Magnetic monopole
A magnetic monopole is a hypothetical elementary
particle in particle physics that is an isolated
magnet with only one magnetic pole . In more
technical terms, 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
does not arise from magnetic monopoles, and
in fact there is no conclusive experimental
evidence that magnetic monopoles exist at
all in the universe.
Some condensed matter systems contain effective
(non-isolated) magnetic monopole quasi-particles,
or contain phenomena that are mathematically
analogous to magnetic monopoles.
Historical background
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
by Ampère's circuital law, not magnetic monopole
fluids. 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 A.M. 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. 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 be created 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
continue to be an area of active research.
(See "Monopoles" in condensed-matter systems
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 a superposition (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 nonzero.
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 ρm, there will also be a "magnetic current
density" variable in the equations, jm.
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 ∇⋅B = 0 (where ∇⋅ is divergence
and B is the magnetic B field).
In Gaussian cgs units
The extended Maxwell's equations are as follows,
in Gaussian cgs units:
In these equations ρm is the magnetic charge
density, jm is the magnetic current density,
and qm is the magnetic charge of a test particle,
all defined analogously to the related quantities
of electric charge and current; v is the particle's
velocity and c 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 c.
In SI units
In SI units, there are two conflicting units
in use for magnetic charge qm: webers (Wb)
and ampere·meters (A·m). The conversion
between them is qm(Wb) = μ0qm(A·m), since
the units are 1 Wb = 1 H·A = (1 H·m−1)·(1
A·m) 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):
Tensor formulation
Maxwell's equations in the language of tensors
makes Lorentz covariance clear. The generalized
equations are:
where
F is the electromagnetic tensor, denotes the
Hodge dual, (so ∗F is the dual tensor to
F),
for a particle with electric charge qe and
magnetic charge qm; v is the four-velocity
and p the four-momentum,
for an electric and magnetic charge distribution;
Je = (ρe, je) is the electric four-current
and Jm = (ρm, jm) 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:
However, this formula is inadequate for a
particle that has both electric and magnetic
charge, and we must add a term involving another
potential "P".
This formula for the fields is often called
the Cabibbo-Ferrari relation, though Shanmugadhasan
proposed it earlier. The quantity εαβγδ
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 ξ, and simultaneously
change the fields and charges everywhere in
the universe as follows (in Gaussian units):
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 ξ = π/2 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 to be 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 qeqm,
and independent of the distance between them.
Quantum mechanics dictates, however, that
angular momentum is quantized in units of
ħ, so therefore the product qeqm 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 qm / r 2 and is directed in the radial
direction, located at the origin. Because
the divergence of B is equal to zero almost
everywhere, except for the locus of the magnetic
monopole at r = 0, one can locally define
the vector potential such that the curl of
the vector potential A equals the magnetic
field B.
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 z greater than 0 above the particle),
and another set of functions for the "southern
hemisphere". These two vector potentials are
matched at the "equator" (the plane z = 0
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 qe of the probe, as well as
to the magnetic charge qm 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 φ of its wave function exp(iφ) must
be unchanged, which implies that the phase
φ added to the wave function must be a multiple
of 2π:
where ε0 is the vacuum permittivity, ħ = h/2π
is the reduced Planck's constant, c is the
speed of light, and ℤ 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 below—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 will have magnetic monopoles anyway.)
If we maximally extend the definition of the
vector potential for the southern hemisphere,
it will be 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.
Topological interpretation
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
1 + iAμdxμ which implies that for finite
paths parametrized by s, the group element
is:
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:
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 e, solenoids with a flux of 2π/e have
no interference fringes, because the phase
factor for any charged particle is e2πi = 1.
Such a solenoid, if thin enough, is quantum-mechanically
invisible. If such a solenoid were to carry
a flux of 2π/e, 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 2π/e. Such
a U(1) gauge group is called compact. Any
U(1) which 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 to be logically independent from
magnetic monopoles. However, the explanation
is essentially the same, because in any GUT
which 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 which 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 N,
the magnetic flux through the sphere is equal
to 2πN/e. This is the Dirac quantization
condition, and it is a topological condition
which 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 which 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
P, which is a lift of the identity. Going
around the loop twice gets you to P2, three
times to P3, 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). In order 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.
String theory
In our 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 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. G is the gauge group,
and it acts on each fiber of the bundle separately.
