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interaction. The counter-terms are expanded in

powers of , and then all graphs involving counter-

term vertices at the chosen order in  are added to the

calculation. The counter-terms are arranged to cancel

all the divergences, so that the UV regulator can be

removed, with m and  held fixed. The counter-terms

cancel the parts of the basic Feynman graphs asso-

ciated with large loop momenta. An algorithmic

specification of the otherwise arbitrary finite parts of

the counter-terms is called a renormalization prescrip-

tion or a renormalization scheme. Thus, it gives a

definite relation between the renormalized and bare

parameters, and hence a definite specification of the

partitioning of L into its three parts.

It has been proved that this procedure works to all

orders in , with corresponding results for other

theories. Even in the absence of fully rigorous

nonperturbative proofs, it appears clear that the results

extend beyond perturbation theory, at least in asymp-

totically free theories like QCD: see the discussion on

Wilsonian RG (see Exact Renormalization Group).

Dimensional Regularization

and Minimal Subtraction

The final result for renormalized graphs does not

depend on the particular regularization procedure.

A particularly convenient procedure, especially in

QCD, is dimensional regularization, where diver-

gences are removed by going to a low spacetime

dimension n. To make a useful regularization method,

n is treated as a continuous variable, n = 4  2.

Great advantages of the method are that it

preserves Poincare´ invariance and many other

symmetries (inc luding the gauge symmetry of

QCD), and that Feynman graph calcu lations are

minimally more complicated than for finite graphs

at n = 4, parti cularly when all the lines are massless,

as in many QCD calculations.

Although there is no such object as a genuine

vector space of finite noninteger dimension, it is

possible to construct an operation that behaves as if

it were an integration over such a space. The

operation was proved unique by Wilson, and

explicit constructions have been made, so that

consistency is assured at the level of all Feynman

graphs. Whether a satisfactory definition beyond

perturbation theory exists remains to be determined.

It is convenient to arrange that the renormalized

coupling is dimensionless in the regulated theory.

This is done by changing the normalization of  with

the aid of an extra parameter, the unit of mass :



¼ 

2

 þ counter-termsðÞ½15

with  and  being held fixed when  !0. (Thus,

the basic interaction in eqn [13] is changed to



2



=4!.) Then for the one-loop graph of eqn [2],

dimensionally regularized Feynman parameter meth-

ods give

i

Iðp; m;Þ¼

i

32

ð4Þ



ðÞ



 p

xð1  xÞi0







½16

A natural renormalization procedure is to subtract

the pole at  = 0, but it is convenient to accompany

this with other factors to remove some universally

occurring finite terms. So

MS renormalization

(‘‘modified minimal subtraction’’) is defined by

using the counter-term

iAðÞ

¼i





32



½17

where S



def

(4 e



)



, with 

= 0.5772 ... being the

Euler constant. This gives a renormalized integral (at

 = 0)



i

32

dx ln

 p

xð1  xÞ





½18

which can be evaluated easily. A particularly simple

result is obtained at m = 0:

i

32

ln

p



þ 2



½19

This formula symptomizes important and very

useful algorithmic simplifications in the higher-

order massless calculations common in QCD.

The

MS scheme amou nts to a de facto standard

for QCD. At higher orders a factor of S



is used in

the counter-terms, with L being the number of

loops.

Coordinate Space

Quantum fields are written as if they are functions

of x, but they are in fact distributions or generalized

functions, with quantum-mechanical operator

values. This indicates that using products of fields

is dangerous and in need of careful definition. The

relation with ordinary distribution theory is simplest

in the coordinate-space version of Feynman graphs.

Indeed in the 1950s, Bogoliubov and Shirkov

formulated renormalization as a problem of

defining products of the singular numeric-valued

distributions in coordinate-space Feynman graphs;

theirs was perhaps the best treatment of renormali-

zation in that era.

402 Renormalization: General Theory

For example, the coordinate-space version of

eqn [5] is



lim

a!0

x d

yfðx; yÞ



Sðx  y; m; aÞ

þ iAðaÞ

ð4Þ

ðx  yÞ

½20

where x and y are the coordinates for the interaction

vertices, f (x, y) is the product of external-line free

propagators, and

S(x  y; m, a) is the coordinate-

space free propagator, which at a = 0 has a

singularity

4

½ðx  yÞ

þ i0

½21

as (x  y)

!0. We see in eqn [20] a version of the

Hadamard finite part of a divergent integral, and

renormalization theory generalizes this to particular

kinds of arbitrarily high-dimension integrals. The

physical realization and justification of the use of

the finite-part procedure is in terms of renormaliza-

tion of parameters in the Lagrangian; this also gives

the procedure a significance that goes beyond the

integrals them selves and involves the full nonpertur-

bative formulation of QFT.

General Counter-Term Formulation

We have written L as a basic Lagrangian density

plus counter-terms, and have seen in an example

how to cancel divergences at one-loop order. In this

section, we will see how the procedure works to all

orders. The central mathematical tool is Bogoliubov’s

R-operation. Here the counter-terms are expanded

as a sum of terms, one for each basic one-particle

irreducible (1PI) graph with a non-negative degree

of divergence. To each basic graph for a Green

function is added a set of counter-term graphs

associated with divergences for subgraphs. The

central theorem of renormalization is that this

procedure does in fact remove all the UV diver-

gences, with the form of the coun ter-terms being

determined by the simple computation of the degree

of divergence for 1PI graphs.

