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symmetries (‘‘orbifold models’’). It then ha ppens

that the invariants not only have the expected

superselection sectors in correspondence with the

representations of the gauge group, but in addition

‘‘twisted’’ sectors appear which, together with the

former, constitute a ‘‘quantum double’’ structure.

The twisted sectors arise by restriction of solitonic

sectors of the original theory, which are in one-to-one

correspondence with the elements of the gauge

group (Mu¨ ger 2005). Solitonic sectors are localiz-

able with respect to two different vacua, and do

not admit an unrestricted composition law.

Special Issues

A particularly simple situation is the case of anyons,

that is, when all sectors have statistical dimension 1.

Then the sectors form an abelian group

G under

fusion, and one can construct a WWW scenario with

global gauge group G the dual of

G. The ensuing

quantum fields satisfy generalized commutation rela-

tions at spacelike separation, given by an abelian

representation of the braid group, where the coeffi-

cients can be arbitrary complex phases (responsible

for the name ‘‘anyons’’). However, it is known that

there can arise an obstruction, which enforces the

‘‘local’’ global gauge transformations (mentioned

before) to be present. In this case, the gauge

symmetry can also be described by a quasiquantum

group. It is noteworthy that free anyon fields have

been constructed in two-dimensional spacetime,

while in three dimensions there can be no (cone-)

localized massive anyon fields which are free in the

sense that they generate only single-particle states

from the vacuum (Mund 1998).

The charge structure of massive quantum field

theories in two dimensions is very different both

from that encountered in conformal quantum field

theories, and from the charge structure in high

dimensions. It has been observed long ago that, in

contrast to four dimensions, the strong locality

property (Haag duality) which is necessary to set

up the DHR analysis of superselection sectors, fails

for the alge bra of invariants under an internal gauge

group in two dimensions. This algebraic feature can

be traced back to the fact that the causal comple-

ment of a point is disconnected in two dimensions,

or, in physical terms, that ‘‘a charge cannot be

transported around a detector’’ without passing

through its region of causal dependence. Mu¨ ger

(1998) has shown that any algebra of observables

which satisfies Haag duality, cannot possess any

nontrivial DHR superselection sectors at all, and

that the only sectors which can exist are solitonic

sectors. This genera l result nicely complies with the

experience with integrable models, as mentioned

before.

There are also some results giving interesting

insight, which can be obtained intrinsically in terms

of the observables. One of them concerns ‘‘central’’

observables (generalized Casimir operators).

Casimir operators in the WWW scenario are

functions of the generators of the internal symmetry

which usually are integrals over densities belonging

to the field algebra F (Noether’s theorem). Since

they also commute with the generators, they can be

approximated by local observables, and are there-

fore defined in each representation of the latter. By

Schur’s lemma, they are multiples of the identity in

each irreducible sector. Since the eigenvalues of

Casimir operators distinguish the representations of

the gauge group, they also distinguish the sectors.

In chiral CFT extended to the circle (see above),

one can find global ‘‘charge measuring operators’’

, one for each sector 

, in the center of the

observable algebra (Fredenhagen et al. 1992) which

have similar properties. They arise as a consequence

of an algebraic obstruction to define the charged

sectors on the circle, related to a nontrivial effect if a

charge is ‘‘transported once around the circle,’’ and

form an operato r representation of the fusion rules

within the global algebra of observables. Under

rather natural conditions clarified by Kawahigashi,

Longo, and Mu¨ ger (2001), the matrix of eigenvalues



) is nondegenerate, that is, the generalized

Casimir operators completely distinguish the super-

selection sectors. In this case, the superselection

category is a modular category (see Braided and

Modular Tensor Categories): the matrix with entries



) and the diagonal matrix with entries 

(U)

(where U is the Mo¨ bius rotation by 2) are multi-

ples of the generators S and T of the ‘‘modular

group’’ PSL(2, Z), in a matrix representation labeled

by the superselection sectors of the chiral observa-

bles. The physical significance of this matrix

representation is that it relates thermal expectati on

values for different values of the temperature (Cardy

1986, Kac and Peterson 1984, Verlinde 1988)

These examples, together with the failure of the

Coleman–Mandula theorem, may illustrate the

intricate relations among spacetime geometry, cov-

ariance, and internal symmetry (charge structure) in

low dimensions. In relativistic quantum field theory,

the link is provided by the principle of locality,

which ‘‘turns geometry into algebra.’’

See also: Algebraic Approach to Quantum Field Theory;

Axiomatic Quantum Field Theory; Braided and Modular

Tensor Categories; Hopf Algebras and q-Deformation

178 Symmetries in Quantum Field Theory of Lower Spacetime Dimensions

Quantum Groups; Integrability and Quantum Field

Theory; Quantum Field Theory: A Brief Introduction;

Quantum Fields with Topological Defects; Symmetries

and Conservation Laws; Symmetries in Quantum Field

Theory: Algebraic Aspects; Symmetry Breaking in Field

Theory; Tomita–Takesaki Modular Theory;

Two-Dimensional Conformal Field Theory and Vertex

Operator Algebras; Two-Dimensional Models.

Further Reading

Bo¨ hm G and Szlacha´nyi K (1996) A coassociative C



-quantum

group with non-integral dimensions. Letters in Mathematical

Physics 35: 437–448.

Coleman S and Mandula J (1967) All possible symmetries of the

S-matrix. Physical Review 159: 1251–1256.

