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Chapter One : The Story of Magnetic Monopoles
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Chapter One : The Story of Magnetic Monopoles
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Chapter One : The Story of Magnetic Monopoles
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Chapter One : The Story of Magnetic Monopoles
Chapter 1
The Story of Magnetic Monopoles
1.1 Introduction
In this chapter, I will present a general overview of the development of magnetic
monopole theory. In doing a literature review of the subject, I have chosen to classify
it in the form of topics, rather than according to strict chronological order. Thus,
point-like monopoles, also known as Dirac monopoles, are discussed first, followed
by monopoles arising in non-Abelian gauge theories. Topological considerations are
given next, and the concept of the stability of monopoles is discussed, followed by a
short discussion of monopoles in other theories. A section on recent research gives
examples of conjectures that are still under debate. Next, a comment is given on the
experimental search of magnetic monopoles. Finally, we discuss the place of
monopoles in a quantized theory. This chapter should convince the reader of the
importance that present day physicists attribute to magnetic monopoles.
Concerning notational convention in this chapter, let me note that I will be freely
changing from ordinary vector analysis to four-dimensional analysis in Minkowski
space. In the first, I will be using a three dimensional metric with the signature
( + + + ). In the second, the four dimensional Minkowski space metric with the
signature ( + − − − ) is employed. Latin indices from the beginning of the alphabet run
from 1 to 3 and signify isospace co-ordinates. Latin indices from the middle of the
alphabet also run from 1 to 3 and denote real space co-ordinates. Greek indices, as
usual, run from 0 to 3 and represent four dimensional space-time. Other notational
conventions will be established as they are needed.
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Chapter One : The Story of Magnetic Monopoles
1.2 Dirac Monopoles
Once, when addressing a group of students, Dirac stressed the point that all
physicists should occupy themselves with the beauty of the equations as one of the
criteria of their validity. Throughout his magnificent career, Dirac was known for his
preoccupation with the abstract prettiness of physical formulae. It was, perhaps, this
obsession that led him to discuss the so-called asymmetry problem of Maxwell’s
Electrodynamic field equations. Dirac must have felt that making the equations more
symmetric would induce further beauty onto them. The asymmetry lay in the fact that
the equations stated, quite bluntly, that ‘Thou shalt not speak of magnetic
monopoles ’, which manifested itself as equation (a) of the Maxwell Equations1:
a ) div B = 0
c) div E = ρ
∂B
=0
∂t
∂E
d ) curl B −
= j,
∂t
b) curl E +
(1.1)
while, of course, electric monopoles were there both in theory - equation (c) - and in
nature; as electrons and other charged ‘point’ particles. In investigating this property,
Dirac ventured (as an abstract exercise in beauty, I imagine) to symmetrize the
equations by introducing magnetic monopoles such that they could be written as
follows [1,2]:
a ) div B = ρ m
c) div E = ρ
∂B
= − jm
∂t
∂E
= j,
d ) curl B −
∂t
b) curl E +
(1.2)
where, obviously, the suffix m stands for ‘magnetic’. Thus the magnetic field is
treated symmetrically with the electric in the field equations, along with a
complementary continuity equation. We see here the origin of the fact that magnetic
charges are odd under parity, manifested in the difference in signs between equations
(1.2-b) and (1.2-d). According to [3], it was actually Heaviside who first published
equations (1.2) as a convenient way of describing another subject entirely [4]. He did
1
Using the Heaviside-Lorentz rationalized units.
4
Chapter One : The Story of Magnetic Monopoles
not really believe in magnetic charges, however, and called them ‘fictitious’.
Following Heaviside, Poincaré discussed some mathematical aspects of equations
(1.2) [5], again without realizing the physical feasibility of magnetic monopoles.
Dirac, on the other hand, was the first to actually propose monopoles as a real
possibility, and introduced (1.2) as the basis of a new concept. Thus, the credit goes
to Dirac for founding monopole theory.
The magnetic field configuration that results from (1.2) can be derived in the
form of a Coulomb-like law that may be written [6]:
B=
v æ 1ö
g
$
r
=
−
g
∇
ç ÷,
è rø
r2
(1.3)
which is, obviously, radial and point-like, i.e with a singularity at r = 0, and g is the
magnetic charge. The magnetic flux resulting from this monopole is
Φ = 4π r 2 B = 4π g .
(1.4)
Now consider an electrically charged particle in the field of this monopole. Its
quantum mechanical wave function in the free state is
ù
éi
ψ = ψ exp ê (p ⋅ r − Et )ú .
û
ëh
(1.5)
In the presence of an electromagnetic field, we make the familiar substitution
e
p → p − A , resulting, for a constant A , in
c
æ ie
ö
ψ → ψ expç − A ⋅ r÷ .
è hc
ø
(1.6)
In other words, the wave’s phase α changes by
α →α−
e
A⋅r .
hc
(1.7)
Now, if we consider a closed path at fixed r and θ around the monopole, with ϕ
ranging from 0 to 2π, and assuming a variable A , the total change of phase is
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Chapter One : The Story of Magnetic Monopoles
∆α
=
e
A ⋅ dl
hc
=
e
curlA ⋅ dS
hc
=
e
B ⋅ dS
hc
=
e
Φ( r , θ ) ,
hc
ò
ò
ò
(1.8)
where Φ( r,θ ) is the flux through the ‘cap’ defined by particular values of r and θ .
Variation in θ results in a change of flux. As θ → 0 , the loop shrinks to a point and
the flux approaches zero. As the loop is lowered over the sphere ( θ → π ), the flux
should approach the value (1.4). However, what happens is that the loop again
shrinks to a point so that the requirement that Φ is finite entails, from (1.8), that A is
singular at θ = π . Since, by varying r, this condition is valid for all possible radii, it
follows that A is singular along a line starting at r = 0, and reaching infinity. This line
is known as the Dirac string. It is clear that in the above argument, the string lies
along the negative z axis, but it may be chosen to coincide with any arbitrary
direction, as we shall see, and it need not be straight, but must, however, be
continuous.
Now, if we insist that B is derivable from A in the familiar fashion, then it
follows that we can imagine that the Dirac monopole looks like a point, out of which
(or into which, depending on whether it is a north- or a south-poled monopole)
emanates lines of magnetic flux. Tracing those lines, we find that they are actually
coming out of (or going into) the string. The string either goes on to infinity, or has
an end that is attached to another monopole of opposite polarity. It is like having a
common household north-south magnet with its mid-section reduced to zero
thickness. From this point of view, the Maxwell equations may be regarded as having
not been really symmetrized after all.
The singularity in A gives rise to the so-called Dirac veto - that the wave
function (1.6) is required to vanish along the string. This can be explained by
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Chapter One : The Story of Magnetic Monopoles
observing that the string has zero radius, but, with lines of magnetic flux passing
through it, it would have an infinite amount of energy per unit length. Thus, Dirac
had to proclaim the string unobservable and prohibit any charged particle from
coming into contact with it, hence the veto. Note that the last sentence contains a
contradiction; why would approach to a region of space (the string) be prohibited if
that region is unobservable anyway? This is but one of the problems the string
concept introduces.
Another problem that arose from the string singularity was that the theory lost
explicit Poincaré invariance, because it featured a preferred direction, the direction of
the string. Yet, the work of Brandt, Neri and Zwanziger [7,8] demonstrated that the
gauge-invariant Green’s functions in the theory are in fact Poincaré invariant, albeit
not explicitly so.
By referring to (1.8), there is no necessity that as θ → π , ∆α → 0 . We must,
however, have ∆α = 2π n in order for ψ to be single-valued. From (1.4) and (1.8)
we then have2
eg =
n
2
n = 1, 2, L .
(1.9)
This is the Dirac quantization condition. Its importance comes from the fact that
it explains the apparent quantization of electric charge. It is often mentioned in the
literature that if at least one Dirac monopole existed in nature, then all electric
charges should carry integer multiples of the same value e, as is, of course, actually
observed. The quantization condition can be traced from the requirement that the
string must be unobservable. It has been shown [9,10] that the converse is also true;
starting from the quantization condition, the conclusion is reached that an
unobservable string must be introduced.
2
I will assume natural units from now on, i.e h = c = 1 .
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Chapter One : The Story of Magnetic Monopoles
Fig. 1.1
The Bohm-Aharonov Experiment. A solenoid (S) is placed next
to a double slit. The passage of electron beams in the vicinity of
the solenoid can be seen to change interference patterns.
