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

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Further Reading

Cherednik I (1996) Basic Methods of Soliton Theory. Advanced

Series in Mathematical Physics, vol. 25. Singapore, New

Jersey, London and Hong Kong: World Scientific.

Date E, Kashiwara M, Jimbo M, and Miwa T (1983) Transfor-

mation groups for soliton equations. In: Jimbo M and Miwa T

(eds.) Proceedings of RIMS Symposium on Non-Linear

Integrable Systems – Classical Theory and Quantum theory,

pp. 39–119. Singapore: World Scientific.

Drinfel’d VG and Sokolov VV (1985) Lie algebras and equations

of Korteweg–de Vries type. JournalofSovietMathematics

30: 1975–2036.

Gardner CS, Greene JM, Kruskal MD, and Miura RM (1967)

Methods for solving the Korteweg–de Vries equation. Physical

Review Letter 19: 1095–1097.

Kac VG (1990) Infinite Dimensional Lie Algebras, 3rd edition.

Cambridge: Cambridge University Press.

Manin YuI (1979) Algebraic aspects of nonlinear differential

equations. Journal of Soviet Mathematics 11: 1–122.

Miwa T, Jimbo M, and Date E (2000) Solitons, (Translated by

Reid, M). Cambridge: Cambridge University Press.

Noumi M (2002) Affine Weyl group approach to Painleve´

equations. In: Tatsien LI (ed.) International Congress of

Mathematicians (2002, Beijing), Proceedings of the Interna-

tional Congress of Mathematicians, August 20–28, 2002,

Beijing, pp. 497–509. Beijing: Higher Education Press.

Novikov SP, Manakov SV, Pitaevskii LP, and Zakharov VE

(1984) Theory of Solitons. The Inverse Scattering Method.

New York and London: Consultants Bureau.

Sato M and Sato Y (1983) Soliton equations as dynamical systems

on infinite-dimensional Grassmann manifold. In: Fujita H,

Lax PD, and Strang G (eds.) Nonlinear Partial Differential

Equations in Applied Science (Tokyo, 1982), North-Holland

Math. Stud., vol. 81, pp. 259–271. Amsterdam: North-

Holland.

Solitons and Other Extended Field Configurations

R S Ward, University of Durham, Durham, UK

Introduction

A soliton is a localized lump (or string or wall, etc.)

of energy, which can move without distortion,

dispersion, or dissipation, and which is stable under

perturbations (and collisions with other solitons). The

word was coined by Zabusky and Kruskal in 1965 to

describe a solitary wave with particle-like properties

(as in electron, proton, etc.). Solitons are relevant to

numerous areas of physics – condensed matter,

cosmology, fluids/plasmas, biophysics (e.g., DNA),

nuclear physics, high-energy physics, etc. Mathema-

tically, they are modeled as solutions of appropriate

partial differential equations.

Systems which admit solitons may be classified

according to the mechanism by which stability is

ensured. Such mechanisms include complete integr-

ability, nontrivial topology plus dynamical balan-

cing, and Q-balls/breathers.

Sometimes the term ‘‘soliton’’ is used in a

restricted sense, to refer to stable localized lumps

which have purely elastic interactions: solitons

which collide without any radiation being emitted.

This is possible only in very special systems, namely,

those that are completely integrable. For these

systems, soliton stability (and the elasticity of

collisions) arises from a number of characteristic

properties, including a precise balance between

dispersion and nonlinearity, solvability by the

inverse scattering transform from linear data, infi-

nitely many conserved quantities, a Lax formulation

(associated linear problem), and Ba¨ cklund transfor-

mations. Examples of such integrable soliton sys-

tems are the sine-Gordon, Korteweg–deVries, and

nonlinear Schro¨ dinger equations.

The category of topological solitons is the most

varied, and includes such examples as kinks,

vortices, monopoles, skyrmions, and instantons.

The requirement of dynamical balancing for these

can be understood in terms of Derrick’s theorem,

which provides necessary conditions for a classical

field theory to admit static localized solutions. The

602 Solitons and Other Extended Field Configurations

Derrick argument involves studying what

happens to the energy of a field when one changes

the scale of space. If one has a scalar field (or

multiplet of scalar fields) , and/or a gauge field F



then the static energy E is the sum of terms such as

VðÞd

x; E

ðD

Þd

where each integral is over (n-dimensional) space

, D

 denotes the covariant spatial derivative of ,

and T

(

) is a real-valued polynomial of degree d.

In particular, for example, we could have T

) =

)(D

), the standard gradient term. Under the

dilation x

7!x

, these functionals transform as

7!

n

; E

7!

dn

; E

7!

4n

In order to have a static solution (critical point of

the static energy functional), one needs to have a zero

exponent on , and/or a balance between positive and

negative exponents. A negative exponent indicates a

compressing force (tending to im plode a localized

lump), whereas a positive exponent indicates an

expanding force; so to have a static lump solution,

these two forces have to balance each other. For

n = 1, a system involving only a scalar field, with

terms of the form E

and E

, can admit static solitons

(e.g., kinks); the scaling argument implies a virial

theorem, which in this case says that E

= E

. For

n = 2, one can have a scalar system with only E

since in this case the relevant exponent is zero (e.g.,

the two-dimensional sigma model). Another n = 2

example is that of vortices in the abelian Higgs model,

where the energy contains terms E

, E

, and E

. For

n = 3, interesting systems have E

together with either

(e.g., skyrmions) or E

(e.g., monopo les). An E

term is optional in these cases; its presence affects, in

particular, the long-range properties of the solitons.

