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mations, φ has to have a zero. This de¬nes the center of the monopole. The
boundary condition
H(0) = 0 , K(0) = 1
in the core of the monopole guarantees continuity of the solution. As in the
abelian Higgs model, the changes in the Higgs and gauge ¬eld are occurring
on two di¬erent length scales. Unlike at asymptotic distances, in the core of
the monopole also charged vector ¬elds are present. The core of the monopole
represents the perturbative phase of the Georgi“Glashow model, as the core of
the vortex is made of normal conducting material and ordinary gauge ¬elds.
With the Ansatz (128) the equations of motion are converted into a coupled
system of ordinary di¬erential equations for the unknown functions H and K
which allows for analytical solutions only in certain limits. Such a limiting case
is obtained by saturation of the Bogomol™nyi bound. As for the abelian Higgs
model, this bound is obtained by rewriting the total energy of the static solutions

12 1 1
(B ± Dφ)2 + V (φ) “ BDφ ,
d3 x B + (Dφ)2 + V (φ) = d3 x
2 2 2

and by expressing the last term via an integration by parts (applicable for covari-
ant derivatives due to antisymmetry of the structure constants in the de¬nition
of D in (83)) and with the help of the equation of motion DB = 0 by the
magnetic charge (125)

d3 xB Dφ = a B dσ = a m.

The energy satis¬es the Bogomol™nyi bound

E ≥ |m| a.

For this bound to be saturated, the strength of the Higgs potential has to ap-
proach zero
V = 0, i.e. » = 0,
and the ¬elds have to satisfy the ¬rst order equation

Ba ± (Dφ)a = 0.

In the approach to vanishing », the asymptotics of the Higgs ¬eld |φ| r’∞ a

must remain unchanged. The solution to this system of ¬rst order di¬erential
equations is known as the Prasad“Sommer¬eld monopole

1 sinh agr
H(agr) = coth agr ’ , K(agr) = .
agr agr
50 F. Lenz

In this limiting case of saturation of the Bogomol™nyi bound, only one length scale
exists (ag)’1 . The energy of the excitation, i.e. the mass of the monopole is
given in terms of the mass of the charged vector particles (119) by

E=M .
As for the Nielsen“Olesen vortices, a wealth of further results have been obtained
concerning properties and generalizations of the ™t Hooft“Polyakov monopole
solution. Among them I mention the “Julia“Zee” dyons [43]. These solutions of
the ¬eld equations are obtained using the Ansatz (128) for the Higgs ¬eld and
the spatial components of the gauge ¬eld but admitting a non-vanishing time
component of the form
A0 = 2 J(agr).
This time component re¬‚ects the presence of a source of electric charge q. Clas-
sically the electric charge of the dyon can assume any value, semiclassical argu-
ments suggest quantization of the charge in units of g [44].
As the vortices of the Abelian Higgs model, ™t Hooft“Polyakov monopoles
induce zero modes if massless fermions are coupled to the gauge and Higgs ¬elds
of the monopole
Lψ = iψγ µ Dµ ψ ’ gφa ψ ψ . (129)
The number of zero modes is given by the magnetic charge |m| (125) [45]. Fur-
thermore, the coupled system of a t™ Hooft“Polyakov monopole and a fermionic
zero mode behaves as a boson if the fermions belong to the fundamental represen-
tation of SU (2) (as assumed in (129)) while for isovector fermions the coupled
system behaves as a fermion. Even more puzzling, fermions can be generated
by coupling bosons in the fundamental representation to the ™t Hooft“Polyakov
monopole. The origin of this conversion of isospin into spin [46“48] is the cor-
relation between angular and isospin dependence of Higgs and gauge ¬elds in
solutions of the form (128). Such solutions do not transform covariantly under
spatial rotations generated by J. Under combined spatial and isospin rotations
(generated by I)
K = J + I, (130)
monopoles of the type (128) are invariant. K has to be identi¬ed with the an-
gular momentum operator. If added to this invariant monopole, matter ¬elds
determine by their spin and isospin the angular momentum K of the coupled
Formation of monopoles is not restricted to the particular model. The Georgi“
Glashow model is the simplest model in which this phenomenon occurs. With
the topological arguments at hand, one can easily see the general condition for
the existence of monopoles. If we assume electrodynamics to appear in the pro-
cess of symmetry breakdown from a simply connected topological group G, the
isotropy group H (69) must contain a U (1) factor. According to the identi-
ties (61) and (40), the resulting non trivial second homotopy group of the coset
Topological Concepts in Gauge Theories 51

π2 (G/[H — U (1)]) = π1 (H) — Z
˜ ˜ (131)
guarantees the existence of monopoles. This prediction is independent of the
group G, the details of the particular model, or of the process of the symme-
try breakdown. It applies to Grand Uni¬ed Theories in which the structure
of the standard model (SU (3) — SU (2) — U (1)) is assumed to originate from
symmetry breakdown. The fact that monopoles cannot be avoided has posed a
serious problem to the standard model of cosmology. The predicted abundance
of monopoles created in the symmetry breakdown occurring in the early universe
is in striking con¬‚ict with observations. Resolution of this problem is o¬ered by
the in¬‚ationary model of cosmology [49,50].