An interesting way of physically explaining the above derivation of the Dirac
condition, which I have found especially enlightening, can be demonstrated by
considering the following gedanken experiment [11]: Imagine that we wish to trick a
gullible experimenter into believing that he has discovered a Dirac monopole. We get
a very long, extremely thin solenoid and place one end of it in the experimenter’s
laboratory, and turn on a current at the other end. The experimenter observes a
magnetic flux emanating from a point, and thinks he has discovered a Dirac
monopole. The only way for the experimenter to realize that it is not a genuine
monopole is to perform a Bohm-Aharonov experiment [12] on it. By placing the
‘monopole’ next to a double slit and passing charged particles through (Fig. 1.1), he
can observe a shift in the interference patterns of the diffracted beams that would
enable him to show that the quantum mechanical probability density of the passage of
the beam changes from ψ 1 + ψ 2
2
2
to ψ 1 + e ieΦψ 2 ; where the suffixes 1 and 2 refer
to the two slits. He is thus able to detect the solenoid. That is, unless the value of eg
conforms with (1.9). Indeed, the solenoid is totally indistinguishable from a true
monopole if and only if the relation between e and g is exactly the Dirac Quantization
Condition.
Much later (38 years later in fact), Schwinger, in developing a theory of a particle
emanating both electric and magnetic fields which he called a dyon [13], derived the
following relationship:
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Chapter One : The Story of Magnetic Monopoles
n = 1, 2, L
e1 g 2 − e2 g1 = n
(1.10)
for a system of two dyons carrying the electric charges e1 and e2 , and the magnetic
charges g1 , and g 2 respectively. Equation (1.10) could be called Schwinger’s
quantization condition, and it states that a point monopole would have twice the
magnetic charge that Dirac’s condition leads to. For some time, Schwinger’s
condition lay in competition with Dirac’s. Nowadays, however, physicists believe
that Dirac’s condition is the correct one. Schwinger himself admitted that he does not
believe in his own condition anymore.
It may be worth noting at this point, for the sake of completeness, that
Schwinger’s theory was originally proposed as a model for quarks, but with the rise
and success of quantum chromodynamics, Schwinger’s formulation dwindled away,
to be only remembered because of [the historical significance of] equation (1.10).
Later, however, his ideas were reintroduced as a model for the possible constituent
particles of both quarks and leptons; dubbed preons [14,15].
Four years after Dirac published his monopole theory, Saha suggested another
way of arriving at the quantization condition [16,17]. He described a system of two
particles, one electrically charged and the other a magnetic monopole, separated by a
vector distance d. By forming the Poynting vector;
P=
1
E×B
4π
(1.11)
which carries electromagnetic momentum, this results in an angular momentum
along d, having the form
L = M + egd$ = r × (p − eA ) + nd$ .
(1.12)
The integral of this angular momentum over all space has a magnitude
independent of d. By quantizing the angular momentum in units of h 2 one
immediately gets the Dirac quantization condition. Quantizing it in values of h leads
to Schwinger’s. More about the angular momentum of such a system later.
Now, let us discuss an interesting feature of equations (1.2) known as duality
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Chapter One : The Story of Magnetic Monopoles
invariance [18]. By inducing a duality rotation on both the fields and the sources;
E → E cos α + B sin α ,
B → − E sin α + B cos α ,
ρ → ρ cos α + ρ m sin α ,
ρ m → − ρ sin α + ρ m cos α ,
j → j cos α + j m sin α ,
j m → − j m sin α + j m cos α ,
(1.13)
and by simple substitution, it can be easily seen that the equations are invariant under
these rotations. In which case, it is conventional to use the term duality vector to
denote ( E, B ) and ( ρ , j ). In fact, e and g also form a duality vector. The duality
invariance indicates that if all particles have the same ratio of electric to magnetic
charge, then it follows that we can always make a duality transformation such that
ρ m = 0 and j m = 0 , taking us back to the Maxwell Equations (1.1).
1.2.1 The Gauge Potential of Dirac Monopoles
In equation (1.6) we introduced the variable A as the vector potential of the field
created by a Dirac monopole, but what is the form of this potential, and how will the
presence of the Dirac string affect its formulation? Dirac asserted that if we require
the string to be along the negative z axis, then A would have the form
A = Aϕ ϕ$
(1.14)
as
Aϕ =
g 1 − cosθ
,
r sin θ
(1.15)
with vanishing values for both the r and θ components. To have the string on the
positive z axis entails
Aϕ = −
g 1 + cosθ
.
r sin θ
(1.16)
The string singularity in A is clearly manifest in these equations, but, as can be
inferred from the discussion in the last section, it is considered unphysical, since the
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Chapter One : The Story of Magnetic Monopoles
only real singularity in A is the one actually at the origin. So, in order to avoid having
to introduce the string, an approach known as the Two Patch method was devised
[19]. It divides the space around the monopole into two regions, one covering all of
space except the negative z axis, and the other covering all of space except the
positive z axis, with an overlapping zone somewhere in the middle. In each region, A
is defined differently;
ArS = AθS = 0 ,
AϕS = −
ArN = AθN = 0 ,
AϕN =
g 1 + cosθ
,
r sin θ
g 1 − cosθ
,
r sin θ
(1.17)
(1.18)
where S and N stand for the southern and northern regions respectively. Obviously,
A S and A N are both finite within their respective domains. In the overlap zone, they
are not the same, but are connected by a gauge transformation;
AϕS = AϕN −
i
2g
= AϕN − S ∇ ϕ S −1 ,
e
r sin θ
(1.19)
with
S = exp(2igeϕ ) .
(1.20)
If we require that S be single-valued, this leads directly to (Surprise!) Dirac’s
quantization condition. By calculating the total flux of the field over all space, it can
be readily shown that (1.17) and (1.18) do indeed represent a magnetic monopole. No
strings, no vetoes, no fuss.
In [20] it was proposed that the use of two distinct potentials would also avoid
strings. In four-vector notation, one can write, for the original unsymmetric Maxwell
equations:
Fµν = ∂ µ Aν − ∂ ν Aµ ,
(1.21)
then by adding a new four-potential M µ , the electromagnetic field tensor is
redefined
11
Chapter One : The Story of Magnetic Monopoles
Fµν = ∂ µ Aν − ∂ ν Aµ + ε µναβ ∂ α M β ,
(1.22)
with ε µναβ being, obviously, the Levi-Civita totally antisymmetric tensor. It has been
shown [21] that (1.22) is equivalent to
and
v
∂A v
E = −∇A0 −
−∇×M,
∂t
(1.23)
v
∂M v
B = −∇M 0 −
+∇×A.
∂t
(1.24)
The two potential approach has a disadvantage, however. It has been claimed
[22,23,24] that, except under certain special conditions, there is no single Lagrangian
from which one can derive equations (1.2). This problem also extends to the
quantized field theory. In other words, the Dirac-Maxwell equations are not derivable
from a Lagrangian [25]. Recently, however, much research was done on refuting this
claim [26]. An effective Lagrangian of QED coupled to dyons was proposed [27].
The resulting generalization of the Euler-Heisenberg Lagrangian contains non-linear
P and T noninvariant, but C invariant (see section 1.6), terms corresponding to virtual
pair creation of dyons.
Also, the Leinard-Weichart potential has been generalized to include magnetic
monopoles [28], that is, a form of retarded potentials was proposed as a solution of
the new Maxwell-Dirac equations (1.2).
1.2.2 An Electrically Charged Particle in a Monopole Field
In considering the motion of a classical electric charge in the field of a monopole,
the problem was found to be completely solvable [29], and its solvability is linked to
the symmetry groups under which the system turns out to be invariant [30].
Firstly, rotational invariance insures the conservation of the total angular
momentum (1.12). Another conserved quantity is the Hamiltonian, that turns out to
coincide with the kinetic energy;
12
Chapter One : The Story of Magnetic Monopoles
H=
(p − eA ) 2
2m
=
1 2
mr& .
2
(1.25)
There also exists a conserved generator of time dilatations:
1
D = tH − m( r ⋅ r&+ r&⋅ r ) ,
4
(1.26)
and a conserved generator of the special conformal transformations:
K = − t 2 H + 2tD +
1 2
mr .
2
(1.27)
The quantum commutation relations of these operators are
[ H , D] = iH
[ D, K ] = iK
[ H , K ] = 2iD.
(1.28)
The use of the above properties in solving the problem of particles in the field of
a monopole is explained in detail in [11,29]. Furthermore, calculations of the
eigenfunctions of the total angular momentum of the problem were demonstrated in
many sources [31]. The final conclusions may be summarized in the following:
1) The effects of the presence of a monopole are surprisingly simple. It merely
slightly changes the centrifugal potential in the radial Shr÷dinger equation. If
we can solve the equation for a given central potential without a monopole, we
can solve it with a monopole.
2) The total quantized angular momentum of the system is
l = eg ,
eg + 1,
eg + 2, L
(1.29)
thus the centrifugal potential is always positive, and the monopole by itself
does not bind charged spinless particles. And because eg is allowed to be a
half integer, two spinless particles, one with an electric charge, and the other
with a magnetic charge, can bind together to make a dyon with half-oddintegral angular momentum.
13
Chapter One : The Story of Magnetic Monopoles
3) A dyon as the one described above obeys Bose-Einstein statistics if eg is
integral and Fermi-Dirac statistics if eg is half-odd-integral [32].