For n = 4, one can have instantons in a pure gauge

theory (term E

only).

It should be noted that if there are no restrictions on

the fields  and A

(such as those arising, e.g., from

nontrivial topology), then there is a more obvious mode

of instability, which will inevitably be present:  7!

and/or A

7!A

,where0   1. In other words, the

fields can simply be scaled away altogether, so that the

height of the soliton (and its energy) go smoothly to

zero. This can be prevented by nontrivial topology.

Another way of preventing solitons from shrink-

ing is to allow the field to have some ‘‘internal’’ time

dependence, so that it is stationary rather than

static. For example, one could allow the complex

scalar field  to have the form  = exp (i!t), where

is independent of time t. This leads to something

like a centrifugal force, which can have a stabilizing

effect in the absence of Skyrme or magnetic terms.

The corresponding solitons are Q-balls.

Kinks and Breathers

The simplest topological solitons are kinks, in

systems involving a real-valued scalar field (x)in

one spatial dimension. The dynamics is governed by

the Lagrangian density

L¼



ð

ð

 WðÞ



where W() is a (fixed) smooth function. The system

can admit kinks if W() has at least two zeros, for

example, W(A) = W(B) = 0withW() > 0forA <

<B. Two well-known systems are: sine-Gordon

(where W() = 2sin(=2), A = 0, and B = 2)and

(where W() = 1  

, A = 1, and B = 1). The corre-

sponding field equations are the Euler–Lagrange equa-

tions for L; for example, the sine-Gordon equation is



 

þ sin  ¼ 0 ½1

Configurations satisfying the boundary condi-

tions  ! A as x !1and  ! B as x !1are

called kinks (and the corresponding ones with

x = 1 and x = 1 interchanged are antikinks).

For kink (or antikink) configurations, there is a

lower bound, called the Bogomol’nyi bound, on the

static energy E[ ]; for kink boundary conditions,

we have

E½¼

1

ð

þ WðÞ

1



 WðÞ½

dx þ

1

WðÞ



WðÞd

with equality if and only if the Bogomol’nyi equation

d

¼ WðÞ½2

is satisfied. A static solution of the Bogomol’nyi

equation is a kink solution – it is a static minimum

of the energy functional in the kink sector. For

example, for the sine-Gordon system, we get E[] 

8, with equality for the sine-Gordon kink

ðxÞ¼4 tan

1

expðx x

while for the 

system, we get E[]  4=3, with

equality for the phi-four kink

ðxÞ¼tanhðx  x

Solitons and Other Extended Field Configurations 603

These kinks are stable topological solitons; the

nontrivial topology corresponds to the fact that the

boundary value of (t, x)atx = 1 is different from

that at x = 1. With trivial boundary conditions

(say  ! A as x !1), stable static solitons are

unlikely to exist, but solitons with periodic time

dependence (which in this context are called breath-

ers) may exist. For example, the sine-Gordon

equation and the nonlinear Schro¨ dinger equation,

both, admit breathers – but these owe their existence

to complete integrability. By contrast, the 

system

(which is not integrable) does not admit breathers; a

collision between a 

kink and an antikink (with

suitable impact speed) produces a long-lived state

which looks like a breather, but eventually decays

into radiation.

In lattice systems, however, breathers are more

generic. In a one-dimensional latti ce system, the

continuous space R is replaced by the lattice Z,so

(t, x) is replaced by 

(t), where n 2 Z. The

Lagrangian is

L ¼



 h

2

ð

nþ1

 

 Wð

where h is a positive parameter, corresponding to the

dimensionless ratio between the lattice spacing and the

size of a kink. The continuum limit is h ! 0. This

system admits kink solutions as in the continuum case;

and for h large enough, it admits breathers as well, but

these disappear as h becomes small.

Interpreted in three dimensions, the kink becomes

a domain wall separa ting two regions in which the

order parameter  takes distinct values; this has

applications in such diverse areas as cosmology and

condensed matter physics.

Sigma Models and Skyrmions

In a sigma model or Skyrme system, the field is a

map  from spacetime to a Riemannian manifold M;

generally, M is taken to be a Lie group or a

symmetric space. The energy density of a static

field can be constructed as follows (the Lorentz-

invariant extension of this gives a relativistic

Lagrangian for fields on spacetime). Let 

be local

coordinates on the m-dimensional manifold M, let

denote the metric of M, and let x

denote the

spatial coordinates on space R

.Anm  m matrix D

is defined by

¼ð@



Þh

ð@



where @

denotes derivatives with respect to the x

Then the invariants E

= tr(D) = j@



and

= (1=2)[(tr D)

 tr(D

)] can be terms in the

energy density, as well as a zeroth-order term

= V(

) not involving derivatives of . A term

of the form E

is called a Skyrme term.