To see the essential difficulty to be solved, consider

a two-loop graph like the first one in Figure 3.Its

divergence is not a polynomial in external momenta,

and is therefore not canceled by an allowed counter-

term. This is shown by differentiation with respect to

external momenta, which does not produce a finite

result because of the divergent one-loop subgraph.

But for consistency of the theory, the one-loop

counter-terms already computed must be themselves

put into loop graphs. Among others, this gives the

second graph of Figure 3, where the cross denotes

that a counter-term contribution is used. The

contribution used here is actually 2/3 of the total

one-loop counter-term, for reasons of symmetry

factors that are not fully evident at first sight. The

remainder of the one-loop coupling renormalization

cancels a subdivergence in another two-loop graph.

It is readily shown that the divergence of the sum of

the first two graphs in Figure 3 is momentum

independent, and thus can be canceled by a vertex

counter-term.

This method is fully general, and is formalized in

the Bogoliubov R-operation, which gives a recursive

specification of the renormalized value R(G)ofa

graph G:

RðGÞ¼

def

G þ

f

;...;



!Cð

½22

The sum is over all sets of nonintersecting 1PI

subgraphs of G, and the notation Gj



!C(

)

denotes

G with all the subgraphs 

replaced by associated

counter-terms C(

). The counter-term C()ofa1PI

graph  has the form

CðÞ¼

def

 T  þ counter-te rmsð

for subdivergencesÞ½23

Here T is an operation that extracts the divergent

part of its argument and whose precise definition

gives the renormalization scheme. For example, in

minimal subtraction we define

TðÞ¼pole part at  ¼ 0of ½24

We formalize the term inside parentheses in eqn

[23] as



RðÞ¼

def

 þ counterterms for subdivergences

¼ þ

f

;...;



!Cð

½25

where the prime on the

denotes that we sum over

all sets of nonintersecting 1PI subgraphs except for

the case that there is a single 

equal to the whole

graph (i.e., the term with n = 1and

=  is

omitted).

Note that, for the

MS scheme, we define the T

operation to be applied to a factor of constant

dimension obtained by taking the appropriate power

of 



outside of the pole-part operation. Moreover,

it is not a strict pole-part operation; instead each

Figure 3 A two-loop graph and its counter-terms. The label B

indicates that it is the two-loop overall counter-term for this graph.

Renormalization: General Theory 403

pole is to be multiplied by S



, where L is the

number of loops, and S



is defined after eqn [17].

Equations [22]–[25] give a recursive construction

of the renormalization of an arbitrary graph. The

recursion starts on one-loop graphs, since they have

no subdivergences, that is, C() = T() for a one-

loop 1PI graph.

Each counter-term C() is implemented as a

contribution to the counter-term Lagrangian. The

Feynman rules ensure that once C() has been

computed, it appears as a vertex in bigger graphs

in such a way as to give exactly the counter-terms

for subdivergences used in the R-operation. It has

been proved that the R-operation does in fact give

finite results for Feynman graphs, and that basic

power counting in exactly the same fashion as at

one-loop determines the relevant operators.

In early treatments of renormalization, a pro blem

was caused by graphs like Figure 4. This graph has

three divergent subgraphs which overlap, rather

than being nested. Within the R-operation approach,

such cases are no harder to deal with than merely

nested divergences.

The recursive specification of R-operation can be

converted to a nonrecursive formulation by the

forest formula of Zavyalov and Stepanov, later

rediscovered by Zimmerman. It is normally the

recursive formulation that is suited to all-orders

proofs.

Whether these results, proved to all orders of

perturbation theory, genuinely extend to the com-

plete theory is not so easy to answer, certainly in a

realistic four-dimensional QFT. One illuminating

case is of a nonrelativistic quantum mechanics

model with a delta-function potential in a two-

dimensional space. Renormalizati on can be applied

just as in field theory, but the model can also be

treated exactly, and it has been shown that the

results agree with perturbation theory.

Perturbation series in relativistic QFTs can at best

be expected to be asymptotic, not convergent. So

instead of a radius of convergence, we should talk

about a region of applicability of a weak-coupling

expansion. In a direct calculation of counter-terms,

etc., the radius of a pplicability shrinks to zero as the

regulator is removed. However, we can deduce the

expansion for a renormalized quantity, whose

expansion is expected to have a nonzero range of

applicability. We can therefore appeal to the

uniqueness of power series expansions to allow the

calculation, at intermediate stages, to use bare

quantities that are divergent as the regulator is

removed.

Renormalizability, Non-Renormalizability,

and Super-Renormalizability

The basic power-counting method shows that if a

theory with conventional fields (at n = 4) has only

operators of dimension 4 or less in its L, then the

necessary counter-term operators are also of dimen-

sion 4 or less. So if we start with a Lagrangian with

all possible such operators, given the field content,

then the theory is renormalizable. This is not the

whole story, as we will see in the discu ssion of gauge

theories.

If we start with a Lagrangian containing operators

of dimension higher than 4, then renormalization

requires operators of ever higher dimension as

counter-terms when one goes to higher orders in

perturbation theory. Therefore, such a theory is said

to be perturbatively non-renormalizable. Some very

powerful methods of cancelation or some nonper-

turbative effects are needed to evade this result.