Finally, the quantized form of the problem of a charged particle-magnetic
monopole system was discussed in [33], and it was reported that it leads to a certain
scattering ambiguity: Either the charged particle must undergo a discontinuous spinflop, or it must exchange charge with the monopole, making it a dyon. If the charged
particle is replaced by a vorton, the ambiguity is resolved. A vorton is a soliton,
smooth at short distances, which can carry all the quantum numbers of ordinary
particles [34]. Generally, a vorton is a semi-classical configuration of generalized
electromagnetic charge satisfying (1.2), i.e it is a kind of magnetic monopole. It is
constructed to be invariant under the O(4) = O(3) × O(3) subgroup of the conformal
group. O(4) is the orthogonal group of rotations in four dimensions. The first O(3) is
the orthogonal group of rotations in three dimensions, whereas the second O(3)
describes toroidal rotations, akin to the vortex rotations of a smoke ring, hence the
name. Thus, the vorton carries two forms of angular momentum; the usual form, plus
a toroidal angular momentum.
1.3 ’t Hooft-Polyakov Monopoles
Magnetic monopoles derivable from non-Abelian gauge theories made their
debut when G. ’t Hooft and A. Polyakov independently researched the subject
[35,36,37]. Magnetic monopole formulations were found to arise from non-Abelian
gauge symmetries with spontaneous symmetry breaking. The theory gained instant
fame, due to the fact that magnetic monopoles follow naturally from the equations,
i.e they are not ‘artificially’ superimposed on the theory as was the case with the
Dirac formulation. It is sometimes said that Dirac introduced the possibility of
magnetic monopoles while ’t Hooft and Polyakov demonstrated the necessity of their
existence. This fact has made most theorists insist that monopoles must exist in
nature, working from the experience that whatever mathematics allowed has indeed
been found.
Let us briefly review the ’t Hooft-Polyakov formulation. Assume a theory with
an O(3) symmetry group and a Lagrangian density
14
Chapter One : The Story of Magnetic Monopoles
1
1
m2 a a
L = − Fµνa F µν a + ( Dµ ϕ a )( D µ ϕ a ) −
ϕ ϕ − λ (ϕ a ϕ a ) 2 ,
4
2
2
(1.30)
where, as usual,
Fµνa = ∂ µ Aνa − ∂ ν Aµa + eε abc Aµb Aνc
Dµ ϕ a = ∂ µ ϕ a + eε abc Aµbϕ c ,
(1.31)
Dµ being the covariant derivative, and ϕ the Higgs field. The required solutions are
static solutions with the gauge potentials having the asymptotic form:
A = −ε iab
a
i
rb
er 2
(r 
→ ∞)
A0a = 0 ,
(1.32)
(1.33)
where i is a spatial index ranging from 1 to 3. The scalar field (Higgs field) is:
ϕa = −
m2 r a
4λ r
(r 
→ ∞) .
(1.34)
This solution has a distinct ‘radial’ form, Polyakov calls it a ‘hedgehog’ solution.
It was shown that there are solutions to the equations of motion, derived from the
Lagrangian (1.30) that fits (1.32, 33, 34) at infinite radial distances.
Now, defining
Aµ =
1 a a
ϕ Aµ ,
ϕ
(1.35)
the field tensor becomes
Fµν = ∂ µ Aν − ∂ ν Aµ −
1
eϕ
3
ε abcϕ a (∂ µ ϕ b )(∂ ν ϕ c ) .
(1.36)
By inserting the asymptotic forms (1.32, 33, 34), the field tensor can be easily
shown to have the components:
15
Chapter One : The Story of Magnetic Monopoles
F0i = 0,
Fij = −
1
ε ijk r k ,
er 3
(1.37)
which correspond to a radial magnetic field
Bk =
rk
,
er 3
(1.38)
with a magnetic flux
1
Φ ∝ ,
e
(1.39)
which ultimately results in a quantization condition:
n = ±1, ± 2, L ,
eg = n
(1.40)
that is, twice the Dirac unit. The conclusion is that the configuration (1.32-34)
displays itself as an Abelian magnetic monopole at radial infinity. Polyakov
calculated that it has finite energy with an estimated mass of 137 M W , where M W is
the mass of the W boson. It has also been shown that ’t Hooft-Polyakov monopoles
admit Fermi statistics in the case of SU(2) which can be reached from SO(3) by
adding an isodoublet. An important feature of the ’t Hooft-Polyakov solution is that it
has finite energy due to the fact that there is no singularity at the origin. The energy is
calculated in the next section.
Monopoles arising in non-Abelian gauge theories are the opposite of Dirac
monopoles in the fact that they have an internal structure, i.e they are ‘unpoint-like’,
or ‘extended’ objects. It has been verified that if their natural ‘extendedness’
parameters tended to zero, Dirac monopoles are recovered, as can be expected [38]. It
was further claimed that they form linear multiplets under the reflection symmetry of
the original gauge group [39]. This implied that as far as the gauge symmetry is kept
unbroken, only the singlet combinations of the monopoles may be admitted in the
physical sector. Calculations of the flux integrals of monopoles were carried out in
[40].
16
Chapter One : The Story of Magnetic Monopoles
1.4 The Bogomol’nyi-Prasad-Sommerfield Model
For estimates of magnetic monopole masses, it is possible to derive a rigorous
lower bound. Although not necessary, the electric field may be excluded by assuming
time-independence and time-reversal invariance, where the time reversal operation is
defined as follows:
T :
A0 (x, t ) 
→ − A0 ( x ,− t )
(1.41)
A ( x, t ) 
→ A (x,− t )
which means that
A0 = 0 ,
∂ 0 Ai = 0 ,
and
(1.42)
resulting in F0i = 0 (i.e, E = 0). For the theory with the Lagrangian (1.30), the energy
may be written in the form:
E=
ò
2ù
1
1
é1
d 3 x ê Bia Bia + Di ϕ a Di ϕ a + λ 2 ϕ a ϕ a − 1 ú .
2
8
ë2
û
(
)(
)
(
)
(1.43)
Integration by parts results in
ò
(
)
E = m d 3 x∂ i Bia ϕ a +
ò
é1
d 3 x ê Bia ± Di ϕ a
ë2
(
)
2
2ù
1
+ λ2 ϕ a ϕ a − 1 ú .
8
û
(
)
(1.44)
The first term, which is actually a quantity known as the topological charge (see
the next section), can be readily evaluated;
E = m4π g +
ò
d 3x é1 a
Bi ± Di ϕ a
ê
4π ë 2
(
)
2
2ù
1
+ λ2 ϕ a ϕ a − 1 ú ,
8
û
(
)
(1.45)
resulting in the inequality
E ≥ 4π g
(1.46)
17
Chapter One : The Story of Magnetic Monopoles
which is the Bogomol’nyi bound for energy [41]. It is possible to saturate the bound
in an appropriate limit, the Prasad-Sommerfield limit. This can be done if we insist
that λ → 0 to get rid of the last term, and that the second term also vanishes, in other
words, if solutions can be found to [42]:
Bia ± Di ϕ a = 0 ,
(1.47)
sometimes known as the Bogomol’nyi equation. The sign in the equation depends on
the sign of g , i.e whether we are discussing monopoles or antimonopoles. It is a first
order equation, and considerably easier to solve than the second order field equations.
Indeed, the equation does have solutions: magnetic monopoles! Not only singlemonopole solutions, but also many-monopole ones. Prasad himself found exact
axially symmetric multi-monopole solutions of arbitrary topological charge of YangMills-Higgs models using (1.47) [43].
The Bogomol’nyi-Prasad-Sommerfield (BPS) model has stimulated wide interest
and inspired much research. The dynamics of BPS monopoles in a weak, constant
electromagnetic field were studied perturbatively, and a Lorentz force law for BPS
dyons was proposed [44]. Interesting relations with Euclidean Yang-Mills theories
have been found linking monopoles with instantons [45,46,47]. Let me discuss this
last concept.
Recall that the main equation satisfying instanton solutions is:
1
Fµν ± ε µναβ F αβ = 0 ,
2
(1.48)
having a static counterpart that is exactly equation (1.47). Hence it appears that BPS
monopole solutions have a specific correspondence with the static solutions of the
free Yang-Mills Euclidean equations. In fact, if an infinite number of instantons are
imagined to lie equally spaced on the time axis, in the so-called ’t Hooft ansatz
[48,49], we can write
ϕ a = −∂ a ln ρ
(1.49)
Aia = −ε iab ∂ b ln ρ + δ ia ∂ 0 ln ρ
(1.50)
18
Chapter One : The Story of Magnetic Monopoles
where
ρ (r , t ) =
∞
år
n =−∞
1
2
+ (t − t n )
2
t n = 2π n .
(1.51)
The summation (1.51) can be readily carried through;
ρ (r , t ) =
1 sinh r
1
.