The boundary condition on field configurations

is that  tends to some constant value 

2 M as

jxj!1in R

. From the topological point of view,

this compactifies R

to S

. In other words,  extends

to a map from S

to M; and such maps are classified

topologically by the homotopy group 

(M). For

topological solitons to exist, this group has to be

nontrivial.

In one spatial dimension (n = 1) with M = S

(say),

the expression E

is identically zero, and we just have

kink-type systems such as sine-Gordon. The simplest

two-dimensional example (n = 2) is the O(3) sigma

model, which has M = S

with its standard metric. In

this system, the field is often expressed as a unit

3-vector field f = (

, 

), with E

= (@

f)  (@

f).

Here the configurations are classified topologically by

their degree (or winding number, or topological

charge) N 2 

) ﬃ Z, which equals

N ¼

4

f  @

f  @

f dx

Instead of f, it is often convenient to use a single

complex-valued function W related to f by the

stereographic projection W = (

þ i

)=(1  

). In

terms of W, the formula for the degree N is

N ¼

2

 W

ð1 þjWj

and the static energy is (with z = x

þ ix

)

E ¼

¼ 8

þjW

ð1 þjWj

¼ 16

ð1 þjWj

x þ 8

jW

ð1 þjWj

¼ 16

ð1 þjWj

x þ 8N

From this, one sees that E satisfies the Bogomol’nyi

bound E  8N, and that minimal-energy solutions

correspond to solutions of the Cauchy–Riemann

equations W

= 0. To have finite energy, W(z) has to

be a rational function, and so solutions with wind-

ing number N correspond to rational meromorphic

functions W(z), of degree jNj. (If N < 0, then W is a

rational function of

z.) The energy is scale invariant

(conformally invariant), and consequently these

solutions are not solitons – they are not quite stable,

since their size is not fixe d. Adding terms E

and E

604 Solitons and Other Extended Field Configurations

to the energy density fixes the soliton size, and the

resulting two-dimensional Skyrme systems admit

true topological solitons.

The three-dimensional case (n = 3), with M being

a simple Lie group, is the original Skyrme model of

nuclear physics. If M = SU(2), then the integer N 2



(SU(2)) ﬃ Z is interpreted as the baryon number.

The (quantum) excitations of the -field correspond

to the pions, whereas the (semiclassical) solitons

correspond to the nucleons. This model emerges as

an effective theory of quantum chromodynamics

(QCD), in the limit wher e the number of colors is

large. If we express the field as a function U(x

)

taking values in a Lie group, then L

= U

1

U takes

values in the corresponding Lie algebra, and E

and

take the form

¼

trðL

¼

tr ½L

; L

½L

; L





The static energy density in the basic Skyrme system

is the sum of these two terms. The static energy

satisfies a Bogomol’nyi bound E  12

jNj, and it is

believed that stable solitons (skyrmions) exist for

each value of N. Classical skyrmions have been

investigated numerically; for values of N up to 25,

they turn out to resemble polyhedral shells. Com-

parison with nucleon phenomenology requires semi-

classical quantization, and this leads to results which

are at least qualitatively correct.

A variant of the Skyrme model is the Skyrme–

Faddeev system, which has n = 3 and M = S

; the

solitons in this case resemble loops which can be

linked or knotted, and which are classified by their

Hopf number N 2 

). In this case, the energy

satisfies a lower bound of the form E  cN

3=4

Numerical experiments indicate that for each N,

there is a minimal-energy solution with Hopf

number N, and with energy close to this topological

lower bound.

Abelian Higgs Vortices

Vortices live in two spatial dimensions; viewed in

three dimensions, they are string-like. Two of their

applications are as cosmic strings and as magnetic

flux tubes in superconductors. They occur as static

topological solitons in the the abelian Higgs model

(or Ginzburg–Landau model), and involve a mag-

netic field B = @

 @

, coupled to a complex

scalar field , on the plane R

. The energy density is

E¼

ðD

ÞðD

Þþ

ð1 jj

½3

where D

 := @

  iA

, and where  is a positive

constant. The boundary conditions are

 ¼ 0; B ¼ 0; jj¼1 ½ 4

as r !1. If we consider a very large circle C on R

so that [4] holds on C, then j

is a map from the

circle C to the circle of unit radius in the complex

plane, and therefore it has an integer winding

number N. Thus configurations are labeled by this

vortex number N.

Note that if E vanishes, then B = 0 and jj= 1: the

gauge symmetry is spontaneously broken, and the

photon ‘‘acquires a mass’’: this is a standard

example of spontaneous symmetry breaking.

The total magnetic flux

B d

x equals 2N;a

proof of this is as follows. Let  be the usual polar

coordinate around C. Becau se jj= 1onC, we can

write  = exp [if ()] for some function f; this f need

not be single-valued, but must satisfy f (2) 

f (0) = 2N with N being an inte ger (in order that

 be single-valued). In fact, this defines the winding

number. Now since D

 = @

  iA

 = 0onC,

we have

¼i

1

 ¼ @

on C. So, using Stokes’ theorem, we get

B d

x ¼

2

d

¼ 2N

If  = 1, then the total energy E =

E d

satisfies the Bogomol’nyi bound E  N; E = N

if and only if a set of partial differential equations

(the Bogomol’nyi equations) are satisfied. Since

like charges repel, the magnetic force between

vortices is repulsive. However, there is also a

force from the Higgs field, and this is attractive.