In the case of dimension-4 interactions, there is

only a finite set of operators given the set of basic

fields, but divergences occur at arbitrarily high

orders in perturbation theory. If, instead, all the

operators have at most dimension 3, then only a

finite number of graphs need counter-terms. Such

theories are called super-renormalizable. The diver-

gent graphs also occur as subgraphs inside bigger

graphs, of course. There is only one such theory in a

four-dimensional spacetime: 

theory, which suf-

fers from an energy density that is unbounded from

below, so it is not physical. In lower spacetime

dimension, where the requirements on operator

dimension are different, there are many more

known super-renormalizable theories, some with a

very rigorous proof of existence.

All the above characterizat ions rely primarily on

perturbative analysis, so they are subject to being

not quite accurate in an exact theory, but they form

a guide to the relevant issues.

Renormalization and Symmetries:

Gauge Theories

In most physical applications, we are interested in

QFTs whose Lagrangian is restricted to obey certain

symmetry requirements. Are these symmetries pre-

served by renormalization? That is, is the Lagran-

gian with all necessary counter-terms still invariant

under the symmetry?

Figure 4 Graph with overlapping divergent subgraphs.

404 Renormalization: General Theory

We first discuss nonchiral symmet ries; these are

symmetries in which the left-handed and right-

handed parts of Dirac fields transform identically.

For Poincare´ invariance and simple global internal

symmetries, it is simplest to use a regulator, like

dimensional regularization, which respects the sym-

metries. Then it is easily shown that the symmetries

are preserved under renormalization. This holds

even if the internal symmetries are spontaneously

broken (as happens with a ‘‘wrong-sign mass term,’’

e.g., negative m

in eqn [1]).

The case of local gauge symmetries is harder. But

their preservation is more important, because gauge

theories contain vector fields which, without a gauge

symmetry, generally give unphysical features to the

theory. For perturbation theory, BRST quantization

is usually used, in which, instead of gauge symme-

try, there is a BRST supersymmetry. This is

manifested at the Green function level by Slavnov–

Taylor identities that are more complicated, in

general, than the Ward identities for simple global

symmetries and for abelian local symmetries.

Dimensional regularization preserves these

symmetries and the Slavnov–Taylor identities. More-

over, the R-operation still produces finite results with

local counter-terms, but cancelations and relations

occur between divergences for different graphs in

order to preserve the symmetry. A simple example is

QED, which has an abelian U(1) gauge symmetry, and

whose gauge-invariant Lagrangian is

L¼



ð0Þ



 @



ð0Þ







i



 e

ð0Þ



 m



½26

At the level of individual divergent 1PI graphs,

we get counter-terms proportional to A



and to



)

, operators not present in the gauge-invariant

Lagrangian. The Ward identities and Slavnov–Taylor

identities show that these counter-terms cancel when

they are summed over all graphs at a given order of

renormalized perturbation theory. Moreover, the

renormalization of coupling and the gauge field are

inverse, so that e

(0)



equals the corresponding

object with renormalized quantities, 





. Natu-

rally, sums of contributions to a counter-term in

L can only be quantified with use of a regulator.

In nonabelian theories, the gauge-invariance proper-

ties are not just the absence of certain terms in L but

quantitative relations between the coefficients of terms

with different numbers of fields. Even so, the argument

with Slavnov–Taylor identities generalizes appropri-

ately and proves renormalizability of QCD, for

example. But note that the relation concerning the

product of the coupling and the gauge field does not

generally hold; the form of the gauge transformation is

itself renormalized, in a certain sense.

Anomalies

Chiral symmetries, as in the weak-in teraction part of

the gauge symmetry of the standard model, are

much harder to dea l with. Chiral symmetries are

ones for which the left-handed and right-handed

components of Dirac field transform independently

under different components of the symmetry group,

local or global as the case may be. Occasionally,

some or other of the left-handed or right-handed

components may not even be present.

In general, chiral symmetries are not preserved by

regularization, at least not without some other

pathology. At best one can adjust the finite parts of

counter-terms such that in the limit of the removal of

the regulator, the Ward or Slavnov–Taylor identities

hold. But in general, this cannot be done consistently,

and the theory is said to suffer from an anomaly. In

the case of chiral gauge theories, the presence of an

anomaly prevents the (candidate) theory from being

valid. A dramatic and nontrivial result (Adler–

Bardeen theorem and some nontrivial generaliza-

tions) is that if chiral anomalies cancel at the

one-loop level, then they cancel at all orders.

Similar results, but more difficult ones, hold for

supersymmetries.

The anomaly cancelation conditions in the standard

model lead to constraints that relate the lepton content

to the quark content in each generation. For example,

given the existence of the b quark, and the  and 



leptons (of masses around 4.5 GeV, 1.8 GeV, and zero

respectively), it was strongly predicted on the grounds

of anomaly cancelation that there must be a t quark

partner of the b to complete the third generation of

quark doublets. This prediction was much later

vindicated by the discovery of the much heavier top

quark with m

’ 175 GeV.

