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| | You consider Santa Claus and decorated trees when you think Christmas, proper? The Jolly 1 probably comes 1st as the number one symbol of the vacation, but there is no mistaking that Christmas trees are nearly just as essential when it comes to celebrating the vacation proper.<br><br>Santa Claus and Christmas trees are so intertwined, so connected, in how we celebrate, it makes 1 feel that Santa maybe even invented the Christmas tree. After all, it is Santa who locations everyone"s presents underneath the tree. Perhaps back in the old days, just before trees, he utilised to place presents beneath children"s beds and in the bathtub. And perhaps sooner or later he got tired of it, and rather invented Christmas trees as a practical and fun place for him to leave presents. Tends to make sense, proper?<br><br>Maybe, but Santa had small to do with the actual cause that Christmas trees came about. Sorry to disappoint you people, but it was really German Christians who started the Christmas tree celebration, way back in the 1500s. To explore more, consider looking at: [http://worncanvas.com/topic.php?id=15126&replies=1 partner sites]. And as for decorating these trees, the story goes that 1 of the leading Christians of them all, Martin Luther, began decorating his family"s tree in the 1500s with lit candles. He got the thought one evening walking property beneath the stars.<br><br>Then the thought truly caught on in the mid-1800s, when the Queen of England and her children began decorating a tree for Christmas. Sketches of the royal family with their decorations got place in all the significant newspapers, and soon every single very good Englishman was beginning a new loved ones tradition: decorating their home and home with colorful decorations and freshly smelling, beautifully green evergreen trees.<br><br>In America, however, we have been a small bit slow, not just with decorating Christmas trees but with celebrating Santa Claus and something else "extra" in their Christmas festivities. We discovered [http://cfcl.com/SemanticMediaWiki/index.php?title=BrittenyGrover777 needs] by searching books in the library. That is since up till the extremely finish of the 1800s, Americans saw Christmas as a very religious vacation. There was no room for chubby males in red suits, or bright decorations and candles, and even a Christmas tree. All Americans did back then was go to church service. Learn supplementary info on [http://www.zhangxiaodong.net/wiki/index.php/Repairing_Pinhole_Leaks_in_Copper_Pipes principles] by visiting our dynamite link. Period.<br><br>But then the Queen of England enjoyed her Christmas tree. And numerous immigrants to the United States, particularly the German ones, celebrated December 25 with trees, lights, and vacation cheer. Ultimately, the fun caught on. In true American fashion, although, Americans not only took on the tradition of Christmas trees and decorations, they enhanced on it. Each and every point had to be larger, greater, and more festive!<br><br>For example, at the turn of the 20th century, Europeans tended to like their trees manageable, at only about the height of your typical sixth grader. But when Americans began catching on to the tree tradition, they decided they liked their trees big, so massive that they touched the ceiling of their homes.<br><br>Then [http://www.Google.Co.uk/search?hl=en&gl=us&tbm=nws&q=Americans&gs_l=news Americans] took it one particular step additional with the invention of electricity in houses. Trees were braided with strings of multicolored lights. Residences have been covered with decorations of each color, blinking lights, and Santa Claus statues with his sleigh of reindeer. Even towns and cities began setting up Santa Claus displays around Town Hall, as well as Christmas trees that reached to the sky..<br><br>Should you adored this informative article and you desire to receive details concerning [http://earlyitem6969.skyrock.com health books] kindly stop by our own site. |
| [[File:CuttingABarMagnet.svg|thumb|200px|It is impossible to make '''magnetic monopoles''' from a [[bar magnet]]. If a bar magnet is cut in half, it is ''not'' the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles. A magnetic monopole cannot be created from normal matter such as [[atom]]s and [[electron]]s, but would instead be a new [[elementary particle]].]]
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| A '''magnetic monopole''' is a hypothetical [[elementary particle]] in [[particle physics]] that is an isolated [[magnet]] with only one magnetic pole (a [[magnet|north pole]] without a [[magnet|south pole]] or vice-versa).<ref name=Hooper>[http://books.google.com/books?id=tGBUvLpgmUMC&pg=PA192 Dark Cosmos: In Search of Our Universe's Missing Mass and Energy, by Dan Hooper, p192]</ref><ref>[http://pdg.lbl.gov/2004/listings/s028.pdf Particle Data Group summary of magnetic monopole search]</ref> In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from [[high-energy physics|particle theories]], notably the [[grand unified theory|grand unified]] and [[superstring theory|superstring]] theories, which predict their existence.<ref>Wen, Xiao-Gang; Witten, Edward, ''Electric and magnetic charges in superstring models'', Nuclear Physics B, Volume 261, p. 651–677</ref><ref>S. Coleman, ''The Magnetic Monopole 50 years Later'', reprinted in ''Aspects of Symmetry''</ref>
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| Magnetism in [[bar magnet]]s and [[electromagnet]]s does not arise from magnetic monopoles, and in fact there is no conclusive experimental evidence that magnetic monopoles exist at all in the universe.
