CER

N-O

PEN

-200

6-00

209

/01/

2006

Instantons in Field Theory

Manuel Lorenzo Sentıs

CERN, CH-1211 Geneva 23, Switzerland

January 6, 2006

Table of Contents

Table of Contents 1

Introduction 2

1 Instantons in a Double Well. Euclidean Space 31.1 The Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Double Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Zero Modes and Collective Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Euclidean Yang-Mills Configurations 112.1 Generalities and Yang-Mills Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Homotopy Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 The Pontryagin Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Valid Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 The Yang-Mills Instantons 193.1 Bogomolny Bound and some Instanton Solutions . . . . . . . . . . . . . . . . . . . . 193.2 The Instanton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 N-Instanton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Symmetries and the Number of Instantons Parameters . . . . . . . . . . . . . . . . . 26

4 Some aspects in QCD 284.1 The Vacuum in a Quantum Non-Abelian Gauge Theory . . . . . . . . . . . . . . . . 284.2 Massless Fermions in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Banks-Casher Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A Semi-Classical Methods in Path Integrals 35

B Another Way to Obtain the Yang-Mill Instanton 38

Bibliography 41

1

Introduction

It appears that all fundamental interactions in nature are of the gauge type. The modern theory of

hadrons - quantum chromodynamics (QCD) - is no exception. It is based on local gauge invariance

with respect to the color group SU(3), which is realized by an octuplet of massless gluons.

The idea of gauge invariance, however, is much older and derives from quantum electrodynamics,

which was historically the first field model in which successful predictions were obtained. By the

end of the forties, theoreticians had already learned how to calculate all observable quantities in

electrodynamics in the form of series in a = 1/137. The first steps in QCD at the end of the

seventies were also made in the framework of perturbation theory. However, it gradually became

clear that, in contrast to electrodynamics, quark-gluon physics was not exhausted by perturbation

theory. The most interesting phenomena - the confinement of colored objects and the formation of

the hadron spectrum - are associated with non-perturbative (i.e., not describable in the framework

of perturbation theory) effects, or rather, with the complicated structure of the QCD vacuum, which

is filled with fluctuations of the gluon field.

In general, there are solutions of the classical, nonlinear equations of motion that exhibit particle-like

behavior that give us powerful insight into the non-perturbative behavior of these theories. Among

these solutions are solitons, monopoles and instantons. These solutions cannot be obtained from

solutions of the corresponding linear part of the field equations and treating the non-linear part

perturbatively.

Instantons are finite-action solutions to the Euclideanized equations of motion. At the classical

level, instantons are not very different from static solutions of Minkowskian equations. This is

obviously because static solutions involve only the spatial coordinates, i.e. the Euclidean subspace

of Minkowskian space-time. Indeed, some of these static solutions can be directly used as instantons

for lower dimensional systems. The only difference is that the requirement of finiteness of energy of

solitons will be replaced by the requirement of finiteness of Euclidean action in the case of instantons.

The name ’instantons’ was coined by ’tHooft (1976) because, unlike solitons of Minkowskian systems

which are not localised in time, these solutions are localised in x4 (the imaginary time coordinate)

as well.

2

Chapter 1

Instantons in a Double Well.

Euclidean Space

1.1 The Euclidean Space

The transition amplitude between two quantum basis states in the Heisenberg representation is

defined as:

〈xf ,T

2|xi,

−T2〉 = 〈xf |e−

i~ HT |xi〉 = N

∫Dxe i

~ S[x] (1.1)

where N is normalization constant, and is defined in Eq. (A.14)(appendix A). If we go from states

with a definite coordinate to states with a definite energy:

H|n〉 = En|n〉 (1.2)

using the completeness relations for the quantum states, we have:

〈xf |e−i~ HT |xi〉 =

∑n

e−i~ EnT 〈xf |n〉〈n|xi〉 (1.3)

In this way we have obtained a sum of oscillating exponentials. If we are interested in the ground state

(in field theory we are always interested in the lowest state, the vacuum), it is much more convenient

to transform the oscillating exponentials into decreasing exponentials by doing the substitution

t→ −it′. Then in the limit t′ →∞ only a single term survives in the sum and this directly tells us

what are the energy E0 and the wave function Ψ0 of the lowest level e−E0t′Ψ0(xf )Ψ+0 (xi).

If the Lagrangian is L = 12mx

2 − V (x) then the action is given by:

S[x] =∫ T

2

−T2

dt [12mx2 − V (x)] (1.4)

3

4

The best way to evaluate the path integral Eq. (1.1) is to go to the Euclidean space, thus by rotating

to imaginary time,

t→ −it′ T → −iT ′ (1.5)

dx

dt= i

dx

dt′(1.6)

L =12mx2 − V (x) → −1

2mx2 − V (x) (1.7)

The action transforms under this Wick rotation:

S =∫ T

2

−T2

Ldt→∫ T ′

2

−T ′2

(−12mx2 − V (x))(−i)dt′ =

∫(12mx2 + V (x)) dt′ = iSE (1.8)

for the transition amplitude we will have:

〈xf |e−i~ HT |xi〉 = N

∫Dx e i

~ S[x] → N

∫Dx e i

~ (iSE [x]) (1.9)

and we obtain in Euclidean space (from now on we will write t,T for the new Euclidean variables):

〈xf |e−1~ HT |xi〉 = N

∫Dx e− 1

~ SE [x] (1.10)

where:

SE [x] =∫ T

2

−T2

dt [12mx2 + V (x)] =

∫ T2

−T2

dtLE(x, x) (1.11)

1.2 Double Well Potential

In this section we will apply the above mentioned formula to the double well potential V (x) =g2

8 (x2 − a2)2. The classical equation which is obtained from the extremum of the action in (1.11)

has the form:δSE [x]δx(t)

= 0 or, mx− V ′(x) = 0 (1.12)

The Euclidean equations, therefore, correspond to a particle moving in an inverted potential other-

wise also known as a double humped potential. The Euclidean energy associated with such a motion

is given by E = 12mx

2 − V (x).

6

- xa−a

V (x)

g2a4

8

Fig.1 Minkowskian potential

- x−a a

g2a4

8

V (x)

Fig.2 Euclidean potential

5

It is easy to find two elemental solutions to the Euclidean classical equation of motion in Eq. (1.12).

Namely, x(t) = ±a (trivial solutions) satisfy the classical equation with minimum energy E = 0.

In this case the particle stays at rest on top of one of the hills. In the Minkowskian space this will

corresponds to the case where the particle executes small oscillations at the bottom of either of the

wells and these small oscillations can be approximated by a harmonic oscillator motion.

There are, however, another interesting solution, one where the particle begins at the top of one

hill at time −T/2, and moves to the top of the other hill at time T/2. Let be:

xcl(t) = ±a tanhω(t− tc)

2(1.13)

where tc is a constant and we have identified mω2 = V ′′(±a) = g2a2; then, it is straightforward

to see that (1.13) are nontrivial solutions of Eq. (1.12) with mxcl − V ′(xcl) = 0 . These solutions

are analogous to the kink and the anti-kink solution of the φ4 model. Notice that this particle-

mechanics example may be thought of as one-dimensional scalar field-theory. Their instantons are

just the static solitons of the corresponding field theory in two (1+1) dimensions. This pattern can

be follow in many other cases when one is looking for instanton solutions.

We also note that, for these solutions Eq. (1.13):

xcl(t→ −∞) = ∓a and xcl(t→ +∞) = ±a

6

- t

−

−

a

−a

tc

Fig.3 Instanton

6

- t

−

−

a

−a

tc

Fig.4 Anti-instanton

As it is seen in Fig.3 and Fig.4 these solutions correspond to the particle starting out on one of the

hill tops at t → −∞ and then moving over to the other hill top at t → ∞. For such solutions we

have:12mx

2cl = 1

2m2mV (xcl) = V (xcl) ⇒ E = 1

2mx2 − V (x) = 0

where E is the Euclidean energy for our system. As in the case of the trivial solution these also

correspond to minimum energy solutions. We can calculate the action corresponding to such a

6

classical motion:

SE [xcl] = S0 =∫ ∞

−∞dt (

12mx2 + V (xcl)) =

∫ ∞

−∞dtmx2

cl = m

∫ ±a

∓a

dxclxcl =

m

∫ ±a

∓a

dxcl(±aω

2cosh−2 ω(t− tc)

2) = m

∫ ±a

∓a

dxcl (∓ω

2a(x2

cl − a2))

= ∓mω2a

(13x3

cl − a2xcl)∣∣∣∣±a

∓a

=mω

2a4a3

3=

23mωa2 =

2m2ω3

3g2

The solutions in (1.13) are, therefore, finite action solutions in the Euclidean space and have

the shape shown in Fig. 1.2 and Fig. 1.2. They are known respectively as the instanton and the

anti-instanton solutions (classical solutions in Euclidean time). If we calculate the Lagrangian for

such a solution, then we find:

LE =12mx2

cl + V (xcl) = mx2cl

=ma2ω2

4cosh−4 ω(t− tc)

2

6

- ttc

Fig.5 Lagrangian of an instanton solution

The graphical representation of this function is shown in Fig.5 , we can deduce that the La-

grangian is fairly localized around t = tc with a size of about ∆t ∼ 1ω =

√m

ga . We say therefore that

instantons are localized solutions in time with a size of about 1ω . The constant tc means the time

when the solution reaches the valley of the Euclidean potential and is arbitrary. This is a result of

the time translation symmetry in the theory.

Just as we can have a one instanton or one anti-instanton solution, we can also have multi-

instanton solutions in such a theory. In this case the particle leaves one top and afterwards comes

back to the same top repeating this cycle several times. However, for simplicity, let us calculate the

contribution to the transition amplitude coming only from the one instanton or one anti-instanton

7

trajectory. From (A.1) and (A.2) (O.I. means One Instanton):

〈+a|e− 1~ HT | − a〉O.I. = N

∫Dx e− 1

~ SE [x] ' (1.14)

N

∫Dη e−

1~ (SE [xcl]+

12

∫ ∫dt1dt2η(t1)

δ2SE [xcl]δxcl(t1)δxcl(t2) η(t2)+O(η3)) = (1.15)

N√det( 1

~δ2SE [xcl]

δxcl(t1)δxcl(t2))e−

1~ (SE [xcl]) = (1.16)

N√det[ 1

~ (−m d2

dt2 + V ′′(xcl))]O.I.

e−1~ S0 (1.17)

Here we have defined x(t) = xcl(t) + η(t), and for the one instanton case xcl(t) = a tanh ω(t−tc)2

as we have already seen in Eq. (1.13).