The balance between the two forces is determ ined

by :if>1, the vor tices repel each other;

whereas if <1, the vortices at tract. In the

critical case  = 1, the force between vortices is

exactly balanced, and there exist static multi-

vortex solutions. In fact, one has the following:

given N points in the plane, there exists an

N-vortex solution of the Bogomol’nyi equations

(and hence of the full field equations) with 

vanishing at the chosen points (and nowhere

else). All static solutions are of this form. These

solutions cannot, however, be written down

explicitly in terms of elementary functions (except

of course for N = 0).

Solitons and Other Extended Field Configurations 605

Monopoles

The abelian Higgs model does not admit three-

dimensional solitons, but a nonabelian generaliza-

tion doe s – such nonabelian Higgs solitons are called

magnetic monopo les. The field con tent, in the

simplest version, is as follows. First, there is a

gauge (Yang–Mills) field F



, with gauge potential



, and with the gauge group being a simple Lie

group G. Second, there is a Higgs scalar field ,

transforming under the adjoint representation of G

(thus  takes values in the Lie algebra of G). For

simplicity, G is taken to be SU(2) in what follows.

So we may write A



= iA





, F



= iF





,and

 = i



, where 

are the Pauli matrices. The

energy of static (@

 = 0 = @

), purely magnetic

= 0) configurations is

E ¼



ðD

Þ

ðD

Þ

ð1 





where B

=(1=2)

jkl

is the magnetic field. The

boundary conditions are B

! 0 and 



! 1as

r !1;so restricted to a large spatial 2-sphere

becomes a map from S

to the unit 2-sphere in the

Lie algebra su(2), and as such it has a degree N 2Z.

An analytic expression for N is

ðD

Þ

x ¼ 2N ½5

At long range, the field resembles an isolated

magnetic pole (a Dirac magnetic monopole), with

magnetic charge 2N. Asymptotically, the SU(2)

gauge symmetry is spontaneously broken to U(1),

which is interpreted as the electromagnetic gauge

group.

In 1974, it was observed that this system admits a

smooth, finite-energy, stable, spherically symmetric

N = 1 solution – this is the ’t Hooft–Polyakov

monopole. There is a Bogomol’nyi lower bound on

the energy E: from 0  (B þ D)

= B

þ (D)

2B  D, we get

E  2N þ

ð1  



x ½6

where [5] has been used. The inequality [6] is

saturated if and only if the Prasad–Sommerfield

limit  = 0 is used, and the Bogomol ’nyi equations

ðD

Þ

¼B

½7

hold. The corresponding solitons are called

Bogomol’nyi–Prasad–Sommerfield (BPS) monopoles.

The Bogomol’nyi equations [7], together with the

boundary conditions described above, form a com-

pletely integrable elliptic system of partial differe n-

tial equations. For any positive integer N, the space

of BPS monopoles of charge N, with gauge freedom

factored out, is parametrized by a (4N  1)-dimen-

sional manifold M

. This is the moduli space of N

monopoles. Roughly speaking, each monopole has a

position in space (three parameters) plus a phase

(one parameter), making a total of 4jNj parameters;

an overall phase can be removed by a gauge

transformation, leaving (4jNj1) parameters. In

fact, it is often useful to retain the overall phase, and

to work with the corresponding 4jNj-dimensional

manifold

. This manifold has a natural metric,

which corresponds to the expression for the kinetic

energy of the system. A point in

represents an

N-monopole configuration, and the slow-motion

dynamics of N monopoles corresponds to geodesics

; this is the geodesic approximation of

monopole dynamics.

The N = 1 monopole is spheri cally symmetric, and

the corresponding fields take a simple form; for

example, the Higgs field of a 1-monopole located at

r = 0is



cothð2rÞ





For N > 1, the expressions tend to be less explicit;

but monopole solutions can nevertheless be char-

acterized in a fairly complete way. The Bogomol’nyi

equations [7] are a dimensional reduction of the self-

dual Yang–Mills equations in R

, and BPS mono-

poles correspond to holomorphic vector bundles

over a certain two-dimensional complex manifold

(‘‘mini-twistor space’’). This leads to various other

characterizations of monopole solutions, for exam-

ple, in terms of certain curves (‘‘spectral curves’’) on

mini-twistor space, and in terms of solutions of a set

of ordinary differential equations called the Nahm

equations. Having all these descriptions enables one

to deduce much about the monopole moduli space,

and to characterize many monopole solutions. In

particular, there are explicit solutions of the Nahm

equations involving elliptic functions, which corre-

spond to monopoles with certain discrete symme-

tries, such as a 3-monopole with tetrahedral

symmetry, and a 4-monopole with the appearance

and symmetries of a cube.