A connection on a G bundle tells you how to
glue F's together at nearby points of M. It
starts with a continuous symmetry group G
which acts on the fiber F, 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 G element associated
to a path act on the fiber F.
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 space time is R4 the space of all possible
connections of the G-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 S2.
A principal G-bundle over S2 is defined by
covering S2 by two charts, each homeomorphic
to the open 2-ball such that their intersection
is homeomorphic to the strip S1×I. 2-balls
are homotopically trivial and the strip is
homotopically equivalent to the circle S1.
So a topological classification of the possible
connections is reduced to classifying the
transition functions. The transition function
maps the strip to G, and the different ways
of mapping a strip into G are given by the
first homotopy group of G.
So in the G-bundle formulation, a gauge theory
admits Dirac monopoles provided G 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 R, 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 which 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's a topological invariant.
This argument for monopoles is a restatement
of the lasso argument for a pure U(1) theory.
It generalizes to d + 1 dimensions with
d ≥ 2 in several ways. One way is to extend
everything into the extra dimensions, so that
U(1) monopoles become sheets of dimension
d − 3. Another way is to examine the type
of topological singularity at a point with
the homotopy group πd − 2(G).
Grand unified theories
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.
Searches for magnetic monopoles
A number of attempts have been made to detect
magnetic monopoles. One of the simpler ones
is to use a loop of superconducting wire to
look for even tiny magnetic sources, a so-called
"superconducting quantum interference device",
or SQUID. Given the predicted density, loops
small enough to fit on a lab bench would expect
to see about one monopole event per year.
Although there have been tantalizing events
recorded, in particular the event recorded
by Blas Cabrera 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 a 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 to be due to a magnetic
monopole could be reproduced by the path followed
by a platinum nucleus decaying first to osmium,
and then to tantalum.
Other experiments rely on the strong coupling
of monopoles with photons, as is the case
for any electrically-charged particle as well.
In experiments involving photon exchange in
particle accelerators, monopoles should be
produced in reasonable numbers, and detected
due to their effect on the scattering of the
photons. The probability of a particle being
created in such experiments is related to
their mass – with heavier particles being
less likely to be created – so by examining
the results of such experiments, limits on
the mass of a magnetic monopole can be calculated.
The most recent such experiments suggest that
monopoles with masses below 600 GeV/c2 do
not exist, while upper limits on their mass
due to the very existence of the universe
– which would have collapsed by now if they
were too heavy – are about 1017  GeV/c2.
The MoEDAL experiment, installed at the Large
Hadron Collider, is currently searching for
magnetic monopoles and large supersymmetric
particles using layers of special plastic
sheets attached to the walls around LHCb's
VELO detector. The particles it is looking
for will damage the sheets along their path,
with various identifying features.
"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 the law ∇⋅B = 0.
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 B-field (i.e.,
they do not violate the law ∇⋅B = 0);
instead, they are sources for other fields,
for example the H-field, or the "B*-field"
(related to superfluid vorticity) 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 Jonathan Morris and Alan Tennant
from the Helmholtz-Zentrum Berlin für Materialien
und Energie (HZB) along with Santiago Grigera
from Instituto de Física de Líquidos y Sistemas
Biológicos (IFLYSIB, CONICET) and other colleagues
from Dresden University of Technology, University
of St. Andrews and Oxford University 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.
Another example is a paper in the February
11, 2011 issue of Nature Physics which 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 B*, related
to superfluid vorticity, which is mathematically
analogous to the magnetic B-field. Because
of the similarity, the field B* is called
a "synthetic magnetic field". In January 2014,
it was reported that monopole quasiparticles
for the B* field were created and studied
in a spinor Bose–Einstein condensate. This
constitutes the first example of a magnetic
monopole observed within a system governed
by quantum field theory.
Further descriptions in particle physics
In physics the phrase "magnetic monopole"
usually denoted a Yang–Mills potential A
and Higgs field ϕ whose equations of motion
are determined by the Yang–Mills action
In mathematics, the phrase customarily refers
to a static solution to these equations in
the Bogomolny–Parasad–Sommerfeld limit
λ → ϕ which realizes, within topological
class, the absolutes minimum of the functional
This means that it in a connection A on a
principal G-bundle over R3 (c.f. also Connections
on a manifold; principal G-object) and a section
ϕ of the associated adjoint bundle of Lie
algebras such that the curvature FA and covariant
derivative DA ϕ satisfy the Bogomolny equations
and the boundary conditions.
