and A

= kI,



e(I) = 1 2 K and



e = 0on

i = 1

. We call ker



e the augmentation ideal of A

and denote by P the projection A ! ker



e onto the

augmentation ideal, P = id  I



e. Furthermore, we

use Sweedler’s notation, (h) =

 h

, for the

coproduct. We define

Aug

ðkÞ

¼ P P

|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

k times



k1

;

A !fker



k

½9

as a map into the k-fold tensor product of the

augmentation ideal. We let A

(k)

= ker Aug

(kþ1)

ker Aug

(k)

, 8 k  1. All bialgebras considered here

are bigraded in the sense that

A ¼

i¼0

ðiÞ

k¼0

ðkÞ

½10

where A

(k)



j = 1

(j)

for all k  1. A

(0)

’ A

(0)

’ K.

The first construction we ha ve to study is the pre-

Lie algebra structure of 1PI graphs.

The Pre-Lie Structure

For each Feynma n graph we have vertices as well as

internal and external edges. External edges are edges

that have an open end not connect ed to a vertex.

They indicate the particles participating in the

scattering amplitud e under consideration and each

such edge carries the quantum numbers of the

corresponding free field. The internal edges and

vertices form a graph in their own right. For an

internal edge, both ends of the edge are connected to

a vertex.

We consider 1PI Feynman graphs. A graph  is

1PI if and only if all graphs, obtained by removal of

any one of its internal edges, are still connected.

Such 1PI graphs are naturally graded by their

number of independent loops, the rank of their

first homology group H

[1]

(, Z). We write jj for

this degree of a graph . Note that jres()j= 0,

where we let res() be the graph obtained when all

edges in 

[1]

int

shrink to a point, as before. Note that

the graph obtained in this manner consists of a

single vertex, to which the edges 

[1]

ext

are attached.

For a 1PI graph , 

[0]

denotes its set of

vertices and 

[1]

=

[1]

int

[ 

[1]

ext

its set of internal

and external edges. In addition, let !

be the

number of spacetime derivatives appearing in the

corresponding monomial in t he Lagrangian.

Having specified free quantum fields and local

interaction terms between them, one im mediately

obtains the set of 1PI graphs. One can then consider

for a given external leg structure

r the set of graphs

with that external leg structure. For a renormaliz-

able theory, we can define a superficial degree of

divergence,

! ¼

r2

½1

int

[

½0

 4jH

½1

ð; ZÞj ½11

for each such external leg structure: !() = !(

)if

res() = res(

); all graphs with the same external leg

structure have the same superficial degree of

divergence, and only for a finite number of distinct

external leg structures

r will this degree indeed

signify a divergence.

This leaves a finite number of external leg structures

to be considered to which we restrict ourselves from

now. Our first observation is that there is a natural

pre-Lie algebra structure on 1PI graphs.

680 Hopf Algebra Structure of Renormalizable Quantum Field Theory

To this end, we define a bilinear operat ion



 



nð

; 

;Þ ½12

where the sum is over all 1PI graphs .Here,

n(

, 

; ) is a section coefficient which counts

the number of ways in which a subgraph 

can be

reduced to a point in  such that 

is obtained. The

above sum is evidently finite as long as 

and 

are finite graphs, and the graphs which contribute

necessarily fulfill jj= j

jþj

j and res() = res(

One then has the following theorem.

Theorem 1 The operation  is pre -Lie:

½

 



 

½

 



¼½

 



 

½

 

½13

which is evident when one rewrites the -product in

suitable gluing operations.

To understand this theorem, note that the

equation claims that the lack of associativity in the

bilinear operation  is invariant under permutation

of the elements indexed 2, 3. This suffices to show

that the antisymmetrization of this map fulfills a

Jacobi identity. Hence, we get a Lie algebra L by

antisymmetrizing this operation:

½

; 

¼

 

 

 

½14

This Lie algebra is graded and of finite dimension in

each degree. Let us look at a couple of examples for

pre-Lie products:

 ¼ ½15

 ¼2

½16

 ¼ ½17

 ¼ 2 ½18

 ¼ ½19

 ¼ ½20

Together with L one is led to consider the dual of its

universal enveloping algebra U(L) using the theorem

of Milnor and Moore. For this we use the above

grading by the loop number.