If we look the determinant for the O.I. case in detail, we see that dxcl

dt = aω2 cosh−2 ω(t−tc)

2 and

from Eq. (A.16): (−m d2

dt2 +V ′′(xcl))dxcl

dt = 0, that is the same as the the zero eigenvalue equation

Eq. (A.11), rotated to imaginary time. We can define the normalized zero eigenvalue solution of this

equation as:

ψ0(t) = (S0

m)−

12dxcl

dt= (

S0

m)−

12aω

2cosh−2 ω(t− tc)

2(1.18)

Using Eq. (A.13), the determinant in Eq. (1.17) can be obtained from this solution simply as:

det(1~(−m d2

dt2+ V ′′(xcl))) ∝ ψ0(

T

2) T →∞ (1.19)

and it is seen that:

limT→∞

ψ0(T

2) → 0 (1.20)

That means that the determinant vanishes. The reason for this is that ψ0(t) happens to be

an exact eigenstate of the operator (−m d2

dt2 + V ′′(xcl)) with zero eigenvalue. (This means that

ψ0(±T2 ) = 0 for T →∞). In this case we say that there is a zero mode in the theory.

1.3 Zero Modes and Collective Coordinates

The presence of a zero mode in the theory will indicate a symmetry in the system. To see this,

let us recall that the determinant in Eq. (1.17) appeared once that we integrated out the Gaussian

fluctuations. The part we have to analyze is:∫Dη e−

12~

∫ ∫dt1dt2 η(t1)

δ2SE [xcl]δxcl(t1)δxcl(t2) η(t2) (1.21)

8

Since ψ0(t) represents a zero mode of the operator δ2SE [xcl]δxcl(t1)δxcl(t2)

, if we make a change of the inte-

gration variable as:

δη(t) = εψ0(t) (1.22)

where ε is a constant parameter, then the Gaussian does not change. That means that the trans-

formation in Eq. (1.22) defines a symmetry of the quadratic action. If we expand the fluctuations

around the classical trajectory in a complete basis of the eigenstates of the operator δ2SE [xcl]δxcl(t1)δxcl(t2)

,

then we can write:

η(t) =∑n≥0

cnψn(t) (1.23)

Dη(t) =∏n≥0

dcn (1.24)

Substituting this expansion into Eq. (1.21), we obtain:

∫Dη e−

12~

∫ ∫dt1dt2 η(t1)

δ2SE [xcl]δxcl(t1)δxcl(t2) η(t2) (1.25)

=∫ ∏

n≥0

dcne− 1

2~∑

n>0λnc2

n

(1.26)

=∫dc0

∫ ∏n>0

dcn e− 1

2~∑

n>0λnc2

n

(1.27)

where λn are the eigenvalues corresponding to the eigenstates ψn. Here we see that the zero mode

disappears of the exponent and therefore there is no Gaussian damping for the dc0 integration.

We have seen in Eq. (1.25) that the integral is divergent. The time translation invariance causes

this divergence actually the position of the center of the instanton can be arbitrary. In the next lines

we will replace the dc0 integration by an integration over the position of the center of the instanton.

Let us expand around the actual trajectory of the instanton:

x(t) = xcl(t− tc) + η(t− tc), or

x(t+ tc) = xcl(t) + η(t) = xcl(t) +∑n≥0

cnψn(t) (1.28)

(Since the trajectory is independent of the center of the instanton trajectory, the fluctuations must

balance out the tc dependence) multiplying the last equation with ψ0(t) and integrating over time,

9

we obtain: ∫ T2

−T2

dt x(t+ tc)ψ0(t) =∫ T

2

−T2

dt [xcl(t) +∑n≥0

cnψn(t)]ψ0(t)

=∫ T

2

−T2

dt ((S0

m)−

12xcl

dxcl(t)dt

) + c0 =12(S0

m)−

12x2

cl(t)∣∣T

2

−T2

+ c0 = c0

The first term drops out because it has the same value at both the limits. We are interested in

large T limits when all these results hold. In this way we see that c0 = c0(tc) and we can change

the c0-integration to an integration over tc. Let us consider an infinitesimal change in the path in

Eq. (1.28) arising from a change in the coefficient of the zero mode namely δη = δc0ψ0(t) where we

assume that δc0 is infinitesimal . In this case, using Eq. (1.18):

δx(t+ tc) = δη = δc0ψ0(t) = δc0(S0

m)−

12dxcl

dt

However, we also note that this is precisely the change in the path, that we would have obtained

to leading order, had we translated the center of the instanton as:

tc → tc + δtc = tc + δc0(S0

m)−

12 (1.29)

In this case:

δx(t+ tc) = x(t+ tc + δtc)− x(t+ tc) ' δtcdx(t+ tc)

dt=

δc0(S0

m)−

12dx(t+ tc)

dt' δc0(

S0

m)−

12dxcl(t)dt

From Eq. (1.29), we see that the Jacobian of this transformation to the leading order is:

dc0(tc)dtc

' (S0

m)

12 (1.30)

We now substitute Eq. (1.30) into Eq. (1.25) to obtain:

∫Dη e−

12~

∫ ∫dt1dt2 η(t1)

δ2SE [xcl]δxcl(t1)δxcl(t2) η(t2) (1.31)

= (S0

m)

12

∫ T2

−T2

dtc

∫ ∏n>0

dcn e− 1

2~∑

n>0λnc2

n

(1.32)

= (S0

m)

12

∫ T2

−T2dtc√

det′( 1~ (−m d2

dt2 + V ′′(xcl)))(1.33)

Here det′ stands for the value of the determinant of the operator without the zero mode. This new

coordinate tc is called collective coordinate. Finally we obtain the form of the transition amplitude

10

in the presence of an instanton:

〈a|e− 1~ HT | − a〉O.I. =

N(S0m )

12 e−

1~ S0√

det′( 1~ (−m d2

dt2 + V ′′(xcl)))

∫ T2

−T2

dtc (1.34)

We emphasize that although we have analyzed only a single specific example with the simplest in-

stanton xcl(t) = ±a tanh ω(t−tc)2 , the method of dealing with zero-frequency modes is in fact general.

Thus, in the Yang-Mills instantons any invariance will generate a zero-frequency mode, and the in-

tegration with respect to the corresponding coefficient must be replaced by integration with respect

to some collective variable.

For the evaluation of the determinant det’ see Coleman [14], we would like to mention that with

a similar procedure, we calculate the transition amplitude for multi-instanton contributions. The

important point is that we would obtain, that the splitting between the two energy levels is:

∆E = E+ − E− = 4√

2m~ω32 ae−

1~ S0 ∝ ω

32 e−ω3const

g2 (1.35)

this quantity cannot be expanded in a series in g2 and therefore it would be not possible to obtain

it by perturbation theory.

Chapter 2

Euclidean Yang-Mills

Configurations

2.1 Generalities and Yang-Mills Equations

In this chapter we will only consider the SU(2) Yang-Mills gauge fields Aaµ, self-coupled to one

another and we will not consider for the moment the ’matter fields’ Φa. To simplify the notation we

will represent the three vector-fields Aaµ, a = 1, 2, 3, by a matrix-valued vector-field Aµ defined by:

Aµ(x) :=∑

a

gσa

2iAa

µ(x) (2.1)

the values of the field are evaluated at each point x, where x denotes the vectorial components

x1, x2, x3, x4. Here g is the coupling constant and σa are the familiar Pauli spin matrices:

σ1 =

(0 1

1 0

)σ2 =

(0 −ii 0

)σ3 =

(1 0

0 −1

)(2.2)

These Pauli matrices σa form the three generators of the two-dimensional representation of the group

SU(2), which is the gauge group of our system. This 2 x 2 anti-Hermitian matrix Aµ(x) represents

for each µ and at each point x, the same information as the three Yang-Mills fields Aaµ(x). In the

same way we define a matrix-valued tensor field:

Fµν(x) :=∑

a

gσa

2iF a

µν(x) (2.3)

where:

Fµν = ∂µAν − ∂νAµ + [Aµ, Aν ]. (2.4)

11

12

The gauge transformations are not formally altered by going to the Euclidean metric. In terms of

the matrix field Aν they become (with U ∈ SU(2)):

Aµ → UAµU−1 + U∂µU

−1 and Fµν → UFµνU−1. (2.5)

We will often use the identity:

∂µ(UU−1) = 0 = (∂µU)U−1 + U∂µ(U−1). (2.6)

Another advantage of this matrix notation is that it reveals in an obvious way the parallel to

electromagnetism where the gauge group is the abelian U(1). In that case the group elements are

U = eiα(x), the fields Aµ(x) are just numbers instead of matrices, and Eq. (2.4) and Eq. (2.5) reduce

to familiar equations in electromagnetism. In this matrix notation, it can easily be seen that this

action and the field equation are invariant under the gauge transformations Eq. (2.5).

Returning to the Euclidean Yang-Mills system, we have four matrix-valued fields Aµ (µ =

1, 2, 3, 4) at each Euclidean space point xµ (µ = 1, 2, 3, 4). As seen in chapter 1 Eq. (1.8), the Eu-

clidean action is obtained from the Minkowskian Lagrangian by the prescription S = iSE , x4 = −ix′4after discarding the scalar field φa, and using the matrix notation for Aµ. This action is:

S = − 12g2

∫d4xTr[FµνFµν ] (2.7)

It is important to notice that the coupling constant g is also present into the scale of the field

Aµ see Eq. (2.1), explicitly it appears only in the factor −1/2g2.

