======================================================================
=                         Magnetic monopole                          =
======================================================================

                            Introduction
======================================================================
In particle physics, a  magnetic monopole is a hypothetical elementary
particle that is an isolated magnet with only one magnetic pole (a
north pole without a south pole or vice versa).
A magnetic monopole would have a net "magnetic charge". Modern
interest in the concept stems from particle theories, notably the
grand unified and superstring theories, which predict their existence.

Magnetism in bar magnets and electromagnets is not caused by magnetic
monopoles, and indeed, there is no known experimental or observational
evidence that magnetic monopoles exist.

Some condensed matter systems contain effective (non-isolated)
magnetic monopole quasi-particles, or contain phenomena that are
mathematically analogous to magnetic monopoles.


Pre-twentieth century
=======================
Many early scientists attributed the magnetism of lodestones to two
different "magnetic fluids" ("effluvia"), a north-pole fluid at one
end and a south-pole fluid at the other, which attracted and repelled
each other in analogy to positive and negative electric charge.
However, an improved understanding of electromagnetism in the
nineteenth century showed that the magnetism of lodestones was
properly explained not by magnetic monopole fluids, but rather by a
combination of electric currents, the electron magnetic moment, and
the magnetic moments of other particles. Gauss's law for magnetism,
one of Maxwell's equations, is the mathematical statement that
magnetic monopoles do not exist. Nevertheless, it was pointed out by
Pierre Curie in 1894 that magnetic monopoles 'could' conceivably
exist, despite not having been seen so far.


Twentieth century
===================
The 'quantum' theory of magnetic charge started with a paper by the
physicist Paul Dirac in 1931. In this paper, Dirac showed that if
'any' magnetic monopoles exist in the universe, then all electric
charge in the universe must be quantized (Dirac quantization
condition). The electric charge 'is', in fact, quantized, which is
consistent with (but does not prove) the existence of monopoles.

Since Dirac's paper, several systematic monopole searches have been
performed. Experiments in 1975 and 1982 produced candidate events that
were initially interpreted as monopoles, but are now regarded as
inconclusive. Therefore, it remains an open question whether monopoles
exist.
Further advances in theoretical particle physics, particularly
developments in grand unified theories and quantum gravity, have led
to more compelling arguments (detailed below) that monopoles do exist.
Joseph Polchinski, a string-theorist, described the existence of
monopoles as "one of the safest bets that one can make about physics
not yet seen". These theories are not necessarily inconsistent with
the experimental evidence. In some theoretical models, magnetic
monopoles are unlikely to be observed, because they are too massive to
create in particle accelerators (see  below), and also too rare in the
Universe to enter a particle detector with much probability.

Some condensed matter systems propose a structure superficially
similar to a magnetic monopole, known as a flux tube. The ends of a
flux tube form a magnetic dipole, but since they move independently,
they can be treated for many purposes as independent magnetic monopole
quasiparticles. Since 2009, numerous news reports from the popular
media have incorrectly described these systems as the long-awaited
discovery of the magnetic monopoles, but the two phenomena are only
superficially related to one another. These condensed-matter systems
remain an area of active research. (See  below.)


               Poles and magnetism in ordinary matter
======================================================================
All matter ever isolated to date, including every atom on the periodic
table and every particle in the standard model, has zero magnetic
monopole charge. Therefore, the ordinary phenomena of magnetism and
magnets have nothing to do with magnetic monopoles.

Instead, magnetism in ordinary matter comes from two sources. First,
electric currents create magnetic fields according to Ampère's law.
Second, many elementary particles have an 'intrinsic' magnetic moment,
the most important of which is the electron magnetic dipole moment.
(This magnetism is related to quantum-mechanical "spin".)

Mathematically, the magnetic field of an object is often described in
terms of a multipole expansion. This is an expression of the field as
the sum of component fields with specific mathematical forms. The
first term in the expansion is called the 'monopole' term, the second
is called 'dipole', then 'quadrupole', then 'octupole', and so on. Any
of these terms can be present in the multipole expansion of an
electric field, for example. However, in the multipole expansion of a
'magnetic' field, the "monopole" term is always exactly zero (for
ordinary matter). A magnetic monopole, if it exists, would have the
defining property of producing a magnetic field whose 'monopole' term
is non-zero.

A magnetic dipole is something whose magnetic field is predominantly
or exactly described by the magnetic dipole term of the multipole
expansion. The term 'dipole' means 'two poles', corresponding to the
fact that a dipole magnet typically contains a 'north pole' on one
side and a 'south pole' on the other side. This is analogous to an
electric dipole, which has positive charge on one side and negative
charge on the other. However, an electric dipole and magnetic dipole
are fundamentally quite different. In an electric dipole made of
ordinary matter, the positive charge is made of protons and the
negative charge is made of electrons, but a magnetic dipole does 'not'
have different types of matter creating the north pole and south pole.
Instead, the two magnetic poles arise simultaneously from the
aggregate effect of all the currents and intrinsic moments throughout
the magnet. Because of this, the two poles of a magnetic dipole must
always have equal and opposite strength, and the two poles cannot be
separated from each other.