Doplicher S and Roberts JE (1990) Why there is a field algebra

with a compact gauge group describing the superselection

structure in particle physics. Communications in Mathematical

Physics 131: 51–107.

Fredenhagen K, Rehren K-H, and Schroer B (1992) Superselection

sectors with braid group statistics and exchange algebras II:

geometric aspects and conformal covariance. Reviews in

Mathematical Physics SI1: 113–157.

Fro¨ hlich J and Kerler T (1993) Quantum Groups, Quantum

Categories, and Quantum Field Theory. Lecture Notes in

Mathematics, vol. 1542. Berlin: Springer.

Fuchs J, Ganchev A, and Vecsernye´s P (1994) On the quantum

symmetry of rational field theories. Theoretical Mathematical

Physics 98: 266–276.

Hadjiivanov LK, Paunov RR, and Todorov IT (1991) Quantum

group extended chiral p-models. Nuclear Physics B 356:

387–438.

Longo R and Rehren K-H (1995) Nets of subfactors. Rev.

Mathematical Physics 7: 567–597.

Mack G and Schomerus V (1992) Quasi Hopf quantum symmetry

in quantum theory. Nuclear Physics B 370: 185–230.

Majid S (1991) Braided groups and algebraic quantum field

theories. Letters in Mathematical Physics 22: 167–175.

Moore G and Seiberg N (1989) Classical and quantum conformal

field theory. Communications in Mathematical Physics 123:

177–254.

Mu¨ ger M (1998) Superselection structure of massive quantum

field theories in 1 þ 1 dimensions. Reviews in Mathematical

Physics 10: 1147–1170.

Mund J (1998) No-go theorem for ‘‘free’’ relativistic anyons in

d = 2 þ 1. Letters in Mathematical Physics 43: 319–328.

Rehren K-H (1997) Weak C



Hopf symmetry. In: Doebner H-D

and Dobrev VK (eds.) Group Theoretical Methods in Physics,

pp. 62–69. (q-alg/9611007). Sofia: Heron Press.

Schomerus V (1995) Construction of field algebras with quantum

symmetries from local observables. Communications in

Mathematical Physics 169: 193–236.

Wick GC, Wightman AS, and Wigner EP (1952) The intrinsic

parity of elementary particles. Physical Review 88: 101–105.

Symmetries in Quantum Field Theory: Algebraic Aspects

J E Roberts, Universita

di Roma ‘‘Tor Vergata,’’

Rome, Italy

Introduction

This article treats the most im portant results and

concepts relating to symmetry and conservation

laws in quantum field theory. It includes such results

as Wigner’s theorem, Goldstone’s theorem, the

Bisognano–Wichmann theorem, the quantum

Noether theorem, and the theorem on the existence

of gauge groups and a field net. It is written within

the framework of algebraic quantum field theory,

this being the simplest setting capable of expressing

all these concepts and results.

Symmetries come in many guises. They are to a

physical system what automorphisms are to a

mathematical theory. In fact, when a physical

system is described in mathematical terms, its

symmetries correspond to the automorphisms of

the mathematical structure and in particular form a

group, its symmetry group. The reader should bear

in mind this simple picture throughout its diverse

variations. Readers unfamiliar with the mathemati-

cal terminology should consult the appendix.

Elementary Quantum Mechanics

Before turning to qua ntum field theory, let us

comment on symmetries in elementary quantum

mechanics. These systems have the density matrices,

that is, positive operators of trace 1, on an infinite-

dimensional separable Hilbert space as states, the

self-adjoint operators as observables. The expecta-

tion value of the bounded observable A in the state

determined by  is given by tr A. Having specified

the mathematical struc ture, the notion of symmetry

follows. With a suggestive notation, it is a pair of

mappings A 7! A,  7! 

1

such that

tr 

1

A ¼ tr A

for all observables A and states .

If we take  and A to be the projections onto C

and C for unit vectors  and , then the above

condition corresponds to the conservation of

Symmetries in Quantum Field Theory: Algebraic Aspects 179

transition probabilities j(, )j

. Thi s formed the

starting point for Wigne r’s analysis, who concluded:

Theorem Every symm etry is of the form A 7!

UAU

1

and  7! UU

1

, where U is a unitary or

antiunitary operator.

As could have be en foreseen from the outset, this

simple result in no way distinguishes one elementary

quantum-mechanical system from another. A more

useful notion of symmetry results if the Hamiltonian

is reckoned as part of the information describing the

system and, therefore, has to be left invariant by a

symmetry. The operator U above must therefore

satisfy the condition UHU

1

= H and it commutes

with the Hamiltonian. As the Hamiltonian is the

generator of time translat ions, U is a constant of

motion. This is the genesis of the relation between

symmetries and conservati on laws.

Quantum Field Theories

The simplest types of quantum field theories can be

described by von Neumann algebras A(O) depend-

ing on double cones O and subject to

O

) AðO

ÞAðO

a structure referred to as the net of observables.

An alternative approach would be to use the

Wightman formalism. This would need a discussion

of pointlike fields and the domains of definition of

unbounded operators, thus complicating a general

exposition of symmetry.

Comparing this description of a quantum field

theory with that of an elementary quantum-

mechanical system, the net clearly substitutes obser-

vables but nothing has yet been said about states.

Since the set of double cones is directed under

inclusion, the union of the A(O)isa-algebra A and

a state of our system is a state on this algebra.

Most states are of no physical relevance. A

characterization of the states of physical rel evance,

even say to elementary particle physics, is not

known although some progress has been made.