This universal enveloping algebra U(L) is built

from the tensor algebra

T ¼

; T

¼LL

|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

k times

½21

by dividing out the ideal generated by the relations

a  b  b  a ¼½a; b2L ½22

Note that in U(L) we have a natural concatenation

product m



. Furthermore, U(L) carries a natural

Hopf algebra structure with this product. For that,

the Lie algebra L furnishes the primitive elements:





ðaÞ¼a  1 þ 1  a; 8 a 2L ½23

It is, by construction, a connected finitely graded

Hopf algebra which is co-commutative but not

commutative. We can then consider its graded

dual, which will be a Hopf algebra H(m, I, ,



that is commutative but not cocommutative. One

finds it upon using a Kronecker pairing



;



> ¼

1;  ¼ 

0; else



½24

The space of primitives of U(L) is in one-to-one

correspondence with the set Indec(H) of indecom-

posables of H, which is the linear span of its

generators. One finds the follow ing theorem.

Theorem 2



Z



Z



Z



;



>¼<Z

½

;



;



> ½25

For example, one finds

Z Z Z ;

¼ Z

Z Z Z ;



 

¼ Z

Z ;

¼2 ½26

H is a graded commutative Hopf algebra which

suffices to describe renormalization theory, as we

see in the next section. We have formulated it for

the superficially divergent 1PI graphs of the theory

with the understanding that the residues of these

graphs are in one-to-one correspondence with the

terms in the Lagrangian of a given theory. Often,

several terms in a Lagrangian correspond to graphs

with the same number and type of external legs, but

correspond to different form-factor projections of

the graph. In such cases, the above approach can be

easily adopted considering suitably colored or

Hopf Algebra Structure of Renormalizable Quantum Field Theory 681

labeled graphs. A similar remark applies if one

desires to incorporate renormalization of super-

ficially convergent Green functions, which requires

nothing more than the consideration of an easily

obtained semidirect product of the Lie algebra of

superficially divergent graphs with the abelian Lie

algebra of superficially convergent graphs.

The Principle of Multiplicative Subtraction

The above algebra structures are available once one

has decided on the set of 1PI graphs of interest. We

now use them toward the renormalization of any

such chosen local quantum field theory.

From the above, 1PI graphs  provide the linear

generators 



of the Hopf algebra H= 

i = 0

where H

lin

= span(



) and their disjoint union

provides the commutative product.

Now let  be a 1PI graph. We find the Hopf

algebra H as described above to have a coproduct

explicitly given as  : H!HH:

ðÞ¼  1 þ 1   þ



  = ½27

where the sum is over all unions of 1PI superficially

divergent proper subgraphs, and we extend this

definition to products of graphs so that we get a

bialgebra.

While the Lie bracket inserted graphs into each

other, the coproduct disentangles them. It is th is

latter operation which is needed in renormalization

theory: we have to render each subgraph finite before

we can construct a local counter-term. That is precisely

what the Hopf algebra structure maps do.

Having a coproduct, two further structure maps

of H are immediate: the counit and the antipode.

The counit



e vanishes on any nontrivial Hopf

algebra element,



e(1) = 1,



e(X) = 0. The antipode is

SðÞ¼ 



SðÞ= ½28

We can work out a few coproducts and antipodes as

follows:

Aug

ð2Þ

ð Þ¼2  ½29

Aug

ð2Þ

ð Þ¼2



½30

Aug

ð2Þ

ð Þ¼  ½31

Aug

ð2Þ

ð Þ¼2  ½32

Aug

ð2Þ

ð Þ¼2



½33

Aug

ð2Þ

ð Þ¼  ½34

We give just one example for an antipode:

Sð

Þ¼ þ2 ½35

Note that for each term in the sum

() =



(i)





(i)

, we have unique gluing data G

such that

 ¼ 

ðiÞ



ðiÞ

; 8 i ½36

These gluing data describe the necessary bijections

to glue the components 

(i)

back into 

(i)

so as to

obtain : using them, we can reassemble the whole

from its parts. Each possible gluing can be inter-

preted as a composition in the insertion operad of

Feynman graphs.

We have by now obtained a Hopf algebra

generated by combinatorial elements, 1PI Feynman

graphs. Its existence is automatic from the above

choices of interactions and free fields. What remains

to be done is a structural analysis of these algebras

for the renormalizable theories we are confronted

with in four spacetime dimensions.