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| Some [[condensed matter]] systems contain effective (non-isolated) magnetic monopole [[quasiparticle|''quasi''-particles]],<ref name=Castelnovo/> or contain phenomena that are mathematically analogous to magnetic monopoles.<ref name=Ray>{{cite journal|last1=Ray|first1=M.W.|last2=Ruokokoski|first2=E.|last3=Kandel|first3=S.|last4=Möttönen|first4=M.|last5=Hall|first5=D. S.|title=Observation of Dirac monopoles in a synthetic magnetic field|journal=[[Nature]]|volume=505|issue=7485|year=2014|pages=657–660|issn=0028-0836|doi=10.1038/nature12954}}</ref>
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| ==Historical background==
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| ===Pre-twentieth century===
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| Many early scientists attributed the magnetism of [[lodestone]]s 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]].<ref>[http://books.google.com/books?id=N1YEAAAAYAAJ&pg=PA352 The encyclopædia britannica, Volume 17, p352]</ref><ref>[http://books.google.com/books?id=6rYXAAAAIAAJ&pg=PA424 Principles of Physics by William Francis Magie, p424]</ref> 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<ref>[[Pierre Curie]], ''Sur la possibilité d'existence de la conductibilité magnétique et du magnétisme libre'' (''On the possible existence of magnetic conductivity and free magnetism''), Séances de la Société Française de Physique (Paris), p76 (1894). {{fr}}[http://www.archive.org/stream/sancesdelasocit19physgoog Free access online copy].</ref> that magnetic monopoles ''could'' conceivably exist, despite not having been seen so far.
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| ===Twentieth century=== | |
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| The [[quantum mechanics|''quantum'']] theory of magnetic charge started with a paper by the [[physicist]] [[Paul A.M. Dirac]] in 1931.<ref>[[Paul Dirac]], "Quantised Singularities in the Electromagnetic Field". Proc. Roy. Soc. (London) '''A 133''', 60 (1931). [http://users.physik.fu-berlin.de/~kleinert/files/dirac1931.pdf Free web link].</ref> In this paper, Dirac showed that if ''any'' magnetic monopoles exist in the universe, then all electric charge in the universe must be [[charge quantization|quantized]].<ref name=littlejohn>[http://bohr.physics.berkeley.edu/classes/221/0708/lectures/Lecture.2007.10.11.pdf Lecture notes by Robert Littlejohn], University of California, Berkeley, 2007–8</ref> The electric charge ''is'', in fact, quantized, which is consistent with (but does not prove) the existence of monopoles.<ref name=littlejohn/>
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| Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975<ref name="PRL-35-487">{{cite journal|title=Evidence for Detection of a Moving Magnetic Monopole|author=P. B. Price|coauthors=E. K. Shirk; W. Z. Osborne; L. S. Pinsky|journal=Physical Review Letters|volume=35|issue=8|pages=487–490|date=August 25, 1975|doi=10.1103/PhysRevLett.35.487|publisher=American Physical Society|bibcode=1975PhRvL..35..487P}}</ref> and 1982<ref name="PRL-48-1378">{{cite journal|author=Blas Cabrera|title=First Results from a Superconductive Detector for Moving Magnetic Monopoles|journal=Physical Review Letters|volume=48|issue=20|pages=1378–1381|date=May 17, 1982|doi=10.1103/PhysRevLett.48.1378|publisher=American Physical Society|bibcode=1982PhRvL..48.1378C}}</ref> produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.<ref>[[#References|Milton]] p.60</ref> Therefore, it remains an open question whether monopoles exist.