2 r cosh r − cos t
(1.52)
The resulting gauge fields are periodic in time, but by performing an appropriate
time-periodic gauge transformation [50];
u = exp( −iτ ⋅ r$θ ) .
(1.53)
æ sin t sinh r ö
θ = tan −1 ç
÷,
è cosh r cos t − 1ø
(1.54)
τ being the isospin operator, we find that the gauge rotated fields are static;
′
æ1
ö
ϕ a = x$a ç − coth r ÷
èr
ø
(1.55)
′
1 ö
æ1
Aia = ε iab x$b ç −
÷.
è r sinh r ø
(1.56)
Equations (1.55, 56) are solutions of the Bogomol’nyi equation, as can be
verified by direct substitution. It is as if a BPS monopole is a kind of ‘instanton
string’. On the other hand, when non-Abelian theories are cast into Abelian ones with
monopoles [51,52], instantons are found to change into closed dyon loops [53]. So it
seems that, surprisingly enough, monopoles and instantons are closely related. Later
on, I shall return to these solutions with more to say.
1.5 Classifying Magnetic Monopoles
There are currently two methods of classifying magnetic monopoles. One is the
dynamical GNO classification due to Goddard, Nuyts, and Olive, and the other is the
topological classification due to Lubkin.
19
Chapter One : The Story of Magnetic Monopoles
1.5.1 The GNO Classification
Let me give a brief review of the way the GNO classification [54] gives a
generalization of magnetic monopole solutions. In this section, we will assume that A
is a matrix, the form of which is derivable from:
Aµ = T a Aµa ,
(1.57)
where the T’s are the set of anti-Hermitian matrices known as the infinitesimal
generators of the group of the theory. They obey the Jacobi identity:
[[T , T ], T ] + [[T , T ], T ] + [[T , T ], T ] = 0 ,
i
j
k
j
k
i
k
i
j
(1.58)
and the commutation relation:
[T , T ] = ε
i
j
ijk
Tk .
(1.59)
The electric field may be excluded by looking for solutions that are timeindependent and time-reversal invariant, as we did in section 1.4; equations
(1.41, 42). Because of (1.42), we are given the freedom to construct gauge
independent transformations, let us do so by setting
Ar = 0 .
(1.60)
Now, we can assume a solution, for large values of r, were A is an expansion in
powers of 1 r and ignore higher order terms, hence:
A=
a (θ , ϕ )
æ 1ö
+ Oç 2 ÷ ,
èr ø
r
(1.61)
which is the required general form of a magnetic monopole solution.
We may write that solution in its covariant components, and in polar form;
A ⋅ dx = Aθ (θ , ϕ )dθ + Aϕ (θ , ϕ )dϕ .
(1.62)
For the field to be nonsingular at the θ = 0 o and θ = 180 o points (north and south
20
Chapter One : The Story of Magnetic Monopoles
poles); Aθ (0, ϕ ) and Aθ (π , ϕ ) must both vanish. Then we induce one final gauge
transformation;
Aθ = 0 .
(1.63)
Using all that together, the field equations take the form;
1
∂ θ Aϕ = 0
sin θ
(1.64)
A = Q(1 − cosθ ) ,
(1.65)
∂ ϕ g F ϕθ + [ Aϕ , g F ϕθ ] = −∂ ϕ Q = 0 ,
(1.66)
∂θ
giving a solution
and
which tells us that Q is constant. For SU(n) theories, Q was found to be
1
Q = − diag (q1 , q 2 ,K q n ) ,
2
(1.67)
with the condition that the sum of the q’s is zero, i.e Q is traceless. If n = 2, i.e SU(2)
theory, then
1
i
Q = − diag (q ,− q ) = − qσ 3 .
2
2
(1.68)
The quantization condition in this case is
e 4π Q = 1 .
(1.69)
Fields having the above properties are generally known as GNO monopoles.
1.5.2 The Lubkin Classification
The second method of classifying magnetic monopoles is topological [55]. The
21
Chapter One : The Story of Magnetic Monopoles
monopole is investigated at large distances from the origin. This is commonly known
as the black box approach; the monopole is inside a black box, we cannot see it. It is
employed in order to avoid complications at the limit r → 0. Now, we are not going
to assume solutions to the field equations, but rather the monopole is going to be
studied from topological considerations, utterly unrelated to any particular theory or
specific field equations.
Lubkin described a topological construction around the monopole that bears his
name. Imagine a sphere surrounding the black box. A closed contour starts at the
north pole, traces an arbitrary path on the surface of the sphere, and ends back at the
north pole. The contour may be any closed path on the sphere, provided that it starts
and ends at the north pole. Associate each contour with a continuously changing
variable τ, which starts at the trivial case τ = 0, i.e when the contour is a point on the
north pole, makes a full rotation on the sphere, and goes back to triviality at τ = 1
(see Fig. 1.2).
Fig. 1.2
The Lubkin Construction. A sphere is imagined to surround the
black box containing a monopole. Shown are examples of possible values of the τ contours.
Now, along each path the gauge field may be integrated to obtain the group
element associated with the path. Thus we have a family of paths in group space
found by:
é
dx µ ù
u(τ ) = a exp ê− Aµ
ds ,
ds úû
ë
ò
22
(1.70)
Chapter One : The Story of Magnetic Monopoles
where s denotes the integration around the loop associated with each value of τ .
Equation (1.70) has the further imposed conditions u(0) = u(1) = 1; the start and the
finish. The topological connection comes from the observation that u is actually a
group element belonging to G, the main group of the theory.
Now, let us make some gauge transformations that will enable us to evaluate the
above
formula.
Analogously to
the
dynamical
classification,
the
gauge
transformations are specially constructed in order to render Aθ = 0 . We are not
worried, however, about Ar because it never enters the construction anyway (we are
working on the surface of a sphere where the radial potential is the same
everywhere). Thus the integral is reduced to
é
u(τ ) = a exp ê −
ë
ò
2 πτ
ù
Aϕ (π , ϕ )dϕ ú .
0
û
(1.71)
To evaluate the integral, a particular form of Aϕ must be assumed. Explicit
values of u(τ ) enables us to calculate the topological class into which a given
monopole field falls. As an example, let us use one of the GNO solutions, equation
(1.65), and assume G to be U(1), the group of ordinary electrodynamics. The result
becomes:
u(τ ) = e i 2π nτ .
(1.72)
As τ goes from 0 to 1, u winds n times around U(1). Thus, for each element of
the first homotopy group of the topological space there is one and only one magnetic
monopole (see the next section).
Examples of topological calculations can be found in [56] and [57].
1.6 Talking Topology
‘Topology is Power ’ [11]. Indeed it is. The ability to deduce deep conclusions
about a wide variety of objects just by studying their topological properties, without
the need to actually solve, or simulate, their field equations, is truly great power. The
possibility, for instance, to deduce the number of magnetic monopoles in a certain
23
Chapter One : The Story of Magnetic Monopoles
theory, just by topological considerations, is indeed very useful. Let me briefly touch
upon how this is done.
A path a in a space X is defined as a continuous spatial function a( s) of a real
parameter s, such that 0 ≤ s ≤ 1 . The point a( 0) corresponds to the beginning of the
path and a(1) to its end. Thus, the sense of the path is important. In fact, a −1 ( s) is
defined as the inverse of a( s) ; being the same path but in the reverse direction. If the
beginning is identified with the end; a( 0) = a(1) , we get a closed path, or loop.
Now, consider two closed paths a( s) and b( s) and assume the existence of a
continuous function L(t , s) such that:
L(t , 0) = a (0) = b( 0)
L(0, s) = a ( s)
L(t ,1) = a (1) = b(1).
L(1, s) = b( s)
(1.73)
The existence of L(t , s) signifies that a( s) and b( s) can continuously transform
into each other. In other words, they are homotopic. To illustrate this notion, consider
Fig. 1.3. It is a schematic representation of loops in a two-dimensional space X.
The shaded circle is a region that is cut out from the plane X. That is, any point
inside the circle is inaccessible to anything that lives in X. I have drawn five closed
paths that share one point amongst them; point P. It is clear, from the definitions
above, that the loops a( s) and b( s) are homotopic to each other and to c −1 ( s) . They
are also homotopic to the point P (a point is formally defined as a null path). In other
words; they can continuously transform into each other and into P. They are not,
however, homotopic to d ( s) and e( s) . That is to say that in order for b( s) to
transform into e( s) for instance, it has to cut through the shaded region, which is
inaccessible by definition. The loops d ( s) and e( s) , however, are homotopic to each
other, but not to the point P, or, for that matter, to any point at all. The shaded region
prevents that. The space X is said to be not simply connected. All the closed paths in
a simply connected space (i.e not containing inaccessible regions) are homotopic to
each other.
24
Chapter One : The Story of Magnetic Monopoles
Fig. 1.3
The two-dimensional space X is not simply connected.
