µ

Equations of Motion. The principle of least action

d4 xL

δS = 0, S=

yields when varying the gauge ¬elds

2

δSY M = ’ d4 x tr {Fµν δF µν } = ’ d4 x tr Fµν [Dµ , δAν ]

ig

d4 x tr {δAν [Dµ , Fµν ]}

=2

40 F. Lenz

the inhomogeneous ¬eld equations

[Dµ , F µν ] = j ν , (84)

with j ν the color current associated with the matter ¬elds

δLm

j aν = . (85)

δAa

ν

For QCD and the Georgi“Glashow model, these currents are given respectively

by

a

¯ ν „ ψ , j aν = gf abc φb (Dν φ)c .

aν

j = g ψγ (86)

2

As in electrodynamics, the homogeneous ¬eld equations for the Yang“Mills ¬eld

strength

˜

Dµ , F µν = 0 ,

with the dual ¬eld strength tensor

1

˜

F µν = µνσρ

Fσρ , (87)

2

are obtained as the Jacobi identities of the covariant derivative

[Dµ , [Dν , Dρ ]] + [Dν , [Dρ , Dµ ]] + [Dρ , [Dν , Dµ ]] = 0 ,

i.e. they follow from the mere fact that the ¬eld strength is represented in terms

of gauge potentials.

Gauge Transformations. Gauge transformations change the color orientation

of the matter ¬elds locally, i.e. in a space-time dependent manner, and are de¬ned

as

„a

U (x) = exp {ig± (x)} = exp ig± (x) a

, (88)

2

with the arbitrary gauge function ±a (x). Matter ¬elds transform covariantly

with U

ψ ’ U ψ , φ ’ U φU † . (89)

The transformation property of A is chosen such that the covariant derivatives

of the matter ¬elds Dµ ψ and Dµ φ transform as the matter ¬elds ψ and φ

respectively. As in electrodynamics, this requirement makes the gauge ¬elds

transform inhomogeneously

1

‚µ U † (x) = AµU ] (x)

Aµ (x) ’ U (x) Aµ (x) + [

(90)

ig

resulting in a covariant transformation law for the ¬eld strength

Fµν ’ U Fµν U † . (91)

Topological Concepts in Gauge Theories 41

Under in¬nitesimal gauge transformations (|g±a (x) | 1)

Aa (x) ’ Aa (x) ’ ‚µ ±a (x) ’ gf abc ±b (x) Ac (x) . (92)

µ µ µ

As in electrodynamics, gauge ¬elds which are gauge transforms of Aµ = 0 are

called pure gauges (cf. (8)) and are, according to (90), given by

1

‚µ U † (x) .

Apg (x) = U (x) (93)

µ

ig

Physical observables must be independent of the choice of gauge (coordinate

system in color space). Local quantities such as the Yang“Mills action density

¯

tr F µν (x)Fµν (x) or matter ¬eld bilinears like ψ(x)ψ(x), φa (x)φa (x) are gauge

invariant, i.e. their value does not change under local gauge transformations.

One also introduces non-local quantities which, in generalization of the trans-

formation law (91) for the ¬eld strength, change homogeneously under gauge

transformations. In this construction a basic building block is the path ordered

integral

s

dxµ

„¦ (x, y, C) = P exp ’ig = P exp ’ig dxµ Aµ .

dσ Aµ x(σ)

dσ C

s0

(94)

It describes a gauge string between the space-time points x = x(s0 ) and y = x(s).

„¦ satis¬es the di¬erential equation

dxµ

d„¦

= ’ig Aµ „¦. (95)

ds ds

Gauge transforming this di¬erential equation yields the transformation property

of „¦

„¦ (x, y, C) ’ U (x) „¦ (x, y, C) U † (y) . (96)

With the help of „¦, non-local, gauge invariant quantities like

tr„¦ † (x.y, C)F µν (x)„¦ (x, y, C) Fµν (y) , ¯

ψ(x)„¦ (x, y, C) ψ(y),

or closed gauge strings “ SU(N) Wilson loops

1

tr „¦ (x, x, C)

WC = (97)

N

can be constructed. For pure gauges (93), the di¬erential equation (95) is solved

by

„¦ pg (x, y, C) = U (x) U † (y). (98)

¯

While ψ(x)„¦ (x, y, C) ψ(y) is an operator which connects the vacuum with meson

states for SU (2) and SU (3), fermionic baryons appear only in SU (3) in which

gauge invariant states containing an odd number of fermions can be constructed.

42 F. Lenz

In SU(3) a point-like gauge invariant baryonic state is obtained by creating three

quarks in a color antisymmetric state at the same space-time point

ψ(x) ∼ abc

ψ a (x)ψ b (x)ψ c (x).

Under gauge transformations,

ψ(x) ’ abc

Ua± (x)ψ ± (x)Ubβ (x)ψ β (x)Ucγ (x)ψ γ (x)

= det U (x) abc ψ a (x)ψ b (x)ψ c (x) .