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| 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".<ref name=Polchinski>[http://arxiv.org/abs/hep-th/0304042 Polchinski, arXiv 2003]</ref> These theories are not necessarily inconsistent with the experimental evidence. In some theoretical [[Scientific modelling|model]]s, magnetic monopoles are unlikely to be observed, because they are too massive to be created in [[particle accelerator]]s (see [[#Searches for magnetic monopoles|below]]), and also too rare in the Universe to enter a [[particle detector]] with much probability.<ref name=Polchinski/>
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| Some [[condensed matter physics|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 [[quasiparticle]]s. Since 2009, numerous news reports from the popular media<ref name=sciencedaily/><ref name=symmetrymagazine/> 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.<ref name=TchernyshyovQuote>[http://physicsworld.com/cws/article/news/40302 Magnetic monopoles spotted in spin ices], September 3, 2009. "Oleg Tchernyshyov at Johns Hopkins University [a researcher in this field] cautions that the theory and experiments are specific to spin ices, and are not likely to shed light on magnetic monopoles as predicted by Dirac."</ref><ref name=GibneyQuote>{{cite journal |doi=10.1038/nature.2014.14612 |author=Elizabeth Gibney |journal=Nature (news section) |date=29 January 2014 |title=Quantum cloud simulates magnetic monopole}} "This is not the first time that physicists have created monopole analogues. In 2009, physicists observed magnetic monopoles in a crystalline material called spin ice, which, when cooled to near-absolute zero, seems to fill with atom-sized, classical monopoles. These are magnetic in a true sense, but cannot be studied individually. Similar analogues have also been seen in other materials, such as in superfluid helium.... Steven Bramwell, a physicist at University College London who pioneered work on monopoles in spin ices, says that the [2014 experiment led by David Hall] is impressive, but that what it observed is not a Dirac monopole in the way many people might understand it. "There’s a mathematical analogy here, a neat and beautiful one. But they’re not magnetic monopoles."</ref> These condensed-matter systems continue to be an area of active research. (See [[#"Monopoles" in condensed-matter systems|"Monopoles" in condensed-matter systems]] below.)
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| ==Poles and magnetism in ordinary matter==
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| {{main|Magnetism}}
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| 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 [[magnet]]s have nothing to do with magnetic monopoles.
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| Instead, magnetism in ordinary matter comes from two sources. First, [[electric current]]s create [[magnetic field]]s 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 [[spin (physics)|quantum-mechanical "spin"]].)
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| 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.
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| 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 [[proton]]s and the negative charge is made of [[electron]]s, 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.
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| ==Maxwell's equations==
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| [[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.<ref>The fact that the electric and magnetic fields can be written in a symmetric way is specific to the fact that space is three-dimensional. When the equations of electromagnetism are extrapolated to other dimensions, the magnetic field is described as being a rank-two [[antisymmetric tensor]], whereas the electric field remains a [[Pseudovector|true vector]]. In dimensions other than three, these two mathematical objects do not have the same number of components.</ref> In fact, symmetric Maxwell's equations can be written when all charges (and hence [[electric current]]s) are zero, and this is how the [[electromagnetic wave equation]] is derived.
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| Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.<ref>http://www.ieeeghn.org/wiki/index.php/STARS:Maxwell%27s_Equations</ref> With the inclusion of a variable for the density of these magnetic charges, say ''ρ''<sub>m</sub>, there will also be a "magnetic current density" variable in the equations, '''j'''<sub>m</sub>.
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| 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 {{nowrap|1=∇⋅'''B''' = 0}} (where ∇⋅ is [[divergence]] and '''B''' is the [[magnetic field|magnetic '''B''' field]]).
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| {{multiple image
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| | align = center
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| | direction = horizontal
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| | footer = The [[electric field|'''E''' field]]s and [[magnetic field|'''B''' field]]s due to [[electric charge]]s (black/white) and [[magnet|magnetic pole]]s (red/blue).<ref>{{cite book |title=McGraw Hill Encyclopaedia of Physics |first1=C.B. |last1= Parker|edition=2nd|publisher=Mc Graw Hill|year=1994|isbn=0-07-051400-3}}</ref><ref>{{cite book |author= M. Mansfield, C. O’Sullivan|title= Understanding Physics|edition= 4th |year= 2011|publisher= John Wiley & Sons|isbn=978-0-47-0746370}}</ref>
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| | image1 = em monopoles.svg
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| | caption1 = '''Left:''' Fields due to stationary [[electric charge|electric]] and magnetic monopoles. '''Right:''' In motion ([[velocity]] '''v'''), an ''electric'' charge induces a '''B''' field while a ''magnetic'' charge induces an '''E''' field. [[Conventional current]] is used.