Proposition 2.1.1. The resulting Euclidean Yang-Mills equation is:

DµFµν = ∂µFµν + [Aµ, Fµν ] = 0 (2.8)

Proof. One requires the action to be stationary with respect to small variations of the gauge fields

Aµ → Aµ + δAµ and its derivative δ(∂νAµ) = ∂ν(δAµ) where δAµ transforms in an adjoint repre-

sentation of the group δA′µ(x) = U(x)δAµ(x)U−1(x). The change of the field strength under the

variation of Aµ can be expressed as follows:

Fµν → Fµν +Dµ(δAν)−Dν(δAµ)

where we have used the definition of covariant derivative Dµφ = ∂µφ+ [Aµ, φ] and the variation of

the Gauge field action is:

δ

∫d4x

12g2

Tr[FµνFµν ] =2g2

∫d4xTr[Fµν(DµδAν)]

and using the relation:

DµTr[(FµνδAν)] = Tr[(DµFµν)δAν ] + Tr[Fµν(DµδAν)]

13

As a consequence of the invariance of Tr[FµνδAν ] under Gauge transformations we can replace the

covariant derivative by the ordinary derivative. By integrating the last equation and neglecting the

surface terms at infinity:

δ

∫dx4 1

2g2Tr[FµνFµν ] = − 2

g2

∫dx4 Tr[(DµFµν)δAν ]

2.2 Homotopy Classification

As in the case of the double well instantons, the Yang-Mills instantons are finite-action solutions

of Eq. (2.8). We first identify the boundary conditions to be satisfied by any finite-action field

configurations including the finite-actions solutions of Eq. (2.8). Based on these boundary conditions,

we shall make a homotopy classification. Actual solutions will be obtained only in the next chapter.

We will consider zero-action configurations as a first step towards identifying finite-action con-

figurations. From Eq. (2.7), we see that S = 0 if and only if Fµν = 0. This allows an infinite set

of possibilities for the parent field Aµ. Note that Fµν = 0 is a gauge-invariant condition, i.e, not

only Aµ = 0 will satisfy it but also any gauge-transformed field obtained from Aµ = 0. Taking into

account Eq. (2.5)these fields are given by:

Aµ(x) = U(x)∂µ(U−1(x)) (2.9)

These fields are called pure gauges and U(x), at each x, is any element of the group SU(2)

in its 2 × 2 representation. It can be directly verified, using Eq. (2.4), that this equation yields

Fµν = 0. The converse, that Fµν = 0 everywhere implies Eq. (2.9), can also be proved. We are

interested in finite-action configurations, it is clear from Eq. (2.7) that Fµν must vanish on the

boundary of Euclidean four-space, i.e. on the three-dimensional spherical surface S∞3 at r = ∞where r := |x| = (x2

1 + x22 + x2

3 + x24)

1/2 is the radius in four dimensions, more precisely Fµν must

tend to zero faster than 1/r2 as r →∞. The fact that Fµν = 0 on S3 means, based on Eq. (2.9) the

following boundary conditions on Aµ:

limr→∞

Aµ = limr→∞

U∂µU−1 (2.10)

where U is some group-element-valued function. Based on Eq. (2.10) we can associate to every finite-

action configuration Aµ a group function U, defined on the surface at infinity, S∞3 . This surface may

be parameterised by three variables α1, α2 and α3, say for example the polar angles in four dimen-

sions. The problem is how to define the radial derivative when U depends only on α1, α2 and α3, a

general Aµ may have a radial component as well. We can use the gauge transformations to overcome

14

this difficulty, we will gauge transform Aµ such that its radial component vanishes everywhere. For

example if we have an Aµ(x) with non zero radial component Ar(x) then we can perform a gauge

transformation, using as a gauge function:

U(x) = P (exp

r∫o

dr′Ar(x′))

The ’path ordering’ prefix P means an ordering of the path of integration.This ordering is specified

by defining U to be a product or the exponentials exp(Ar dr′) over each infinitesimal element of the

path, arranged sequentially as we go from the starting point(x=0) to the end-point x.

A′r(x) = UArU−1 − (∂rU U

−1)

= U(Ar −Ar)U−1 = 0

Given a point x, consider a path, which in this case runs from the origin to x along the radius vector.

The component of Aµ parallel to this path is Ar. Thus we have a line-integral of ’path ordered’

exponentials of Aµ along this radial path. It is important to note that Aµ(x′) at each x’ is a matrix

and these matrices at different x’ will not commute in general and their ordering has to be defined.

We can write the boundary condition as (since the action is gauge invariant, if Aµ has finite

action, so does A′µ, with A′r ≡ 0):

(A′µ(x))S∞3= U(α1, α2, α3)∂µU

−1(α1, α2, α3) (2.11)

where U(α1, α2, α3) has to be defined only on S∞3 . The infinite different possibilities of choosing

the boundary condition, that is to say U(α1, α2, α3), can be related by homotopy considerations.

Since the matrices U form a two-dimensional, representation of SU(2), the function U(α1, α2, α3)

represents a mapping of S∞3 into the group space of SU(2). By definition of SU(2), the matrices U

are the set of all 2 × 2 unitary unimodular matrices. In order to classify the mappings of S∞3 into

SU(2), let us study the topology of SU(2). The elements of SU(2) can be written uniquely in the

form:

U =4∑

µ=1

aµsµ (2.12)

where s4 = I, is the unit 2 × 2 matrix, s1,2,3 = iσ1,2,3, and aµ are any four real numbers satisfy-

ing Σµaµaµ = 1. The group is thus parametrised by these four real variables aµ subject to this

constraint. The group space is therefore the three-dimensional surface of a unit sphere in four di-

mensions. We will call this surface Sint3 . In this way we see that the function U(α1, α2, α3) is a

mapping of S∞3 into Sint3 .

15

2.2.1 The Pontryagin Index

We can define homotopy classes (a discrete number of classes) in the mappings S∞3 → Sint3 , each

characterised by an integer Q. This integer is often called the Pontryagin index. The important point

is that mappings from one class cannot be continuously deformed into a mapping from another class,

we can say that they are topologically separated. Any given sector corresponds to a given homotopy

class of the U(α1, α2, α3) occurring in the boundary condition Eq. (2.11) for Aµ. A field Aµ(x) in

four-space belonging to a given sector Q cannot be continuously deformed so as to fall into another

sector without violating finiteness of action. For, if this were possible, the U(α1, α2, α3) related to

the boundary value of Aµ(x) would have also deformed from one homotopy class to another, which

cannot happen by definition. The expression for the Pontryagin index Q is:

Q ≡∫Q(x)d4x = − 1

16π2

∫d4xTr[FµνFµν ] (2.13)

where:

Fµν =12εµνρσFρσ. (2.14)

and where Q(x) is the topological charge density. Now we will show that the expression Eq. (2.13)

gives the winding number i.e. the number of times the group SU(2) is wrapped around the surface

at infinity S∞3 . We will show it stepwise with the following propositions:

Proposition 2.2.1. The integrand of the topological charge Q is a total 4-divergence i.e. Q(x) =

∂µjµ, where:

jµ = −(1/8π2)εµναβTr[Aν(∂αAβ +23AαAβ)] (2.15)

Proof. We begin with an identity:

DµFµν ≡ ∂µFµν + [Aµ, Fµν ] = εµναβ{∂µ(∂αAβ +AαAβ) + [Aµ, (∂αAβ +AαAβ)]} = 0 (2.16)

where the last equality follows simply from expanding out the terms and using the antisymmetry of

εµναβ . We can write:

−16π2Q(x) = Tr[FµνFµν ] = Tr[(∂µAν − ∂νAµ)Fµν + (AµAν −AνAµ)Fµν ] =

= Tr{(∂µAν − ∂νAµ)Fµν +Aµ[Aν , Fµν ]} = Tr[(∂µAν − ∂νAµ)Fµν −Aµ∂νFµν ] =

= Tr[∂µAνFµν − ∂ν(AµFµν)]

where the cyclic property of the trace as well asDµFµν = 0 have been used. And finally by expanding

out Fµν :

−16π2Q(x) = Tr{εµναβ [(∂µAν)(∂αAβ +AαAβ)− ∂ν(Aµ∂αAβ +AµAαAβ)]} =

= Tr{εµναβ [2∂µ(Aν∂αAβ +23AνAαAβ)]}

16

where we have used: Tr[εµναβ(∂µAν)AαAβ ] = 13Tr[εµναβ∂µ(AνAαAβ)] which follows from the

cyclicity of trace and the antisymmetry of εµναβ

With the help of this proposition we can express Q as an integral over S∞3 as follows:

Q =∫Q(x)d4x =

∮S∞3

dσαjα (2.17)

On the surface at infinity S∞3 , our finite-action configurations have Fµν = 0 and therefore:

εµναβ∂αAβ = −εµναβAαAβ (2.18)

and using Eq. (2.15) we have:

Q =1

24π2

∮S∞3

dσµεµναβTr[AνAαAβ ] (2.19)

And inserting the asymptotic behavior of the fields Aµ Eq. (2.11):

Q = − 124π2

∮S∞3

dσµεµναβTr[(∂νU)U−1(∂αU)U−1(∂βU)U−1]. (2.20)

In Eq. (2.20) we have the integrand in terms of the group-element-valued function U on S∞3 . Now

we have to discuss how to measure the group volume, and thus we can deduce how many times the

group has been wrapped around S∞3 by a given gauge function U. We can parametrise SU(2) by

three independent variables ξ1, ξ2 and ξ3. The group measure may be written as:

dµ(U) = ρ(ξ1, ξ2, ξ3)dξ1dξ2dξ3

the function ρ(ξ1, ξ2, ξ3) called density has to be such that the measure dµ(U) is invariant un-

der group translations. This would correspond in the case of finite group with the fact that

the number of elements cannot change when we multiply them by some given element. For ex-

ample with U ′ = UU we have (ξ1, ξ2, ξ3) → (ξ′1, ξ′2, ξ

′3) and the density should be such that:

dµ(U) = ρ(ξ1, ξ2, ξ3)dξ1dξ2dξ3 = dµ(U ′) = ρ(ξ′1, ξ′2, ξ

′3)dξ

′1dξ

′2dξ

′3

Proposition 2.2.2. The density:

ρ(ξ1, ξ2, ξ3) = εijkTr(U−1 ∂U

∂ξiU−1 ∂U

∂ξjU−1 ∂U

∂ξk) (2.21)

satisfies translation invariance.

Proof.