Renormalization Schemes

A precise definition of the counter-terms entails

a specification of the renormalization prescription

(or scheme), so that the finite parts of the counter-

terms are determined. This apparently induces extra

arbitrariness in the results. However, in the 

Lagrangian (for example), there are really only two

independent parameters. (A scaling of the field does

not affect any observables, so we do not count Z as

a parameter here.) Thus, at fixed regulator para-

meter a or , renormalization actually just gives a

reparametrization of a two-parameter collection of

theories. A renormalization prescription gives the

Renormalization: General Theory 405

change of variables between bare and renormalized

parameters, a rather singular transformation when

the regulator is removed. If we have two different

prescriptions, we can deduce a transformation

between the renormalized parameters in the two

schemes. The renormalized mass and coupling m

and 

in one scheme can be obtained as functions

of their values m

and 

in the other scheme, with

the bare parameters, and hence the physics, being

the same in both schemes. Since these are renorma-

lized parameters, the removal of the regulator leaves

the transformation well behaved.

Generalization to all renormalizable theori es is

immediate.

Renormalization Group and Applications

and Generalizations

One part of the choice of renormalization scheme is

that of a scale parameter such as the unit of mass  of

the

MS scheme. The physical predictions of the theory

are invariant if a change of  is accompanied by a

suitable change of the renormalized parameters, now

considered as -dependent parameters ()andm().

These are called the effective, or running, coupling and

mass. The transformation of the parametrization of

the theory is called an RG transformation.

The bare coupling and mass 

and m

are RG

invariant, and this can be used to obtain equations

for the RG evolution of the effective parameters

from the perturbatively computed counter-terms.

For example, in 

theory, we have (in the

renormalized theory after removal of the regulator)

d

dln

¼ ðÞ½27

with () = 3

=(16

) þ O(

). As exemplified in

eqns. [18] and [19], Feynman diagrams depend

logarithmically on . By choosing  to be comparable

to the physical external momentum scale, we remove

possible large logarithms in this and higher orders.

Thus, provided that the effective coupling at this scale

is weak, we get an effective perturbation expansion.

This is a basic technique for exploiting perturba-

tion theory in QCD, for the strong interact ions,

where the interactions are not automatically weak.

In this theory the RG  function is negative so that

the coupling decreases to zero as  !1; this is the

asymptotic freedom of QCD.

A closely related method is that associated with

the Callan–Symanzik equation, which is a formula-

tion of a Ward identity for anomalously broken

scale invariance. However, RG methods are the

actually used ones, normally, even if sometimes an

RG equation is incorrectly labeled as a Callan–

Symanzik equation.

The elementary use of the RG is not sufficient for

most interesting processes, which involve a set of

widely different scales. Then more powerful theo-

rems come into play. Typical are the factorization

theorems of QCD (see Quantum Chromodynamics).

These express differential cross sections for certain

important reactions as a product of quantities that

involve a single scale:

d ¼CQ;;ðÞðÞfm;;ðÞðÞ

þ small correction ½28

The product is typically a matrix or a convolution

product. The factors obey nontrivial RG equations,

and these enable different values of  to be used in

the diff erent factors. Predictions arise because some

factors and the kernels of the RG equation are

perturbatively calculable, with a weak effective

coupling. Other factors, such as f in eqn [28], are

not perturbative. These are quantities with names

like ‘‘parton distribution functions,’’ and they are

universal between many different processes. Thus,

the nonpe rturbative functions can be measu red in a

limited set of reactions and used to predict cross

sections for many other reactions with the aid of

calculations of the perturbative factors.

Ultimately, this whole area depends on physical

phenomena associated with renormalization.

Concluding Remarks

The actual ability to remove the divergences in

certain QFTs to produce consistent, finite, and

nontrivial theories is a quite dramatic result. More-

over, associated with the integrals that give the

divergences is behavior of the kind that is analyzed

with RG methods and generalizations. So the

properties of QFTs associated with renormalization

get tightly coupled to many interesting consequences

of the theories, most notably in QCD.

QFTs are actually very abstruse and difficult

theories; only certain aspects currently lend them-

selves to practical calculations. So the reader should

not assume that all aspects of their rigorous

mathematical treatment are perfect. Experience,

both within the theories and in their comparison

with experiment, indicates, nevertheless, that we

have a good approximation to the truth.

When one examines the mathematics associated

with the R-operation and its generalizations with

factorization theorems, there are clearly present

some interesting mathematical structures that are

not yet formulated in their most general terms. Some

406 Renormalization: General Theory

indications of this can be seen in the work by

Connes and Kreim er (see Hopf Algebra Structure of

Renormalizable Quantum Field Theory), where it is

seen that renormalization is associated with a Hopf

algebra structure for Feynman graphs.

With such a deep subject, it is not surprising that

it lends itself to other approaches, notably the

Connes–Kreimer one and the Wilsonian one (see

Exact Renormalization Group). Readers new to the

subject should not be surprised if it is difficult to get

a fully unified view of these differen t approaches.

Notes on Bibliography

Reliable textbo oks on quantum field theory

are Sterman (1993) and Weinberg (1995). A clear

account of the foundations of perturbative QCD

methods is given by Sterman (1996).

A pedagogical account of renormalization and

related subjects may be found in Collins (1984).