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| | width1 = 350
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| | image2 = em dipoles.svg
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| | caption2 = '''Top:''' '''E''' field due to an [[electric dipole moment]] '''d'''. '''Bottom left:''' '''B''' field due to a ''mathematical'' [[magnetic dipole]] '''m''' formed by two magnetic monopoles. '''Bottom right:''' '''B''' field due to a natural [[magnetic dipole moment]] '''m''' found in ordinary matter (''not'' from monopoles).
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| | width2 = 300
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| }}
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| ===In Gaussian cgs units===
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| The extended Maxwell's equations are as follows, in [[Gaussian units|Gaussian cgs]] units:<ref name="moulin-2001">{{cite journal |title=Magnetic monopoles and Lorentz force |author=F. Moulin |volume=116 |issue=8 |pages=869–877 |year=2001 |arxiv=math-ph/0203043
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| |journal=[[Nuovo Cimento B]] |bibcode = 2001NCimB.116..869M }}</ref>
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| :{| class="wikitable" style="text-align: center;"
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| |+Maxwell's equations and Lorentz force equation with magnetic monopoles: Gaussian cgs units
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| |-
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| ! scope="col" width="200px" | Name
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| ! Without magnetic monopoles
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| ! With magnetic monopoles
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| |-
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| ! [[Gauss's law]]
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| |colspan="2"| <math>\nabla \cdot \mathbf{E} = 4 \pi \rho_{\mathrm e} </math>
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| |-
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| ! [[Gauss's law for magnetism]]
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| | <math>\nabla \cdot \mathbf{B} = 0 </math>
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| | <math>\nabla \cdot \mathbf{B} = 4 \pi \rho_{\mathrm m} </math>
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| |-
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| ! [[Faraday's law of induction]]
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| | <math>-\nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}</math>
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| | <math>-\nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} + \frac{4 \pi}{c}\mathbf{j}_{\mathrm m}</math>
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| |-
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| ! [[Ampère's law]] (with Maxwell's extension)
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| |colspan="2"| <math>\nabla \times \mathbf{B} = \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t} + \frac{4 \pi}{c} \mathbf{j}_{\mathrm e} </math>
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| |-
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| ![[Lorentz force]] law<ref name="moulin-2001"/><ref>{{cite journal |title=Relativity and electromagnetism: The force on a magnetic monopole |author=Wolfgang Rindler |publisher=American Journal of Physics |date=November 1989 |volume=57 |issue=11
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| |pages=993–994 |doi=10.1119/1.15782 |journal=American Journal of Physics |bibcode = 1989AmJPh..57..993R }}</ref>
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| |<math>\mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) </math>
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| |<math>\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)</math>
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| |-
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| |}
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| In these equations ''ρ''<sub>m</sub> is the ''magnetic charge density'', '''j'''<sub>m</sub> is the ''magnetic current density'', and ''q''<sub>m</sub> 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 [[Planck units#Nondimensionalization of fundamental physical equations|nondimensionalized]] form, remove the factors of ''c''.
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| ===In SI units===
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| In [[SI]] units, there are two conflicting units in use for magnetic charge ''q''<sub>m</sub>: [[Weber (unit)|webers (Wb)]] and [[ampere]]·meters (A·m). The conversion between them is ''q''<sub>m</sub>(Wb) = μ<sub>0</sub>''q''<sub>m</sub>(A·m), since the units are 1 Wb = 1 H·A = (1 H·m<sup>−1</sup>)·(1 A·m) by [[dimensional analysis]] (H is the [[Henry (unit)|henry]] – the SI unit of [[inductance]]).