                        Maxwell's equations
======================================================================
Maxwell's equations of electromagnetism relate the electric and
magnetic fields to each other and to the motions of electric charges.
The standard equations provide for electric charges, but they posit no
magnetic charges. Except for this difference, the equations are
symmetric under the interchange of the electric and magnetic fields.
In fact, symmetric Maxwell's equations can be written when all charges
(and hence electric currents) are zero, and this is how the
electromagnetic wave equation is derived.

Fully symmetric Maxwell's equations can also be written if one allows
for the possibility of "magnetic charges" analogous to electric
charges. With the inclusion of a variable for the density of these
magnetic charges, say , there is also a "magnetic current density"
variable in the equations, .

If magnetic charges do not exist - or if they do exist but are not
present in a region of space - then the new terms in Maxwell's
equations are all zero, and the extended equations reduce to the
conventional equations of electromagnetism such as  (where  is
divergence and  is the magnetic  field).


In Gaussian cgs units
=======================
The extended Maxwell's equations are as follows, in Gaussian cgs
units:

Maxwell's equations and Lorentz force equation with magnetic
monopoles: Gaussian cgs units
Name    Without magnetic monopoles      With magnetic monopoles
Gauss's law     |colspan="2"| \nabla \cdot \mathbf{E} = 4 \pi
\rho_{\mathrm e}
Gauss's law for magnetism       \nabla \cdot \mathbf{B} = 0     \nabla \cdot
\mathbf{B} = 4 \pi \rho_{\mathrm m}
Faraday's law of induction      -\nabla \times \mathbf{E} =
\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}      -\nabla \times
\mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} +
\frac{4 \pi}{c}\mathbf{j}_{\mathrm m}
Ampère's law (with Maxwell's extension)        |colspan="2"| \nabla \times
\mathbf{B} = \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t} +
\frac{4 \pi}{c} \mathbf{j}_{\mathrm e}
!Lorentz force law      |\mathbf{F}=q_{\mathrm
e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right)
|\mathbf{F}=q_{\mathrm
e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) +
q_{\mathrm
m}\left(\mathbf{B}-\frac{\mathbf{v}}{c}\times\mathbf{E}\right)

In these equations  is the 'magnetic charge density',   is the
'magnetic current density', and  is the 'magnetic charge' of a test
particle, all defined analogously to the related quantities of
electric charge and current;  is the particle's velocity and  is the
speed of light. For all other definitions and details, see Maxwell's
equations. For the equations in nondimensionalized form, remove the
factors of .


In SI units
=============
In SI units, there are two conflicting units in use for magnetic
charge : webers (Wb) and ampere·meters (A·m). The conversion between
them is , since the units are  by dimensional analysis (H is the henry
- the SI unit of inductance).

Maxwell's equations then take the following forms (using the same
notation above):
Maxwell's equations and Lorentz force equation with magnetic
monopoles: SI units
Name    rowspan=2 | Without magnetic  monopoles colspan=2 | With
magnetic monopoles
Weber convention        Ampere·meter convention
Gauss's Law     colspan="3" | \nabla \cdot \mathbf{E} =
\frac{\rho_{\mathrm e}}{\varepsilon_0}
Gauss's Law for magnetism       \nabla \cdot \mathbf{B} = 0     \nabla \cdot
\mathbf{B} = \rho_{\mathrm m}   \nabla \cdot \mathbf{B} =
\mu_0\rho_{\mathrm m}
Faraday's Law of induction      -\nabla \times \mathbf{E} = \frac{\partial
\mathbf{B}} {\partial t}        -\nabla \times \mathbf{E} = \frac{\partial
\mathbf{B}} {\partial t} + \mathbf{j}_{\mathrm m}       -\nabla \times
\mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} +
\mu_0\mathbf{j}_{\mathrm m}
Ampère's Law (with Maxwell's extension)        colspan="3" | \nabla \times
\mathbf{B} = \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t} +
\mu_0 \mathbf{j}_{\mathrm e}
Lorentz force equation  \mathbf{F}=q_{\mathrm
e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)    \begin{align}
\mathbf{F} ={} &q_{\mathrm
e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) +\\
&\frac{q_{\mathrm m}}{\mu_0}\left(\mathbf{B}-\mathbf{v}\times
\frac{\mathbf{E}}{c^2}\right)   \end{align}     \begin{align}     \mathbf{F}
={} &q_{\mathrm
e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) +\\
&q_{\mathrm
m}\left(\mathbf{B}-\mathbf{v}\times\frac{\mathbf{E}}{c^2}\right)
\end{align}


Tensor formulation
====================
Maxwell's equations in the language of tensors makes Lorentz
covariance clear. The generalized equations are:

Maxwell equations       Gaussian units  SI units (Wb)   SI units (A�
m)
Faraday-Gauss law       \partial_\alpha F^{\alpha\beta} =
\frac{4\pi}{c}J^\beta_{\mathrm e}       \partial_\alpha F^{\alpha\beta} =
\mu_0 J^\beta_{\mathrm e}       \partial_\alpha F^{\alpha\beta} = \mu_0
J^\beta_{\mathrm e}
Ampère-Gauss law       \partial_\alpha {\tilde F^{\alpha\beta}} =
\frac{4\pi}{c} J^\beta_{\mathrm m}      \partial_\alpha {\tilde
F^{\alpha\beta}} = \frac{1}{c} J^\beta_{\mathrm m}      \partial_\alpha
{\tilde F^{\alpha\beta}} = \frac{\mu_0}{c} J^\beta_{\mathrm m}
Lorentz force law       \frac{dp_\alpha}{d\tau} = \left[ q_{\mathrm e}
F_{\alpha\beta} + q_{\mathrm m} {\tilde F_{\alpha\beta}} \right]
\frac{v^\beta}{c}       \frac{dp_\alpha}{d\tau} = \left[ q_{\mathrm e}
F_{\alpha\beta} + \frac{q_{\mathrm m}}{\mu_0 c} {\tilde
F_{\alpha\beta}} \right] v^\beta        \frac{dp_\alpha}{d\tau} = \left[
q_{\mathrm e} F_{\alpha\beta} + \frac{q_{\mathrm m}}{c} {\tilde
F_{\alpha\beta}} \right] v^\beta

where
* is the electromagnetic tensor,  is the dual electromagnetic tensor,
*for a particle with electric charge  and magnetic charge ;  is the
four-velocity and  the four-momentum,
*for an electric and magnetic charge distribution;  is the electric
four-current and  the magnetic four-current.

For a particle having only electric charge, one can express its field
using a four-potential, according to the standard covariant
formulation of classical electromagnetism:

{{block indent|1=F_{\alpha \beta} = \partial_{\alpha} A_{\beta} -
\partial_{\beta} A_{\alpha} \,}}

However, this formula is inadequate for a particle that has both
electric and magnetic charge, and we must add a term involving another
potential .

{{block indent|1=F_{\alpha \beta} = \partial_{\alpha} A_{\beta} -
\partial_{\beta} A_{\alpha} \
+\partial^{\mu}(\varepsilon_{\alpha\beta\mu\nu}P^{\nu}),}}

This formula for the fields is often called the Cabibbo-Ferrari
relation, though Shanmugadhasan proposed it earlier.  The quantity  is
the Levi-Civita symbol, and the indices (as usual) behave according to
the Einstein summation convention.


Duality transformation
========================
The generalized Maxwell's equations possess a certain symmetry, called
a 'duality transformation'. One can choose any real angle , and
simultaneously change the fields and charges everywhere in the
universe as follows (in Gaussian units):
Charges and currents    Fields
\begin{pmatrix} \rho_{\mathrm e} \\     \rho_{\mathrm m}
\end{pmatrix}=\begin{pmatrix}   \cos \xi & -\sin \xi \\     \sin \xi
& \cos \xi \\       \end{pmatrix}\begin{pmatrix}    \rho_{\mathrm e}' \\
\rho_{\mathrm m}'       \end{pmatrix}   |\begin{pmatrix}        \mathbf{E} \\
\mathbf{H}      \end{pmatrix}=\begin{pmatrix}   \cos \xi & -\sin \xi \\
\sin \xi & \cos \xi \\      \end{pmatrix}\begin{pmatrix}    \mathbf{E'} \\
\mathbf{H'}     \end{pmatrix}
\begin{pmatrix} \mathbf{J}_{\mathrm e} \\       \mathbf{J}_{\mathrm m}
\end{pmatrix}=\begin{pmatrix}   \cos \xi & -\sin \xi \\     \sin \xi
& \cos \xi \\       \end{pmatrix}\begin{pmatrix}    \mathbf{J}_{\mathrm e}'
\\      \mathbf{J}_{\mathrm m}' \end{pmatrix}   \begin{pmatrix} \mathbf{D} \\
\mathbf{B}      \end{pmatrix}=\begin{pmatrix}   \cos \xi & -\sin \xi \\
\sin \xi & \cos \xi \\      \end{pmatrix}\begin{pmatrix}    \mathbf{D'} \\
\mathbf{B'}     \end{pmatrix}

where the primed quantities are the charges and fields before the
transformation, and the unprimed quantities are after the
transformation. The fields and charges after this transformation still
obey the same Maxwell's equations. The matrix is a two-dimensional
rotation matrix.

Because of the duality transformation, one cannot uniquely decide
whether a particle has an electric charge, a magnetic charge, or both,
just by observing its behavior and comparing that to Maxwell's
equations. For example, it is merely a convention, not a requirement
of Maxwell's equations, that electrons have electric charge but not
magnetic charge; after a  transformation, it would be the other way
around. The key empirical fact is that all particles ever observed
have the same ratio of magnetic charge to electric charge. Duality
transformations can change the ratio to any arbitrary numerical value,
but cannot change the fact that all particles have the same ratio.
Since this is the case, a duality transformation can be made that sets
this ratio at zero, so that all particles have no magnetic charge.
This choice underlies the "conventional" definitions of electricity
and magnetism.