The best account of renormalization theory before

the 1970s is given by Bogoliubov and Shirkov

(1959); the viewpoint is very modern, including a

coordinate-space distribution-theoretic view. A

full account of the Wilsonian method as applied

to renormalization is given by Polchinski (1984).

Manuel and Tarrach (1994) give an excellent account

of renormalization for a theory with a non-relativistic

delta-function potential in 2 space dimensions, which

provides a fully tractable model.

Tkachov (1994) reviews a systematic application

of distribution theoretic methods to asymptotic

problems in QFT. Finally, Weinzierl (1999) provides

a construction of dimensional regularization with the

aid of K-theory using an underlying vector space of

the physical integer dimension. Other constructions,

referred to in this paper, follow Wilson and use an

infinite-dimensional underlying space.

Acknowledgments

The author would like to thank Professor K Goeke

for hospitality and support at the Ruhr-Universita¨t-

Bochum. This work was also supported by the US DOE.

See also: Anomalies; BRST Quantization; Effective

Field Theories; Electroweak Theory; Euclidean Field

Theory; Exact Renormalization Group; High T

Superconductor Theory; Holomorphic Dynamics; Hopf

Algebra Structure of Renormalizable Quantum Field

Theory; Lattice Gauge Theory; Operator Product

Expansion in Quantum Field Theory; Perturbation Theory

and its Techniques; Perturbative Renormalization Theory

and BRST; Quantum Chromodynamics; Quantum Field

Theory: A Brief Introduction; Singularities of the Ricci

Flow; Standard Model of Particle Physics; Supergravity.

Further Reading

Bogoliubov NN and Shirkov DV (1959) Introduction to the

Theory of Quantized Fields. New York: Wiley-Interscience.

Collins JC (1984) Renormalization. Cambridge: Cambridge

University Press.

Kraus E (1998) Renormalization of the electroweak standard

model to all orders. Annals of Physics 262: 155–259.

Manuel C and Tarrach R (1994) Perturbative renormalization in

quantum mechanics. Physics Letters B 328: 113–118.

Polchinski J (1984) Renormalization and effective Lagrangians.

Nuclear Physics B 231: 269–295.

Sterman G (1993) An Introduction to Quantum Field Theory.

Cambridge: Cambridge University Press.

Sterman G (1996) Partons, factorization and resummation. In:

Soper DE (ed.) QCD and Beyond, pp. 327–406. (hep-ph/

9606312). Singapore: World Scientific.

Tkachov FV (1994) Theory of asymptotic operation. A summary of

basic principles. Soviet Journal of Particles and Nuclei 25: 649.

Weinberg S (1995) The Quantum Theory of Fields, Vol. I.

Foundations. Cambridge: Cambridge University Press.

Weinzierl S (1999) Equivariant dimensional regularization, hep-

ph/9903380.

Zavyalov OI (1990) Renormalized Quantum Field Theory.

Dordrecht: Kluwer.

Renormalization: Statistical Mechanics and Condensed Matter

M Salmhofer, Universita

t Leipzig, Leipzig, Germany

Renormalization Group

and Condensed Matter

Statistical mechanical systems at critical points

exhibit scaling laws of order parameters, susceptibi-

lities, and other observables. The exponents of these

laws are universal, that is, independent of most

details of the system. For example, the liquid–gas

transition for real gases has the same exponents as

the magnetization transition in the three-dimensional

Ising model.

The renormalization group (RG) was developed

by Kadanoff, Wilson, and Wegner, to understand

these critical phenomena (Domb and G reen 1976).

The central idea is that the system becomes scale

invariant at the critical point, which makes it

natural to average over degrees of freedom on

increasing length scales successively in the

Renormalization: Statistical Mechanics and Condensed Matter 407

calculation of the partition function. This leads to a

map between effective interactions associated to

different length scales. Thus, the focus shifts from

the analysis of a single interaction to that o f a flow

on a space of interactions. This space is in general

much larger than the original formulation of the

model would suggest: the description of long-

distance or low-energy properties may be in terms

of variables that were not even present in the

original formulation of the system. Phenomeno-

logically, this corresponds to the emergence of

collective degrees of freedom.

Condensed matter theory is itself already an

effective theory, and its ‘‘microscopic’’ formulation

gets inputs from the underlying theories, which

determine in particular the statistics of the particles

and their interactions at the scale of atomic energies.

At much lower-energy scales, which are relevant for

low-temperature phenomena in condensed matter,

collective excitations of different, sometimes exotic,

statistics may emerge, but the starting point is given

naturally in terms of fermionic and bosonic parti-

cles. For this reason, the discussion given below will

be split in these two cases.

A major difference between high-energy and

condensed matter systems is that the latter have a

well-defined Hamiltonian which can be used to

define the finite-volume ensembles of quantum

statistical mechanic s and which determines the time

evolution, as well as various analyticity properties.

The relevant spatial dimensions in condensed

matter are d  3, but some results in higher

dimensions relevant for the development of the

method will also be discussed below. The cases

d = 1 and d = 2 have always been of mathematical

interest but in recent years have become important

for the theory of new materials.

Some interesting topics cannot be covered here

due to space restrictions, notably the application of

renormalization methods to membrane theory (see

Wiese (2001)) and renormalization methods for

operators (see Bach et al. (1998)).