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| Maxwell's equations then take the following forms (using the same notation above):<ref>For the convention where magnetic charge has units of webers, see [[#References|Jackson 1999]]. In particular, for Maxwell's equations, see section 6.11, equation (6.150), page 273, and for the Lorentz force law, see page 290, exercise 6.17(a). For the convention where magnetic charge has units of ampere-meters, see (for example) [http://arxiv.org/abs/physics/0508099v1 arXiv:physics/0508099v1], eqn (4).</ref>
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| :{| class="wikitable" style="text-align: center;"
| |
| |+Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units
| |
| |-
| |
| ! scope="col" width="200px" | Name
| |
| ! Without magnetic monopoles
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| ! Weber convention
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| ! Ampere·meter convention
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| |-
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| !Gauss's Law
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| |colspan="3"| <math>\nabla \cdot \mathbf{E} = \frac{\rho_{\mathrm e}}{\epsilon_0}</math>
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| |-
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| !Gauss's Law for magnetism
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| || <math>\nabla \cdot \mathbf{B} = 0</math>
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| || <math>\nabla \cdot \mathbf{B} = \rho_{\mathrm m}</math>
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| || <math>\nabla \cdot \mathbf{B} = \mu_0\rho_{\mathrm m}</math>
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| |-
| |
| !Faraday's Law of induction
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| || <math>-\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t}</math>
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| || <math>-\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} + \mathbf{j}_{\mathrm m}</math>
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| || <math>-\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} + \mu_0\mathbf{j}_{\mathrm m}</math>
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| |-
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| !Ampère's Law (with Maxwell's extension)
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| |colspan="3"| <math>\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} + \mu_0 \mathbf{j}_{\mathrm e}</math>
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| |-
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| !Lorentz force equation
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| || <math>\mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
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| || <math>\begin{align}
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| \mathbf{F}&=q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\\
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| & + \frac{q_{\mathrm m}}{\mu_0}\left(\mathbf{B}-\mathbf{v}\times \frac{\mathbf{E}}{c^2}\right)
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| \end{align}</math>
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| || <math>\begin{align}
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| \mathbf{F}&=q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\\
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| & + q_{\mathrm m}\left(\mathbf{B}-\mathbf{v}\times\frac{\mathbf{E}}{c^2}\right)
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| \end{align}</math>
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| |}
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| ===Tensor formulation===
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| Maxwell's equations in the language of [[tensor]]s makes [[Lorentz covariance]] clear. The generalized equations are:<ref>{{cite news
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| | author = J.A. Heras, G. Baez
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| | year = 2009
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| | location = Mexico
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| | publisher =
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| | title = The covariant formulation of Maxwell's equations expressed in a form independent of specific units
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| | arxiv = 0901.0194
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| | url = http://arxiv.org/abs/0901.0194
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| }}</ref><ref>{{cite news
| |
| | author = F. Moulin
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| | year = 2002
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| | location = Cachan, France
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| | publisher =
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| | title = Magnetic monopoles and Lorentz force
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| | arxiv = math-ph/0203043
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| | url = http://arxiv.org/abs/math-ph/0203043
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| }}</ref>
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| | |
| :{| class="wikitable"
| |
| |-
| |
| ! Maxwell equations
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| ! Gaussian units
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| ! SI units (Wb)
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| ! SI units (A⋅m)
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| |-
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| ! Faraday-Gauss law
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| | <math>\partial_\alpha F^{\alpha\beta} = \frac{4\pi}{c}J^\beta_{\mathrm e}</math>
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| | <math>\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta_{\mathrm e}</math>
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| | <math>\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta_{\mathrm e}</math>
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| |-
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| ! Ampère-Gauss law
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| | <math>\partial_\alpha {\star F^{\alpha\beta}} = \frac{4\pi}{c} J^\beta_{\mathrm m}</math>
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| | <math>\partial_\alpha {\star F^{\alpha\beta}} = \frac{\mu_0}{c} J^\beta_{\mathrm m}</math>
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| | <math>\partial_\alpha {\star F^{\alpha\beta}} = \frac{1}{c} J^\beta_{\mathrm m}</math>
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| |-
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| ! Lorentz force law
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| | <math>\frac{dp_\alpha}{d\tau} = \frac{1}{c}\left[ q_{\mathrm e} F_{\alpha\beta}v^\beta + q_{\mathrm m} {\star F_{\alpha\beta}}v^\beta \right]</math>
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| | <math>\frac{dp_\alpha}{d\tau} = q_{\mathrm e} F_{\alpha\beta}v^\beta + q_{\mathrm m} {\star F_{\alpha\beta}}v^\beta </math>
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| | <math>\frac{dp_\alpha}{d\tau} = q_{\mathrm e} F_{\alpha\beta}v^\beta + \frac{q_{\mathrm m}}{\mu_0} {\star F_{\alpha\beta}v^\beta} </math>
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| |}
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| | |
| where
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| *''F'' is the [[electromagnetic tensor]], <math>\star</math> denotes the [[Hodge dual]], (so ∗''F'' is the dual tensor to ''F''),
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| *for a particle with electric charge ''q''<sub>e</sub> and magnetic charge ''q''<sub>m</sub>; ''v'' is the [[four-velocity]] and ''p'' the [[four-momentum]],
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| *for an electric and magnetic charge distribution; ''J''<sub>e</sub> = (''ρ''<sub>e</sub>, '''j'''<sub>e</sub>) is the electric [[four-current]] and ''J''<sub>m</sub> = (''ρ''<sub>m</sub>, '''j'''<sub>m</sub>) the magnetic four-current.
