Dunajski M. Solitons, Instantons, and Twistors

Подождите немного. Документ загружается.

96 5: Lagrangian formalism and ﬁeld theory

5.3.1 Topology and Bogomolny equations

All ﬁnite energy ﬁeld conﬁgurations approach vacuum at ±∞. Asymptotic

values cannot change (Figures 5.2–5.4). Topological conserved currents are

properties of a ﬁnite energy continuous ﬁeld. For (5.2.13) with D = 1 topolog-

ical conserved quantities are

−

= lim

x→−∞

φ and φ

= lim

x→∞

φ.

If φ

= φ

−

then the ﬁeld φ(x) can be continuously deformed into the zero-

energy vacuum φ = φ

. If on the other hand φ

= φ

−

then φ cannot be continu-

ously deformed into a vacuum, which is the reason for topological stability of

kinks.

Figure 5.2 Kink, N =1

Figure 5.3 Anti-kink, N = −1

Figure 5.4 Kink–anti-kink pair, N =0

5.3 Scalar kinks 97

The associated conserved current

N = φ

− φ

−



is an integral of a total derivative which depends only on boundary conditions.

It is conserved because we insisted on the ﬁniteness of the energy. Note that

the ﬁeld equations have not entered the discussion at this stage. Topological

conserved currents are in this sense different from the Noether currents which

result from continuous symmetries of the Lagrangian.

We shall look for minimal energy conﬁgurations. If U ≥ 0 we can always

ﬁnd W(φ) such that

U(φ)=



dW(φ)

dφ



Now

E =



dx(φ

+ φ

+ W



+(φ

± W

)

∓ 2φ



+(φ

± W

)

∓ [W(φ(∞)] − W[φ(−∞)].

Therefore

E ≥|W [φ(∞)] − W [φ(−∞)]|. (5.3.20)

This is the Bogomolny bound. It depends only on the topological data at ±∞.

Say

φ(∞)=φ

>φ(−∞)=φ

The minimum energy conﬁgurations satisfy φ

= 0 (the static condition), E =

W(φ

) − W(φ

) and

dφ

(5.3.21)

which is the Bogomolny equation. Its solutions are kinks (5.3.18).

The ﬁeld equation (5.3.17) is a second-order PDE, and its special solutions

arise form the ﬁrst-order ODE (the Bogomolny equation). A time-dependent

solution to (5.3.17) can be found by the Lorentz boost. In general (any dimen-

sion and Lagrangian) the full ﬁeld equations are usually not integrable, but the

Bogomolny equations are often integrable (and have lower order).

Example. U = λ

(φ

− a

)

/2 gives W = λ(a

φ − φ

/3). The Bogomolny

equations (5.3.21) yield

= λ(a

− φ

98 5: Lagrangian formalism and ﬁeld theory

and φ is the static kink solutions (5.3.19) with energy

E = W(a) − W(−a)=

λa

This kink is stable as it would take inﬁnite energy to change this solution

into a constant vacuum solution φ = 0. In general kinks minimize the energy

within their topological class. The absolute minimum is of course 0 which

corresponds to φ

= φ

Example. U =1− cos βφ. This theory has inﬁnitely many vacua,

2πn

parameterized by n ∈ Z, and kinks interpolate between adjacent vacua. The

ﬁeld equation (5.3.17) is the Sine-Gordon equation.

− φ

+ β sin βφ = 0 (5.3.22)

which is essentially equivalent to (2.1.2) as the parameter β can be set to 1

by scalings of (x, t,φ).

The kink solution with φ(−∞)=0,φ(∞)=2π/β is given by

φ =

arctan

−1

β(x−x

)

The solution is multivalued and we get all possible kinks depending on which

branch we choose. In this case time-dependent solutions to the full equations

(multi-kinks and solitons) can be constructed using the integrability of the

Sine-Gordon equation. The simplest solution generating technique is the

Bäcklund transformation described in Section 2.1.2.

The Sine-Gordon equation admits time-dependent solutions such that φ

tends to the same limit at ±∞. These so-called breathers have trivial topolog-

ical charge and owe their stability to the complete integrability of (5.3.22),

and the existence of an inﬁnite number of conservation laws preventing

annihilation into radiation. By contrast U = λ

(φ

− a

)

/2 does not posses

such solutions, as the corresponding ﬁeld equations are not integrable.

5.3.2 Higher dimensions and a scaling argument

Can there exist ﬁnite-energy static critical points of (5.2.12) in more than

one spatial dimension? In this section we shall examine a scaling argument,

originally due to Derrick [37] and rule out all dimensions higher than 2. In

Section 6.1.1 we shall return to Derrick’s argument in the context of gauge

theory where the spatial dimensions three and four are also allowed.