                        Dirac's quantization
======================================================================
One of the defining advances in quantum theory was Paul Dirac's work
on developing a relativistic quantum electromagnetism. Before his
formulation, the presence of electric charge was simply "inserted"
into the equations of quantum mechanics (QM), but in 1931 Dirac showed
that a discrete charge naturally "falls out" of QM. That is to say, we
can maintain the form of Maxwell's equations and still have magnetic
charges.

Consider a system consisting of a single stationary electric monopole
(an electron, say) and a single stationary magnetic monopole.
Classically, the electromagnetic field surrounding them has a momentum
density given by the Poynting vector, and it also has a total angular
momentum, which is proportional to the product , and independent of
the distance between them.

Quantum mechanics dictates, however, that angular momentum is
quantized in units of , so therefore the product  must also be
quantized. This means that if even a single magnetic monopole existed
in the universe, and the form of Maxwell's equations is valid, all
electric charges would then be quantized.

What are the units in which magnetic charge would be quantized?
Although it would be possible simply to integrate over all space to
find the total angular momentum in the above example, Dirac took a
different approach. This led him to new ideas. He considered a
point-like magnetic charge whose magnetic field behaves as  and is
directed in the radial direction, located at the origin. Because the
divergence of  is equal to zero almost everywhere, except for the
locus of the magnetic monopole at , one can locally define the vector
potential such that the curl of the vector potential  equals the
magnetic field .

However, the vector potential cannot be defined globally precisely
because the divergence of the magnetic field is proportional to the
Dirac delta function at the origin. We must define one set of
functions for the vector potential on the "northern hemisphere" (the
half-space  above the particle), and another set of functions for the
"southern hemisphere". These two vector potentials are matched at the
"equator" (the plane  through the particle), and they differ by a
gauge transformation. The wave function of an electrically charged
particle (a "probe charge") that orbits the "equator" generally
changes by a phase, much like in the Aharonov-Bohm effect. This phase
is proportional to the electric charge  of the probe, as well as to
the magnetic charge  of the source. Dirac was originally considering
an electron whose wave function is described by the Dirac equation.

Because the electron returns to the same point after the full trip
around the equator, the phase  of its wave function  must be
unchanged, which implies that the phase  added to the wave function
must be a multiple of :

Units   Condition
Gaussian-cgs units               2 \frac{q_{\mathrm e} q_{\mathrm m}}{\hbar c}
\in \mathbb{Z}
SI units (weber convention)              \frac{q_{\mathrm e} q_{\mathrm m}}{2
\pi \hbar} \in \mathbb{Z}
SI units (ampere·meter convention)              \frac{q_{\mathrm e} q_{\mathrm
m}}{2 \pi \varepsilon_0 \hbar c^2} \in \mathbb{Z}

where  is the vacuum permittivity,  is the reduced Planck's constant,
is the speed of light, and  is the set of integers.

This is known as the Dirac quantization condition. The hypothetical
existence of a magnetic monopole would imply that the electric charge
must be quantized in certain units; also, the existence of the
electric charges implies that the magnetic charges of the hypothetical
magnetic monopoles, if they exist, must be quantized in units
inversely proportional to the elementary electric charge.

At the time it was not clear if such a thing existed, or even had to.
After all, another theory could come along that would explain charge
quantization without need for the monopole. The concept remained
something of a curiosity. However, in the time since the publication
of this seminal work, no other widely accepted explanation of charge
quantization has appeared. (The concept of local gauge invariance�see
Gauge theory�provides a natural explanation of charge quantization,
without invoking the need for magnetic monopoles; but only if the U(1)
gauge group is compact, in which case we have magnetic monopoles
anyway.)

If we maximally extend the definition of the vector potential for the
southern hemisphere, it is defined everywhere except for a
semi-infinite line stretched from the origin in the direction towards
the northern pole. This semi-infinite line is called the Dirac string
and its effect on the wave function is analogous to the effect of the
solenoid in the Aharonov-Bohm effect. The quantization condition comes
from the requirement that the phases around the Dirac string are
trivial, which means that the Dirac string must be unphysical. The
Dirac string is merely an artifact of the coordinate chart used and
should not be taken seriously.

The Dirac monopole is a singular solution of Maxwell's equation
(because it requires removing the worldline from spacetime); in more
complicated theories, it is superseded by a smooth solution such as
the 't Hooft-Polyakov monopole.


Dirac string
==============
A gauge theory like electromagnetism is defined by a gauge field,
which associates a group element to each path in space time. For
infinitesimal paths, the group element is close to the identity, while
for longer paths the group element is the successive product of the
infinitesimal group elements along the way.