Yang–Mills Instantons

Consider gauge fields in four-dimensional Euclidean

space R

, with gauge group G. For simplicity, in

what follows, G is taken to be SU(2); one can extend

much of the structure to more general groups, for

example, the simple Lie groups. Let A



and F



606 Solitons and Other Extended Field Configurations

denote the gauge potential and gauge field. The

Yang–Mills action is

S ¼

tr F





x ½8

where we assume a boundary condition, at infinity

in R

, such that this integral converges. The Euler–

Lagrange equations which describe critical points of

the functional S are the Yang–Mills equations





¼ 0 ½9

Finite-action Yang–Mills fields are called instantons.

The Euclidean action [8] is used in the pa th-integral

approach to quantum gauge field theory; therefore,

instantons are crucial in understanding the path

integral.

The dual of the field tensor F



F







The gauge field is self-dual if F



= F



, and anti-

self-dual if F



= F



. In view of the Bianchi

identity D



 F



= 0, any self-dual or anti-self-dual

gauge field is automatically a solution of the Yang–

Mills equations [9]. This fact also follows from the

discussion below, where we see that self-dual

instantons give local minima of the action.

The Yang–Mills action (and Yang–Mills equa-

tions) are conformally invariant; any finite-action

solution of the Yang–Mills equations on R

extends

smoothly to the conformal compactification S

Gauge fields on S

, with gauge group SU(2), are

classified topologically by an integer N, namely, the

second Chern number

N ¼ c

¼

8

tr F



 F





x ½10

From [8] and [10] a topological lower bound on the

action is given as follows:

0 

tr F



F







F





¼ 8S  16

and so S  2

N, with equality if and only if the

field is self-dual. If N < 0, we get S  2

jNj, with

equality if and only if F is anti-self-dual. So the self-

dual (or anti-self-dual) fields minimize the action in

each topological class.

For the remainder of this section, we restrict to self-

dual instantons with instanton number N > 0. The

space (moduli space) of such instantons, with gauge

equivalence factored out, is an (8N  3)-dimensional

real manifold. In principle, all these gauge fields can

be constructed using algebraic-geometry (twistor)

methods: instantons correspond to holomorphic vector

bundles over complex projective 3-space (twistor

space). One large class of solutions which can be

written out explicitly is as follows: for N = 1and

N = 2 it gives all instantons, while for N  3itgivesa

(5N þ 4)-dimensional subfamily of the full (8N  3)-

dimensional solution space. The gauge potentials in

this class have the form



¼ i





log  ½11

where the 



are constant matrices (antisymmetric

in ) defined in terms of the Pauli matrices 



¼ 



¼ 



¼ 



The real-valued function  = (x



) is a solution of

the four-dimensional Laplace equation given by

ðx



Þ¼

k¼0



ðx



 x



Þðx



 x



where the x



are N þ 1 distinct points in R

, and the



are N þ 1 positive constants: a total of 5N þ 5

parameters. It is clear from [11] that the overall

scale of  is irrelevant, leaving a (5N þ 4)-parameter

family. For N = 1 and N = 2, symmetries reduce the

parameter count further, to 5 and 13, respe ctively.

Although  has poles at the points x = x

, the gauge

potentials are smooth (possibly after a gauge

transformation).

Finally, it is worth noting that (as one might

expect) there is a gravitational analog of the gauge-

theoretic structures described here. In other words,

one has self-dual gravitational instantons – these are

four-dimensional Riemannian spaces for which the

conformal-curvature tensor (the Weyl tensor) is

self-dual, and the Ricci tensor satisfies Einstein’s

equations R



=g



. As before, such spaces

can be constructed using a twistor-geometrical

correspondence.

Q-Balls

A Q-ball (or nontopological soliton) is a soliton

which has a periodic time dependence in a degree of

freedom which corresponds to a global symmetry.

The simplest class of Q-ball systems involves a

complex scalar field , with an invariance under the

constant phase transformation  7!e

i

; the Q-balls

are soliton solutions of the form

ðt; xÞ¼e

i!t

ðxÞ½12

Solitons and Other Extended Field Configurations 607

where (x) is a complex scalar field depending only

on the spatial variables x. The best-known case is

the 1-soliton solution

ðt; xÞ¼a

ﬃﬃﬃ

expðia

tÞsechðaxÞ

of the nonlinear Schro¨ dinger equation i

þ 

jj

= 0.

More generally, consider a system (in n spatial

dimensions) with Lagrangian

L¼

ð@



Þð@



ÞUðjjÞ

where (x



) is a complex-valued field. Associated

with the global pha se symmetry is the conserved

Noether charge Q =

Im(





x. Minimizing the

energy of a configuration subject to Q being fixed

implies that  has the form [12]. Without loss of

generality, we may take !  0. Note that Q = !I,

where I =

j j

x. The energy of a configuration

of the form [12] is E = E

þ E

, where

Uðj jÞd

Let us take U(0) = 0 = U

(0), with the field satisfying

the boundary condition ! 0asr !1.

A stationary Q-lump is a critical point of the

energy funct ional E[ ], subject to Q having some

fixed value. The usual (Derrick) scaling argument

shows that any stationary Q-lump must satisfy

ð2  nÞE

 nE

þ nE

¼ 0 ½13

For simplicity, in what follows, let us take n  3.

Define m > 0byU

(0) = m

; then, near spatial

infinity, the Euler–Lagrange equations give r



 !