5.3 Scalar kinks 99

If φ(x) is a static solution to (5.2.13) in D spatial dimensions, then

∇

φ =

dφ

and φ is a critical point of the energy functional

E(φ)=





|∇φ|

+ U(φ)



= E

grad

+ E

Consider a one-parameter family of conﬁgurations φ

(c)

(x)=φ

(1)

(cx), where

(x) is a static ﬁnite-energy solution. Then

E[φ

(c)

D−2

grad

Since E(φ

) is a minimum of E we have

dE[φ

(c)

]/dc|

c=1

which implies

(D − 2)E

grad

+ DE

=0. (5.3.23)

D = 1. Static solutions are possible with E

grad

= E

. These are the kinks

(5.3.21).

D = 2. Static solutions are possible with E

= 0. This can still lead to non-

linear ﬁeld theories if the target space is a manifold without a linear

structure, φ : R

2+1

→ . These so-called sigma models will be studied in

Section 5.4.

D = 3. Finite-energy static solutions do not exist. Adding a Skyrme term

|∇φ|

to the Lagrangian density allows static solutions via the scaling

argument. See [114] for a complete discussion of the Skyrme model.

Although the D = 1 kinks do not generalize to solitons in D > 1, they can

be trivially lifted to translationally invariant solutions of scalar ﬁeld theory

(5.2.13) in any dimension. These lifted solutions have inﬁnite energy as a result

of the integration along the (D − 1) spatial directions on which the kinks do

not depend. The energy is however ﬁnite per unit volume. This type of solution

is called a domain wall. Finally we note that the Derrick argument breaks down

for time-dependent conﬁgurations.

5.3.3 Homotopy in ﬁeld theory

Any static, smooth ﬁeld conﬁguration φ : R

D+1

−→ R

is topologically trivial,

as it can be transformed to zero by a homotopy (1 − τ )φ (see Appendix A for

discussion of homotopy). Non-trivial ﬁeld conﬁgurations appear if we assume

100 5: Lagrangian formalism and ﬁeld theory

that the energy density

E =

∇φ

·∇φ

+ U(φ

,...,φ

)

of a static ﬁeld decays as r −→ ∞. This condition alone does not necessarily

lead to ﬁnite energy, but it gives rise to a topological classiﬁcation. Let M ⊂ R

be a submanifold of the target space implicitly deﬁned by

U(φ

,...,φ

)=U

mi n

where U

mi n

is the minimal value of U. At spatial inﬁnity φ = φ

∞

must take its

values in M, or the density E will not vanish. Therefore

∞

: S

D−1

−→ M,

and smooth ﬁeld conﬁgurations are classiﬁed by elements of the homotopy

group π

D−1

(M).

Later we will meet other ways of classifying smooth ﬁeld conﬁgurations: In

sigma models U = 0, and φ : R

D+1

−→ . Fields are classiﬁed by elements of

() as any static ﬁeld with ﬁnite E must smoothly extend to the one-point

compactiﬁcation S

of R

. In pure gauge theories gauge ﬁelds are classiﬁed by

Chern numbers which arise from integrating various powers of the ﬁeld tensor.

This will be discussed in Section 6.4. Finally in gauge theories with Higgs ﬁelds,

the Higgs ﬁelds at inﬁnity carry all topological information (see Section 6.3.1).

5.4 Sigma model lumps

Sigma models are non-linear in a fundamental way: the target space is not a

linear space. Consider a ﬁeld φ : R × R

−→ S

N−1

, with components φ

) ∈

which satisfy the non-linear relation



a=1

=1. (5.4.24)

The kinetic Lagrangian density

L =

∂

gives rise to a non-linear equation. To see this introduce a Lagrange multiplier

λ(x

), and consider the Euler–Lagrange equations of



= L − (1/2)λ(x

)(1 −



5.4 Sigma model lumps 101

This yields

φ

− λφ

=0, where  = η

µν

∂

and the constraint (5.4.24). Multiplying the relation above by φ

, summing

over a, and eliminating λ yield the non-linear ﬁeld equations

φ

− (φ

φ

)φ

=0. (5.4.25)

The Lagrangian and the equation (5.4.25) are invariant with respect to global

O(N) rotations of the ﬁeld φ. This is an example of a model with an internal

symmetry.