The Renormalization Group

In this section we briefly describe the setup of two

important versions of the RG, namely the block spin

RG and the RG based on scale decompositions of

singular covariances.

Block spin RG

Let  be a finite lattice, for example, a finite subset

of Z

. For the following, it is convenient to take 

to be a cube of side-length L

for L > 1 and some

large K. Let T be a set and 



= { :  !T } be the

set of spin configurations. Common examples for

the target space T are T = {1, 1} for the Ising

model, T = S

N1

for the O(N) model, and T = R

for unbounded spins. Let S



: 



!R,  7!S



()be

an interaction and

Zð; S



Þ¼

x2

dðxÞe

S



ðÞ

½1

In the unbounded case, S



is assumed to grow

sufficiently fast for jj!1, so that Z exists; for the

case of a finite set T, the integral is replaced by a

sum. Denote the corresponding Boltzmann factor by

(, S



ð; S



ÞðÞ¼

Zð; S



S



ðÞ

½2

The block spin transformation consists of an

integration step and a rescaling step. Divide the

lattice into cubic blocks of side-length L and define

a new lattice 

by associating one lattice site of the

new lattice to each L-block of the old lattice. For

any 

: 

!T, let



ð

Þ¼

x2

dðxÞPð

;Þe

S



ðÞ

½3

where P(

, )  0 and

2

d

)P(

, ) = 1

for all , so that 

remains a probability distribu-

tion. Since 

is positive, one defines



ð

Þ¼log 

ð

Þ½4

By construction, the partition function is invariant:

Z(

, S



) = Z(, S



). The new lattice 

has spacing L;

now rescale to make it a unit lattice. This completes

the RG step in finite volume.

In an algorithmic sense, the ‘‘blocking rule’’

P(

, ) can be viewed as a transition probability of

a configuration  to a configuration 

. P may be

deterministic, that is, simply fix 

as a function

of . From the intuition of averaging over local

fluctuations, 

is often taken to be some average of

(x)atx in a block around x

, hence the name.

Obviously, the thus defined RG transformation

often cannot be iterated arbitrarily, since in every

application, the number of points of the lattice shrinks

byafactorL

,sothatafterK iterations, a lattice with

only a single point is left over. It is necessary to take the

infinite-volume limit L !1 to obtain a map that

operates from a space to itself. However, [4] can

become problematic in that limit: Gibbs measures 

can map to measures 

whose large-deviation proper-

ties differ from those of Gibbs measures. The discus-

sion of this problem and its solution is reviewed in

Bricmont and Kupiainen (2001). The problem can be

408 Renormalization: Statistical Mechanics and Condensed Matter

solved in different ways, relaxing conditions on Gibbs

measures or, in the Ising model, changing the descrip-

tion from the spins to the contours. The crucial point is

that the difficulties arise only because [4] is applied

globally, that is, to every 

.Thesetofbad

has very

small probability.

Block spin methods have been used in mathema-

tical construction of quantum field theories, for

example, in the work of Gawedzki and Kupiainen

(1985) and Balaban (1988) (see the subsection

‘‘Field theory and statistical mechanics’’). The

above-mentioned problem was avoided there by

not taking a logarithm in the so-called large-field

region (which has very small probab ility).

Scale Decomposition RG

The generating functionals of quantum field theory

and quantum statistical mechanics can be cast into

the form

ZðC; V;Þ¼

d

ð

Þe

Vð

þÞ

½5

Here d

denotes the Gaussian measure with covar-

iance C,andV is the two-body interaction between the

particles. The field variables are real or complex for

bosons and Grassmann-valued for fermions. Differ-

entiating log Z with respect to the external field 

generates the connected amputated correlation func-

tions. The covariance determines the free propagation

of particles; the interaction their collisions.

In most cases, such functional integrals are a priori

ill-defined, even if V is small (and bounded from

below) because the covariance C is singular. That is,

the integral kernel C(X, X

)oftheoperatorC either

diverges as jx  x

j!0 (ultraviolet (UV) problem) or

C(X, X

) has a slow decay as jx  x

j!1 (infrared

(IR) problem). In our notational convention, X may,

in addition to the configuration variable x,also

contain discrete indices of the fields, such as a spin or

color index. The dependence of C on x and x

assumed to be of the form x  x

. A typical example

is the massless Gaussian field in d dimensions, where

C is the inverse Fourier transform of

C(k) = 1=k

k 2 R

, which has both a UV and an IR problem, or

its lattice analog,

DðkÞ¼

i¼1

ð1  cosðak

1

with a the lattice constant, which has only an IR

problem. A typical interaction is of the type

VðÞ¼

dX dY



ðXÞðXÞvðX; YÞ



ðYÞðYÞ½6

Again, we assume that the potential v depends on x

and y only via x  y, so that translation invariance

holds. In both UV and IR cases, naive perturbation

theory fails even as a formal power series. That is,

writing V = V

, with a coupling constant  which is

treated as a formal expansion parameter, the singu-

larity of C leads to termwise divergences in the series.

The theory is called perturbatively renormalizable if

all divergences can be removed by posing counter-

terms of certain types, which are fixed by physically

sensible renormalization conditions. Identifying the

UV renormalizable theories was a breakthrough in

high-energy physics. The IR renormalization problem

is different, and in some respects harder, because

there is almost no freedom to put counter-terms: the

microscopic model is given from the start. This will

be discussed in more detail below for an example.