) = 0. So, in order to satisfy the boundary

condition ! 0asr !1, we need !<m.

It is clear from [13] that if U  (1=2)m

j j

everywhere, then there can be no solution. So

K = min[2U (j j)=j j

] has to satisfy K < m

. Also,

we have

U 

KI ¼ðK=!

ÞE

> ðK=!

ÞE

½14

where the final inequality comes from [13].Asa

consequence, we see that !

is restricted to the range

K <!

< m

½15

An example which has been studied in some detail is

U(f ) = f

[1 þ (1  f

)

]; here m

= 4 and K = 2, so

the range of frequency for Q-balls in this system is

ﬃﬃﬃ

<!<2. The dynamics of Q-balls in systems

such as these turns out to be quite complicated.

See also: Abelian Higgs Vortices; Homoclinic

Phenomena; Integrable Systems: Overview; Instantons:

Topological Aspects; Noncommutative Geometry from

Strings; Sine-Gordon Equation; Topological Defects and

Their Homotopy Classification.

Further Reading

Atiyah MF and Hitchin NJ (1988) The Geometry and Dynamics

of Magnetic Monopoles. Princeton: Princeton University

Press.

Coleman S (1988) Aspects of Symmetry. Cambridge: Cambridge

University Press.

Drazin PG and Johnson RS (1989) Solitons: An Introduction.

Cambridge: Cambridge University Press.

Goddard P and Mansfield P (1986) Topological structures in field

theories. Reports on Progress in Physics 49: 725–781.

Jaffe A and Taubes C (1980) Vortices and Monopoles. Boston:

Birkha¨user.

Lee TD and Pang Y (1992) Nontopological solitons. Physics

Reports 221: 251–350.

Makhankov VG, Rubakov YP, and Sanyuk VI (1993) The Skyrme

Model: Fundamentals, Methods, Applications. Berlin: Springer.

Manton NS and Sutcliffe PM (2004) Topological Solitons.

Cambridge: Cambridge University Press.

Rajaraman R (1982) Soliton and Instantons. New York: North-

Holland.

Rebbi C and Soliani G (1984) Solitons and Particles. Singapore:

World Scientific.

Vilenkin A and Shellard EPS (1994) Cosmic Strings and Other

Cosmological Defects. Cambridge: Cambridge University Press.

Ward RS and Wells RO Jr. (1990) Twistor Geometry and Field

Theory. Cambridge: Cambridge University Press.

Zakrzewski WJ (1989) Low Dimensional Sigma Models. Bristol:

IOP.

608 Solitons and Other Extended Field Configurations

Source Coding in Quantum Information Theory

N Datta, University of Cambridge, Cambridge, UK

T C Dorlas, Dublin Institute for Advanced Studies,

Dublin, Republic of Ireland

Introduction

Two key issues of classical and quantum informa-

tion theory are storage and transmission of informa-

tion. An information source produces some outputs

(or signals) more frequently than others. Due to this

redundancy, one can reduce the amount of space

needed for its storage without compromising on its

content. This data compression is done by a suitable

encoding of the output of the source. In contrast, in

the transmission of information through a channel,

it is often advantageous to add redundancy to a

message, in order to combat the effects of noise.

This is done in the form of error-correcting codes.

The amount of redundancy which needs to be added

to the original message depends on how much noise

is present in the channel (see, e.g., Nielson and

Chuang (2000)). Hence, redundancy plays comple-

mentary roles in data compression and transmission

of data through a noisy channel. In this review we

focus only on data compression in quantum infor-

mation theory.

In classical information theory, Shannon showed

that there is a natural limit to the amount of

compression that can be achieved. It is given by

the Shannon entropy. The analogous concept in

quantum information theory is the von Neumann

entropy. Here, we review some of the main results

of quantum data compression and the significance of

the von Neumann entropy in this context.

The review is structured as follows. We first give

a brief introduction to the Shannon entropy and

classical data compression. This is followed by a

discussion of quantum entropy and the idea behind

quantum source coding. We elaborate on data

compression schemes for three different classes of

quantum sources, namely memoryless sources,

ergodic sources, and sources modeled by Gibbs

states of quantum spin systems. In the bulk of the

review, we concentrate on source-dependent, fixed-

length coding schemes. We conclude with a brief

discussion of universal and variable-length coding.

We would like to point out that this review article

is by no means complete. Due to a restriction on its

length, we had to leave out various important

aspects and developments of quantum source

coding.

Classical Data Compression

Entropy and Source Coding

A simple model of a classical information source

consists of a sequence of discrete random variables

, X

, ..., X

, whose values represent the output of

the source. Each random variable X

,1 i  n,

takes values x

from a finite set, the source alphabet

X. Hence,

(n)

:= (X

, ..., X

) takes values x

(n)

, ..., x

) 2X

. We recall the definition of entropy

(or information content) of a source.

