Magnetic monopole

From Wikipedia, the free encyclopedia

In physics, a magnetic monopole is a hypothetical particle that may be loosely described as "a magnet with only one pole" (see electromagnetic theory for more on magnetic poles). In more accurate terms, it would have net magnetic charge. Interest in the concept stems from particle theories, notably Grand Unified Theories and superstring theories that predict either the existence or the possibility of magnetic monopoles.

1 Background
2 Maxwell's Equations
3 Dirac's quantization
4 Mathematical approach to Dirac monopole
5 Grand Unified Theories
6 Attempts to find monopoles
7 See also
8 References
9 External links

[edit] Background

Image:Question dropshade.svg

Unsolved problems in physics: Are there any particles that carry "magnetic charge", and if so, why are they so difficult to detect?

Magnets exert forces on one another; similar to electric charges, like poles will repel each other and unlike poles will attract. When a magnet, that is, an object conventionally described as having a north and a south pole, is cut in half across the axis joining those "poles", the resulting pieces are two normal (albeit smaller) magnets each with its own north pole and south pole, rather than two separate north-only and south-only pieces.

The modern understanding of magnetism posits that all magnetic effects are actually due to the motion of charged particles; that is, all magnets are in fact electromagnets. The magnetic force is actually due to the finite speed of a disturbance of the electric field, the speed of light, which gives rise to forces that appear to be acting along a line at right angles to the charges. In effect, the magnetic force is the portion of the electric force directed to where the charge used to be.

Even atoms have a tiny field. In the Bohr model of an atom, the electrons orbit the nucleus and thus have a change in motion giving rise to a magnetic field. Permanent magnets have measurable magnetic fields because the atoms (and molecules) are arranged in a way that their individual tiny fields align and add up.

In this model, the lack of a single pole makes intuitive sense; cutting a bar magnet in half does nothing to the arrangement of the molecules within, and you end up with two bars with the same arrangement, and thus the same field. This also explains how heating or simply hitting a magnet made from a soft material will degauss it, as the molecules within are moved about.

Since all known forms of magnetic phenomena involve the motion of electrically charged particles, and since no theory suggests that "pole" is, in that context, a thing rather than a convenient fiction, it may well be that nothing that could be called a magnetic monopole exists or ever did or could.

[edit] Maxwell's Equations

Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The equations are very nearly symmetric under interchange of electric and magnetic field; in fact symmetric equations could be written if one allowed for the possibility of "magnetic charges" exactly analogous to the observed electric charges.

With the inclusion of these magnetic charges, say <math>\rho_m</math>, there will also be "magnetic current", <math>J_m</math>. The new Maxwell's equations would change as follows.

Name	Without Magnetic Monopoles	With Magnetic Monopoles
Gauss's law:	<math>\nabla \cdot \mathbf{D} = \rho_e</math>	<math>\nabla \cdot \mathbf{D} = \rho_e</math>
Gauss' law for magnetism:	<math>\nabla \cdot \mathbf{B} = 0</math>	<math>\nabla \cdot \mathbf{B} = \rho_m</math>
Faraday's law of induction:	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}</math>	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} - \mathbf{J}_m</math>
Ampère's law (with Maxwell's extension):	<math>\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}} {\partial t} + \mathbf{J}_e</math>	<math>\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}} {\partial t} + \mathbf{J}_e</math>

If indeed magnetic monopoles do exist, the inclusion of the magnetic current would validate the monopoles' existence as well as maintain the validity of Maxwell's Equations at the macroscopic level.

When no magnetic charges are present in a region, these symmetric equations reduce to the conventional equations of electromagnetism, that is, ( ∇·B = 0 ).

So, classically, the question is "Why does the magnetic charge always seem to be zero?" Nothing in Maxwell's equations suggests that it has to be, nor has any other development based on these original formulations. This has been a curiosity for a long time, but it has become more of a problem in recent years, when new theories of physics seem to predict the existence of magnetic monopoles.

[edit] 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 is simply "inserted" into QM, but in 1931 Dirac showed that a discrete charge naturally "falls out" of QM. This is because a system consisting of a single stationary electric monopole (an electron, say) and a single stationary magnetic monopole would have angular momentum. The magnitude of the angular momentum would depend only on the product of the electric charge (e) and "magnetic charge" (g) and would be independent of the distance between them. Quantum mechanics requires that angular momentum be quantized in units of ħ. This means that if even a single magnetic monopole existed in the universe, all electric charges would then be quantized.