A much more ambitious, and largely open, project

is to do this renormalization nonperturbatively, that

is, to treat  as a real (typically, small) parameter.

Some results will be discussed below.

The RG is set up by a scale decomposition

C =

. In the example of the massless Gaussian

field, one would take each

to be a C

function

supported in the region {k 2 R

: M

 k

 M

jþ1

where M > 1 is a fixed constant, a nd the summation

over j runs over Z.

The scale decomposition of C leads to a represen-

tation of [5] by an iteration of Gaussian convolution

integrals with covariances C

, hence a sequence of

effective interactions V

, defined recursively by

V

ðÞ

d

jþ1

ð

Þe

V

jþ1

ð

þÞ

; V

¼ V ½7

For a singular covariance, the scale decomposition is

an infinite sum. A formal object like [5] is now

regularized by starting with a finite sum, that is,

imposing a UV and IR cutoff, which is mathemati-

cally well defined, and then taking limits of the thus

defined objects. Again, in condensed matter applica-

tions, imposing an IR cutoff is an operation that

needs to be justified, for example, by showing that

taking the limit as the cutoff is removed commutes

with the infinite-volume limit.

Note that the RG map, which is the iteration

7!V

j1

, goes to lower and lower j, corresponding

to longer and longer length scales. The convention

that the iteration starts at some fixed j, for example,

j = 0, is appropriate for IR problems. In UV

problems, the iteration would start at some large

, which defines a UV cutoff and is taken to

infinity, to remove the cutoff, at the end.

A variant using a continuous scale decomposition,

C =

, originally due to Wegner and Hought on,

became very popular after Polchinski (1984) used it

Renormalization: Statistical Mechanics and Condensed Matter 409

to give a short argument for perturbative renorma-

lizability. Polchinski’s equation, the analog of the

recursion [7], reads

¼



V



V 

V



;

V





½8

Here







; C







denotes the Laplacian in field space associated to the

covariance C. Polchinski’s argument has been devel-

oped into a mathematical tool that applies to many

models. For an introduction to perturbative renor-

malization using this method, see Salmhofer (1998).

Equations of the type [8] have also been very useful

beyond perturbation theory: much work has been

done based on the beautiful representation of Mayer

expansions found in Bryd ges and Kennedy (1987)

using RG equations.

Mathematical Structure and Difficulties

The RG flow is thus, depending on the implementa-

tion, either a sequence or a continuous flow of

interactions. Setting up this flow in mathematical

terms is not easy and indeed part of the mathema-

tical RG analysis is to find a suitable space of

interactions that is left invariant by the succe ssive

convolutions, and then to control the RG iteration.

A serious problem is the proliferation of interac-

tions: already a single application of the RG

transformation [7] maps a simple interaction, such

as [6], to a nonlocal functional of the fields,

ðÞ¼

m0

dX

 v

ðjÞ

ðX

; ...; X

ÞðX

ÞðX

Þ½9

Already for perturbative renormalization, one needs

to extra ct local terms, calculate their flow more

explicitly, and control the power counting of the

remainder. The convergence of the series is not an

issue in formal perturbation theory because in every

finite order r in , the sum over m is finite.

For nonperturbative renormalization, however,

the problem is much more serious. For bosonic

systems, the expansion in powers of the fields in

[9] is divergent, and one needs a split into small-

field and large-field regions and cluster expansions

to obtain a well-defined sequence of effective

actions (Gawedzki and Kupiainen 1985, Feldman

et al. 1987, Rivassean 1993). That is, the local

parts are extracted and treated explicitly only in

the small-field region, and this is combined with

estimates on t he rareness of large-field regions

using cluster expansions. For fermions, the expan-

sion in powers of the fields can be proved to

converge for regular, summable c ovariances, which

leads to substantial technical simplifications.

The spatial proliferation of interactions is absent

only in certain one-dimensional and in specially

constructed higher-dimensional models, the so-

called ‘‘hierarchical models.’’ In these models, the

search for an RG fixed point is still a nonlinear

fixed-point problem, whose treatment leads to

interesting mathematical results.

This article will be restricted t o the mathema-

tical use of the R G both in perturbative and

nonperturbative quantum field theory of con-

densed matter systems. Many nonrigorous but

very interesting applications have also come out

of this method, showing that it also works well in

practice, but they will not be reviewed here. Before

discussing condensed matter systems, the pioneer-

ing works done on the mathematical RG, which

were largely motivated by high-energy physics,

will be reviewed briefly, as they laid the founda-

tion of much of the technique used later in the

condensed matter case.

Field Theory and Statistical Mechanics

Because of the close connection between quantum

field theory and statistical mechanics given by

formulas of the Feynman–Kac type, a significant

amount of work on the mathematical RG focused

on models of classical statistical mechanics in

connection with field theories and gauge theories.

Here we mention some of the pioneering results in

that field.

The scale decomposition method was developed

in a mathematical form and applied to perturbative

UV renormalization of scalar field theories, as well

as nonperturbative analysis of some models, by

Gallavotti and Nicolo`(Gallavotti 1985).