If the discrete random variables X

, ..., X

which

take values from a finite alphabet X have joint

probabilities

PðX

¼ x

; ...; X

¼ x

Þ¼p

ðx

; ...; x

then the Shannon entropy of this source is defined by

HðX

; ...; X

¼



ðx

; ...; x

 log p

ðx

; ...; x

Þ½1

Here and in the following, the logarithm is taken to

the base 2. This is because the fundamental unit of

classical information is a ‘‘bit,’’ which takes two

values 0 and 1. Notice that H(X

, ..., X

) in fact

only depends on the (joint) probability mass func-

tion (p.m.f.) p

and can also be denoted as H(p

There are several other concepts of entropy, for

example, relative entropy, conditional entropy, and

mutual information. See, for example, Cover and

Thomas (1991) and Nielson and Chuang (2000).It

is easy to see that

1. 0  H(X

, ..., X

)  n log jXj, where jXj denotes

the number of letters in the alphabet X. Two

other important properties are as follows:

2. H(X

, ..., X

) is jointly concave in X

, ..., X

and

3. H(X

,..., X

) H(X

,..., X

) þH(X

mþ1

,..., X

)

for m < n.

The latter property is called subadditivity.

In the next section, analogous quantities are

introduced for quantum information and the corre-

sponding properties are stated.

Suppose that the random variables X

, X

, ..., X

are independent and identically distributed (i.i.d.).

Then the entropy of each random variable modeling

the source is the same and can be denoted by H(X).

From the point of view of classical information

theory, the Shannon entropy has an important

operational definition. It quantifies the minimal

Source Coding in Quantum Information Theory 609

physical resources needed to store data from a

classical information source and provides a limit to

which data can be compressed reliably (i.e., in a

manner in which the original data can be recovered

later with a low probability of error). Shannon

showed that the original data can be reliably

obtained from the compressed version only if the

rate of compression is greater than the Shannon

entropy. This result is formulated in Shannon’s

noiseless channel coding theorem (Shannon 1918,

Cover and Thomas 1991, Nielson and Chuang

2000) given later.

The Asymptotic Equipartition Property

The main idea behind Shannon’s noiseless channel

coding theorem is to divide the possible values

, x

, ..., x

of random variables X

, ..., X

into

two classes – one consisting of sequences which have

a high probability of occurrence, known as ‘‘typical

sequences,’’ and the other consisting of sequences

which occur rarely, known as ‘‘atypical sequences.’’

The idea is that there are far fewer typical sequences

than the total number of possible sequences, but

they occur with high probability. The existence

of typical sequences follows from the so-called

‘‘asymptotic equipartition property’’:

Theorem 1 (AEP). If X

, X

, ... are i.i.d.

random variables with p.m.f. p(x), then



log p

ðX

; ...; X

Þ!

HðXÞ½2

where H(X ) is the Shannon entropy for a single

variable, and p

, ..., X

) denotes the random

variable taking values p

, ..., x

) =

i = 1

p(x

)

with probabilities p

, ..., x

This theorem has been generalized to the case of

sequences of dependent variables (X

)

n2Z

which are

ergodic for the shift transformation defined below.

It is easiest to formulate this for an information

stream which extends from 1 to þ1:

Definition Asequence(X

)

n2Z

is called ‘‘stationary’’

if for any n

< n

and any x

, ..., x

2X,

PðX

¼x

; ...; X

¼x

¼ PðX

þ1

¼ x

; ...; X

þ1

¼ x

We define the shift transformation  by

 ðx

n2Z



¼ðx

n2Z

; x

¼ x

n1

½3

Then (X

)

n2Z

is called ‘‘ergodic’’ if it is stationary

and if every subset A X

such that (A) = A has

probability 0 or 1, that is, P((X

)

n2Z

2 A) = 0or1.

It is known that (X

)

n2Z

is ergodic if and only if

its probability distribution is extremal in the set of

invariant probability measures. The generalization

of Theorem 1 (McMillan 1953, Breiman 1957) now

reads:

Theorem 2 (Shannon–McMillan–Breiman theo-

rem). Suppose that the sequence (X

)

n2Z

ergodic. Then

lim

n!1



log p

ðX

; ...; X



¼ h

with probability 1

½4

where h

is the Kolmogorov–Sinai entropy defined by

¼ lim

n!1

HðX

;...;X

Þ¼inf

HðX

;...;X

Þ½5

Remark. It follows from the subadditivity property

(3) above that the sequence (1=n)H(p

) is decreas-

ing, and it is obviously bounded below by 0.

We now define the set of typical sequences (or more

precisely, -typical sequences) as follows:

Definition Let X

, ..., X

be i.i.d. random vari-

ables with p.m.f. p(x). Given >0, -typical set T

(n)



is the set of sequences (x

...x

) for which

nðHðXÞþÞ

 pðx

...x

Þ2

nðHðXÞÞ

½6

In the case of an ergodic sequence, H(X) is replaced

by h

in [6].

Let jT

(n)



j denote the total number of typical

sequences and P{T

(n)



} denote the probability of the

typical set. Then the following is an easy conse-

quence of Theorem 1.

Theorem 3 (Theorem of typical sequences). For

any >0 9n

() > 0 such that 8n  n

() the follow-

ing hold:

(i) P{T

(n)



} > 1   and

(ii) (1  )2

n(H(X))

jT

(n)



j2

n(H(X)þ)

Shannon’s Noiseless Channel Coding Theorem

Shannon’s noiseless channel coding theorem is a

simple application of the theorem of typical

sequences and says that the optimal rate at which

one can reliably compress data from an i.i.d.

classical information source is given by the Shannon

entropy H(X) of the source.