The assertion underlying perturbation theory is

the fact that meaningful approximations to physical

observable quantities can be found by evaluating

these graphs using Feynman rules.

First, as disjoint scattering processes give rise to

independent amplitudes, one is led to the study of

characters of the Hopf algebra, maps  : H!V such

that   m = m

(  ).

Such maps assign to any element in the Hopf

algebra an element in a suitable target space V.

The study of tree-level amplitudes in lowest-order

perturbation theory justifies assigning to each edge

a propagator and to each elementary scattering

process a vertex, which define the F eynman rules

(res()) and the underlying Lagrangian, on the

level of residues of these very graphs. Graphs are

constructed from edges and vertices which are

provided precisely by the residues of those diver-

gent graphs, hence one is led to assign to each

Feynman graph an evaluation in terms of an

integral over the continuous quantum numbers

assigned to edges or vertices, which leads to the

familiar integrals over momenta in closed loops

mentioned before.

Then, with the Feynman rules providing a

canonical charac ter , we will have to make one

further choice: a renormalization scheme. The need

for such a choice is no surprise: after all we are

eliminating short-distance singularities in the graphs,

682 Hopf Algebra Structure of Renormalizable Quantum Field Theory

which renders their remaining finite part a mbiguous,

albeit in a most interesting manner.

Hence, we choose a map R : V ! V, from which

we obviously demand that it does not modify the

UV-singular structure, and furthermore that it obeys

RðxyÞþRðxÞRðyÞ¼RðRðxÞyÞþRðxRðyÞÞ ½37

which guarantees the multiplicativity of renormali-

zation and is at the heart of the Birkhoff decom-

position, which emerges below: it tells us that

elements in V split into two parallel subalgebras

given by the image and kernel of R. Algebras for

which such a map exists are known as Rota–Baxter

algebras. The role Rota–Baxter algebras play for

associative algebras is similar to the role Yang–

Baxter algebras play for Lie algebras. The structure

of these algebras allows one to connect renormaliza-

tion theory to integrable systems. In addition, most

of the results obtained initially for a specific

renormalization scheme, such as minimal subtrac-

tion, can also be obtained, in general, upon a

structural analysis of the corresponding Rota–Bax ter

algebras.

To see how all the above comes together in

renormalization theory, we define a further char-

acter S



that deforms   S slightly and delivers the

counter-term for  in the renormalization scheme R:



ðÞ¼Rm

ðS



   PÞ

¼R½ðÞ  R





ðÞð=Þ

½38

which should be compared with the undeformed

  S ¼ m

ðS      PÞ

¼ðÞ



  SðÞð=Þ½39

The fact that R is a Rota–Baxter map ensures that



is an element of the charac ter group G of the

Hopf algebra, S



2 Spec(G). Note that we have now

determined the modified Lagrangian:

¼ S



ð

Þ½40

The classical results of renormalization theory

follow immediately using this group structure: we

obtain the renormalization of  by the application

of a renormalized character



?ðÞ¼m

ðS



 Þ ½41

and Bogoliubov’s



R operation as



RðÞ¼m

ðS



 Þðid  PÞðÞ

¼ ðÞþ





ðÞð=Þ½42

so that



?ðÞ¼



RðÞþS



ðÞ½43

Here, S



?is an element in the group of characters

of the Hopf algebra, with the group law given by the

convolution



?

¼ m

ð

 

Þ ½44

so that the coproduct, counit, and coinverse (the

antipode) give the product, unit, and inverse of this

group, as befits a Hopf algebra. This Lie group has

the previous Lie algebra L of graph insertions as its

Lie algebra: L exponentiates to G.

What we have achieved above is a local renorma-

lization of quantum field theory. Let M

be a

monomial in the Lagrangian L of degree !

¼ D

fg½45

Then one can prove, using the Hochschild cohomol-

ogy of H:

Theorem 3 (Locality)

fg¼D

fg½46

that is, renormalization commutes with infinitesimal

spacetime variations of the fields .

We can now work out the renormalization of a

Feynman graph :

ð

Þ¼ I þI 

þ2  ½47



ð

Þ¼ð Þþ2S



ð Þð Þ½48

¼ ð

Þ2R ð Þ

ð Þ½49



ð Þ¼R



ð Þ



½50



ð ÞS



?