In electrodynamics, the group is U(1), unit complex numbers under
multiplication. For infinitesimal paths, the group element is  which
implies that for finite paths parametrized by , the group element is:

{{block indent|1=\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} \, ds
\right) = \exp \left( ie\int A\cdot dx \right) . }}

The map from paths to group elements is called the Wilson loop or the
holonomy, and for a U(1) gauge group it is the phase factor which the
wavefunction of a charged particle acquires as it traverses the path.
For a loop:

{{block indent|1=e \oint_{\partial D} A\cdot dx  = e \int_D (\nabla
\times A) \, dS = e \int_D B \, dS.}}

So that the phase a charged particle gets when going in a loop is the
magnetic flux through the loop. When a small solenoid has a magnetic
flux, there are interference fringes for charged particles which go
around the solenoid, or around different sides of the solenoid, which
reveal its presence.

But if all particle charges are integer multiples of , solenoids with
a flux of  have no interference fringes, because the phase factor for
any charged particle is . Such a solenoid, if thin enough, is
quantum-mechanically invisible. If such a solenoid were to carry a
flux of , when the flux leaked out from one of its ends it would be
indistinguishable from a monopole.

Dirac's monopole solution in fact describes an infinitesimal line
solenoid ending at a point, and the location of the solenoid is the
singular part of the solution, the Dirac string. Dirac strings link
monopoles and antimonopoles of opposite magnetic charge, although in
Dirac's version, the string just goes off to infinity. The string is
unobservable, so you can put it anywhere, and by using two coordinate
patches, the field in each patch can be made nonsingular by sliding
the string to where it cannot be seen.


Grand unified theories
========================
In a U(1) gauge group with quantized charge, the group is a circle of
radius . Such a U(1) gauge group is called compact. Any U(1) that
comes from a grand unified theory is compact - because only compact
higher gauge groups make sense. The size of the gauge group is a
measure of the inverse coupling constant, so that in the limit of a
large-volume gauge group, the interaction of any fixed representation
goes to zero.

The case of the U(1) gauge group is a special case because all its
irreducible representations are of the same size - the charge is
bigger by an integer amount, but the field is still just a complex
number - so that in U(1) gauge field theory it is possible to take the
decompactified limit with no contradiction. The quantum of charge
becomes small, but each charged particle has a huge number of charge
quanta so its charge stays finite. In a non-compact U(1) gauge group
theory, the charges of particles are generically not integer multiples
of a single unit. Since charge quantization is an experimental
certainty, it is clear that the U(1) gauge group of electromagnetism
is compact.

GUTs lead to compact U(1) gauge groups, so they explain charge
quantization in a way that seems logically independent from magnetic
monopoles. However, the explanation is essentially the same, because
in any GUT that breaks down into a U(1) gauge group at long distances,
there are magnetic monopoles.

The argument is topological:

# The holonomy of a gauge field maps loops to elements of the gauge
group. Infinitesimal loops are mapped to group elements
infinitesimally close to the identity.
# If you imagine a big sphere in space, you can deform an
infinitesimal loop that starts and ends at the north pole as follows:
stretch out the loop over the western hemisphere until it becomes a
great circle (which still starts and ends at the north pole) then let
it shrink back to a little loop while going over the eastern
hemisphere. This is called 'lassoing the sphere'.
# Lassoing is a sequence of loops, so the holonomy maps it to a
sequence of group elements, a continuous path in the gauge group.
Since the loop at the beginning of the lassoing is the same as the
loop at the end, the path in the group is closed.
# If the group path associated to the lassoing procedure winds around
the U(1), the sphere contains magnetic charge. During the lassoing,
the holonomy changes by the amount of magnetic flux through the
sphere.
# Since the holonomy at the beginning and at the end is the identity,
the total magnetic flux is quantized. The magnetic charge is
proportional to the number of windings , the magnetic flux through the
sphere is equal to . This is the Dirac quantization condition, and it
is a topological condition that demands that the long distance U(1)
gauge field configurations be consistent.
# When the U(1) gauge group comes from breaking a compact Lie group,
the path that winds around the U(1) group enough times is
topologically trivial in the big group. In a non-U(1) compact Lie
group, the covering space is a Lie group with the same Lie algebra,
but where all closed loops are contractible. Lie groups are
homogenous, so that any cycle in the group can be moved around so that
it starts at the identity, then its lift to the covering group ends at
, which is a lift of the identity. Going around the loop twice gets
you to , three times to , all lifts of the identity. But there are
only finitely many lifts of the identity, because the lifts can't
accumulate. This number of times one has to traverse the loop to make
it contractible is small, for example if the GUT group is SO(3), the
covering group is SU(2), and going around any loop twice is enough.
# This means that there is a continuous gauge-field configuration in
the GUT group allows the U(1) monopole configuration to unwind itself
at short distances, at the cost of not staying in the U(1). To do this
with as little energy as possible, you should leave only the U(1)
gauge group in the neighborhood of one point, which is called the core
of the monopole. Outside the core, the monopole has only magnetic
field energy.

Hence, the Dirac monopole is a topological defect in a compact U(1)
gauge theory. When there is no GUT, the defect is a singularity - the
core shrinks to a point. But when there is some sort of short-distance
regulator on space time, the monopoles have a finite mass. Monopoles
occur in lattice U(1), and there the core size is the lattice size. In
general, they  are expected to occur whenever there is a
short-distance regulator.