Infrared 

theory in four dimensions was

constructed using block spin methods (Gawedzki

and Kupiainen 1985) and scale decomposition RG

(Feldman et al. 1987). An essential feature of the 

model is its IR asymptotic freedom, meaning that

the local part of the effective quartic interaction

tends to zero in the IR limit.

Block spin methods were used by Balaban (1988)

to construct gauge theories in three and four

dimensions. For gauge theories, the block spin RG

has the major advantage that it allows to define a

gauge-invariant RG flow. The scale decomposition

violates gauge invariance, which creates substantial

technical problems (Rivass eau 1993).

410 Renormalization: Statistical Mechanics and Condensed Matter

Condensed Matter: Fermions

Starting with the seminal work of Feldman and

Trubowitz (1990, 1991) and Benfatto and

Gallavotti (1995), this field has become one of the

most successful applications of the mathematical

RG. We use this example to discuss the scale

decomposition method in a bit more detail.

We shall mainly focus on models in d  2

dimensions (the case d = 1 is described in detail in

Benfatto and Gallavotti (1995)). The system is put

into a finite (very large) box  of side-length L. For

simplicity we take periodic boundary conditions.

The Hilbert space for spin-1/2 electrons is the

fermionic Fock space F =

n0

(, C

). The

grand canonical ensemble in finite volume is given

by the density operator  = Z

1

(HN)

, with the

Hamiltonian H and the number operator N,inthe

usual second quantized form. The parameter

 = T

1

is the inverse temperature and the chemical

potential  is an auxiliary parameter used to fix the

average particle number.

The grand canonical trace defining the ensemble

can be rewritten in functional-integral form. It takes

the form [5], but now d

stands for a Grassmann

Gaussian ‘‘measure,’’ which is really only a linear

functional (for definitions, see, e.g., Salmhofer

(1998, chapter 4 and appendix B)). A two-body

interaction corresponds to a quartic interaction

polynomial V,asin[6]. The covariance is (in the

infinite-volume limit L !1)

Cð;xÞ¼



!2M

ð2Þ

iðkx!Þ

Cð!; kÞ

Cð!; kÞ¼

i!  eðkÞ

½10

where  2 (0, ] is a Euclidian time v ariable and k

is the spatial momentum. The summation over !

runs over the set of fermionic Matsubara frequen-

cies M

= T(2Z þ 1). The function e(k) = "(k)  ,

where "(k) is the band function given by the single-

particle term in the Hamiltoni an. For a lattice

system, k 2B

, the momentum space torus (e.g.,

for the lattice Z

, B

= R

=2Z

); for a continuous

system, k 2 R

, hence there is a spatial UV

problem. Electrons in a crystal have a natural

spatial UV cutoff (see Salmhofer (1998, chapter 4)

for a discussion) so we assume in the following

that there is either a UV cutof f or that the system is

on a lattice. A nonperturbative definition of the

functional integral involves a limit from discrete

times (by the Trotter product formula); see, for

example, Salmhofer (1998) or Feldman et al.

(2003, 2004).

Perturbative Renormalization

Renormalization of the Fermi surface at zero

temperature In the limit T !0, the Matsubara

frequency ! becomes a real variable, hence the

propagator has a singulari ty at ! = 0 and k 2 S,

where S = {k : e(k) = 0}, a codimension-1 subset of

, is the Fermi surface. The existence of a Fermi

surface which does not degenerate to a point is a

characteristic feature of systems showing metallic

behavior.

The singularity implies that

C 62 L

(R B

) for

any p  2. Because terms of the type

dkFð!; kÞ

Cð!; kÞ



p1

i¼1

ð!; kÞ

Cð!; kÞ



½11

appear for all p  1 in the formal perturbation

expansion, with functions T

and F that do not

vanish on the singularity set of C, the perturbation

expansion for observables is termwise divergent.

The deeper reason for these problems is that the

interaction shifts the Fermi su rface so that the true

propagator has a singularity of the form

G(!, k) = (i!  e(k)  (!, k))

1

. If the self-energy 

is a sufficiently regular function, G has the same

integrability properties as C, but the singularity of G

is on the set

S = {k : e(k) þ (0, k) = 0} (the singular-

ity in ! remains at ! = 0).

Let 1 =

j0



(!, k)beaC

partition of unity

such that

for j < 0 supp 

fð!; kÞ: 

j2

ji!  eðkÞj  

g½12

where M > 1 and 

is a fixed constant (an energy

scale determined by the global properties of the

function e; see Salmhofer (1998, chapter 4)). The

corresponding covariances



have the prop-

erties that for j < 0, k

 const.M

and k



const.M

j

. Using these bounds and expanding

(j)

r1

(j)

m, r



, one can derive estimates for the

coefficient functions v

(j)

m, r

Of course, the scale decomposition by itself does

not solve the problem of the moving singularity. It

only allows us to pinpoint the proble matic terms in

the expansion. To construct the self-energy ,as

well as all higher Green functions, a two-step

method is used (Feldman and Trubowitz 1990,

1991, Feldman et al. 1996, 2000). First, a counter-

term function K which modifies e is introduced, so

that all two-point insertions T

get subtracted on

the Fermi surface, hence replaced by

(!, k) =

(!, k)  T

(0, k

), with k

obtained from k by a

Renormalization: Statistical Mechanics and Condensed Matter 411