U(ξ1, ξ2, ξ3) → U ′(ξ′1, ξ′2, ξ

′3) = UU(ξ1, ξ2, ξ3); U = U−1U ′; U−1 = (U ′)−1U and

ρ(ξ1, ξ2, ξ3) = εijkTr((U ′)−1U U−1 ∂U′

∂ξ′p

∂ξ′p∂ξi

(U ′)−1U U−1 ∂U′

∂ξ′q

∂ξ′q∂ξj

(U ′)−1U U−1 ∂U′

∂ξ′r

∂ξ′r∂ξk

) =

= εijkTr((U ′)−1 ∂U′

∂ξ′p(U ′)−1 ∂U

′

∂ξ′q(U ′)−1 ∂U

′

∂ξ′r

∂ξ′p∂ξi

∂ξ′q∂ξj

∂ξ′r∂ξk

) but

εijk

∂ξ′p∂ξi

∂ξ′q∂ξj

∂ξ′r∂ξk

= εpqrDet|∂ξ′

∂ξ|

17

Proposition 2.2.3. The expression of Q Eq. (2.13) reduces to an integral over the group measure

Proof. We can write Eq. (2.20) as:

Q = − 124π2

∮S∞3

dσµ[εµναβTr(U−1 ∂U

∂ξiU−1 ∂U

∂ξjU−1 ∂U

∂ξk)∂ξi∂xν

∂ξj∂xα

∂ξk∂xβ

] (2.22)

Instead of considering the hypersphere S∞3 at infinity, we will consider an infinite hypercube with

cartesian coordinates xµ. The eight sides of this cube are the surfaces xµ = ±∞, for µ = 1, 2, 3, 4.

The last equation can be written for example for the hypercube’s surface at x4 = −∞:

Q = − 124π2

∫dx1dx2dx3 εlmnTr(U−1 ∂U

∂ξiU−1 ∂U

∂ξjU−1 ∂U

∂ξk)∂ξi∂xl

∂ξj∂xm

∂ξk∂xn

(2.23)

But we have: εlmn∂ξi

∂xl

∂ξj

∂xm

∂ξi

∂xndx1dx2dx3 = εijkdξ1dξ2dξ3 and therefore the contribution from the

surface x4 = −∞ is:

− 124π2

∫dξ1dξ2dξ3 εijkTr(U−1 ∂U

∂ξiU−1 ∂U

∂ξjU−1 ∂U

∂ξk) (2.24)

A similar contribution comes from all eight surfaces of the hypercube, and finally we can say taking

into account Eq. (2.21) that:

Q ∝∫dξ1dξ2dξ3 ρ(ξ1, ξ2, ξ3) (2.25)

Q ∝∫dµ(U). (2.26)

2.2.2 Valid Gauge Transformations

We still have to solve the fact that the boundary condition Eq. (2.11) is not gauge invariant, in fact:

U∂µ(U−1) −→ U ′(U∂µU−1)(U ′)−1 + U ′∂µ(U ′)−1 = (U ′U)∂µ(U ′U)−1 (2.27)

On the other hand we have just seen that Tr[FµνFµν ] as well as Q are gauge invariant.

If we could choose U ′ = U−1, so that Aµ = 0 on S∞3 then Q could be reduced to zero and that

would violate the gauge invariance of Q. In Eq. (2.27) we have to distinguish between the behavior of

U and of U ′. The former appears in the boundary condition and therefore it has to be non-singular

only at S∞3 . The latter transforms the field throughout space and has to be non-singular at all x.

We can express U ′ in polar coordinates as U ′ = U ′(|x|, α1, α2, α3), due to the non-singular

behavior of U ′ at all x, U ′ must be a continuous function for all the values of |x|, α1, α2 and α3.

Therefore at |x| = 0, U ′(0, α1, α2, α3) must be independent of the αi in order to be non-singular.

18

That is, it must be a constant matrix. Any constant SU(2) matrix can be continuously obtained

from the identity matrix. U ′ must be continuously deformable to the identity matrix for all |x|.Letting |x| → ∞, the boundary value of U ′ on S∞3 must also be continuously deformable to the

identity matrix, i.e. it must lie in the Q=0 sector. If U ′ were equal to U−1 on S∞3 then U also would

belong to Q=0. Conversely, when U belongs to Q 6= 0, it cannot be continued all the way down to

|x| = 0 without encountering a singularity.

From group and homotopy theory we know that if U belongs to the index Q1 and U ′ to Q2 then the

function U ′U belongs to Q1 + Q2. We can conclude by saying that if U ′(x) is to be a valid gauge

transformation, its boundary value on S∞3 must correspond to Q2 = 0. Thus Q1 +Q2 = Q1 and the

homotopy index will not be affected by gauge transformations, consistent with the gauge-invariant

expression Eq. (2.13).

Chapter 3

The Yang-Mills Instantons

3.1 Bogomolny Bound and some Instanton Solutions

At the classical level, instantons are not very different from static solutions of Minkowskian equations

(solitons). This is obviously because static solutions involve only the spatial coordinates, i.e. the

Euclidean subspace of Minkowskian space-time. In this chapter we will look, among finite-action

configurations in any given homotopy sector Q for some configurations that actually solve the field

equation. The exact Q=+1 (or -1) solution we obtain will be called the instanton (or anti-instanton).

We will also get some exact solutions with Q = N (or -N), with |N | > 1. These are the multi-

instanton (or anti-instanton) solutions. We begin with the trivial identity:

−∫d4xTr[(Fµν ± Fµν)2] ≥ 0 (3.1)

Using Tr[FµνFµν ] = Tr[FµνFµν ], this gives:

−∫d4xTr[FµνFµν + FµνFµν ± 2FµνFµν ] ≥ 0 this gives (3.2)

−∫d4xTr[FµνFµν ] ≥ ∓

∫d4xTr[FµνFµν ] (3.3)

Using the expression of the Euclidean action S Eq. (2.7) and the expression of the Pontryagin

homotopy index Q Eq. (2.13), we can convert this inequality in the Bogomolny bound:

S ≥ (8π2

g2)|Q| (3.4)

We obtained the Yang-Mills equation Eq. (2.8) by extremising the action S through the variational

principle δS[Aµ] = 0, in fact Eq. (2.8) is just an explicit form of this principle. Since such small

variations will keep the fields within the same sector, we see that after the variation we can still keep

19

20

in the same homotopy sector. In this way we can find solutions of Eq. (2.8) falling in a homotopy

sector and these fields will extremise the action in this homotopy sector. From Eq. (3.2) we see that

the absolute minimum value of S i.e.:

S =8π2

g2|Q| (3.5)

is attained in any given sector Q when:

Fµν =12εµν%σF%σ = ±Fµν (3.6)

Thus, selfdual and anti-selfdual configurations extremise S, and hence solve Eq. (2.8). Of course,

there can be other other extrema apart from the absolute minima of S. In other words there can

be other solutions of Eq. (2.8) different from the selfdual or the anti-selfdual solution. We will

content ourselves with the selfdual and anti-selfdual solutions, in which case we need to solve only

Eq. (3.6) rather than the (possibly) more general Eq. (2.8). That fields satisfying Eq. (3.6) also

satisfy Eq. (2.8) can be seen in a straightforward way:

DµFµν = ∂µFµν + [Aµ, Fµν ]

=12εµν%σ{∂µ(∂%Aσ +A%Aσ) + [Aµ, (∂%Aσ +A%Aσ)]} = 0

This is satisfied by any Fµν of the form Eq. (2.4), obviously when Fµν = ±Fµν the last equation

reduces to the field equation Eq. (2.8). The alternate proof using Eq. (3.6) brings out the added

fact that such solutions in fact give the absolute minimum of the action S in any given Q-sector.

Since F = ±F corresponds to an extremum S, any selfdual or anti-selfdual F is a solution. An

anti-instanton solution is a anti-selfdual F having Q = −1. An instanton solution is a selfdual F

having Q = 1. The corresponding actions are SI = −8π2/g2 and SI = +8π2/g2. Let us make the

following ansatz for the gauge field:

Aµ(x) = iΣµν∂ν(lnφ(x)) (3.7)

where φ(x) is a scalar function to be determined and:

Σµν =12

0 +σ3 −σ2 −σ1

−σ3 0 +σ1 −σ2

+σ2 −σ1 0 −σ3

+σ1 +σ2 +σ3 0

(3.8)

Or:

Σµν = ηiµνσi/2; i=1,2,3 with: (3.9)

ηiµν = −ηiνµ =

{εiµν for µ, ν = 1, 2, 3

−δiµ for ν = 4(3.10)

21

We can also define:

Σµν = ηiµνσi/2; i=1,2,3 with: (3.11)

ηiµν = −ηiνµ =

{εiµν for µ, ν = 1, 2, 3

δiµ for ν = 4(3.12)

Σµν =12

0 +σ3 −σ2 σ1

−σ3 0 +σ1 σ2

+σ2 −σ1 0 σ3

−σ1 −σ2 −σ3 0

(3.13)

and:

Σµν = Σµν (3.14)

Σµν fulfill the properties:

[Σνσ, Σνρ] = i[δµνΣσρ + δρσΣµν − δµρΣσν − δνσΣνρ] (3.15)

εµναβΣβσ = δµσΣνα + δνσΣαµ + δασΣµν (3.16)

˜Σµν =12εµναβΣαβ = −Σµν (3.17)

Thus we obtain for the tensor field:

Fµν = iΣνσ(∂µ∂σ lnφ− ∂µ(lnφ)∂σ(lnφ))− iΣµσ(∂ν∂σ lnφ− ∂ν(lnφ)∂σ(lnφ))− iΣµν(∂σ lnφ)2

and for the dual tensor field:

Fµν = iεµναβ [Σβσ(∂α∂σ lnφ− ∂α(lnφ)∂σ(lnφ))− 12Σαβ(∂σ lnφ)2]

= iΣνα(∂α∂µ lnφ− ∂α(lnφ)∂µ(lnφ))− (µ↔ ν) + iΣµν∂σ∂σ(lnφ)

Self-duality yields:

∂µ(∂σ lnφ)− (∂µ lnφ)(∂σ lnφ) = ∂σ(∂µ lnφ)− (∂σ lnφ)(∂µ lnφ) (3.18)

and also:

∂σ∂σ(lnφ) + (∂σ lnφ)2 = 0 (3.19)

the first is evident, and the second is:2φ

φ= 0 (3.20)

When φ is non-singular Eq. (3.20) reduces to 2φ = 0, with the only solution φ = constant and

Aµ = 0. When we consider singular φ we obtain more interesting solutions. For example, with

φ(x) = 1/|x2|. At x 6= 0 we have:2φ

φ=

1φ∂σ(

2xσ

|x|4) = 0 (3.21)

22

At x = 0, 2(1/|x|2) = −4π2δ4(x). Thus even at x = 0 2φφ = −4π2|x|2δ4(x) = 0. Therefore

φ(x) = 1/|x2| solves Eq. (3.20) and so does the more general form:

φ(x) = 1 +N∑

i=1

λ2i

|xµ − aiµ|2µ = 1, 2, 3, 4 (3.22)

where aiµ and λi are any real constants, which are respectively the center and the size of the

instanton. The existence of solutions of arbitrary sizes is a necessary consequence of the scale

invariance of the classical field theory.