Solving the constraint (5.4.24) yields φ

= ±



1 − φ

, where p, q, r =1,

...,N − 1, and allows us to write the Lagrangian density as

L =

(φ)η

µν

∂

, (5.4.26)

where

= δ

1 −



is the metric on S

N−1

induced by the Euclidean inner product in R

. Lagrange

densities of the form (5.4.26) where η and g are arbitrary metrics on a space

time and a target space, respectively, deﬁne more general sigma models. For

example, superstring theory can be viewed as a sigma model where η is a

metric on a ‘string world-sheet’ which is a Riemann surface, and g is a metric

on a 10-dimensional target which plays the role of ‘space time’.

From now on we restrict our attention to (5.4.25) with N = 3. The corre-

sponding model describes the Heisenberg ferromagnet in low temperatures,

when the local magnets line up [66]. The system is characterized by the

direction of the spin vector, that is, a unit vector φ

We will be interested in time-independent solutions with a ﬁnite-energy func-

tional



x. This condition implies that r|∇φ

|→0asr →∞, therefore

φ(x

) tends to a constant ﬁeld φ

∞

at spatial inﬁnity, which we choose to be the

north pole (0, 0, 1). This means that the ﬁnite-energy static solutions extend

to S

. This sphere has inﬁnite radius, and is a one-point compactiﬁcation of

. The Laplacian  in two dimensions is conformally invariant is the sense

that

c

= 

This is not strictly true, as the ﬁniteness of the L

norm of ∇φ does not imply that ∇φ −→ 0.

There could exist ﬁnite-energy maps which do not extend to S

. Sacks and Uhlenbeck [143]

show that this does not happen. Their proof uses the equations of motion and their conformal

invariance.

102 5: Lagrangian formalism and ﬁeld theory

for two conformally related metrics g and c(x

)g. This can be veriﬁed from the

deﬁnition



= |g|

−1/2

∂

(|g|

1/2

∂

Therefore the static equations

φ

− (φ

φ

)φ

= 0 (5.4.27)

are satisﬁed on S

Another consequence of the conformal invariance is that the spatial rescaling

does not change the energy, and can be used to shrink any static solution down

to zero. Therefore the solitons we are just about to describe do not fully deserve

their name, and are called lumps by some authors.

Continuous maps φ : S

−→ S

are classiﬁed by their topological degree (in

this context also called topological charge) given by (A7)

Q = deg φ =

8π



abc

∂

which partially characterizes static solutions. A ﬁeld with a given Q cannot be

continuously deformed into a ﬁeld with different Q. It is now clear why we

have focused on N = 3. The spheres with N > 3 as target spaces would not

lead to non-trivial topological conﬁgurations, as the relevant homotopy group

N−1

) vanishes.

The degree Q carries only global information and ﬁelds can have different

energies within one topological sector.

Proposition 5.4.1 The energy

E =



∂

x ≥ 4π|Q| (5.4.28)

is bounded from below with equality when the ﬁrst-order Bogomolny

equations

∂

= ±ε

abc

∂

(5.4.29)

are satisﬁed.

Proof 1 Consider the identity



(∂

± ε

abc

∂

)(∂

± ε

abc

∂

x ≥ 0

and use the relations

= δ

,ε

abc

ade

= δ

− δ

, and φ

∂

to deduce (5.4.28) and (5.4.29). 

5.4 Sigma model lumps 103

The solutions to the Bogomolny equations (5.4.29) are critical points of the

energy functionals, and so are also solutions to the second-order static-ﬁeld

equations (5.4.27). It can be shown [166] that all ﬁnite-energy solutions to

(5.4.27) are solutions to the Bogomolny equations. In the context of the

Heisenberg ferromagnet the solutions of (5.4.29) are called spin waves. The

degree of φ describes how often the spins aligned along some axis twist around

this axis. More generally, the degree can be interpreted as the number of lumps,

because generically the energy density is concentrated in Q localized regions.

The Bogomolny equations (5.4.29) can be easily solved with the help of

complex numbers. Identify S

with a complex projective line CP

(the cor-

responding model is sometimes called the CP

model, or O(3) model). Let

f : R

2,1

−→ CP

be given by

+ iφ

2 f

1+| f |

and φ

| f |

− 1

| f |

, so that f =

+ iφ

1 − φ

(5.4.30)

The Bogomolny equations now imply that f is holomorphic, or anti-

holomorphic in z = x

+ ix

. The total energy (5.4.28) of static ﬁelds in terms

of f is given by

E =



df ∧ df

(1 + | f |

)

and the rational function

f = c

(z − p

) ···(z − p

)

(z − r

) ···(z −r

)

gives ﬁnite-energy solutions which saturate the Bogomolny bound with

deg(φ)=Q. This is because only the rational functions give Q-fold coverings

of an extended complex plane with ﬁnite Q. The overall factor c can be set to 1

by a global rotation of the ﬁeld. The space of static solutions is isomorphic to

the space-based rational maps because lim

z→∞

φ = φ

∞

. It is the complement in

of a hypersurface where the enumerator and denominator have common

poles.