6 Quantization of Yang“Mills Theory
Gauge Copies. Gauge theories are formulated in terms of redundant variables.
Only in this way, a covariant, local representation of the dynamics of gauge
degrees of freedom is possible. For quantization of the theory both canonically
or in the path integral, redundant variables have to be eliminated. This procedure
is called gauge ¬xing. It is not unique and the implications of a particular choice
are generally not well understood. In the path integral one performs a sum over
all ¬eld con¬gurations. In gauge theories this procedure has to be modi¬ed by
making use of the decomposition of the space of gauge ¬elds into equivalence
classes, the gauge orbits (72). Instead of summing in the path integral over
formally di¬erent but physically equivalent ¬elds, the integration is performed
over the equivalence classes of such ¬elds, i.e. over the corresponding gauge
orbits. The value of the action is gauge invariant, i.e. the same for all members
of a given gauge orbit,
S A[U ] = S [A] .

Therefore, the action is seen to be a functional de¬ned on classes (gauge orbits).
Also the integration measure

d A[U ] = d [A] dAa (x) .
, d [A] = µ

is gauge invariant since shifts and rotations of an integration variable do not
change the value of an integral. Therefore, in the naive path integral

d [A] eiS[A] ∝
Z= dU (x) .

a “volume” associated with the gauge transformations x dU (x) can be fac-
torized and thereby the integration be performed over the gauge orbits. To turn
this property into a working algorithm, redundant variables are eliminated by
imposing a gauge condition
f [A] = 0, (132)
52 F. Lenz

which is supposed to eliminate all gauge copies of a certain ¬eld con¬guration
A. In other words, the functional f has to be chosen such that for arbitrary ¬eld
con¬gurations the equation
f [A [ U ] ] = 0
determines uniquely the gauge transformation U . If successful, the set of all gauge
equivalent ¬elds, the gauge orbit, is represented by exactly one representative.
In order to write down an integral over gauge orbits, we insert into the integral
the gauge-¬xing δ-functional
N 2 ’1
δ [f a (A (x))] .
δ [f (A)] =
x a=1

This modi¬cation of the integral however changes the value depending on the
representative chosen, as the following elementary identity shows
δ (x ’ a)
δ (g (x)) = , g (a) = 0. (133)
|g (a) |
This di¬culty is circumvented with the help of the Faddeev“Popov determinant
∆f [A] de¬ned implicitly by

d [U ] δ f A[U ]
∆f [A] = 1.

Multiplication of the path integral Z with the above “1” and taking into account
the gauge invariance of the various factors yields

˜ d [A] eiS[A] ∆f [A] δ f A[U ]
Z= d [U ]
[U ]
d [A] eiS [A ] ∆f A[U ] δ f A[U ]
= d [U ] = d [U ] Z.

The gauge volume has been factorized and, being independent of the dynamics,
can be dropped. In summary, the ¬nal de¬nition of the generating functional for
gauge theories
d4 xJ µ Aµ
d [A] ∆f [A] δ (f [A] ) eiS[A]+i
Z [J] = (134)

is given in terms of a sum over gauge orbits.

Faddeev“Popov Determinant. For the calculation of ∆f [A], we ¬rst consider
the change of the gauge condition f a [A] under in¬nitesimal gauge transforma-
tions . Taylor expansion
δfx [A] b

a [U ] a 4
fx A fx [A] + dy δA (y)
δAb (y) µ

d4 y M (x, y; a, b) ±b (y)
= fx [A] +
Topological Concepts in Gauge Theories 53

with δAa given by in¬nitesimal gauge transformations (92), yields

δfx [A]
b,c bcd
M (x, y; a, b) = ‚µ δ + gf (y) . (135)
δAc (y)

In the second step, we compute the integral

∆’1 [A] = d [U ] δ f A[U ]

by expressing the integration d [U ] as an integration over the gauge functions ±.
We ¬nally change to the variables β = M ±

∆’1 [A] = | det M |’1 d [β] δ [f (A) ’ β]

and arrive at the ¬nal expression for the Faddeev“Popov determinant

∆f [A] = | det M | . (136)

• Lorentz gauge

= ‚ µ Aa (x) ’ χa (x)
fx (A) µ

M (x, y; a, b) = ’ δ ab 2 ’ gf abc Ac (y) ‚y δ (4) (x ’ y)

• Coulomb gauge

= divAa (x) ’ χa (x)
fx (A)
M (x, y; a, b) = δ ab ∆ + gf abc Ac (y) ∇y δ (4) (x ’ y) (138)

• Axial gauge

= nµ Aa (x) ’ χa (x)
fx (A) µ


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