Let us begin with N = 0, which corresponds to φ(x) = 1 and Aµ = 0. This is the trivial solution

and belongs to the Q = 0 sector.

3.2 The Instanton Solution

We consider N = 1 in Eq. (B.7). With yµ = (xµ − a1µ) and y2 ≡ yµyµ, we have φ(x) = 1 + λ21/y

2

and:

Aµ(x) = −2iλ21Σµν

yν

y2(y2 + λ21)

(3.23)

This solution is singular at y = 0. The singularity can be removed by a correspondingly singu-

lar gauge transformation. Since the self-duality condition and the equation of motion are gauge

covariant, the resulting function will continue to satisfy them. The required gauge function is :

U1(y) = (y4 + iyjσj)/|y| (3.24)

this is because of appendix B Eq. (B.7) and from the definitions Eq. (3.9), Eq. (3.11)

U1(y) ∂∂yµ

[U1(y)]−1 = −2iΣµν(yν/y2), (in appendix B: Σµν = − 1

2σµν Σµν = − 12 σµν) and:

U1(y)−1 ∂

∂yµ[U1(y)] = −2iΣµν(yν/y

2). (3.25)

Thus we can write Eq. (3.23) as:

Aµ(x) = U1(y)−1 ∂

∂yµ[U1(y)]

λ21

y2 + λ21

(3.26)

By gauge transforming with U1(y):

Aµ(x) → A′µ(x)

= [U1(y)][Aµ + ∂µ][U1(y)]−1

= (∂µU1)U−11 (

λ21

y2 + λ21

− 1)

= −(∂µU1)U−11

y2

y2 + λ21

= U1(∂µU−11 )

y2

y2 + λ21

23

and finally:

A′µ(x) = −2iΣµνyν

y2 + λ21

= −2iΣµν(x− a1)ν

|(x− a1)|2 + λ21

(3.27)

This solution has no singularities at any x for any given λ1 6= 0. The corresponding tensor field,

using Eq. (2.4):

F ′µν = 4iΣµνλ2

1

[|x− a1|2 + λ21]2

∝ 1x4

(3.28)

This solution can be compared with Eq. (B.12).

Proposition 3.2.1. The Pontryagin index of the instanton solution is Q = 1.

Proof.

Ui(x) =(x4 + ixjσj)

r=∑

µ

xµsµ (3.29)

On comparing it with the general representation Eq. (2.12),this gauge function correspond to aµ =

xµ. That is, every point on S∞3 is mapped on to the corresponding point, at the same polar angles,

on S(int)3 . Thus the homotopy index Q must be equal to unity

Q = − 124π2

∫S3dS εijkTr(λiλjλk) (3.30)

where λk = U∂kU To calculate Q, we note that the surface integral extends over values of x on S3.

Going from one value of x to another on S3 is a rotation of S3, and such a rotation can always be

undone by a continuous change of U, which does not affect Q. Under a continuous change of U, the

integrand of Eq. (3.30) changes by an 4-divergence ∂µXµ therefore, we can write the integrand as its

value at any fixed x, plus ∂µXµ, and the latter gives no contribution upon integration. Therefore,

to calculate Q it suffices to evaluate the integrand at any fixed x, and multiply the result by 2π2,

the volume of S3. It is most convenient to evaluate the integrand at (x4 = 1,x = 0). In the

neighborhood of this point, we can set up 3-dimensional Cartesian axes x1,x2,x3 lying in S3. Then

we have, at this point,

U = 1,

∂kU−1 = ∂k(x4 + ix · σ

r) =

iσk

r− xk

r3(x4 + ix · σ) = iσk

therefore λk = iσk

and finally Q = (2π2)(− i

24π2εijkTr(σiσjσk)) = 1

Another way to prove that is to see that the action for this field reduces to:

S = − 12g2

∫d4xTr[F ′µνF

′µν ] =

48λ41

g2

∫d4y

(y4 + λ21)4

=8π2

g2(3.31)

24

and then knowing that for the selfdual solution Fµν = Fµν and comparing with Eq. (3.5) which

we write again:

S =8π2

g2|Q| (3.32)

we see that Q=1.

3.3 N-Instanton Solutions

We obtain these solutions by inserting Eq. (B.7) in Eq. (3.7):

Aµ(x) = iΣµν∂ν [ln(1 +N∑

i=1

λ2i

|yi|2)] (3.33)

= −2iΣµν(∑

i

λ2i yiν

|yi|4)/(1 +

∑j

λ2j

|yj |2) (3.34)

Where (yi)µ ≡ (x− ai)µ, using Eq. (3.25) we can write it as:

Aµ(x) =N∑

i=1

U−11 (yi)∂µU1(yi)fi(x) (3.35)

where:

fi(x) = (λ2

i

(x− ai)2)/(1 +

∑j

λ2j

(x− aj)2) (3.36)

and Ui(yi) is the same kind of Q = 1 gauge function given by Eq. (3.24). This field Aµ(x) is a

solution of the Yang-Mills equation since it has been derived from the selfduality condition. But it

is also singular, it has N singularities, at the points x = ai, i = 1 · · ·N

fj(x) → δij as x→ ai (3.37)

Aµ → U−11 (x− ai)∂µ[U1(x− ai)] as x→ ai (3.38)

Thus, near any one of its singularities the N-instanton solution behaves like a pure gauge, just as

did the one-instanton solution near its center. Therefore, the N-instanton solution can be thought

of as a collection of N individual single instantons. Near the core of the ith instanton (x ' ai), the

effect of the other instantons is negligible, and the N-instanton solution is dominated by that single

instanton. From Eq. (3.37), we see that the singularities can be removed if we find a gauge function

that reduces to U1(x− ai) when x approaches any of the ai. We take:

UN (x) = (Z4 + iZ · σ)/√Z2 (3.39)

with:

Zµ(X) =N∑

i=1

βi(yiµ

y2i

) (3.40)

25

and yi = x− ai where βi are constants. As yi → 0 for some i, Zµ → βiyiµ/y2i , Z

2 → β2i /y

2i and:

UN (x) → βi(yi4 + iσ · yi

y2i

)yi

βi= U1(yi)

Gauge transformation of Aµ(x) by UN (X) will remove all the singularities at x = ai:

A′µ(x) = UN (x)(Aµ + ∂µ)U−1N and

A′µ(x) −→︸︷︷︸x→ai

U1(yi)[U1(yi)−1∂µU1(yi)]U−11 (yi) + U1(yi)∂µU

−11 (yi) + possible finite terms

= ∂µ[U1(yi)U−11 (yi)] + finite terms

The first term vanishes, and the singularity at x = ai is removed. This holds for each ai, i = 1 · · ·N .

However, there is the danger that new singularities appear because of the gauge function UN (x)

Eq. (3.39), that has another singularities at Zµ = 0. We still have the arbitrary real constants βi in

the definition of Zµ, to play with.

Proposition 3.3.1. The Pontryagin index of the N-instanton solution is Q = N .

Proof. From the Eq. (2.13)

Q = − 116π2

∫d4xTr[F ′µνF

′µν ]

where F ′µν is the tensor field corresponding to A′µ. Since the integrand is non-singular, we can

exclude from the integration region N small spheres with centers at x = ai and radii ε each, where

ε → 0. For any finite small ε, the integration is over Vε, which stands for all space minus these N

spheres. In Vε even the original solution Aµ(x) is non-singular and furthermore:

Tr[FµνFµν ] = Tr[F ′µνF′µν ] (3.41)

where Fµν corresponds to Aµ, since Aµ and A′µ are related by gauge transformations. Hence:

Q = limε→0

− 116π2

∫d4xTr[F ′µνF

′µν ] (3.42)

The singularities of Aµ are excluded from Vε so we can use Gauss’s theorem:

Q = limε→0

(∮

S∞

jµdσµ −∮

σ1

jµdσµ −∮

σ2

jµdσµ · · · −∮

σN

jµdσµ) (3.43)

where jµ is the current. As |x| → ∞, Aµ(x) behaves as 1/|x3| and hence∮

S∞jµdσµ → 0. On the

tiny spheres (x → ai) Aµ becomes a pure gauge, with gauge function U−11 (x − ai). When Aµ is a

pure gauge, we can use the result of a former proposition to obtain∮jµdσµ with gauge function U1,

in our case the gauge function is U−11 and the value of the integral in each tiny sphere is -1, and we

have:

Q = limε→0

[(0− (−1)− (−1) · · · − (−1)] = N (3.44)

26

Since these N-instanton solutions are selfdual, their action is S = 8π2Q/g2 = 8π2N/g2. In

a Minkowskian space a static solution of N-solitary waves separated by finite distances, would be

expected generally to have a different energy from N such isolated solitary waves. The difference

in energy would be attributed to the interaction between these solitary waves. For our Euclidean

system, the action functional S, which is somewhat analogous, has the remarkable property:

S(N instatons) = NS(one instaton) (3.45)

Anti-selfdual and N-anti-instanton solutions are obtained in the same way, by replacing Σµν by Σµν

in the ansatz Eq. (3.7).

3.4 Symmetries and the Number of Instantons Parameters

It has to be noticed that in a 4-dimensional space our coupling constant g has dimension zero (con-

stant), what does not occur for dimensions other than 4. This produces many problems in connection

with infrarot-divergence above all for large instantons.

The Yang-Mills Instanton possesses symmetry under the full 4-dimensional conformal group,

which has 15 generators corresponding to:

- Scale or dilatation transformations

- The four translations

- The six O(4) rotations

- The four special conformal transformations: xµ → ( xµ−cµx2

1−2cµxµ+x2c2 ) all this in addition to gauge

transformations.