Pure mathematical advances in the theory of
monopoles from the 1980s onwards have often
proceeded on the basis of physically motived
questions.
The equations themselves are invariant under
gauge transformation and orientation-preserving
symmetries. When γ is large, ϕ/||ϕ|| defines
a mapping from a 2-sphere of radius γ in
R3 to an adjoint orbit G/k and the homotopy
class of this mapping is called the magnetic
charge. Most work has been done in the case
G = SU(2), where the charge is a positive
integer k. The absolute minimum value of the
functional is then 8πk and the coefficient
m in the asymptotic expansion of ϕ/||ϕ||
is k/2.
The first SU(2) solution was found by E. B.
Bogomolny, J. K. Parasad and C. M. Sommerfield
in 1975. It is spherically symmetric of charge
1 and has the form
In 1980, C.H.Taubes showed by a gluing construction
that there exist solutions for all large k
and soon after explicit axially-symmetric
solutions were found. The first exact solution
in the general case was given in 1981 by R.S.Ward
for k = 2 in terms of elliptic function.
There are two ways of solving the Bogomolny
equations. The first is by twistor methods.
In the formulation of N.J. Hitchin, an arbitrary
solution corresponds to a holomorphic vector
bundle over the complex surface TP1, the tangent
bundle of the projective line. This is naturally
isomorphic to the space of oriented straight
lines in R3.
The boundary condition show that the holomorphic
bundle is an extension of line bundles determined
by a compact algebraic curve of genus (k − 1)2
(the spectral curve) in TP1, satisfying certain
constraints.
The second method, due to W.Nahm, involves
solving an eigen value problem for the coupled
Dirac operator and transforming the equations
with their boundary conditions into a system
of ordinary differential equations, the Nahm
equations.
where Ti(s) is a k×k -matrix valued function
on (0,2).
Both constructions are based on analogous
procedures for instantons, the key observation
due to N.S.Manton being of the self-dual Yang–Mills
equations (c.f. also Yang–Mills field) in
R4.
The equivalence of the two methods for SU(2)
and their general applicability was established
in (see also). Explicit formulas for A and
are difficult to obtain by either method,
despite some exact solutions of Nahm's equations
in symmetric situations.
The case of a more general Lie group G, where
the stabilizer of ϕ at infinity is a maximal
torus, was treated by M.K.Murray from the
twistor point of view, where the single spectral
curve of an SU(2)-monopole is replaced by
a collection of curves indexed by the vortices
of the Dynkin diagram of G. The corresponding
Nahm construction was designed by J.Hustubise
and Murray.
The moduli space (c.f. also Moduli theory)
of all SU(2) monopoles of charge k up to gauge
equivalence was shown by Taubes to be a smooth
non-compact manifold of dimension 4k − 1.
Restricting to gauge transformations that
preserve the connection at infinity gives
a 4k-dimensional manifold Mk, which is a circle
bundle over the true moduli space and carries
a natural complete hyperKähler metric (c.f.
also Kähler–Einstein manifold). With suspected
to any of the complex structures of the hyper-Kähler
family, this manifold is holomorphically equivalent
to the space of based rational mapping of
degree k from P1 to itself.
The metric is known in twistor terms, and
its Kähler potential can be written using
the Riemann theta functions of the spectral
curve, but only the case k = 2 is known in
a more conventional and usable form (as of
2000). This Atiyah–Hitchin manifold, the
Einstein Taub-NUT metric and R4 are the only
4-dimensional complete hyperKähler manifolds
with a non-triholomorphic SU(2) action. Its
geodesics have been studied and a programme
of Manton concerning monopole dynamics put
into effect. Further dynamical features have
been elucidated by numerical and analytical
techniques.
A cyclic k-fold conering of Mk splits isometrically
is a product, where is the space of strongly
centred monopoles. This space features in
an application of S-duality in theoretical
physics, and in G.B.Segal and A.Selby studied
its topology and the L2 harmonic forms defined
on it, partially confirming the physical prediction.
Magnetic monopole on hyperbolic three-space
were investigated from the twistor point of
view b M.F.Atiyah (replacing the complex surface
TP1 by the comoplement of the anti-diagonal
in P1 × P1) and in terms of discrete Nahm
equations by Murray and M.A.Singer.