Now, if we define a family of homotopic loops, like a( s) , b( s) , and c( s) , such
that they all share the same starting point P, then we can define a multiplication
process between the loops. Visually, multiplying two loops together means moving
on the first loop then on the second. Thus, having a common point P is essential. It is
clear that the set of all loops homotopic to a( s) form a group that contains the
multiplication products and inverses of its components. This group is known as the
fundamental group, or the first homotopy group of the space X, denoted by π 1 ( X ) .
This concept was introduced by Poincaré in 1895. Higher homotopy groups were
discovered by Hurewicz in 1939. For a simply connected space, π 1 ( X ) is trivial;
π 1 ( X ) = 1 . Another example is the case of the group of electrodynamics; U(1), the
space of which is the circle S 1 . The first homotopy group of it; π 1 (U 1 ) , is
isomorphic to, i.e has a one-to-one correspondence with, the group of integers Z
( )
under addition. That is π 1 (U 1 ) = π 1 S 1 = Z . It is because of this that the BohmAharonov effect, for example, is possible. The solenoid creates an inaccessible region
in space [6].
Now, in field theory, it has been shown that the boundary conditions of solitons,
magnetic monopoles and other topological objects fall into distinct classes, of which
the vacuum belongs only to one. These boundary conditions are characterized by a
particular correspondence (mapping) between the group space and co-ordinate space,
and because these mappings fall into different homotopy classes that they are
25
Chapter One : The Story of Magnetic Monopoles
topologically distinct. These mappings are characterized by different values of the
winding number n, an example of which appears in equation (1.72), that labels the
homotopy classes. Thus, the connection with Lubkin’s construction is clarified.
Let me, at this point, describe an analogy in co-ordinate space that has been
useful for my own understanding of topology. Imagine a strip of paper, narrow in
width and very long. If we take both ends of the strip and glue them together to make
a cylinder, the strip can be described as a two dimensional space inside which a
‘traveler’ can make a full circle around his or her universe and return to the starting
point without noticing anything unusual. Now, if the strip is twisted 180° before
gluing its ends together, the shape that is formed is the so-called M÷bius Strip. It is a
two dimensional space with one surface. Meaning that the traveler (perhaps an insect
restricted on the surface on the strip and cannot fly) after rotating full circle around
the ‘universe’ would come back to the starting point as his or her own mirror image.
He or she has to rotate twice in order to regain his or her original self. We can call the
strip a topologically interesting space with winding number equal to 1 corresponding
to the one twist we made in the strip before gluing the two ends together. The twist
itself is called a kink in the space, which could be identified as a soliton in the field
theoretic sense of the word. The simplest form of kinks is that which results by
solving the so-called Klein-Gordon equation. Generally, any topological object is,
basically, a kink, including monopoles and instantons [6]. It is noted that the stability
of solitons comes from the topology of space. Indeed, the kink in the M÷bius strip
cannot decay or resolve itself without cutting the space in two and destroying its
topology. Thus, the stability of topological objects is a direct result of topological
stability.
Classical vacuum configurations in four-space have a potential Aµ with a gauge
transformation that can be defined as
Aµ → UAµ U ∗ − i (∂ U )U ∗ ,
( )
(1.74)
U x µ being the gauge function. In this case the winding number can be calculated as
follows:
26
Chapter One : The Story of Magnetic Monopoles
n(U ) =
1
24π 2
ò ε tr[U (∂ U )U (∂ U )U (∂ U )]d x .
∗
ijk
i
∗
j
∗
3
k
(1.75)
Assuming that U ( x)  → 1, U ( x) is actually a map from S 3 into the gauge
x →∞
group, classified by the winding number n(U ) . The topological charge, also known
as the topological index, or the Pontryagin index, can be calculated as follows:
u=
ò
1
~
Tr Fµν F µν d 4 x .
2
16π
(1.76)
This index has certain applications. In instanton theory, for example, it denotes
the degree of the mapping of the group space S 3 of SU(2) and the co-ordinate space
boundary S 3 [6]. The winding number labels the homotopy classes onto which those
mappings fall. It has also been shown [58,59,60] that the topological charge is a
conserved quantity.
Now, it has been established that in a Yang-Mills quantum theory the vacuum is
infinitely degenerate. It follows that the true ground state of Hilbert space may be
written:
∞
vac
θ
=
åe
inθ
vac n ,
(1.77)
n =−∞
characterized by a particular value of θ . Vacua of the type (1.77) are generally
known as θ -vacua, and they have several important consequences. If θ ≠ 0 , the
vacuum state is complex, and time reversal invariance is violated. From the CPT
theorem, this results in the combined violation of charge conjugation and parity. This
is known as the strong CP problem. Another way of putting it is to say that adding a
θ term to the Lagrangian of a theory such as iθ Fµν F µν results in CP-violating
expectation values [61]. Although such a term is formally a total divergence, it has
been shown [62,63] that it cannot be neglected. This interaction contributes directly
to the neutron electric dipole moment d n . The extremely good experimental limits on
d n requires θ < 10 −5 [64]. A satisfactory explanation of why θ is so small yet not
zero is yet to be found. This problem results in the famous baryon number violation
problem. Sometimes, magnetic monopoles are involved with this phenomena, see,
27
Chapter One : The Story of Magnetic Monopoles
for instance, the calculations of the S-matrix of the scattering of quarks and leptons
from magnetic monopoles that has yielded statistical estimates of baryon number
violating scattering cross sections [65].
Now, the point is that the existence of magnetic monopoles has been argued to be
a solution to the strong CP problem. Non-perturbative analysis’ of the problem were
performed [66,67,68], and it was claimed that the gauge orbit space with gauge
potentials and well-defined gauge transformations topologically restricted on the
space boundary in non-Abelian theories with a θ term has a monopole structure if
and only if there is a magnetic monopole in ordinary space. Dirac’s quantization
condition was reported to ensure that θ must be quantized as follows:
θ = 0,
2π
u
u = 1, 2, L ,
(1.78)
u being the topological charge of the monopole field.
1.7 The Stability of Magnetic Monopoles
Now, let us investigate what happens to a monopole under small perturbations
[69]. These calculations, known as the Brandt-Neri stability analysis, give indications
as to the stability of the monopole, which is going to be of interest to us later on. We
discuss SO(3) theory. Since SU(2) monopoles are a subset of their SO(3)
counterparts, then the following also takes care of SU(2) monopoles.
We work in the temporal gauge and write the gauge field as an SO(3) monopole
plus a small perturbation;
1
A = − iqσ 3 A D + δ A ,
2
(1.79)
where A D is the Dirac monopole potential defined by
A D ⋅ dx = (1 − cosθ )dϕ ,
and
q = 0,
1
,L .
2
28
(1.80)
Chapter One : The Story of Magnetic Monopoles
Let us write the perturbation in matrix form;
ié φ
δA = − ê ∗
2ë ψ
ψù
,
−φ úû
(1.81)
where φ is a real vector and ψ is a complex one. Linearizing the field equations in
the perturbation, they become
v v
−∂ 02 φ = ∇ × ∇ × φ
(
)
−∂ 02 ψ = D × (D × ψ ) +
(1.82-a)
iqr
× ψ = H$ψ ,
r3
(1.82-b)
where the covariant derivative D is, of course,
v
D = ∇ − iqA D ,
(1.83)
and q here plays the role of a coupling constant.
The stability is determined by calculating the eigenvalue spectrum of the operator
H$ . If it has any negative values at all, then the solution is unstable. H$ has a set of
eigenmodes with zero eigenvalues. This is a consequence of the invariance of the
temporal-gauge equations of motion under time independent gauge transformations
of the form:
i éγ (x) ζ (x) ù
ξ ( x) = 1 − ê ∗
+ higher order terms ,
2 ëζ (x) −γ (x) úû
under which,
ψ
→ ψ + Dζ K .
(1.84)
(1.85)
Since ψ = 0 is a solution of (1.82-b), so must be its gauge transformation Dζ .
Thus for an arbitrary ζ :
H$( Dζ ) = 0 ,
(1.86)
which are the trivial (zero eigenvalued) eigenmodes. The physical modes are the ones
orthogonal to the gauge modes;
29
Chapter One : The Story of Magnetic Monopoles
ò
(
)
dx ψ ∗ ⋅ Dζ = 0 ,
(1.87)
for any ζ . Equivalently
D⋅ψ = 0.
(1.88)
In order to simplify the calculations, let us, as before, consider rotational
invariance. H$ commutes with the angular momentum
J = L + S,
(1.89)
)
L = − i r × D − qr ,
(1.90)
where [11]
and S is the spin operator defined by
(a ⋅ S)b = ia × b
(1.91)
for any two vectors a and b. The orbital angular momentum takes the quantum
values:
l = q , q + 1, q + 2, L ,
(1.92)
as before, and the total angular momentum takes the values
j = q − 1, q, q + 1, L
= q, q + 1, L
q≥1
q = 0, or
1
.