We would like to know the number of physically distinct solutions we can obtain from a particular N-

instanton solution by exploiting the full symmetries of the system. For the case of the one-instanton

solution. We have already seen five parameters, λ and aµ, corresponding to dilatation and translation

symmetry. The effect of the remaining generators of the conformal group ( O(4) rotations and special

conformal transformations) can be undone by gauge transformations and translations (see Jackiw

and Rebbi [35], [36], [37], [38]). Local gauge transformations are not the same kind of symmetries as

spatial rotations. Their action produces not a physically distinct system, but the same system in a

different (gauge) convention. One fixes the gauge by some gauge condition and then derives physical

consequences. For the global gauge transformations (i.e. x-independent) however, it has been proved

(see Coleman [15]) that these are symmetries like rotations since they lead to conservation laws. In

our case the global gauge transformations are elements of SU(2) therefore global symmetry gives

three more degrees of freedom, adding these 3 parameters to the 5 corresponding to dilatation and

translation symmetry we obtain a total of eight free parameters for the single instanton. The solution

27

expressed by Eq. (3.27) reveals only five of these eight free parameters for the single instanton. In

order to exploit the full symmetry of the system we have to substitute the specific gauge function

U1(x) in Eq. (3.24) by a global element in the group space. We would expect 8N parameters in an

N-instanton solution. We have already seen that the N-instanton solution looks like a one-instanton

solution near each of the ai. Equivalently, when the sizes λi of the individual instantons are small

compared with their separations, the N-instanton solution will look like a collection of N(almost)

non-overlapping single instantons. This count of 8N degrees of freedom could be reduced to 8N − 3

if we decide on physical grounds to ignore global transformations as symmetries. In Eq. (3.33) we

have only 5N parameters because of the limitations of the ansatz Eq. (3.7) (for a rigorous derivation

see Atiyah and Ward [1]).

Chapter 4

Some aspects in QCD

4.1 The Vacuum in a Quantum Non-Abelian Gauge Theory

In this section we will continue with the non-abelian Yang-Mills theory, we will work with SU(2),

but our analysis will also hold for self-coupled gauge fields of any higher SU(N) group in (3+1) di-

mensions, with minor modifications and this is because of the Bott’s Theorem, that states that any

continuous mapping of S∞3 into any general simple Lie group G can be continuously deformed into

a mapping of S∞3 into an SU(2) subgroup of G. QCD is essentially an SU(3) theory, with fermions

added.

Let us recall the equations of our system in Minkowskian space:

S =2

2g2

∫d4xTr[FµνF

µν ] =1g2

∫d4xTr[(∂0A +∇A0 − [A0,A]) ·E +

12(E2 + B2)] = (4.1)

=2g2

∫d4xTr[∂0A ·E +

12(E2 + B2)−A0(∇ ·E + [A,E])] (4.2)

we see that the canonical variable are A and 2/g2E here we have introduced the analogs of the

electric and magnetic fields: Eia = F i0

a and Bia = − 1

2εijkFjka i, j, k = 1, 2, 3

The first step is to identify the minimum-energy configurations (the classical vacua) of this

system, so that a suitable vacuum state may be constructed around them. The set of classical vacua

of this system are the pure gauges satisfying Fµν = 0, or equivalently according to Eq. (2.9):

Aµ(xµ) = e−α(xµ)∂µeα(xµ)

where α(xµ) is any traceless anti-hermitian (2 × 2) matrix function, α is time-dependent, we can

remove it by partially fixing the gauge so that A0(xµ) = 0 and then we have α(xi) where i = 1, 2, 3.

Further, we can restrict ourselves to those α which satisfy eα = 1 at all points |x| = ∞. We will

28

29

show later that quantum tunnelling as calculated by the appropriate Euclidean functional integral

will connect the α = 0 configuration with only those classical vacua which satisfy [eα(xi)]x→∞ = 1.

Therefore three-dimensional space is compacted into the surface of a hypersphere S∞3 . Each such

function gives a mapping of this S∞3 into the group space of SU(2). Such mappings can be classified

into homotopy sectors π3(SU(2)) = Z and we can adapt Eq. (2.20) to this situation (with N instead

of Q):

N =1

24π2

∫d3x εijkTr[(eα∇ie

−α)(eα∇je−α)(eα∇ke

−α)]

Consequently the set of classical vacua:

Ai(x) = e−α(x)∇ieα(x) (4.3)

can be divided into a discrete infinity of sectors, each sector associated with a given homotopy class N

of the corresponding gauge group element e−α(x). For convenience, we have taken eα = 1 as x→∞,

an other constant than 1 is also possible. The extent of gauge equivalence under time-independent

transformations (which are the only ones left in the A0 = 0 gauge) is fixed by Gauss’s law. As

generalised to Yang-Mills theory this law states that:

DiF0i = ∂iE

i + [Ai, Ei] = 0 (4.4)

Here Ei ≡ F 0i is the Yang-Mills ”electric” field in 2x2 matrix form, this equation has no time

derivatives and is therefore a constraint. It has to be imposed as a condition on physical states which

are required to satisfy DiEi|Ψ〉phys = 0. Under infinitesimal transformations e−δΛ(x) ' 1−δΛ(x) we

have Ai → Ai + [Ai, δΛ] +∇i(δΛ) = Ai +Di(δΛ), where Di is the covariant derivative. We consider

for a moment only those gauge functions Λ(x) which satisfy Λ(x) → 0 as x → ∞, the operator

generating the gauge transformations e−Λ(x) is:

U = exp(2ig2

∫d3xTr((DiΛ)Ei)) = exp(− 2i

g2

∫d3xTr(ΛDiE

i)) (4.5)

By virtue of Gauss’s law Eq. (4.4):

UΛ|Ψ〉phys = |Ψ〉phys (4.6)

Thus Gauss’s law forces gauge equivalence only under gauge transformations, which satisfy Λ(x →∞) = 0. By contrast, consider the transformation g1(x) which takes the N=n sector into the N=n+1

sector. The corresponding gauge function Λ1(x) does not satisfy Λ1(x→∞) = 0. Therefore, states

in different homotopy sectors are not forced to be gauge equivalent by Gauss’s law.

We therefore have a distinct topological vacuum |N〉 in each sector. These are not the true vacua

since they will tunnel into one another. We can anticipate this, at least in a semiclassical expansion,

30

since the Yang-Mills system permits finite-action instanton solutions. The correct vacua will be

linear combinations of the topological vacua:

|θ〉 =∞∑

N=−∞eiNθ|N〉 (4.7)

Consider the unitary operator U(g1) which executes the gauge tranformation g1 and gives U(g1)|N〉 =

|N + 1〉. Since [U(g1),H] = 0 and U(g1) is unitary, the eigenstates of H must satisfy U(g1)|θ〉 =

e−iθ|θ〉, which is fulfilled by Eq. (4.7). Finally, if B is any gauge-invariant operator,[B,U(g1)] = 0.

Hence

0 = 〈θ|[B,U(g1)]|θ′〉 = (e−iθ′ − e−iθ)〈θ|B|θ′〉

Therefore, 〈θ|B|θ′〉 = 0 if θ 6= θ′. Hence each |θ〉 is the vacuum of a separate sector of states,

unconnected by any gauge-invariant operators. Let us turn to the Euclidean functional integral

which gives the tunnelling amplitude. The fields Aµ to be integrated over must approach vacuum

(pure gauge) configurations at the boundary of Euclidean space-time. This boundary is a surface

S∞3 . Since a pure gauge field has the form Aµ = e−α∂µeα, where e−α is a element of SU(2), each

given boundary condition is a mapping of S∞3 → SU(2). Such mappings are again classifiable into

homotopy sectors with an associated Pontryagin Index Eq. (2.13):

Q = − 116π2

∫d4xTr[FµνFµν ] =

124π2

∮S∞3

dσµεµναβTr[AνAαAβ ]

The Euclidean functional integral in the sector Q will be:

limτ→∞

[G(τ)]Q = limτ→∞

Tr(e−HT

~ ) =∫{D[Aµ]}Qexp[−SE + Sgf ] (4.8)

where Sgf are the gauge-fixing terms [17].

Fig.6 Instanton

III

I

II

x4 time 6

?

�

31

Next, consider the A0 = 0 gauge, and distort the boundary of space-time into a large closed

cylinder (Fig.6). The two flat surfaces of the cylinder stand for all of three-dimensional space, at

τ = ±∞ respectively, while the curved surface stands for the boundary of space x → ∞ at all τ .

In the A0 gauge, the curved surface will make no contribution to the surface integral for Q. Hence

Q = N+−N−, where N± = 124π2

∫dx3 εijkTr[AiAjAk]τ=±∞; furthermore, one can use the freedom

of time-independent gauge transformations to set α(x) = 0 (Ai(x) = 0) at τ = −∞. Then N− = 0

and Q = N+; the functional integral can be interpreted, in this gauge, as the Euclidean transition

amplitude connecting the N=0 sector to the N=Q sector. Finally, consider the curved surface

of the cylinder where, in the A0 = 0 gauge, 0 = A0(x → ∞, τ) = [e−α(x,τ)(∂/∂τ)eα(x,τ)]x→∞.

Hence [eα(x,τ)]x→∞ is independent of τ . Since in the initial configuration at τ = −∞ we have

chosen eα(x,−∞) = 1, therefore in the final configuration at τ = ∞, again eα(x→∞,∞) = 1. Thus

transition takes place only to those classical vacua where [eα(x,τ)]x→∞ = 1. Our homotopy analysis

of the classical vacua, which we based on this restriction, is hence justified. Finally the Euclidean

functional integral would be (τ →∞):

〈θ|e−Hτ

~ |θ〉 =∑N,Q

e−iQθ〈N +Q|e−Hτ

~ |N〉 = 2πδ(0)∑Q

e−iQθ

∫D[Aµ]Qe−SEuc =

= 2πδ(0)∫D[Aµ]allQ exp(−SEuc +

iθ

16π2

∫Tr(FµνFµν)d4x)

We have an extra term in the Minkowskian Lagrangian ∆Lθ = θ16π2Tr(FµνFµν), this term is a total

divergence and will not affect the classical Yang-Mills equations. The problem is that with this

additional term the system violates P and T symmetry. The exception is the θ = 0 case when the

extra term vanishes.