A ‘‘compression scheme’’ C

of rate R maps

possible sequences

x = (x

, ..., x

) to a binary string

of length dnRe: C

: x 7!y = (y

, ..., y

dnRe

), where

2X; jXj= d and y

2 {0, 1} 81  i dnRe. The

corresponding decompression scheme takes the dnRe

610 Source Coding in Quantum Information Theory

compressed bits and maps them back to a string of n

letters from the alphabet X: D

: y 2 {0, 1}

dnRe

7!x

, ..., x

). A compression–decompression scheme

is said to be ‘‘reliable’’ if the probability that

6¼ x

tends to 0 as n !1. Shannon’s noiseless channel

coding theorem (Shannon 1918, Cover and Thomas

1991) now states

Theorem 4 (Shannon). Suppose that {X

} is an i.i.d.

information source, with X

 p(x) and Shannon

entropy H(X). If R > H(X) then there exists a

reliable compression scheme of rate R for the

source. Conversely, any compression scheme with

rate R < H(X) is not reliable.

Proof (sketch). Suppose R > H(X). Choose >0

such that H(X) þ <R. Consider the set T

(n)



typical sequences. The method of compression is

then to examine the output of the source, to see if it

belongs to T

(n)



. If the output is a typical sequence,

then we compress the data by simply storing an

index for the particular sequence using dnRe bits in

the obvious way. If the input string is not typical,

then we compress the string to some fixed dnRe bit

string, for example, (00 ...000). In this case, data

compression effectively fails, but, in spite of this, the

compression–decompression scheme succeeds with

probability tending to 1 as n !1,sincebyTheorem 3

the probability of atypical sequences can be made

small by choosing n large enough.

If R < H(X), then any compression scheme of rate

R is not reliable. This also follows from Theorem 3

by the following argument. Let S(n) be a collection

of sequences

(n)

of size jS(n)j2

dnRe

. Then the

subset of atypical sequences in S(n) is highly

improbable, whereas the corresponding subset of

typical sequences has probability bounded by

nH(X)

!0asn !1.

Quantum Data Compression

Quantum Sources and Entropy

In quantum information processing systems, infor-

mation is stored in quantum states of physical

systems. The most general description of a quantum

state is provided by a density matrix.

A ‘‘density matrix’’  is a positive semidefinite

operator on a Hilbert space H ,withtr = 1, and the

expected value of an operator A on H is given by

ðAÞ¼tr ðAÞ½7

The functional  on M= B(H), the algebra of linear

operators on H, is positive (i.e., (A)  0, if A  0)

and maps the identity I 2Mto 1. Such a functional

is also called a state. Conversely, given such a state

on a finite-dimensional algebra M, there exists a

unique density matrix 



such that [7] holds, so the

concepts can be used interchangeably. (This is not

true in the infinite-dimensional case.)

The quantum analog of the Shannon entropy is

called the von Neumann entropy. For any quantum

state  (or equivalently 



), it is defined by

SðÞSð



Þ:¼tr 



log 





½8

Here we use log to denote log

and define 0 log

0  0, as for the Shannon entropy. Let the density

matrix 



have a spectral decomposition





i¼1



j½9

Here {j

i} is the set of eigenvectors of 



. They

form an orthonormal basis of the Hilbert space H.

By the fact that 



is positive definite and has trace 1,

the eigenvalues 

of 



determine a probability

distribution. When expressed in terms of the 

, the

von Neumann entropy of  reduces to the Shannon

entropy corresponding to this probability distribu-

tion (henceforth, the subscript  of 



will be

omitted): S() = H(

), where  = {

, ..., 

The von Neumann entropy has properties analo-

gous to H(X

, ..., X

), in particular (Ohya and Petz

1993, Nielson and Chuang 2000)

1. 0  S()  log(dim (H));

2. S() is concave in ;and

3. if  is a state on H= H

H

then S() S(

) þ

S(

)if

and 

are the restrictions of  to

 I and I H

respectively.

A ‘‘quantum information source’’ in general is

defined by a sequence of density matrices 

(n)

Hilbert spaces H

of increasing dimensions N

given

by a decomposition



ðnÞ

j

ðnÞ

ih

ðnÞ

j½10

where the states j

(n)

i are interpreted as the signal

states, and the numbers p

(n)

 0 with

(n)

= 1, as

their probabilities of occurrence. The vectors j

(n)

need not be mutually orthogonal.

Compression–Decompression

Scheme and Fidelity

To compress data from such a source one encodes

each signal state j

(n)

i by a state



(n)

2B(

) where

dim

= d

(n) < N

. Thus, a compression scheme

is a map C

(n)

: j

(n)

ih

(n)

j7!



(n)

2B(

). The state



(n)

is referred to as the compressed state. A

corresponding decompression scheme is a map

(n)

: B(

) 7!B(H

). Both C

(n)

and D

(n)

must be

Source Coding in Quantum Information Theory 611