¼½id  R



ð



½51

The formulas [47]–[51] are given in their recursive

form. Zimmermann’s original forest formula solving

this recursion is obtain ed when we trace our

considerations back to the fact that the coproduct

can be written in nonrecursive form as a sum over

forests, and similarly for the antipod e.

Hopf Algebra Structure of Renormalizable Quantum Field Theory 683

Diffeomorphisms of Physical Parameters

In the above, we have effectively obtained a Birkhoff

decomposition of the Feynman rules  2 Spec(G)

into two characters – 

= S



?2 Spec(G)and





= S



2 Spec(G) – for any Rota–Baxter map R.

Thanks to Atkinson’s theorem, this is possible for

any renormalization scheme R. For the minimal

subtraction scheme, it amounts to the decomposi-

tion of the Laurent series ()(), which has poles of

finite order in the regulator , into a part holo-

morphic at the origin and a part holom orphic at

complex infini ty. This has a particularly nice

geometric interpretation upon considering the

Birkhoff decomposition of a loop around the origin,

providing the clutching da ta for the two half-spheres

defined by that very loop.

Whilst in this manner a satisfying understanding

of perturbative renormalization is obtained, the

character group G remains rather poorly under-

stood. On the other hand, renormalization can be

captured by the study of diffeomorphisms of

physical parameters as, by definition, the range of

allowed modification in renormalization theory is

determined by the variation of the coefficients of

monomials

(

r) of the underlying Lagrangian

L ¼

ð

rÞ½52

Thus, one desires to obtain the whole Birkhoff

decomposition at the level of diffeomorphisms of the

coupling constants.

The crucial step toward that goal is to realize the

role of a standard quantum field-theoretic formula

of the form

new

¼ g

old

½53

where

e2resðvÞ

½1

ext

ﬃﬃﬃﬃﬃﬃ

½54

for some vertex v, which obtains the new coupling

in terms of a diffeomorphism of the old. This

formula provides, indeed, a Hopf algebra homo-

morphism from the Hopf algebra of diffeomorph-

isms to the Hopf algebra of Feynman graphs,

regarding Z

(a series over counter-terms for all

1PI graphs with the external leg structure corre-

sponding to the coupling g), in two different ways: it

is, at the same time, a formal diffeomorphism in the

coupling constant g

old

and a formal series in Feyn-

man graphs. As a consequence, there are two

competing coproducts acting on Z

. That both give

the same result defines the required homomorphism,

which transposes to a homomorphism from the

largely unknown group of characters of H to the

one-dimensional diffeomorphisms of this coupling.

In summary, one finds that a couple of basic

facts enable one to make a transition from the

abstract group of characters of a Hopf algebra of

Feynman graphs (which, incidentally, equals the Lie

group assigned to the Lie algebra with universal

enveloping algebra the dual of this Hopf algebra) to

the rather concrete group of diffeomorphisms of

physical observables. These steps are given as

follows:



Recognize that Z factors are given as cou nter-

terms over a formal series of graphs starting with

1, graded by powers of the coupling, hence

invertible.



Recognize the series Z

as a formal diffeomorph-

ism, with Hopf algebra coefficients.



Establish that the two competing Hopf algebra

structures of diffeomorphisms and graphs are

consistent in the sense of a Hopf algebra

homomorphism.



Show that this homomorphism transposes to a Lie

algebra and hence Lie group homom orphism.

The effective coupling g

eff

(") now allows for a

Birkhoff decomposition in the space of formal

diffeomorphisms.

Theorem 4 Let the unrenormalized effective cou-

pling constant g

eff

(") viewed as a formal power

series in g be considered as a loop of formal

diffeomorphisms and let g

eff

(") = (g

eff



)

1

(") g

eff

(")

be its Birkhoff decomposition in the group of formal

diffeomorphisms. Then the loop g

eff



(") is the bare

coupling constant and g

eff

(0) is the renormalized

effective coupling.

The above results hold as they stand for any

massless theory which provides a single coupling

constant. If there are multiple interaction terms

in the Lagrangian, one finds similar results relat-

ing the group of characters of the corresponding

Hopf algebra to the group of formal diffeomorph-

isms in the multidimensional space of coupling

constants.