The net structure is the hallmark of a field theory

and allows us to distinguish two important classes of

symmetries. An internal symmetry  satisfies the

condition

ðAðOÞÞ ¼ AðOÞ

for all double cones O. By contrast, a spacet ime

symmetry is an automorphism 

implementing a

Poincare´ transformation L and hence satisfying the

condition



ðAðOÞÞ ¼ AðL OÞ

for every double cone O. It is usually the case that

internal symmetries commute with spacetime

symmetries.

The state of prime relevance to elementary particle

physics is the vacuum state !

. The corresponding

Gelfand–Naimark–Segal (GNS) representation 

called the vacuum representation. Now the vacuum

state of a quantum field theory is typically unique

and as such invariant under a symmetry of the system



1

= !

Spacetime Symmetries

Since the vacuum state is invariant, we have a

unitary representation of the Poincare´ group imple-

menting the spacetime symmetries in the vacuum

representation. To illustrate the role of repre senta-

tions up to a factor, we take instead the GNS

representation of a pure state corresponding to a

particle of half-integral spin. Here we need a unitary

representation of the covering group of the Poincare´

group, inhomogeneous SL(2, C) to implement the

symmetries. The situation for the subgroup of

rotations is the same.

The most important property of these representa-

tions is positivity of the energy. More precisely, in a

representation of relevance to elementary particle

physics such as the vacuum representation, the

generator P

of time translations is a positive

operator P

 0. Expressed in a frame-independent

way, the spectrum of spacetime translations is

contained in the closed forward light cone. It is

one of the basic principles to be exploited in

applying quantum field theory to elementary particle

physics. Notice that the princ iple is no longer valid

for an equilibrium state.

A similar situation arises in conformal field

theory. Here the role of double cones in Minkowski

space is played by intervals on the circle and that of

the Poincare´ group by the Mo¨ bius group on the

circle PSL(2, R). Again, the Mo¨ bius group cannot

always be unitarily implemented and conformal

invariance is defi ned via a continuous unitary

representation of its covering group. Most impor-

tantly, there is an analog of positivity of the energy.

The generator of rotations of the circle is a positive

operator.

A remarkable aspect of spacetime symmetries was

discovered by Bisognano and Wichmann in an

application of modular theory in the field-theoretical

context looking not at double cones but at wedges.

A wedge W is a Poincare´ transform of the standard

wedge x

> jx

j. They found that the modular

automorphisms of A(W) and the vacuum vector 

182 Symmetries in Quantum Field Theory: Algebraic Aspects

have a geometric significance. For the standard

wedge, they got the following result.

Theorem If the net is derived from Wightman

fields, the modular operator is e

2K

, where K is the

generator of boosts in the 1-direction and the

modular conjugation is ZR, where  is the TCP-

operator, R is the rotation through  about the

1-axis, and Z is the unitary operator equal to 1 on

the Bose subspace and i on the Fermi subspace.

The modular data for A(O) and 

also admit a

geometric interpretation for the free massless scalar

field.

These facts enhance our understanding of space-

time symmetries. The ideas have meanwhile been

applied to curved spacetime to select a state with

vacuum-like properties using the principle of the

geometric action of the modular conjugation.

Gauge Symmetry

Gauge symmetries do not fit into our scheme in that

they act trivially on the observable algebra A.To

exhibit a gauge symmetry we need a larger net F

called the field net. The gauge group will be the

group of automorphisms of F leaving the subnet A

pointwise fixed and A the subnet of F of fixed

points under G. This has the merit of indicating the

mathematical framework for gauge symmetry but

otherwise begs important questions. A priori one

does not know what properties F should have nor

how it should be constructed.

The right approach is to understand what intrinsic

structure of A governs the existence of a nontrivial

gauge group. This brings us back to the states or

representations relevant to elementary particle phy-

sics. A condition for selecting some of these relevant

representations is that asymptotically they be like

the vacuum in spacelike directions. More precisely,

 must be unitarily equivalent to the vacuum

representation 

on the spacelike complement of

every double cone.

The resulting theory of superselection sectors

hinges on the property of Haag duality that, for

each double cone O,

AðOÞ ¼ AðO

where O

denotes the spacelike complement of O.It

implies that every representation satisfying the

selection criterion is unitarily equivalent to one of

the form 

, where  is an endomorphism of A

localized in some fixed but arbitrary double cone,

that is, (A) = A if A 2 A(O

). The endomorphisms

thus obtained are closed under composition and

hence the objects of a full tensor subcategory T of

the category of all endomorphisms and their inter-

twiners. There is a dimension function d defined on

the objects of T , d() = 1, 2, ..., 1.IfT

denotes

the full subcategory whose objects have finite

dimension, then the following result holds.

Theorem T

is equivalent to the tensor category of

finite-dimensional continuous unitary representa-

tions of a canonical compact group G. There is a

canonical field net F with Bose–Fermi commutation

relations extending A such that G is the group of

automorphisms of F leaving A pointwise fixed.

The first step in the proof is to define and analyze

the statistics of the representations in question. The

statistics of an irreducible representation  can be

classified as being para-Bose or para-Fermi of order

d(). The second step is to show that each  of finite

dimension has a well-defined conjugate up to

equivalence. The third and most difficult step is

showing that T

can be embedded in the tensor

category of Hilbert spaces.