In more recent years, a new class of theories has also suggested the
existence of  magnetic monopoles.

During the early 1970s, the successes of quantum field theory and
gauge theory in the development of electroweak theory and the
mathematics of the strong nuclear force led many theorists to move on
to attempt to combine them in a single theory known as a Grand Unified
Theory (GUT). Several GUTs were proposed, most of which implied the
presence of a real magnetic monopole particle. More accurately, GUTs
predicted a range of particles known as dyons, of which the most basic
state was a monopole. The charge on magnetic monopoles predicted by
GUTs is either 1 or 2 'gD', depending on the theory.

The majority of particles appearing in any quantum field theory are
unstable, and they decay into other particles in a variety of
reactions that must satisfy various conservation laws. Stable
particles are stable because there are no lighter particles into which
they can decay and still satisfy the conservation laws. For instance,
the electron has a lepton number of one and an electric charge of one,
and there are no lighter particles that conserve these values. On the
other hand, the muon, essentially a heavy electron, can decay into the
electron plus two quanta of energy, and hence it is not stable.

The dyons in these GUTs are also stable, but for an entirely different
reason. The dyons are expected to exist as a side effect of the
"freezing out" of the conditions of the early universe, or a symmetry
breaking. In this scenario, the dyons arise due to the configuration
of the vacuum in a particular area of the universe, according to the
original Dirac theory. They remain stable not because of a
conservation condition, but because there is no simpler 'topological'
state into which they can decay.

The length scale over which this special vacuum configuration exists
is called the 'correlation length' of the system. A correlation length
cannot be larger than causality would allow, therefore the correlation
length for making magnetic monopoles must be at least as big as the
horizon size determined by the metric of the expanding universe.
According to that logic, there should be at least one magnetic
monopole per horizon volume as it was when the symmetry breaking took
place.

Cosmological models of the events following the Big Bang make
predictions about what the horizon volume was, which lead to
predictions about present-day monopole density. Early models predicted
an enormous density of monopoles, in clear contradiction to the
experimental evidence. This was called the "monopole problem". Its
widely accepted resolution was not a change in the particle-physics
prediction of monopoles, but rather in the cosmological models used to
infer their present-day density. Specifically, more recent theories of
cosmic inflation drastically reduce the predicted number of magnetic
monopoles, to a density small enough to make it unsurprising that
humans have never seen one. This resolution of the "monopole problem"
was regarded as a success of cosmic inflation theory. (However, of
course, it is only a noteworthy success if the particle-physics
monopole prediction is correct.) For these reasons, monopoles became a
major interest in the 1970s and 80s, along with the other
"approachable" predictions of GUTs such as proton decay.

Many of the other particles predicted by these GUTs were beyond the
abilities of current experiments to detect. For instance, a wide class
of particles known as the X and Y bosons are predicted to mediate the
coupling of the electroweak and strong forces, but these particles are
extremely heavy and well beyond the capabilities of any reasonable
particle accelerator to create.


String theory
===============
In the universe, quantum gravity provides the regulator. When gravity
is included, the monopole singularity can be a black hole, and for
large magnetic charge and mass, the black hole mass is equal to the
black hole charge, so that the mass of the magnetic black hole is not
infinite. If the black hole can decay completely by Hawking radiation,
the lightest charged particles cannot be too heavy. The lightest
monopole should have a mass less than or comparable to its charge in
natural units.

So in a consistent holographic theory, of which string theory is the
only known example, there are always finite-mass monopoles. For
ordinary electromagnetism, the upper mass bound is not very useful
because it is about same size as the Planck mass.


Mathematical formulation
==========================
In mathematics, a (classical) gauge field is defined as a connection
over a principal G-bundle over spacetime.  is the gauge group, and it
acts on each fiber of the bundle separately.

A 'connection' on a -bundle tells you how to glue fibers together at
nearby points of . It starts with a continuous symmetry group  that
acts on the fiber , and then it associates a group element with each
infinitesimal path. Group multiplication along any path tells you how
to move from one point on the bundle to another, by having the
element associated to a path act on the fiber .

In mathematics, the definition of bundle is designed to emphasize
topology, so the notion of connection is added on as an afterthought.
In physics, the connection is the fundamental physical object. One of
the fundamental observations in the theory of characteristic classes
in algebraic topology is that many homotopical structures of
nontrivial principal bundles may be expressed as an integral of some
polynomial over any connection over it. Note that a connection over a
trivial bundle can never give us a nontrivial principal bundle.

If spacetime is  the space of all possible connections of the -bundle
is connected. But consider what happens when we remove a timelike
worldline from spacetime. The resulting spacetime is homotopically
equivalent to the topological sphere .