4.2 Massless Fermions in QCD

When massless spin-1/2 fields are coupled to the Yang-Mills fields, they lead to further modifications

in the behaviour of the vacuum state. The system of massless fermions coupled to non-abelian gauge

fields has some relevance to hadron physics. Quantum chromodynamics (QCD) is a candidate to

describe hadron physics and is a non-abelian gauge theory in (3+1) dimensions. The gauge group

(’colour’ group ) is SU(3), and therefore calls for eight gauge fields Aaµ a = 1, 2 . . . 8. These can

be represented collectively by an anti-hermitian traceless 3 × 3 matrix, analogous to the Yang-

Mills system. Let λa/2 be the hermitian traceless generators of SU(3) in the fundamental 3 × 3

representation. They satisfy [λa/2, λb/2] = ifabcλc/2 where fabc are the structure constants of

the group SU(3), and Tr(λaλb) = 2δab finally Aµ(x, t) = g2i

∑λaAa

µ(x, t), where g is the coupling

constant we have also the Eq. (2.4): Fµν = ∂µAν − ∂νAµ + [Aµ, Aν ]. In QCD, these gauge fields

32

couple to several species of spin-1/2 quarks. Each species is represented by the Dirac field Ψfα, where

f is a flavour index, α is a colour index and the spin index has been suppressed. For a given flavour

(up, down, strange, charm, top, bottom) Ψfα (α = 1, 2, 3), transforms according to the fundamental

representation of the colour SU(3) group. The quarks have masses mf , these in general may depend

on flavour but not on colour, which is an exact gauge symmetry of the model. Our gauge-invariant

Lagrangian density for the model is:

L(x, t) =∑f,α,β

Ψαf (iγµDµ −mf )αβΨβ

f +1

2g2Tr[FµνF

µν ] (4.9)

where (Dµ)αβ = (∂µ +Aµ)αβ We can do the decomposition: Ψ = ΨL + ΨR where:

ΨL,R =1± γ5

2Ψ

Indirect evidence on quark masses suggests that the masses of the up and down quarks are much

smaller than the masses of quarks of other flavours. When mf = 0 the model Eq. (4.9) enjoys further

symmetries beyond SU(3):

L(x, t) =∑

f

ΨL(iγµDµ)ΨL +∑

f

ΨR(iγµDµ)ΨR +1

2g2Tr[FµνF

µν ]

=∑

f

Ψ′L(iγµDµ)Ψ′L +∑

f

Ψ′R(iγµDµ)Ψ′R +1

2g2Tr[FµνF

µν ].

Where:

Ψ′iL = RijΨ

jL

and R ∈ SU(f), therefore SU(f)L × SU(f)R is a symmetry of L when m = 0.

Let the Dirac operator be iD/ = iDµγµ = i(∂µ + Aµ)γµ, we can determine the spectrum of

this operator iD/ ψλ = λψλ we have D/ + = −D/ , therefore all the eigenvalues of D/ are {iλk,−iλk}.Quarks modify the weight in the euclidean partition function by the fermionic determinant:∏

f

det[iD/ (Aµ)−mf ] (4.10)

Using the fact that non-zero eigenvalues come in pairs, the determinant for each flavour can be

written as:

det[iD/ −m] = mν∏λ6=0

(iλ−m) = mν∏λ≥0

(iλ−m)(−iλ−m) = mν∏λ≥0

(λ2 +m2) (4.11)

where ν is the number of zero modes. Note that the integration over fermions gives a determinant in

the numerator. This means that (as m→ 0) the fermion zero mode makes the tunneling amplitude

33

vanish and individual instantons cannot exist. In the QCD vacuum, however, chiral symmetry

is spontaneously broken and the quark condensate 〈qq〉 is non-zero. The quark condensate is the

amplitude for a quark to flip its chirality, so we expect that the instanton density is not controlled by

the current masses, but by the quark condensate, which does not vanish as m→ 0. The experimental

value for the quark condesate is 〈qq〉 ' −(230Mev)3. The perturbations theory gives no explanation

to this value.

4.2.1 Banks-Casher Relation

The quark propagator P (x, y) = −〈x|(iD/ )−1|y〉 is given by:

P (x, y) = −∑

λ

ψλ(x)ψ+λ (y)

λ+ im(4.12)

using the fact that the eigenfunctions are normalized:∫d4xTr(P (x, x)) = −

∑λ

1λ+ im

〈ΨΨ〉 = limm→0

limV→∞

1VTr(

1D/ +m

) =

= limm→0

limV→∞

1V

∑λi>0

(1

iλi +m+

1−iλi +m

) = limm→0

∫ ∞

0

dλ ρ(λ)2m

λ2 +m2

= limm→0

12

∫ ∞

−∞dλ ρ(λ)

2mλ2 +m2

= limm→0

122πρ(im)

in the last step we have performed a complex integration over the positive imaginary axis and we

have finally:

〈ΨΨ〉 = πρ(λ = 0) (4.13)

this is known as the Banks-Casher relation (1980). The interpretation of this relation is (’t Hooft):

In instanton field, it exists an ψ0(x) such that D/ ψ0(x) = 0, the equation for this zero-mode is:

ψ0(x) =1π

ρ

r(r2 + ρ2)32x · γφ

in singular gauge, where φαβ = 1√2εαβ ; α, β = 1, 2 (Dirac& colour) i.e. φ has only 2 non-zero

components. This zero-mode is localized around instanton (|ψ|2 ∼ r6) and is chiral i.e. 1+γ52 ψ0(x) =

0, 1−γ52 ψ0(x) = ψ0(x).

Each Instanton contributes with f zero-modes (one per flavour) to ρ(λ = 0) Eq. (4.13). The result

〈ΨΨ〉 = −∫ ∞

0

dλ ρ(λ)2m

λ2 +m2

34

shows that the order in which we take the chiral and thermodynamic limits is crucial. In a finite

system, the integral is well behaved in the infrared and the quark condensate vanishes in the chiral

limit. This is consistent with the observation that there is no spontaneous symmetry breaking in

a finite system. A finite spin system, for example, cannot be magnetized if there is no external

field. If the energy barrier between states with different magnetization is finite and there is no

external field that selects a preferred magnetization, the system will tunnel between these states

and the average magnetization is zero. Only in the thermodynamic limit can the system develop a

spontaneous magnetization. However, if we take the thermodynamic limit first, we can have a finite

density of eigenvalues arbitrarily close to zero. In this case, the λ integration is infrared divergent

as m → 0 and we get a finite quark condensate Eq. (4.13). Studying chiral symmetry breaking

requires an understanding of quasi-zero modes, the spectrum of the Dirac operator near λ = 0. If

there is only one instanton the spectrum consists of a single zero mode, plus a continuous spectrum

of non zero-modes. In the chiral limit, fluctuations of the topological charge are suppressed, so one

can think of the system as containing as many instantons as anti-instantons. In this way we can

explain the non-vanishing value of the quark condensate by introducing an instanton density, further

calculations give 〈qq〉 ' −(240Mev)3, in concordance with the phenomenological value. There is no

explanation of this value in the framework of perturbation theory.

Acknowledgements

I would like to thank Dr. Philippe de Forcrand for the very stimulating discussions about physics

and for the suggestions to improve this document.

Appendix A

Semi-Classical Methods in Path

Integrals

Let us consider a general action S[x] which is not necessarily quadratic. The transition amplitude

associated with this action, is given by:

〈xf ,T

2|xi,

−T2〉 = N

∫Dxe i

~ S[x] (A.1)

This integral is oscillatory. However, we know that, in such a case, we can rotate to the Euclidean

space and the integrand become well behaved. We will continue to use the real time description

keeping in mind the fact that in all our discussions, we are assuming that the actual calculations

are always done in the Euclidean space and the results are rotated back to Minkowski space. The

classical trajectory satisfies the equation:

δSE [x]δx(t)

∣∣∣x=xcl

= 0 (A.2)

Therefore, it provides an extremum of the exponent in the path integral. Furthermore, the action

is a minimum for the classical trajectory. Thus, we can expand the action around the classical

trajectory . Namely x(t) = xcl(t) + η(t) then we have:

S[x] = S[scl + η] (A.3)

= S[xcl] +12

∫ ∫dt1dt2η(t1)

δ2S[xcl]δxcl(t1)δxcl(t2)

η(t2) +O(η3) (A.4)

and the transition amplitude becomes:

〈xf ,T

2|xi,

−T2〉 = N

∫Dη exp

[ i~(S[xcl]+

12

∫ ∫dt1dt2η(t1)

δ2S[xcl]δxcl(t1)δxcl(t2)

η(t2)+O(η3))]

(A.5)

35

36

' NK√det( 1

~δ2S[xcl]

δxcl(t1)δxcl(t2))e

i~ (S[xcl]) (A.6)

where we have used that: ∫Dηei

∫ tfti

dt η(t)O(t)η(t) =K√

detO(t)(A.7)

When det( 1~

δ2S[xcl]δxcl(t1)δxcl(t2)

) = 0, the path integral has to be evaluated more carefully using the

method of collective coordinates.

For a general action:

S[x] =∫ tf

ti

dt(12mx2 − V (x)) (A.8)

we have:δ2S[xcl]

δxcl(t1)δxcl(t2)= −(m

d2

dt21+ V ′′(x))δ(t1 − t2) (A.9)

Therefore, we see that:

det(1~

δ2S[xcl]δxcl(t1)δxcl(t2)

) ∝ det(1~((m

d2

dt2+ V ′′(x))) (A.10)

We are interested in evaluating determinants of the form det( 1~ ((m d2

dt2 + W (x))) in the space of

functions where η(ti) = 0 = η(tf ).