The Local Implementation

of Symmetries

Gauge symmetry has its associated conservation

laws in that the different sectors of the last section

are labeled by conserved quantities such as baryon

number, lepton number, or electric charge, gener-

ically called charges. The theory is built round the

idea of creating charge and elements of the field net

carry charges. But there should be a dual approach

based on measuring charges. One would like to

prove the existence of local conserved currents

corresponding to these charges. This has not proved

possible but there is a good substitute, described

below, which can be regarded as a weak version of a

quantum Noether theorem.

If O

O

is a strict inclusion of double cones,

then the theory is said to satisfy the split property if

there is a type I factor M such that

AðO

ÞMAðO

where a type I factor is a von Neumann algebra

isomorphic to some B(H). In this case M can be

chosen in a canonical fashion and there is an

isomorphism called the universal localizing map

of B(H) onto M, where H is the underlying Hilbert

space. We have (A) = A for A 2 A(O

Theorem If U is an implementing representation of

the internal symmetry group G, (U) will be a

representation of G in M that continues to imple-

ment the symmetry on A(O

). If G is a Lie group

Symmetries in Quantum Field Theory: Algebraic Aspects 183

then the infinitesimal generators in the representa-

tion are an analog of locally integrated current

densities.

Spontaneously Broken Symmetry

The standard physical example of a spontaneously

broken symmetry is magnetization. Despite the

overall rotational symmetry, a magnet picks out a

preferred direction as its direction of magnetization.

The chosen state breaks the symmetry.

The phenomenon of spontane ously broken sym-

metry involves an interplay of symmetries and

certain classes of states, vacuum states, ground

states, or equilibrium states. If such an ! is induced

by a vector cyclic and separating for a local algebra

A(O), then, as explained in the appendix, given O,

modular theory yields a canonical unitary represen-

tation V of the internal symmetry grou p G:

gA ¼ V



; A 2 AðOÞ

The results concern the breaking of a one-

parameter group  7! 



of symmetries. More

precisely, one asks whether ! = 0 or not, where 

is the infinitesimal generator of  7!



ðFÞ¼lim

!0



1

ð



ðFÞFÞ

where norm convergence is understood and holds on a

dense domain. , the derivation, is an infinitesimal

symmetry. Goldstone first showed that the sponta-

neous breaking of such symmetries requires the

presence of massless bosons. The following result is

taken from a more modern treatment. O

here denotes

the double cone whose base is the ball in t = 0ofradius

R centered on the origin and D the domain of .

Theorem Let  be a derivation on a field net F in

s > 1 spatial dimensions such that for F 2 F

) \D

Fjc

R;"

ðkFkþkF



kÞþ "kFk

(i) If lim inf

R!1

R,"

(s1)=2

= 0, then !

 = 0.

(ii) If lim inf

R!1

R,"

(s1)=2

< 1, then !

 6¼ 0 is

only possible if the spectrum of the translations

coincides with the forward light cone V

and the

boundary @V

={0} has non-trivial spectral mea-

sure (i.e. , there are massless particles in the

theory).

(iii) If c

R,"

is polynomially bounded in R, then

 6¼0 is only possible if the spectrum of

translations coincides with V

but there are

not necessarily any massless particles.

Symmetries of the S-matrix

Scattering theory not only allows one to construct

the multiparti cle scattering states but also shows

that internal symmetries and spacetime symmetries

continue to act on these states and are therefore

symmetries of the S-matrix. We can, however, ask

what are all the symmetries of the S-matrix. An

answer was provided by Coleman and Mandula,

who showed that, when there is nontrivial scatter-

ing, there are no further symmetries of the S-matrix.

Appendix

In an effort to make this article more self-contained,

this appendix collects together a few simple perti-

nent concepts and results from the theory of

operator algebras. A C



-algebra is a -algebra A

with a norm kkmaking it into a Banach algebra

and satisfying



Ak¼kAk

for every A 2A. Any C



-algebra can be realized as a

norm closed -subalgebra of the C



-algebra B(H)of

all bounded operators on a Hilbert space H.Avon

Neumann algebra R is a C



-algebra that is the dual

space of a Banach space. This Banach space R



, the

predual of R, is intrinsically defined. The topology

on R determined by duality with R



is called the

-topology. B (H) is a von Neumann algebra and its

predual is the set of trace class operators. Any

von Neumann algebra can be realized as a -closed

unital -subalgebra of some B(H).

A state on a C



-algebra A is a positive linear

functional ! of norm 1. If A has a unit I the

normalization condition can be expressed as

!(I) = 1. Of fundamental importance is the relation

between representations an d states. A representation

of A on a Hilbert space H is just a structure-

preserving mapping or morphism of A into B(H).

For simplicity, we suppose that A has a unit. Given

a state !, there is an a ssociated representation 

defined by a vector  such that 

(A) is dense in

the Hilbert space in question, that is, it is a cyclic

vector for the representation and

!ðAÞ¼ð;

ðAÞÞ; A 2A

that is, the cyclic vector implements the given state.

This is referred to as the GNS construction. Given

any two such representations, there is a unique

unitary operator mapping the one cyclic vector onto

the other and realizing the equivalence of the

representations.

184 Symmetries in Quantum Field Theory: Algebraic Aspects

A state of a von Neumann algebra is said to be

normal if it is continuous in the -topology. If ! is

normal, then 

(R)is-closed.

An inclusion of unital von Neumann algebras has

the split property if there is an intermediate type I

factor, that is, if it has the form R

B(H) R

The following elementary observation is often

used in treating symmetries. If  is an automorphism

of A with !