A principal -bundle over  is defined by covering  by two charts, each
homeomorphic to the open 2-ball such that their intersection is
homeomorphic to the strip . 2-balls are homotopically trivial and the
strip is homotopically equivalent to the circle .  So a topological
classification of the possible connections is reduced to classifying
the transition functions. The transition function maps the strip to ,
and the different ways of mapping a strip into  are given by the first
homotopy group of .

So in the -bundle formulation, a gauge theory admits Dirac monopoles
provided  is not simply connected, whenever there are paths that go
around the group that cannot be deformed to a constant path (a path
whose image consists of a single point). U(1), which has quantized
charges, is not simply connected and can have Dirac monopoles while ,
its universal covering group, is simply connected, doesn't have
quantized charges and does not admit Dirac monopoles. The mathematical
definition is equivalent to the physics definition provided
that�following Dirac�gauge fields are allowed that are defined only
patch-wise, and the gauge field on different patches are glued after a
gauge transformation.

The total magnetic flux is none other than the first Chern number of
the principal bundle, and depends only upon the choice of the
principal bundle, and not the specific connection over it. In other
words, it is a topological invariant.

This argument for monopoles is a restatement of the lasso argument for
a pure U(1) theory. It generalizes to  dimensions with  in several
ways. One way is to extend everything into the extra dimensions, so
that U(1) monopoles become sheets of dimension . Another way is to
examine the type of topological singularity at a point with the
homotopy group .


                  Searches for magnetic monopoles
======================================================================
Experimental searches for magnetic monopoles can be placed in one of
two categories: those that try to detect preexisting magnetic
monopoles and those that try to create and detect new magnetic
monopoles.

Passing a magnetic monopole through a coil of wire induces a net
current in the coil. This is not the case for a magnetic dipole or
higher order magnetic pole, for which the net induced current is zero,
and hence the effect can be used as an unambiguous test for the
presence of magnetic monopoles. In a wire with finite resistance, the
induced current quickly dissipates its energy as heat, but in a
superconducting loop the induced current is long-lived. By using a
highly sensitive "superconducting quantum interference device" (SQUID)
one can, in principle, detect even a single magnetic monopole.

According to standard inflationary cosmology, magnetic monopoles
produced before inflation would have been diluted to an extremely low
density today. Magnetic monopoles may also have been produced
thermally after inflation, during the period of reheating. However,
the current bounds on the reheating temperature span 18 orders of
magnitude and as a consequence the density of magnetic monopoles today
is not well constrained by theory.

There have been many searches for preexisting magnetic monopoles.
Although there have been tantalizing events recorded, in particular
the event recorded by Blas Cabrera Navarro on the night of February
14, 1982 (thus, sometimes referred to as the "Valentine's Day
Monopole"), there has never been reproducible evidence for the
existence of magnetic monopoles. The lack of such events places an
upper limit on the number of monopoles of about one monopole per 1029
nucleons.

Another experiment in 1975 resulted in the announcement of the
detection of a moving magnetic monopole in cosmic rays by the team led
by P. Buford Price. Price later retracted his claim, and a possible
alternative explanation was offered by Alvarez. In his paper it was
demonstrated that the path of the cosmic ray event that was claimed
due to a magnetic monopole could be reproduced by the path followed by
a platinum nucleus decaying first to osmium, and then to tantalum.

High energy particle colliders have been used to try to create
magnetic monopoles. Due to the conservation of magnetic charge,
magnetic monopoles must be created in pairs, one north and one south.
Due to conservation of energy, only magnetic monopoles with masses
less than half of the center of mass energy of the colliding particles
can be produced. Beyond this, very little is known theoretically about
the creation of magnetic monopoles in high energy particle collisions.
This is due to their large magnetic charge, which invalidates all the
usual calculational techniques. As a consequence, collider based
searches for magnetic monopoles cannot, as yet, provide lower bounds
on the mass of magnetic monopoles. They can however provide upper
bounds on the probability (or cross section) of pair production, as a
function of energy.

The ATLAS experiment at the Large Hadron Collider currently has the
most stringent cross section limits for magnetic monopoles of 1 and 2
Dirac charges, produced through Drell-Yan pair production. The search
uses the fact that magnetic monopoles are long lived (they don't
quickly decay), as well as their characteristic interaction with
matter, which is highly ionizing, and so they would leave a very
specific signal in the detector. In 2019 the search for magnetic
monopoles in the ATLAS detector reported its first results from data
collected from the LHC Run 2 collisions at center of mass energy of 13
TeV, which at 34.4 fb-1 is the largest dataset analyzed to date.

The MoEDAL experiment, installed at the Large Hadron Collider, is
currently searching for magnetic monopoles and large supersymmetric
particles using nuclear track detectors and aluminum bars around
LHCb's VELO detector. The particles it is looking for damage the
plastic sheets that comprise the nuclear track detectors along their
path, with various identifying features. Further, the aluminum bars
can trap sufficiently slowly moving magnetic monopoles. The bars can
then be analyzed by passing them through a SQUID.

The Russian astrophysicist Igor Novikov claims the fields of
macroscopic black holes are potential magnetic monopoles, representing
the entrance to an Einstein-Rosen bridge.