Dirac considered a point-like magnetic charge whose magnetic field behaves as <math>\mu/r^2</math> and is directed in the radial direction. Because the divergence of <math>B</math> is equal to zero almost everywhere, except for the locus of the magnetic monopole at <math>r=0</math>, one can locally define the vector potential such that the curl of the vector potential <math>A</math> equals the magnetic field <math>B</math>.

However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the delta function at the origin. We must define one set of functions for the vector potential on the Northern hemisphere, and another set of functions for the Southern hemispheres. These two vector potentials are matched at the equator, and they differ by a gauge transformation. The wave function of an electrically charged particle (a probe) that orbits the equator generally changes by a phase, much like in the Aharonov-Bohm effect. This phase is proportional to the electric charge <math>q_e</math> of the probe, as well as to the magnetic charge <math>q_m</math> 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 <math>\exp(i\phi)</math> of its wave function must be unchanged, which implies that the phase <math>\phi</math> added to the wave function must be a multiple of <math>2\pi</math>:

This is known as the Dirac quantization condition. In certain units, the condition is exactly given by the relation above. 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 inverse 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.

[edit] Mathematical approach to Dirac monopole

Classically, gauge theory is described by a connection over a principal G-bundle over spacetime. Ordinary spacetime has the topology of R⁴, which is topologically trivial. So, the space of all possible connections over the principal G-bundle is connected. But let's see what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S². So, it suffices to classify the connected components of the space of all connections over a principal G-bundle over S². To do this, consider 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, which is given by the first homotopy group of G. In other words, a gauge theory can only admit Dirac monopoles provided G is not simply connected. For instance, 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 even in principle.

It is easy to see how this argument generalizes to d + 1 dimensions with <math>d\geq 2</math>. We look at the homotopy group π_d−2(G).

[edit] Grand Unified Theories

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

In the early 1970s, the successes of quantum field theory and gauge theory in the development of electroweak and 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, or GUT. Several GUTs were proposed, most of which had the curious feature of suggesting 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 is a monopole. The charge on magnetic monopoles predicted by GUTs is either 1 or 2gD, depending on the theory.

The majority of particles appearing in any quantum field theory are unstable, and decay into other particles in a variety of reactions that have to conserve various values. Stable particles are stable because there are no lighter particles to decay into that still conserve these values. For instance, the electron has a lepton number of 1 and an electric charge of 1, 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 and is therefore not stable.

The dyons in these same theories 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 symmetry breaking. In this model the dyons arise due to the vacuum configuration 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 for them to decay to.

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.

This leads to a direct prediction of the amount of monopoles in the universe today, which is about 10¹¹ times the critical density of our universe. The universe appears to be close to critical density, so monopoles should be fairly common.

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 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. Monopoles, on the other hand, are expected to be stable and have a "reasonable" mass, and therefore should be detectable not only in modern accelerators, but also "in the wild" by looking for them in appropriate detectors.

For this reason, monopoles became a major interest in the 1970s and 80s, along with the other "approachable" prediction of GUTs, proton decay.

[edit] Attempts to find monopoles

A number of attempts have been made to detect magnetic monopoles. One of the simplest is to use a loop of superconducting wire that can look for even tiny magnetic sources, a so-called "superconducting quantum interference detector", or SQUID. Given the predicted density, loops the size of a soup can 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 1 monopole per 10²⁹ nucleons.

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 -- heavier particles are less likely to be created -- so by examining such experiments limits on the mass 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 existence of the universe (which would have collapsed by now if they were too heavy) is about 10¹⁷ GeV/c².

Non-inflationary Big Bang cosmology suggests that monopoles should be plentiful, and the failure to find magnetic monopoles is one of the main problems that led to the creation of cosmic inflation theory. In inflation, the visible universe was much smaller in the period before inflation, and despite the very short time before inflation, it would have been small enough for the whole visible universe to have been within the horizon, and thus not requiring many monopoles, perhaps only one. At the moment, versions of inflation seem to be the most likely cosmological theories.

Interestingly, the other major prediction of GUTs, proton decay, has also not been observed. The absence of these two key pieces of evidence has generally led to a decline in work on "classic" GUTs, and the introduction of more "radical" proposals, such as superstrings.