1

= !, there is a unique unitary

operator leaving the cyclic vector  invariant and

inducing  in the represent ation 

. In other words,

U= and

U

ðAÞU

1

¼ 

ðAÞ

If we apply the above lemma to a group G of

symmetries leaving a state invariant, it yields a

group U(g) of unitaries satisfying the condition

UðghÞ¼UðgÞUðhÞ; g; h 2 G

since U(g) is uniquely defined by the above

conditions.

When there is no invariant state, the situation is

more complicated. Suppose there is a group G of

symmetries and a representation  of A where each

g is unitarily implemented. Thus, there is a unitary

U(g)with

UðgÞðAÞUðgÞ

1

¼ ðgAÞ; A 2 A

All we can now conclude is that

UðghÞ¼Zðg; hÞUðgÞUðhÞ

where Z(g, h) is a unitary in A

, the commutant of

A, satisfying the 2–cocycle identity

Zðgh; kÞZðg; hÞ¼Zðg; hkÞ

Zðh; kÞ

where

X = U(g)XU(g)

1

. U is said to be a repre-

sentation up to a factor. It can be chosen to be a

representation if the cocycle Z is a coboundary, that

is, if there is a unitary Y(g)inA

such that

YðgÞ

YðhÞ¼Yðgh ÞZðg; hÞ

In general, little is known about solving problems

of this kind, but there are a number of results when

 is irreducible and the unitary group of its

commutant reduces to the circle.

We turn now to consider the modular theory of

von Neumann algebras. A vector  is said to be

separating for a von Neumann algebra R if A=0

and A 2R implies A = 0. If  is both cyclic and

separating, there is a uniquely de termined closed

antilinear involution S with SA=A



 for A 2R.

If S = J

1=2

is the polar decomposition of S, then the

unitary operators 

induce automorphisms 

of R

and JRJ = R

. J is called the modular conjugation, 

the modular operator, and 

the modular auto-

morphisms. The closure of {

1=4

A: A 2R, A  0}

is a cone, called the natural cone. Every normal state

of R is implemented by a unique vector in the

natural cone. If  is an automorphism of R, there is

therefore a unique vector 



in the natural cone

such that, for every A 2R,

ð;

1

ðAÞÞ¼ð



; A



There is now a canonical unitary operator V



defined by



A ¼ ðAÞ



maps the natural cone into itself and  7! V



is an

implementing representation of the group of auto-

morphisms of R. Under these circumstances, we do

not have to deal with representations up to a factor.

See also: Algebraic Approach to Quantum Field Theory;

Axiomatic Quantum Field Theory; Boundary Conformal

Field Theory; Current Algebra; Quantum Fields with

Topological Defects; Supergravity; Symmetries in

Quantum Field Theory of Lower Spacetime Dimensions;

Two-Dimensional Models.

Further Reading

Bisognano JJ and Wichmann EH (1976) On the duality condition

for quantum fields. Journal of Mathematical Physics 17:

303–321.

Borchers HJ and Buchholz D (1985) The energy–momentum

spectrum in local field theories with broken Lorentz symme-

try. Communications in Mathematical Physics 97: 169–185.

Buchholz D (1982) The physical state space of quantum electro-

dynamics. Communications in Mathematical Physics 85: 49–71.

Buchholz D, Doplicher S, and Longo R (1986 ) On Noether’s

theorem in quantum field theory. Annals of Physics 170 :

1–17.

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Symmetries in Quantum Field Theory: Algebraic Aspects 185

Symmetry and Symmetry Breaking in Dynamical Systems

I Melbourne, University of Surrey, Guildford, UK

Introduction

The same symmetries may underlie diverse contexts

such as phase trans itions of crystals (Landau

theory), fluid dynamics, and problems in biology

and chemical engine ering. Hence, seemingly unre-

lated systems may exhibit similar phenomena in

regard to symmetries of patterns and trans itions

between patterns (spontaneous symmetry breaking).

It is natural to focus attention on aspects of pattern

formation that are universal or model independent –

aspects depending on underlying symmetries rather

than model-specific details.

The general framework is that the underlying

system is governed by an evolution equation

x ¼ f ðxÞ½1

with symmetry group . To avoid techni calities, we

assume that [1] is an ordinary differential equation

(ODE), the vector field f : R

! R

is as smooth as

desired, and  is a compact Lie group acting linearly

on R

. An inner product may be chosen so that

 acts orthogonally. The vector field in [1] is

-equivariant if

f ðxÞ¼f ðxÞ for all x 2 R

;2  ½2

Equivalently, if x(t) is a solution and  2 , then

x(t) is a solution.

In this article, we are interested in the dynamics to

be expected for equivariant vector fields, and

transitions that arise as parameters are varied. The

symmetry group  is taken as given, whereas f is a

general -equivariant vector field. (Other features

such as energy conservation or time reversibility

must be built into the general setup, but are

excluded in this article.)

Isotropy Subgroups and Commuting

Linear Maps

Let  be a compact Lie group acting linearly on R

The isotropy subgroup of x 2 R

is defined to be



¼f 2 : x ¼ xg

Note that 

x

= 



1

for all x 2 R

,  2 .

Given an isotropy subgroup   , define the

fixed-point subspace

Fix  ¼fy 2 R

: y ¼ y for all  2 g

If f : R

is a  -equivariant vector field, then

f (Fix )  Fix  for each isotropy subgroup .