The Role of Hochschild Cohomology

The Hochs child cohomology of the combinatorial

Hopf algebras which we discuss here plays three

major roles in quantum field theory:

1. it allows one to prove locality from the accom-

panying filtration by the augmentation degree

coming from the kernels ker Aug

(k)

;

684 Hopf Algebra Structure of Renormalizable Quantum Field Theory

2. it allows one to write the quantum equations of

motion in terms of the Hopf algebra primitives,

elements in H

lin

\ { ker Aug

(2)

=ker Aug

(1)

}; and

3. it identifies the relevant sub-Hopf algebras

formed by time-ordered products.

Before we discuss these properties, let us first

introduce the relevant Hochs child cohomology.

Hochschild Cohomology of Bialgebras

Let (A, m, I, , ) be a bialgebra, as before. We

regard linear maps L : A !A

n

as n-cochains and

define a coboundary map b, b

= 0by

bL :¼ðid  LÞ þ

i¼1

ð1Þ



 L

þð1Þ

nþ1

L  I ½55

where 

denotes the coproduct applied to the ith

factor in A

n

, which defines the Hochschild coho-

mology of A.

For the case n = 1, for L : A !A, [55] reduces to

bL ¼ðid  LÞ    L þ L  I ½56

The category of objects (A, C), which consists of

a commutative bialgebra A and a Hochschild

1-cocycle C on A, has an initial object (H

, B

where H

is the Hopf algebra of (nonplanar) rooted

trees, and the closed but nonexact 1-cocycle B

grafts a product of rooted trees together at a new

root as described below.

The higher (n > 1) Hochschild cohomology of H

vanishes, but in what follows, the closedness of B

will turn out to be crucial.

The Hopf Algebra of Rooted Trees

A rooted tree is a simply connected contractible

compact graph with a distinguished vertex, the root.

A forest is a disjoint union of rooted trees.

Isomorphisms of rooted trees or forests are iso-

morphisms of graphs preserving the distinguished

vertex/vertices. Let t be a rooted tree with root o.

The choice of o determines an orientation of the

edges of t, away from the root, say. Forests are

graded by the number of vertices they contain.

Let H

be the free commutative algebra generated

by rooted trees. The commutative product in H

corresponds to the disjoint union of trees, such

that monomials in H

are scalar multiples of forests.

We demand that the linear operator B

on H

defined by

ðIÞ¼ ½57

ðt

...t

Þ¼

½58

is a Hochschild 1-cocycle, which makes H

a Hopf

algebra. The resulting coproduct can be described as

follows:

ðtÞ¼I  t þ t  I þ

adm c

ðtÞR

ðtÞ½59

where the sum goes over all admissible cuts of the

tree t. Such a cut of t is a nonempty set of edges of t

that are to be removed. The forest which is

disconnected from the root upon removal of those

edges is denoted by P

(t) and the part which remains

connected to the root is denoted by R

(t). A cut c(t)

is admissible if, for each vertex l of t, it contains at

most one edge on the path from l to the root.

This Hopf algebra of nonplanar rooted trees is the

universal object after which all such commutative

Hopf algebras H providing pairs (H, B), for B a

Hochschild 1-cocycle, are formed.

Theorem 5 The pair (H

, B

), unique up to

isomorphism, is universal among all such pairs. In

other words, for any pair (H, B) where H is a

commutative Hopf algebra and B a closed nonexact

1-cocycle, there exists a unique Hopf algebra

morphism H



H such that B   =   B

This theorem suggests that we investigate the

Hochschild cohomology of the Hopf algebras of 1PI

Feynman graphs. It clarifies the struc ture of 1PI

Green functions.

The Roles of Hochschild Cohomology

The Hochschild cohomology of the Hopf algebras of

1PI graphs sheds light on the structure of 1PI Green

function in at least four different ways:



it gives a coherent proof of locality of counter-

terms – the very fact that

½Z

; D

¼0 ½60

means that the coefficients in the Lagrangian

remain independent of momenta, and hence the

Lagrangian remains a polynomial expression in

fields and their derivatives;



the quantum equation of motions takes a very

succinct form, identifying the Dyson kernels with

the primitives of the Hopf algebra;



sub-Hopf algebras emerge from the study of the

Hochschild cohomology, which connects the repre-

sentation theory of these Hopf algebras to the

structure of theories with internal symmetries; and



these Hopf algebras are intimately connected to

the structure of transcendental functions, such as

Hopf Algebra Structure of Renormalizable Quantum Field Theory 685