2
(1.93)
We can use (1.91) to reach
S⋅r
2
H$ = −(S ⋅ D) + q 2 .
r
The following form of H$ can hence be found:
30
(1.94)
Chapter One : The Story of Magnetic Monopoles
é 2 2
j ( j + 1) − q 2 ù
$
Hψ = ê −∂ r − ∂ r +
ú ψ + X ( D ⋅ ψ ) + D( Y ⋅ ψ ) ,
r
r2
ë
û
(1.95)
where the X and Y terms do not contribute to the matrix element of H$ between
physical modes, so we are not interested in their explicit form. Now, for q ≥ 1, j can
be q − 1 , and j ( j + 1) − q = −q , which, again, signifies that the centrifugal force is
attractive. If we calculate the expectation value of a radial function such as
Z
=
for r < R
0
r
1æ
ç r − Re − a ö÷
ø
rè
=
for r ≥ R,
(1.96)
where R and a are positive numbers, we get
∞
H$
=
ò
0
2
qù
é
r 2 dr ê−∂ 2r − ∂ r − 2 ú Z
r
r û
ë
∞
=
ò [
dr r 2 (∂ r Z ) − qZ 2
2
]
0
æ1
ö
= ç − q 2 ÷ ln a + L .
è4
ø
For any fixed value of R,
(1.97)
H$ becomes negative for large values of a. The
conclusion that may be reached from this sort of calculations is that all GNO
monopoles with fields q ≥ 1 are unstable under arbitrarily small perturbations (see
[70] for an example of applying the Brandt-Neri method). The way this instability is
manifested is going to be our main issue in the following chapters. It is generally
assumed in the literature that monopoles evolve in time in two possible ways. The
first is by the emission of non-Abelian radiation. The second is that the monopoles
break up into smaller monopoles with smaller charges, and this chain continues until
stability is reached. Recently, semi-classical calculations of this last conjecture have
31
Chapter One : The Story of Magnetic Monopoles
argued that monopoles and dyons decay on curves of marginal stability in N = 2
supersymmetric theories, but was reported to be absent in N = 4 models [71].
1.8 Recent Research
Let me, in this section, give a brief overview of some of the research conducted
in magnetic monopole theory that is still debatable. The reported results in this
section differ from the rest of the chapter in the fact that they still lie within the
shadowy region of controversy.
Lately, there has been a sudden interest in Dirac’s quantization condition. It has
been severely criticized [72] in the context of extending the theory to include quarkmonopole interactions. On the other hand, some very recent research claimed
rediscovery of Schwinger’s condition. It was reported that a U(1) Hamiltonian
reduction of a four dimensional isotropic quantum oscillator resulted in a bound
system of two spinless Schwinger dyons, with equation (1.10) representing the
quantization condition [73,74]. Some interesting research [75,76] argued that Dirac
magnetic monopoles must exist and are, in fact, quite common. The reason that they
have not been found so far is that we have been using Dirac’s theory, which,
according to the researchers, may not be valid. Instead, a different theory was
presented, and a new approach at designing experiments was suggested. The
Japanese physicist Hirata, however, deduced the Dirac condition from a brand new
approach. It resulted as a by-product of a Hamiltonian formulation of
electrodynamics with monopoles that he discovered [77]. Hirata’s formulation has
the curious property that despite its use of Dirac strings, it is free of Dirac’s veto. It
describes the interactions between electrically charged particles, monopoles, and
photons.
The Dirac-Maxwell equations (1.2) have been the incentive for some
controversies. Take as an example the debate started by the ansatz [78] that a
satisfactory condition for the existence of solutions for transients in lossy media and
the calculation of the signal velocities of which is the modification of Maxwell’s
equations to include magnetic monopoles, i.e equations (1.2). This claim was
challenged by one researcher as unnecessary [79], and commended by another as
32
Chapter One : The Story of Magnetic Monopoles
theoretically and practically appropriate [80]. The debate is still on. Another example
is the argument that the presence of magnetic monopoles should induce the existence
of a second type of photon in de Broglie’s light theory [81].
Recently, a new model, which I have found particularly interesting, was
published [82]. Magnetic monopole solutions within finite range electrodynamics, i.e
with non-zero photon mass, were constructed. Its interactions with a charged particle
were discussed. Since gauge invariance is lost (because of the presence of massive
photons), use of angular momentum algebra was attempted, and it was argued that
the construction of such an algebra is impossible, hence a quantization condition is
non-existent. This pointed to the conclusion that Dirac monopoles and massive
photons cannot exist in the same theory. Another electrodynamical model of massless
bosons and fermions incorporating magnetic monopoles was reported to contain an
infrared logarithm [83]. So, when the number of massless matter in such a model is
sufficiently large, monopoles are suppressed and in the weak-coupling limit charged
particles are unconfined. This result provided a mechanism by which interlayer
tunneling of excitations with one unit of the ordinary electric charge can be
suppressed while that of a doubly charged object is allowed.
Much research was conducted on the properties of the charged particle-monopole
system. For example, the scattering amplitude of the system at center of mass
energies much larger than the masses of the particles was calculated [84], and, in the
limit of low-momentum transfer, was reported to be proportional to the monopole
field’s strength, assuming Dirac’s condition. The spontaneous emission rate of the
system was reported to display significant increase due to the monopole field [85],
the radiative corrections of which were discussed within the framework of the selffield QED and the Pauli approximation [86]. The system’s energy spectrum as a
bound state has been calculated in the general case of the charged particle being any
charged fermion and the monopole being a generalized Dirac dyon. Its spectrum is
divided into many lines and displays a Zeeman effect [87]. Relativistic calculations
of the system showed that classical considerations give the correct single-particle
energy formula not including quantum corrections [88].
33
Chapter One : The Story of Magnetic Monopoles
A curious feature of monopoles in an electromagnetic field is that like-charges
were found to be attracted to each other [89], as the slow decrease of the distance
between two monopoles causes an increase in the field energy. Exact calculations of
the energy spectrum for a relativistic Dirac electron in the presence of a Coulomb
field, a 1 r scalar potential, a Dirac monopole field, and a Bohm-Aharonov potential
were carried out in [90,91]. Dyon-Fermion bound states were investigated in both the
relativistic and the non-relativistic limits [92], and it was reported that the fermion
moves on a cone with its apex at the dyon and axis along its angular momentum. It
was also claimed that exact solutions of the Dirac equation for the system are not
possible due to the presence of terms vanishing more rapidly than 1 r in the potential
of the system.
Further treatments include the research done on monopoles in electric currents
[93], and the effect of monopole fields on the hyperfine structure of the atom [94].
Other investigations include the lattice calculations done by Crea [95] of quarks
interacting with monopole condensates in SU(2) gauge theory. He argued that the
monopoles behave just as in lattice electrodynamics in the maximal Abelian gauge. It
was also argued that the monopole condensate is a parameter in quark confinement
[96,97]. Further analytical and numerical calculations of monopole fields in SU(2)
were carried out in [98]. Also in connection with Dirac monopoles in QCD, superheavy bosons of masses 328 GeV and 11.3 TeV were predicted to exist [99]. Monte
Carlo lattice calculations of this model were performed in [100], and the monopole
masses were calculated on the lattice in [101].
It has long been realized that strongly coupled non-Abelian theories have to be
treated non-perturbatively if they were to show their most interesting features,
magnetic monopoles included. With the development of fast-running digital
computers, and their subsequent availability to all researchers, it was natural for
approximate lattice calculations on computing machines to take the lead. For
instance, generalized lattice calculations of the relationship between SO(3) and SU(2)
monopoles were done in [102]. Further lattice calculations resulted in the formulation
of both point and composite monopole configurations with consequences that
showed the difference between the density of quenched monopoles from the high and
34
Chapter One : The Story of Magnetic Monopoles
low energy phases at the phase transition between O(3) and O(2) gauge theories
[103].
Morris, in [104], proposed a solution of generalized Yang-Mills systems in the
case of vanishing external sources. A radially symmetric solution was constructed,
signaling the presence of a point magnetic monopole with unit magnetic charge. This
solution solved the ’t Hooft-Polyakov ansatz for the spontaneously broken GeorgiGlashow model in the limit of a vanishing Higgs field. Modifications of the model in
which ’t Hooft-Polyakov monopoles cease to exist were found [105], with one
shortcoming; the theory becomes non-renormalizable.
It was demonstrated that the Weinberg-Salam model [106,107] exhibited
unstable field solutions that behaved like magnetic monopoles [108]. Spherically
symmetric monopole-like and dyon-like configurations were reported to have been
found within the model [109], and were interpreted as a non-trivial hybrid between
Abelian and non-Abelian monopoles. Further calculations of dyonic solutions in the
electroweak model were carried out in [110]. Scattering amplitudes of nonrelativistic
W-bosons with ’t Hooft-Polyakov monopoles were calculated in [111]. Using
perturbative methods, a series of bound modes interpreted as states of the gauge
bosons were found within interacting fields of ’t Hooft-Polyakov monopoles, dyons
and electric charges [112].