Hence Fix  is flow invariant.

The normalizer N() = { 2  : 

1

=} is the

largest subgroup of  that acts on Fix  ,and



= f j

Fix

is (N()=)-equivariant.

An isotropy subgroup  is axial if dim Fix =1,

and then N()= ﬃ Z

or 1. More generally,  is

maximal if there are no isotropy subgroups T with

  T   other than T = and T =. Then

N()= acts fixed-point freely on Fix  and the

connected component of the identity (N(=)

ﬃ 1,

SO(2) or SU(2). Correspondingly  is called real,

complex, or quaternionic. In the complex case

dim Fix  is even; in the quaternionic case dim Fix

  0 mod 4.

The dihedral group =D

of ord er m is the

symmetry group of the regular m-gon, m  3. Its

standard action on R

is generated by

 ¼

cos 2=m sin 2=m

sin 2=m cos 2=m

 ¼

0 1

For m even, the isotropy subgroups up to conjugacy

are

; Z

ðÞ; Z

ðÞ; 1

where Z

(g) denotes the cyclic group of order j

generated by g. The maximal isotropy subgroups

=Z

(), Z

() are axial with N( )= ﬃ Z

. For

m odd, Z

() is conjugate to Z

() leaving three

conjugacy classes of isotropy subgroups, and

=Z

() is axial with N()==1.

The space of commuting linear maps

Hom



ðR

Þ¼ L : R

linear:

L ¼ L for all  2 g

is completely described representation-theoretically.

Recall that  acts irreducibly on R

if the only

-invariant subspaces of R

are R

and {0}. Then

Hom



) is a real division ring (skew field) DﬃR,

C or H. The representation is called absolutely

irreducible when D= R and nonabsolutely irreduci-

ble when D= C or H.

184 Symmetry and Symmetry Breaking in Dynamical Systems

If the action of  is not irreducible, write R

V

(nonuniquely) as a sum of irreducible

subspaces. Summing together irreducible subspaces

that are isomo rphic to form isotypic components W

gives the (unique) isotypic decomposition

= W

W

‘

.IfL 2 Hom



), then

L(W

)  W

for each j, hence Hom



) =

Hom



) Hom



‘

). Each W

consists of

isomorphic copies of an irreducible representation

with division ring D

. Let M

(D) denote the space of

k  k matrices with entries in D. The n

Hom



ðR

ÞﬃM

ðD

ÞM

‘

ðD

‘

Þ½3

Spectral properties of commuting linear maps can be

recovered from the decomposition [3], paying due

attention to multiplicity and complex conjugates of

eigenvalues.

Equivariant Dynamics

The dynamics of equivariant systems includes

(relative) equilibria and periodic solutions, robust

heteroclinic cycles/net works, and symmetric chaotic

attractors.

Equilibria

Consider the ODE [1] with -equivariant vector

field f satisfying [2].Ifx(t)  x

is an equilibrium,

f (x

) = 0, then there is a group orbit x

equilibria.

Let =

be the isotropy subgroup of x

.If

dim =dim , then generically (for an open dense

set of -equivariant vector fields), the eigenvalues of

(df )

have nonzero real part, hence x

is hyperbolic.

If the eigenvalues all have negative real part, then x

is asymptotically stable. If at least one eigenvalue

has positive real part, then x

is unstable. Hyper-

bolic equilibria are isolated and persist under

perturbations of f; the perturbed equilibria continue

to have isotropy . Since (df )

2Hom



decomposition [3] for the action of  on R

facilitates stability compu tations for x

If dim  < dim , then x

is a continuous group

orbit of equilibria. Generically, dim ker (df )

dim dim  and ker (df )

= {x

:  2 L}, where

L is the Lie algebra of . The remaining k = n 

dim  þ dim  eigenvalues generically have nonzero

real part so x

is normally hyperbolic. If all k

eigenvalues have nonzero real part, then x

asymptotically stable. If at least one has positive real

part, then x

is unstable. When N()= is finite,

generically x

is an isolated equilibrium in Fix  and

persists as an equilibrium with isotropy  under

perturbation.

Relative Equilibria and Skew Products

A point x

2 R

(or the corresponding group orbit

x

) is a relative equilibrium if f (x

) 2 T

x

Lx

.Ifx

has isotropy ,thenx

is a relati ve

equilibrium if f (x

) 2 LD



,whereD



= (N()=)

Write f (x

) = x

, where  2 LD



. The closure of

the one-parameter subgroup exp(t) is a maximal

torus in D



for almost every . All maximal tori are

conjugate with common dimension d = rank D



The solution x(t) = exp(t)x

is typically a

d-dimensional quasiperiodic motion. ‘‘Typically’’

holds in both the topological and probabilistic sense

and there is no phase-locking. When d = 1, x(t)is

periodic, often called a rotating wave.

Choose a -invariant local cross section X to the

group orbit x

at x

. There is a  -invariant

neighborhood of x

that is -equivariantly diffeo-

morphic to (  X)=, where  acts freely on   X

 ð; xÞ¼ð

1

;xÞ

and  acts by left multiplication on the first

factor. The -equivariant ODE on (  X)= lifts

to a (  )-equivariant skew product on   X

 ¼ ðxÞ;

x ¼ hðxÞ½4

where  : X !L, h : X !X satisfy the -equivariance

conditions

ðxÞ¼Ad



ðxÞ¼ðxÞ

1

hðxÞ¼hðxÞ

and h(x

) = 0.