Exercises

1. Starting with the Lagrangian of the Sine-Gordon theory

L =

(φ

− φ

) − (1 − cos βφ)

104 5: Lagrangian formalism and ﬁeld theory

derive the Sine-Gordon equation. Find a kink solution of the Sine-Gordon

theory, and use the Bogomolny bound to ﬁnd its energy. How many types

of kinks are there?

2. The Sine-Gordon equation is

− φ

=sinφ, where φ = φ(x, t).

Set τ =(x + t)/2,ρ =(x − t)/2 and consider the Bäcklund transformations

∂

(φ

− φ

)=2b sin



+ φ



and ∂

(φ

+ φ

)=2b

−1

sin



− φ



where b = const and φ

,φ

are functions of (τ,ρ). Take φ

= 0 and construct

the one-soliton (kink) solution φ

3. Let φ = φ(x, t) be a scalar ﬁeld and let U = U(φ) ≥ 0. Deﬁne the energy of

solutions to the Euler–Lagrange equations with the Lagrangian density

L =

−

− U(φ).

Assume that U =(1/2)φ

(φ

− β

)

, where β ∈ R. How many static kink

solutions are there? Find a moving kink solution for the model if β =0.

4. The Lagrangian density for a complex scalar ﬁeld φ on the two-dimensional

Minkowski space R

1,1

L =

|φ

−

|φ

−

−|φ|

)

, a ∈ R.

Find the ﬁeld equations, and verify that the real kink φ

(x)=a tanh (λax)

is a solution. Now consider a small pure imaginary perturbation φ(x, t)=

(x)+iη(x, t) with η real and ﬁnd the linear equation satisﬁed by η.

By considering η =sech (αx)e

ωt

show that the kink is unstable.

5. Let φ : R

2,1

→ S

. Set

+ iφ

2 f

1+| f |

and φ

| f |

− 1

| f |

and deduce that the Bogomolny equations

∂

= ±ε

abc

∂

and φ

imply that f is holomorphic or anti-holomorphic in z = x

+ ix

. Find an

expression for the total energy

E =



∂

in terms of f .

Gauge field theory

In Chapter 5 we have given examples of Lagrangians invariant under the action

of symmetry groups. These symmetry transformations were identical at every

point of space time. This is referred to as global symmetry in the physics

literature. In this chapter we shall introduce a concept of gauge symmetry,

where the symmetry transformation is allowed to depend on a space-time

point. This is what physicists call local symmetry. This type of symmetry

is already present in Maxwell’s electrodynamics. The kinetic term i

ψγ

∂

in the Lagrangian involving the matter ﬁeld (electron) ψ is unchanged if we

replace ψ by e

ieθ

ψ, where e is the electric charge of an electron and γ

are the

Dirac matrices. If θ is a constant, we talk about global symmetry. Gauging, or

localizing, this symmetry comes down to allowing θ = θ(x

). The Lagrangian

is no longer invariant, unless we replace the ordinary derivative ∂/∂x

by a

covariant derivative D

= ∂

− ie A

, where the gauge potential A

transforms

as A

→ A

+ ∂

θ.

The matter Lagrangian is then complemented by adding a gauge term

−(1/4)F

µν

where F

µν

= ∂

− ∂

is the gauge ﬁeld. The whole

Lagrangian is invariant under the local gauge transformations and the gauge

potential is promoted to a dynamical variable. It corresponds to a gauge boson

which in the case of electrodynamics is identiﬁed with a photon. The symmetry

group which has been gauged in this example is U(1). Thus electrodynamics

is a U(1) gauge theory. The abelian nature of U(1) implies that there are no

interactions between the photons.

The breakthrough made by Yang and Mills [188] was to replace U(1) by a

non-abelian Lie group G. In the gauging process one needs to introduce one

gauge boson for each generator of G. The bosons are particles which ‘carry

interactions’ between the matter ﬁelds, and the form of the interactions is

dictated by the gauge symmetry. The bosons take values in the Lie algebra of G.

If G = SU(3) there are eight gauge bosons generalizing one photon. They

are called gluons. The matter ﬁelds generalizing the electron are called quarks.

The quarks are charged with colour, which generalizes the electric charge. The

quantum SU(3) gauge theory is called quantum chromodynamics (QCD). It is

a theory of strong nuclear interactions.