The O(5) × U(1) electroweak model with two particle generations of quarks and
leptons has been shown to display SO(3) ’t Hooft-Polyakov monopoles after the
theory spontaneously breaks down to the O(3) level [113]. The existence of
monopoles triggers the so-called Cabibbo rotation of d and s quarks along with the
ν e and ν µ flavors which in turn result in neutrino oscillations.
Monopole solutions with spherical symmetry were reported for SU(3) gauge
theory as it breaks down to U(2) and U(1) × U(1) [114]. A new computational
method for determining the monopole currents in these theories was proposed in
[115].
New theories include the one developed in [116]. It is a theory of dyons in both
the Abelian and non-Abelian limits with a gauge potential described in terms of two
35
Chapter One : The Story of Magnetic Monopoles
time-like forms. It was shown that for non-Abelian models exhibiting dyonic
solutions a complete set of fermionic zero modes can be found [117], due to which,
chiral symmetry breaking occurs, only if the QCD vacuum is represented as a gas of
dyons. Further elaborations on this point were performed in [118] and its connection
with topological considerations was reported in [119]. Discussions of non-Abelian
quantization conditions can be found in many sources; see, for example, [120].
Some researchers choose to treat monopole theory in a decidedly heavy
mathematical form. Examples of this include the argument that the presence of
monopoles in non-Abelian gauge theories is the source of the so-called confining
terms in correlators when the most general form of non-Abelian Stokes theorem is
calculated [121]. Discussions of the cohomology of magnetic monopoles may be
found in [122]. In [123] variational methods of spontaneously broken theories with
monopoles were proposed.
Topological methods were applied many times with different goals. In [124] a
topological quantum field theory of magnetic monopoles in SU(n) Yang-Mills-Higgs
background was devised. It was obtained by gauge-fixing the topological action
defining the monopole charge. The Bogomol’nyi equations were used as gauge
conditions. Consequently, the geometrical equations were recovered as ghost
equations of motion. In the limit n → ∞, interesting phenomena arise: the functional
integration is forced to cover only the moduli space and the role of the ghosts
stemming from the gauge-fixing process provides a smooth semi-classical
approximation. Other research proposed that an equivalence between topological
quantum field theory of magnetic monopoles and the so-called stochastic partition
function of a two dimensional BF-type topological field existed [125].
Recently, a new type of magnetic monopoles has taken the theoretical scene by
storm. It was found by Lee and Weinberg [126]. The solution showed the presence of
non-singular finite energy monopoles within a class of purely Abelian gauge theories
containing charged vector mesons. It was reported to be a non-topological solution.
However, it was lately reinterpreted as a topological solution of a non-Abelian
gauged Higgs model [127]. Much research has been done on this newly discovered
36
Chapter One : The Story of Magnetic Monopoles
monopole. A Bogomol’nyi bound was found for it, and its fit in the BPS model was
demonstrated [128].
Calculations of electric charge at finite energies for monopoles in spontaneously
broken gauge theories with a CP violating θ term were recently attempted [129].
These calculations were later extended to include fermions coupled with ’t HooftPolyakov monopoles [130].
The ground state structure and topological properties of CP prime fields with
finite action and a fraction charge were considered in a 2+1 system of strongly
correlated particles [131]. It was argued that at low energies there is an electrically
charged Boson condensate, in the intermediate energy region there exists a Coulomb
phase of topological charges and phases of oblique confinement of dyons.
1.9 Magnetic Monopoles in Exotic Theories
By ‘exotic theories’ I mean theories lying in an, as yet, unproven area, such as
Grand Unified Theories (GUTs) and String theories, and theories in other closely
related fields, such as astrophysics and cosmology. GUT monopoles are sometimes
called GUMs, short for Grand Unified Monopoles. It was shown that, due to the
confinement hypothesis of SU(3), the non-Abelian magnetic field does not leave the
surface of the monopole, but the ordinary U(1) magnetic field does. Thus, such a
monopole could be detected in experiments as an unpoint-like Dirac monopole; a
Dirac monopole with a composite structure (the size of GUMs is usually of the order
of the nucleon size, with masses in the 1015 GeV region!).
One of the simplest GUTs is the so called Georgi-Glashow model based on the
SU(5) group, spontaneously breaking down to SU(3)×SU(2)×U(1) [132]. Magnetic
monopoles arise from this theory quite naturally, and are in fact complex particles
and extremely heavy. A complete set of invariants for the bosonic sector of the model
was constructed in [133]. Among the invariants there are magnetic vortices which
confine the monopoles already in the theory. Numerical investigations on the lattice
of the vacuum sector of the model resulted in distributions and correlations of
magnetic fluxes as well as correlations between the fluxes and other local
37
Chapter One : The Story of Magnetic Monopoles
observables [134]. The quantization of monopole fields in this model was discussed
[135], being based on the order-disorder duality existing between the monopole
operator and the Lagrangian fields (see section 1.11). Studies of the chaotic behavior
in monopole theories can be found in [136].
Monopole solutions in heterotic string theory were discussed in toroidal
compactifications to space-time [137], and the existence of supersymmetric bound
states between pure monopoles and dyons was investigated [138]. Monopole
solutions were found in compact manifolds [139]. A conformal field theory
representing a four dimensional classical solution of string theory has been reported
to have dyonic properties in the U(1) limit [140]. Recent Studies of the
supersymmetric SO(Nc) gauge theories showed that they had monopole and dyon
solutions [141]. Supersymmetric N = 2 SU(2) Yang-Mills models were shown to
display an entire spectrum of dyonic solutions [142,143]. Another dyonic spectrum
was found in the N = 4 supersymmetric Yang-Mills model with a gauge group SU(3)
spontaneously breaking down to U(1) × U(1) [144]. Extending monopoles to string
theories may be able to resolve many problems that monopoles were found to create
in conventional point-particle quantum field theories; like, for example, the violation
of the Jacobi identity found in some theories as a direct consequence of the presence
of magnetic charge, which was solved by considering the string theoretic form of the
model [145].
In cosmology, the use of magnetic monopole theory is heavily relied upon.
Suffice to know that the masses of magnetic monopoles, which are very heavy as
mentioned before, are thought to help in explaining the problem of the missing mass
of the universe [146], which will decide once and for all whether the universe is open
or closed. Pessimistic estimates have indicated that monopoles must have been
produced so copiously in the very early universe and annihilated so inefficiently
subsequently that they would, at the current epoch, form the dominant contribution to
the mass of the universe [147].
Calculations of the effects of the forces exerted on test particles outside a
relativistically rotating massive body, such as galactic nuclei and quasars, with a high
saturation of magnetic monopoles are a typical example of the involvement of
38
Chapter One : The Story of Magnetic Monopoles
monopoles in astrophysics [148,149]. Cosmological research has established the
presence and effect of magnetic monopole solutions in a Yang-Mills-Higgs SO(3)
model against the background of gravitational fields in an expanding universe,
admitting static and non-static BPS solutions [150]. In fact, one exact solution of the
Einstein-Maxwell geometrodynamic equations has been reported to exhibit the
properties of a rotating massive radiating dyon [151]. It was argued that using a
duality transformation, any classical general relativity solution describing a massive
charged object can be generalized to include both electric and magnetic charges
simultaneously [152]. Einstein-Yang-Mills-Higgs models exhibit monopole solutions
with static anti-gravitating properties that has been demonstrated to be equivalent to a
class of string theoretic BPS monopole solutions discovered earlier by Harvey and
Liu [153,154].
An interesting result was found when monopole solutions of the Yang-MillsHiggs theory were studied against the vacuum expectation value η of the Higgs
field. It was numerically calculated that above a certain critical value of η , of the
order of the Planck mass, the monopole expands exponentially and creates a
wormhole structure in the space surrounding it [155]. If the expectation value was
around the critical value, three types of solutions can occur: the monopole either
expands as stated, collapses into a black hole, or assumes a stable configuration. This
magnetic monopole-general relativity connection is being currently exploited to the
most, and many physicists feel that magnetic monopoles may play an important role
in the final unified theory of all interactions.
General relativistic fields (gravity) coupled to GUTs were considered in many
papers. One of these with a bearing on our subject is [156], where the GUT used was
the SU(5) model. An entire family of solutions interpreted as antisymmetric
monopole and dyon black holes was proposed. They have a magnetic charge equal to
3
± e with a space-time metric taking the Kerr-Neuman form (i.e a rotating
2
Schwarzchild singularity). Another closely related solution of the same model
exhibited the same value of magnetic charge with a different space-time metric; the
so-called Reissner-Nordstrom metric [157]. This solution represents a black hole
39
Chapter One : The Story of Magnetic Monopoles
only over a certain limit. Other solutions of dyonic black holes were reported in
[158].