Thus, dynamics near the relative equilibrium

x

 R

reduces to dynamics near the ordinar y

equilibrium x

2 X for the -equivariant vector

h : X !X, coupled with  drifts. In particular, the

stability of x

is determined by (dh)

Periodic Solutions

A nonequilibrium solution x(t)isperiodicifx(t þ T) =

x(t)forsomeT > 0. The least such T is the (absolute)

period. The spatial symmetry group  is the isotropy

subgroup of x(t) for some, and hence all, t 2 R.The

periodic solution P = {x(t): 0  t < T} lies inside

Fix . Define the spatiotemporal symmetry group

={ 2  : P = P}. Note that  is a normal subgroup

of  and either = ﬃ S

(P is a rotating wave) or

= ﬃ Z

and P is called a standing wave or a discrete

rotating wave. For each  2 ,thereexistsT



2 [0, T)

such that x(t) = x(t þ T



). The relative period of x(t)

is the least T > 0suchthatx(T) 2 x

If dim =dim , then generically P is hyperbolic,

hence isolated, the stability of P is determined by its

Symmetry and Symmetry Breaking in Dynamical Systems 185

Floquet exp onents, and P persists under perturba-

tion as a periodic solution with spatial symmetry 

and spatiotemporal symmetry . For  infinite and

N()= finite, generically P is isolated in Fix  and

the neutral Floquet exponent has multiplicity

dim   dim  þ 1.

Relative Periodic Solutions

A solution x(t) is a relative periodic solution if it is

not a relative equilibrium and x(T) 2 x(0) for some

T > 0. The least such T is the relative period. The

spatial symmetry group =

x(t)

for some, hence

all, t. The spa tiotemporal symmetry group  is the

closed subgroup of  generated by  and , where

x(T) = x(0), and generically = ﬃ T

 Z

is a

maximal topologically cyclic (Cartan) subgroup of

N()= containing . Then x(t)isa(d þ 1)-

dimensional quasiperiodi c motion.

The dynamics near the relative periodic solution

is again governed by a skew product. There exists

n  1 such that 

= exp (n), where  2 LZ()

and Z()   is the centralizer of . Define

 = exp(). Form a semidirect product  o Z

by adjoining to  an element Q of order 2n such

that QQ

1

= 

1

for  2 .

In a comoving frame with velocity , a neighbor-

hood of the relative periodic orbit is -equivariantly

diffeomorphic to (  X  S

)= o Z

, where X is

a  o Z

-invariant cross section, S

= R=2nZ and

 o Z

acts on   X  S

 ð; x;Þ¼ð

1

;x;Þ

Q ð; x;Þ¼ð

1

; Qx;þ 1Þ

The -equivariant ODE on (  X  S

)= o Z

lifts to a   ( o Z

)-equivariant skew product

 ¼ ðx;Þ;

x ¼ hðx;Þ;

 ¼ 1 ½5

where  : X  S

!L, h : X  S

!X satisfy appro-

priate  o Z

-equivariance conditions.

Robust Heteroclinic Cycles

Heteroclinic cycles, degenerate in systems without

symmetry, arise robustly in equiva riant systems. Let

, ..., x

2 R

be saddles with W

)  {x

} 

W

iþ1

) (where m þ 1 = 1). If 

, ..., 

  are

isotropy subgroups, W

)  Fix 

, and x

iþ1

is a

sink in Fix 

, then saddle–sink connections from x

to x

iþ1

persist for nearby -equivariant flows. The

union

i = 1

W

) forms a robust heteroclinic cycle

(see the subsection ‘‘Dynamics’’ for an example). Such

cycles, when asymptotically stable, are a mechanism

for intermittency or bursting, notably in rotating

Rayleigh–Be´nard convection (where rolls disappear

and reorient themselves at approximately 60



), and

provide a possible intrinsic explanation for irregular

reversals of the Earth’s magnetic field.

Asymmetric perturbations (deterministic or noisy)

destroy the cycles, but the perturbed attractors

inherit the bursting behavior.

Establishing the existence of heteroclinic connec-

tions is often straightforward when dim Fix 

= 2

and nontrivial with dim Fix 

3. Criteria for

asymptotic stability of heteroclinic cycles are given

in terms of real parts of eigenval ues of (df )

,and

depend on the geometry of the representation of .

Robust cycles exist also between more complicated

dynamical states such as periodic solutions or chaotic

sets (cycling chaos). When W

) connects to two or

more distinct states, the collection of unstable

manifolds forms a heteroclinic network leading to

competition between various subnetworks.

Symmetric Attractors

Suppose that  is a finite group acting linearly on R

A closed subset A  R

has symmetry groups =

{ 2 : x = x for all x 2 A}, ={ 2 : A = A}.

Here,  is an isotropy subgroup and     N().

In applications,  corresponds to instantaneous

symmetry and  to symmetry on average.

If A is an attractor (a Lyapunov stable !-limit set)

for a -equivariant vector field f : R

, then 

fixes a connected component of Fix   L , where L

is the union of proper fixed-point spaces in Fix .

Provided dim Fix   3, all pairs ,  satisfying

the above restrictions arise as symmetry groups of a

nonperiodic attractor A. If dim Fix   5, then A is

realized by a uniformly hyperbolic (Axiom A)

attractor.

If dim Fix   3 and  fixes a connected compo-

nent of Fix   L, then A is realized by a periodic

sink provided = is cyclic. If dim Fix =2, then in

addition either = or =N().