              "Monopoles" in condensed-matter systems
======================================================================
Since around 2003, various condensed-matter physics groups have used
the term "magnetic monopole" to describe a different and largely
unrelated phenomenon.

A true magnetic monopole would be a new elementary particle, and would
violate Gauss's law for magnetism . A monopole of this kind, which
would help to explain the law of charge quantization as formulated by
Paul Dirac in 1931, has never been observed in experiments.

The monopoles studied by condensed-matter groups have none of these
properties. They are not a new elementary particle, but rather are an
emergent phenomenon in systems of everyday particles (protons,
neutrons, electrons, photons); in other words, they are
quasi-particles. They are not sources for the -field (i.e., they do
not violate ); instead, they are sources for other fields, for example
the -field, the "-field" (related to superfluid vorticity), or various
other quantum fields. They are not directly relevant to grand unified
theories or other aspects of particle physics, and do not help explain
charge quantization�except insofar as studies of analogous situations
can help confirm that the mathematical analyses involved are sound.

There are a number of examples in condensed-matter physics where
collective behavior leads to emergent phenomena that resemble magnetic
monopoles in certain respects, including most prominently the spin ice
materials. While these should not be confused with hypothetical
elementary monopoles existing in the vacuum, they nonetheless have
similar properties and can be probed using similar techniques.

Some researchers use the term magnetricity to describe the
manipulation of magnetic monopole quasiparticles in spin ice, in
analogy to the word "electricity".

One example of the work on magnetic monopole quasiparticles is a paper
published in the journal 'Science' in September 2009, in which
researchers described the observation of quasiparticles resembling
magnetic monopoles. A single crystal of the spin ice material
dysprosium titanate was cooled to a temperature between 0.6 kelvin and
2.0 kelvin. Using observations of neutron scattering, the magnetic
moments were shown to align into interwoven tubelike bundles
resembling Dirac strings. At the defect formed by the end of each
tube, the magnetic field looks like that of a monopole. Using an
applied magnetic field to break the symmetry of the system, the
researchers were able to control the density and orientation of these
strings. A contribution to the heat capacity of the system from an
effective gas of these quasiparticles was also described.


This research went on to win the 2012 Europhysics Prize for condensed
matter physics.

In another example, a paper in the February 11, 2011 issue of 'Nature
Physics' describes creation and measurement of long-lived magnetic
monopole quasiparticle currents in spin ice. By applying a
magnetic-field pulse to crystal of dysprosium titanate at 0.36�K, the
authors created a relaxing magnetic current that lasted for several
minutes. They measured the current by means of the electromotive force
it induced in a solenoid coupled to a sensitive amplifier, and
quantitatively described it using a chemical kinetic model of
point-like charges obeying the Onsager-Wien mechanism of carrier
dissociation and recombination. They thus derived the microscopic
parameters of monopole motion in spin ice and identified the distinct
roles of free and bound magnetic charges.

In superfluids, there is a field , related to superfluid vorticity,
which is mathematically analogous to the magnetic -field. Because of
the similarity, the field  is called a "synthetic magnetic field". In
January 2014, it was reported that monopole quasiparticles for the
field were created and studied in a spinor Bose-Einstein condensate.
This constitutes the first example of a quasi-magnetic monopole
observed within a system governed by quantum field theory.


                              See also
======================================================================
* Bogomolny equations
* Dirac string
* Dyon
* Felix Ehrenhaft
* Flatness problem
* Gauss's law for magnetism
* Ginzburg-Landau theory
* Halbach array
* Horizon problem
* Instanton
* Magnetic monopole problem
* Meron
* Soliton
* 't Hooft-Polyakov monopole
* Wu-Yang monopole


                           External links
======================================================================
* [https://arxiv.org/abs/hep-ex/0302011 Magnetic Monopole Searches
(lecture notes)]
* [http://pdg.lbl.gov/2004/listings/s028.pdf Particle Data Group
summary of magnetic monopole search]
* [http://www.vega.org.uk/video/programme/56 'Race for the Pole' Dr
David Milstead] Freeview 'Snapshot' video by the Vega Science Trust
and the BBC/OU.
*
[http://www.drillingsraum.com/magnetic_monopole/magnetic_monopole.html
Interview with Jonathan Morris] about magnetic monopoles and magnetic
monopole quasiparticles. Drillingsraum, April 16, 2010
* [http://www.nature.com/news/2009/090903/full/news.2009.881.html
'Nature', 2009]
* [https://www.sciencedaily.com/releases/2009/09/090903163725.htm
'Sciencedaily', 2009]
*
*


License
=========
All content on Gopherpedia comes from Wikipedia, and is licensed under CC-BY-SA
License URL: http://creativecommons.org/licenses/by-sa/3.0/
Original Article: http://en.wikipedia.org/wiki/Magnetic_monopole

You are viewing proxied material from sdf.org. The copyright of proxied material belongs to its original authors. Any comments or complaints in relation to proxied material should be directed to the original authors of the content concerned. Please see the disclaimer for more details.