Operators that create ¬nite size baryonic states must contain appropriate gauge

strings as given by the following expression

ψ(x, y, z) ∼ [„¦(u, x, C1 )ψ(x)]a [„¦(u, y, C2 )ψ(y)]b [„¦(u, z, C3 )ψ(z)]c .

abc

The presence of these gauge strings makes ψ gauge invariant as is easily veri¬ed

with the help of the transformation property (96). Thus, gauge invariance is

enforced by color exchange processes taking place between the quarks.

Canonical Formalism. The canonical formalism is developed in the same way

as in electrodynamics. Due to the antisymmetry of Fµν , the Lagrangian (80)

does not contain the time derivative of A0 which, in the canonical formalism,

has to be treated as a constrained variable. In the Weyl gauge [33,34]

a = 1....N 2 ’ 1,

Aa = 0, (99)

0

these constrained variables are eliminated and the standard procedure of canon-

ical quantization can be employed. In a ¬rst step, the canonical momenta of

gauge and matter ¬elds (quarks and Higgs ¬elds) are identi¬ed

δLY M δLmq δLmH

= iψ ± † ,

a = ’E

ai

= πa .

, ± a

‚0 Ai ‚0 ψ ‚0 φ

By Legendre transformation, one obtains the Hamiltonian density of the gauge

¬elds

1

HY M = (E 2 + B 2 ), (100)

2

and of the matter ¬elds

10i

Hm = ψ † γ γ Di + γ 0 m ψ,

QCD : (101)

i

12 1

Hm =

π + (Dφ)2 + V (φ) .

Georgi“Glashow model : (102)

2 2

The gauge condition (99) does not ¬x the gauge uniquely, it still allows for time-

independent gauge transformations U (x), i.e. gauge transformations which are

generated by time-independent gauge functions ±(x) (88). As a consequence the

Hamiltonian exhibits a local symmetry

H = U (x) H U (x)† (103)

Topological Concepts in Gauge Theories 43

This residual gauge symmetry is taken into account by requiring physical states

|¦ to satisfy the Gauß law, i.e. the 0-component of the equation of motion (cf.

(84))

[Di , E i ] + j 0 |¦ = 0.

In general, the non-abelian Gauß law cannot be implemented in closed form

which severely limits the applicability of the canonical formalism. A complete

canonical formulation has been given in axial gauge [35] as will be discussed

below. The connection of canonical to path-integral quantization is discussed in

detail in [36].

5 ™t Hooft“Polyakov Monopole

The t™ Hooft“Polyakov monopole [37,38] is a topological excitation in the non-

abelian Higgs or Georgi“Glashow model (SU (2) color). We start with a brief

discussion of the properties of this model with emphasis on ground state con¬g-

urations and their topological properties.

5.1 Non-Abelian Higgs Model

The Lagrangian (82) and the equations of motion (84) and (85) of the non-

abelian Higgs model have been discussed in the previous section. For the follow-

ing discussion we specify the self-interaction, which as in the abelian Higgs model

is assumed to be a fourth order polynomial in the ¬elds with the normalization

chosen such that its minimal value is zero

1

»(φ2 ’ a2 )2 ,

V (φ) = » > 0. (104)

4

Since φ is a vector in color space and gauge transformations rotate the color

direction of the Higgs ¬eld (89), V is gauge invariant

V (gφ) = V (φ) . (105)

We have used the notation

gφ = U φU † , g ∈ G = SU (2).

The analysis of this model parallels that of the abelian Higgs model. Starting

point is the energy density of static solutions, which in the Weyl gauge is given

by ((100), (102))

1 1

(x) = B2 + (Dφ)2 + V (φ). (106)

2 2

The choice

A = 0, φ = φ0 = const. , V (φ0 ) = 0 (107)

minimizes the energy density. Due to the presence of the local symmetry of

the Hamiltonian (cf. (103)), this choice is not unique. Any ¬eld con¬guration

44 F. Lenz

connected to (107) by a time-independent gauge transformation will also have

vanishing energy density. Gauge ¬xing conditions by which the Gauß law con-

straint is implemented remove these gauge ambiguities; in general a global gauge

symmetry remains (cf. [39,35]). Under a space-time independent gauge transfor-

mation

a

a„

g = exp ig± , ± = const , (108)

2

applied to a con¬guration (107), the gauge ¬eld is unchanged as is the modulus

of the Higgs ¬eld. The transformation rotates the spatially constant φ0 . In such

a ground-state con¬guration, the Higgs ¬eld exhibits a spontaneous orientation

analogous to the spontaneous magnetization of a ferromagnet,

|φ0 | = a .

φ = φ0 ,

This appearance of a phase with spontaneous orientation of the Higgs ¬eld is

a consequence of a vacuum degeneracy completely analogous to the vacuum

degeneracy of the abelian Higgs model with its spontaneous orientation of the