Suppose A is an attractor and  2   . Then

A \ A = ;. Varying a parameter, A may undergo a

symmetry-increasing bifurcation: A grows until it

collides with A producing a larger attractor with

symmetry on average generated by  and .

Determining symmetries of an attractor by inspec-

tion is often infeasible. A detective is a -equivariant

polynomial  : R

!V where every subgroup of  is

an isotropy subgroup for the action on V, and each

component of  is nonzero. Suppose that A  R

an attract or with physical (Sinai–Ruelle–Bowen)

measure . By ergodicity, the time average

¼ lim

T!1

ðxðtÞÞdt 2 V

186 Symmetry and Symmetry Breaking in Dynamical Systems

is well defined for almost every trajectory x(t)in

supp . Generically, 

=

so computing the

symmetry of A reduces to computing the symmetry

of a point.

If  is an infinite compact Lie group, and A is an

!-limit set containing points of trivial isotrop y, then

A cannot be uniformly hyperbolic. Hence partially

hyperbolic flows arise naturally in systems with

continuous symmetry. Consider the skew product

[4] where =1 and h : X !X possesses a hyperbolic

basic set   X with equilibrium measure  (for a

Ho¨ lder potential). Let  denote Haar measure on .

Then    is partially hyperbolic, and    is

ergodic (even Bernoulli) for an open dense set of

equivariant flows. Such stably ergodic flows possess

strong statistical properties (rapid decay of correla-

tions, central-limit theorem); a possible explanation

for hypermeander (Brownian-like motion) of spiral

waves in planar excitable media.

Forced Symmetry Breaking

In applicat ions, symmet ry is not perfect and account

should be taken of 

-equivariant perturbations of

[1] for 

a subgroup of  (including 

= 1). This

topic is not discussed in this article, except in the

subsec tions ‘‘R obust heterocl inic cycles’’ an d

‘‘Branc hing pa tterns an d finite deter minacy. ’’

Equivariant Bifurcation Theory

Consider families of ODEs

x = f (x, ), with bifurca-

tion parameter  2 R and vect or field f : R



R !R

satisfying f (0, 0) = 0 and the -equivariance

condition

f ðx;Þ¼f ðx;Þ

for all x 2 R

;2 R;2 

A local bifurcation from the equilibrium x = 0

occurs if (df )

0, 0

is nonhyperbolic. The center sub-

space E

is the sum of generalized eigenspaces

corresponding to eigenvalues on the imaginary

axis, and is -invariant. By center manifold theory,

local dynamics ((x, ) near (0, 0)) are captured by the

center manifold W

. After center manifold reduction

(or Lyapunov–Schmidt reduction if the focus is on

equilibria), it may be assumed that R

= E

If ( df )

0, 0

possesses zero eigenvalues, then there is

a steady-state bifurcation. Generically, (df )

0, 0

= 0

and E

is absolutely irreducible. There are two

subcases.

If  acts trivially on R

, then n = 1 and generically

there is a saddle–node (or limit point) bifurcation

where the zero sets of f (x, )andx

  are

diffeomorphic for (x, ) near (0, 0). Higher-order

degeneracies can be treated using singularity theory.

The equilibria and their stability determines the

local dynamics. All bifurcating equilibria have

isotropy , so there is no symmetry breaking.

From now on, consider the remaining subcase

where  acts absolutely irreducibly and nontrivially

on R

. Then Fix ={0}, f (0, )  0, and (df )

0, 

c()I

where generically c

(0) 6¼ 0. Assume that

(0) > 0, so the ‘‘trivial solution’’ x = 0 is asympto-

tically stable subcritically (<0) and unstable

supercritically (>0). Bifurcating solutions lie out-

side Fix  and hence there is spo ntaneous symmetry

breaking.

Axial Isotropy Subgroups

The ‘‘equivariant branching lemma’’ gu arantees

branches of equilibria with isotropy  for each

axial isotropy subgroup. There are three associated

branching patterns, see Figure 1 .

If N()==Z

, then f



is odd. Generically,



(0, 0) 6¼ 0, since ( x

þþx

)x is -equivar-

iant, and there are two branches of equilibria

bifurcating supercritically or subcritically together,

and lying on the same group orbit. The branches

form a symmetric pitchfork whose direction of

branching is determined by sgn @



(0, 0).

If N()= ﬃ 1, then generically f



is even. If all

quadratic -equivariant maps vanish on Fix , then

the bifurcation is sub/supercritical depending on

sgn @



(0, 0) but the branches lie on distinct group

orbits. This is an asymmetric pitch fork.

If @



(0, 0) 6¼ 0, then the equilibria exist tran-

scritically: for <0and>0.

The natural actions of D

on R

are absolutely

irreducible. The axial branches are symmetric

pitchforks for m  4 even, asymmetric pitchforks

for m  5 odd, and transcri tical for m = 3.

The actions of D

, m  5 odd, provide the

simplest instances of hidden symmetries, where

certain N()=-equivariant mappings on Fix  do

not extend to smooth -equivariant mappings on R

Nonaxial Maximal Isotropy Subgroups

For  a real maximal isotropy subgroup, dim Fix 

odd, there exist branches of equilibria with isotropy

(a) (b) (c)

Figure 1 Axial branches: (a) supercritical symmetric pitchfork,

(b) supercritical asymmetric pitchfork, and (c) transcritical

branches.

Symmetry and Symmetry Breaking in Dynamical Systems 187