Let us write the eigenvalue problem for the following operator:

1~((m

d2

dt2+W (x))ψ(λ)

W = λψ(λ)W (A.11)

with the initial value conditions ψ(λ)W (ti) = 0 and ψ(λ)

W (ti) = 1, then:

det( 1~ (m d2

dt2 +W1(x))− λ)

det( 1~ (m d2

dt2 +W2(x))− λ)=ψ

(λ)W1

(tf )

ψ(λ)W2

(tf )(A.12)

where W1 and W2 are two bounded potentials. We note that for λ coinciding with an eigenvalue

of one of the operators, the left hand side will have a zero or a pole. But so will the right hand

side for the same value of λ because in such a case ψλ would correspond to an energy eigenfunction

satisfying the boundary conditions at the end points. Since both the left and the right hand side of

the above equation are meromorphic functions of λ with identical zeroes and poles, they must be

equal. It follows from this result that:

det( 1~ (m d2

dt2 +W1(x)))

ψ(0)W1

(tf )=det( 1

~ (m d2

dt2 +W2(x)))

ψ(0)W2

(tf )= constant (A.13)

That is, this ratio is independent of the particular form of the potential W (x) and can be used to

define the normalization constant in the path integral. For example we can define a quantity N by:

det( 1~ (m d2

dt2 +W (x)))

ψ(0)W (tf )

= π~N2 (A.14)

37

As a consequence of the above discussion N is independent of W, and we have the desired formula

for evaluating Gaussian functional integrals:

NK√det( 1

~δ2S[xcl]

δxcl(t1)δxcl(t2))e

i~ (S[xcl]) = [π~ψ(0)

W (tf )]12 e

i~ (S[xcl]) (A.15)

We note here that the classical equations following from our action have the form:

md2xcl

dt2+ V ′(xcl) = 0 =

d

dt(m

d2xcl

dt2+ V ′(xcl))

0 = md2

dt2(dxcl

dt) + V ′′(xcl)

dxcl

dt

or:

(md2

dt2+ V ′′(xcl))

dxcl

dt= 0 (A.16)

We can see that:

ψ(0)V ′′(t) ∝

dxcl

dt∝ p(xcl) (A.17)

and finally:

〈xf ,T

2|xi,

−T2〉 =

N√p(xf )

ei~ (S[xcl]) (A.18)

Appendix B

Another Way to Obtain the

Yang-Mill Instanton

Let us define for µ, ν = 1, 2, 3, 4:

σµ = (~σ,−i) σ+µ = (~σ, i), (B.1)

σµν = i(σµσ+ν − δµν) (B.2)

It is straightforward to proof:

σµν =12εµναβσαβ = σµν (B.3)

(x · σ)(x · σ+) = r2 (B.4)

Let us make also the following definition:

U = ix · σr

U−1 = −ix · σ+

r(B.5)

We can deduce the following expression for the vector field in the pure gauge Eq. (2.9):

Aµpure−gauge = U∂µU−1 = i

σµνxν

r2(B.6)

For a finite-energy solution with Q = 1, let us make the following ansatz:

Aµ = iσµνxν

r2f(r2) (B.7)

f(0) = 0 (regularity at r = 0), f(∞) = 1 (finite energy, Q = 1). To make Eq. (B.7)this a solution,

we only need to choose f such that Fµν = Fµν (selfdual condition for the instantons).

38

39

Proposition B.0.1. From Eq. (B.7) we can derive the following expressions for the tensor field:

Fµν = −2i{f(1− f)r2

σµν +1r2

[f ′ − f(1− f)r2

](σµαxαxν − σνβxαxν)}, (B.8)

Fµν = −2i{f(1− f)r2

σµν +1r2

[f ′ − f(1− f)r2

]εµναβ(σαλxλxβ − σβλxλxα)} (B.9)

Proof. We will proof only the first identity Eq. (B.8):

Fµν = ∂µAν − ∂νAµ + (AµAν −AνAµ) =

i(σνµf(r2)r2

+ σνβxβ2xµf

′

r2− σνβxβ

2xµf

r4− σµν

f(r2)r2

− σµαxα2xνf

′

r2+ σµαxα

2xν

r4f) +

+i21r4f2σµαxασνβxβ − i2

1r4f2σνβxβσµαxα =

= i[(σνµ − σµν)f

r2+

2f ′

r2(σνβxµxβ − σµαxνxα) +

2fr4

(σµαxνxα − σνβxµxβ)−

−f2

r4xαxβ(σµασνβ − σνβσµα)

The latter parenthesis is:

(σµασνβ − σνβσµα) = (σµα(−1)σβν − σνβ(−1)σαµ) =

= i(σµσ+α − δµα)(−1)i(σβσ

+ν − δβν)− i(σνσ

+β − δνβ)(−1)i((σασ

+µ − δµα =

= σµσ+α σβσ

+ν − σνσ

+β σασ

+µ + σνσ

+β δµα − δµασβσ

+ν + δνβσασ

+µ − σµσ

+α δβν

and finally using (x · σ)(x · σ+) = r2 we have:

−f2

r4xαxβ(σµασνβ − σνβσµα) =

−f2

r4[(σµσ

+ν − σνσ

+µ )r2 + xµxβ(σνσ

+β − σβσ

+ν ) + xαxν(σασ

+µ − σµσ

+α )] =

and now using σνσ+β = σνβ

i + δνβ , we obtain:

−f2

r4xαxβ(σµασνβ − σνβσµα) == 2i[

σµν

r2f2 + xµxβ

σνβ

r4f2 − xαxν

σµα

r4f2]

and altogether:

∂µAν − ∂νAµ + ig(AµAν −AνAµ) =

= 2i[−σµνf

r2+ σµν

f2

r4+ (

f ′

r2− f

r4+f2

r4)(σνβxβxµ − σµαxαxν)]

40

Thus Fµν is self-dual if the second term vanishes:

f ′ − f(1− f)r2

= 0 (B.10)

the general solution satisfying the required boundary conditions is:

f(r2) =r2

ρ2 + r2(B.11)

where ρ is an arbitrary scale parameter, the translational invariance permits to displace the center

at an arbitrary point r0, for which it is necessary to replace r by r − r0 and

Fµν = Fµν = −2iρ2

(r2 + ρ2)2σµν (B.12)

which is the field tensor for an instanton with Q = 1.

Bibliography

[1] M. Atiyah, A. Ward, Comm. Math. Phys. 55, 117.

[2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz, Y.S. Tyupkin, Pseudoparticle solutions of the

Yang-Mills equations, 1975, Phys. Let. Vol.59B, 1, pg.85.

[3] A.A. Belavin, A.M. Poliakov, 1977, Nucl. Phys. B123, 429.

[4] N.N. Bogoliubov, D.V. Shirkov, Introduction to the theory of quantised fields, Interscience

Publishers Inc. New York 1959.

[5] J.D. Bjorken, S.D. Drell, Relativistic quantum fields, McGraw-Hill, New York.

[6] C. G. Callan, R. F. Dashen, D. J. Gross, (1976), Phys. Lett. 63B, 334.

[7] C. G. Callan, R. F. Dashen, D. J. Gross, (1977), Phys. Lett. 66B, 375.

[8] C. G. Callan, R. F. Dashen, D. J. Gross, (1977), Phys. Rev. D16, 2526.

[9] C. G. Callan, R. F. Dashen, D. J. Gross, (1978), Phys. Rev. D17, 2717.

[10] C. G. Callan, R. F. Dashen, D. J. Gross, (1979), Phys. Rev. D19, 1826.

[11] C. G. Callan, R. F. Dashen, D. J. Gross, (1979), Phys. Rev. D20, 3279.

[12] C. G. Callan, R. F. Dashen, D. J. Gross, (1980), Phys. Rev. Lett. 44, 435.

[13] C. G. Callan, R. F. Dashen, D. J. Gross, F. Wilczek, A. Zee, (1978), Phys. Rev. D18, 4684.

[14] S. Coleman, Aspects of Symmetry, Cambridge univerty press, 1995.

[15] S. Coleman, The uses of instantons, Erice Lectures, 1977.

[16] S. Coleman, 1973, Comm. Math. Phys. 31, 259.

[17] S. Coleman, 1975, Laws of Hadronic Matter, Ed. A. Zichichi, Academic press, New York.

41

42

[18] S. Coleman, 1975, Phys. Rev. D11, 2088.

[19] S. Coleman, 1977 Comm. Math. Phys. 55, 113.

[20] S. Coleman, 1977, Classical lumps and their quantum descendants, 1975, Erice Lectures in New

phenomena in sub-nuclear physics, Ed. A. Zichichi, plenum press, New York.

[21] S. Coleman, J. Mandula, 1969, Phys. Rev. 159, 1251.

[22] L. Cheng, Gauge theory of elementary particle physics.

[23] A. Das, Field theory, world scientific lecture notes in physics, 1993.

[24] P. de Forcrand, Finite temperature field theory, ETH lecture 2002.

[25] J. Froehlich, Quantum field theory, ETH lecture 2002.

[26] G.M. Graf, Group theory, ETH lecture 2002.

[27] G. ’t Hooft, Symmetry breaking through Bell-Jackiw anomalies, 1976, Phys. Rev. Lett. Vol.37,

1, pg.8.

[28] G. ’t Hooft, M. Veltman, Nucl. Phys. B144,189 (1972); B162, 444 (1973).

[29] G. ’t Hooft, Nucl. Phys. B75, 461, (1974).

[30] G. ’t Hooft, Nucl. Phys. B79, 276, (1974).

[31] G. ’t Hooft, Nucl. Phys. B138, 1, (1978).

[32] G. ’t Hooft, Phys. Rev. D 14, 3432 (1976).

[33] G. ’t Hooft, The whys of subnuclear physics, Ed. A. Zichichi, Plenum press, New York 1979.

[34] C. Itzykson, J.B. Zuber, Quantum field Theory, McGraw-Hill 1980.

[35] R. Jackiw, C. Rebbi, (1976a), Phys. Rev. Lett. 36, 1116.

[36] R. Jackiw, C. Rebbi, (1976b), Phys. Rev. D14, 517.

[37] R. Jackiw, C. Rebbi, (1976c), Phys. Rev. D13, 3398.

[38] R. Jackiw, C. Rebbi, (1976d), Phys. Rev. Lett. 37, 172.

[39] A.M. Polyakov, Nucl. Phys. B120, 429 (1977).

[40] R. Rajamaran, Solitons and instantons, North-Holland, Amsterdam, 1982.

43

[41] R. Rosenfelder, Solitonen, Instantonen und Monopole, ETH lecture 2001.

[42] L. H. Ryder, Quantum Field Theory, Cambridge university press, 2005.

[43] B. Sakita, Quantum theory of many variable systems and fields, World scientific publishing1985.

[44] T. Schaefer, E.V. Shuryak, Instantons in QCD, arXiv:hep-ph/9610451 v1 23 Oct. 1996.

[45] A.S.Schwarz, Topology for Physicists, Springer, 1996.

[46] A.S.Schwarz, Quantum field theory and topology, Springer, 1993.

[47] M. Shiftman et al, Sov. Phys. Usp.25, 195 (1982).

[48] M. Shiftman A.I. Vainshtein, V.I. Zakharov Nucl. Phys. B163, 46 1980; 165, 45 (1980).

[49] S. Weinberg, The quantum theory of Fields, Cambridge univerty press 1999-2000.

[50] A.I. Vainshtein, V.I. Zakharov, V.A. Novikov, M.A. Shifman, ABC of Instantons Vainshtein,

Sov.Phys. 25(4), pg. 195 April 1982.

[51] F. Wilczek, 1977, Quark confinement and field theory, Ed. D. Stump, D. Weingarten, John

Wiley and Sons, New York.

[52] J. Zinn-Justin Quantum field theory and critical phenomena, Oxford Univ. Press.