One of the most famous unification theories is the geometrical theory of Kaluza
and Klein. It was proposed in the 1920’s as a generalization of General Relativity to
include the electromagnetic field. With the development of quantum field theory and
the discovery of the other two major fields; the strong nuclear and the weak nuclear,
the Kaluza-Klein model fell out of favor. Then, in the 1970’s, it was revived as a
possibly correct way of unifying the electromagnetic and the gravitational fields after
quantization. The significance of this theory in our discussion is manifest in the fact
that the Kaluza-Klein theory was shown to admit soliton solutions that could be
interpreted as magnetic monopoles [159]. These monopoles have the curious
property that they seem to violate the principle of equivalence in the sense that their
inertial mass is approximately the Planck mass, but their gravitational mass vanishes.
What consequences this incredible result will yield are yet to be seen.
1.10 Magnetic Monopoles in the Lab
This work being of a theoretical nature, we will not linger much on the issue of
the experimental search for monopoles. It is, however, essential for us to note that, so
far, magnetic monopoles have not been positively identified in experiment. The
reason for this is recognized to be that monopoles are extremely massive, in other
words, in order to detect them in scattering experiments for instance, we need to go
into an energy domain that is much higher than what is available in present day
accelerators. In fact, it is doubtful that monopoles could be created in man-made
experiments due to their great mass, the hope then rests on finding them ready-made
in nature.
The history of the search for monopoles, since the early 1930’s to the late 60’s is
reviewed in [160]. In [3], a beautiful review is given of some of the most common
techniques used to detect magnetic monopoles. Further reviews may be found in
40
Chapter One : The Story of Magnetic Monopoles
[161]. These methods include induction techniques [162,163,164], the higher order
gradiometer method, the series-parallel gradiometer, the track-etch method [165],
ionization techniques, acoustic detection, detection by superconducting phase
changes, optical pumping magnetometry and others. One of the most interesting
methods of finding monopoles is the water tank method. It is based on the principle
of Cherenkov radiation that results from nucleon decays catalyzed by magnetic
monopoles. The method is described in many sources; for example, see [166].
Typical results of the water tank method may be found in [167]. Other instruments
include the so-called MACRO detector described in [168]. The Japanese Tokamak
operation based on monopole circuitry is described in [169]. Other methods include
the use of a new plastic material that goes by the name CR-39 with a small amount of
antioxidants added. Results of the appropriateness of this method were compared to
the use of pure CR-39 [170]. More traditional methods include the use of ordinary
scintillator chambers [171].
An important advance in experimental monopole theory was reached when a
finite element method for calculating complex magnetostatic fields excited by both
axial and radial currents was formulated [172,173]. The method was called the
Fictitious Magnetic Monopole Model (FMMM). In it, a single scalar potential is used
to derive a precise solution of three dimensional anisotropic nonlinear rotational
fields. FMMM was used to calculate the magnetostatic field of monopoles; essential
in monopole search experiments [174]. Further properties and applications of
FMMM were reported in [175].
Monopole searches are generally conducted in two major domains; cosmic rays,
and rock ores. The most famous monopole candidate was discovered by Professor
Blas Cabrera of Stanford. He published data indicating that slow moving heavy
monopoles may have been detected in cosmic rays [176]. He noted, however, that he
could not rule out the possibility that it may have been some sort of background noise
that he had failed to take into consideration; in other words, the statistical probability
of the data representing a monopole was not high enough. Using this candidate event,
and assuming the data is actually noise, he managed to set an upper limit on the
. × 10 −10 cm-2s-1sr-1. Even though this experiment is not
cosmic ray monopole flux of 61
considered as proof of monopole discovery, interest in magnetic monopoles, both
41
Chapter One : The Story of Magnetic Monopoles
theoretical and experimental, boomed after Cabrera’s paper. In fact, the decade
following it resulted in a revival of interest in monopole theory. Cabrera himself
seems to have dedicated his life to the search for monopoles. In [177], he reported
450 days of experiment time and 6600 hours of computer time on Stanford’s eightloop superconducting detector, also to no avail. Other experiments inspired by
Cabrera’s method were performed. One was reported in 1984 [178], having lasted
310 days, again with negative results.
Slowly moving monopoles have also been sought in a two year water tank
experiment resulting in an upper flux limit of 5.6 × 10 −15 cm-2s-1, for a velocity range
10 −4 ≤ β ≤ 4 × 10 −3 , where, of course, β is the particle velocity divided by the speed
of light [179].
Another limit of monopole flux was set at 10 −15 cm-2s-1sr-1, also from
astrophysical considerations [180]. This last, known as the Parker Limit, is
considered a standard figure in the literature today. More recent searches using a
water tank include the three year experiment described in [181], setting an upper flux
limit of 8.7 × 10 −5 cm-2s-1, for a velocity β > 2 × 10 −3 . LEP searches for Dirac
monopoles conducted by Kinoshita resulted in the following charge data:
0.2 g D < g < 3.6 g D , where the theoretical Dirac magnetic charge unit is g D = 58.5e ,
as calculated from the quantization condition [182]. A mass upper limit of 44.9 GeV
was also found. Other LEP limits of Dirac monopoles [183] are : 0.9 g D < g < 3.6 g D ,
with a mass m < 45 GeV.
Searches of ’t Hooft-Polyakov monopoles include the one reported in [184]
resulting in values of the so-called suppression factor: F 2 ~ 10 −30 − 10 −150 , which,
experimenters tell us, indicate that ’t Hooft-Polyakov monopoles are not likely to be
produced in man-made accelerators, it must be discovered in natural phenomena.
That is, not even the Superconducting Super Collider (SSC) would have yielded
results in that respect, had it been built.
Further searches include the one reported in [185], which resulted in the
following limits:
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Chapter One : The Story of Magnetic Monopoles
for β ~ 10 −3
Flux ~ 2.7 × 10 −15 cm-2s-1sr-1,
for σ ~ 10 −25
Flux < 10 −15 cm-2s-1sr-1.
Searches for monopoles in rock ores are less frequent than the ones conducted
for monopoles in flight. Due to the great momentum of the heavier monopoles, they
have a high penetrability of matter. It has been estimated that GUMs with β ~ 10 −3
can penetrate the entire Earth. It has been shown, however, that the magnetic binding
forces between a monopole and an atom or a molecule with a large magnetic
moment, like magnetite iron ore, could overcome the Earth’s gravitational attraction
of slowly moving monopoles [186,187,188]. Hence, it is conceivable that over the
millennia, GUMs may have become trapped in the Earth’s crust. One group has
reported searching for magnetic monopoles in schist, ferromanganese nodules, iron
ores, and meteorites [189]. They used a superconducting induction coil connected to
a superconducting quantum interference device. A total of 331 kilograms of matter
were tested, including 112 kilograms of meteorites, i.e rocks of extraterrestrial origin.
Again, no monopoles were found. An overall ratio of monopoles to nucleons in the
. × 10 −29 with a 90% confidence level. The
samples was calculated to be less than 12
results of the search so far, as the figures show, are quite disappointing.
1.11 Magnetic Monopoles in a Quantized Theory
It will be noted that mostly we have been and are going to be working within the
framework of classical field theory. Still, quantum deductions are being made. There
is no contradiction. According to earlier research, it has been shown that many
quantum conclusions may be reached from purely classical considerations, as is
adequately reviewed in [190]. For instance, we can talk about the classical stability of
a field and deduce conclusions about the stability of its particles after quantization.
Thus the necessity of quantization does not display itself until we come to the point
(which we will not in this work) of actually requiring such figures as scattering cross
sections and the like. Let me, however, insert a note about the place of magnetic
monopoles in today’s quantum theory.
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Chapter One : The Story of Magnetic Monopoles
The quantum field theory of electrodynamics incorporating Dirac magnetic
monopoles (i.e the quantized version of the Dirac-Maxwell theory) has been dubbed
QEMD (quantum electromagnetodynamics) in [191], which also gives a most
exhaustive review of the problem of quantizing Dirac monopoles, including deriving
Feynman rules and the renormalization of the theory. It has been proposed that,
within an approximation, QEMD is the effective theory for ’t Hooft-Polyakov
monopoles [192]. The theory was founded by Cabibbo and Ferrari [20], Schwinger
[193], and Zwanziger [194]. Other formulations were later introduced, for example;
the one potential formulation [195,196], the path-dependent formulation [197], and
the string formulation [198,199]. Recently, the so-called order-disorder method was
devised in [200]. An interesting property of quantized Dirac monopole fields is its
ability to induce pair production of electrically charged scalar particles. Path integral
calculations of such a situation indicated that a magnetic charge g creates pairs with
charge e, orbital angular momentum l < eg − 1 2 and azimuthal component m = 0
[201]. Updates on pair production rates were reported in [202], for both point-like
and composite monopoles. The latest QEMD was very recently formulated in [203].
Some of these methods were found to be mathematically equivalent, but others stand
independent, like Cabibbo’s. With no experimental data to guide us, it is yet to be
seen how magnetic monopoles can be convincingly incorporated in quantum field
theory.
44