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| For a particle having only electric charge, one can express its field using a [[Electromagnetic four-potential|four-potential]], according to the standard [[covariant formulation of classical electromagnetism]]:
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| :<math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\gamma} \,</math>
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| 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".<ref name=Can>Shanmugadhasan, S. "The Dynamical Theory of Magnetic Monopoles", ''[[Canadian Journal of Physics]]'' Vol. 30, p. 218. (1952).</ref><ref name=Found>Fryberger, D. "[http://www.slac.stanford.edu/pubs/slacpubs/4000/slac-pub-4237.pdf On Generalized Electromagnetism and Dirac Algebra]", ''[[Foundations of Physics]]'' Vol. 19, p. 125 (1989).</ref>
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| | |
| :<math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \ +\partial^{\mu}(\epsilon_{\alpha\beta\mu\nu}P^{\nu}),</math>
| |
| | |
| This formula for the fields is often called the [[Nicola Cabibbo|Cabibbo]]-Ferrari relation, though Shanmugadhasan proposed it earlier.<ref name=Found /> The quantity ε<sup>αβγδ</sup> is the [[Levi-Civita symbol]], and the indices (as usual) behave according to the [[Einstein summation convention]].
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| | |
| ===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):<ref name=Jackson611>[[#References|Jackson 1999]], section 6.11.</ref>
| |
| :{| class="wikitable"
| |
| |-
| |
| ! Charges and currents
| |
| ! Fields
| |
| |-
| |
| | <math>\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}</math>
| |
| |<math>\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}</math>
| |
| |-
| |
| | <math>\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}</math>
| |
| | <math>\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}</math>
| |
| |-
| |
| |}
| |
| | |
| 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 (mathematics)|matrix]] is a [[Two-dimensional space|two-dimensional]] [[rotation matrix]].
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| 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.<ref name=Jackson611/> 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.<ref name=Jackson611/>
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| | |
| ==Dirac's quantization==
| |
| One of the defining advances in [[quantum mechanics|quantum theory]] was [[Paul Dirac]]'s work on developing a [[special relativity|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.
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| 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 ''q''<sub>e</sub>''q''<sub>m</sub>, and independent of the distance between them.
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| Quantum mechanics dictates, however, that angular momentum is quantized in units of ''ħ'', so therefore the product ''q''<sub>e</sub>''q''<sub>m</sub> 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 [[charge quantization|quantized]].
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| | |
| What are the units in which magnetic charge would be quantized? Although it would be possible simply to [[integration (mathematics)|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 ''q''<sub>m</sub> / ''r''<sup> 2</sup> 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 (mathematics)|curl]] of the vector potential ''A'' equals the magnetic field ''B''.
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| | |
| 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'' > 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 ''q''<sub>e</sub> of the probe, as well as to the magnetic charge ''q''<sub>m</sub> of the source. Dirac was originally considering an [[electron]] whose wave function is described by the [[Dirac equation]].
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| | |
| 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 2''π'':
| |
| | |
| :{| class="wikitable"
| |
| |-
| |
| ! Units
| |
| ! Condition
| |
| |-
| |
| | [[Gaussian units|Gaussian-cgs units]]
| |
| || <math>2 \frac{q_{\mathrm e} q_{\mathrm m}}{\hbar c} \in \mathbb{Z}</math>
| |
| |-
| |
| | [[SI units]] ([[Weber (unit)|weber]] convention)<ref>[[#References|Jackson 1999]], section 6.11, equation (6.153), page 275</ref>
| |
| || <math>\frac{q_{\mathrm e} q_{\mathrm m}}{2 \pi \hbar} \in \mathbb{Z}</math>
| |
| |-
| |
| | SI units ([[ampere]]·meter convention)
| |
| || <math>\frac{q_{\mathrm e} q_{\mathrm m}}{2 \pi \epsilon_0 \hbar c^2} \in \mathbb{Z}</math>
| |
| |-
| |
| |}
| |
| | |
| where ''ε''<sub>0</sub> 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 [[integer]]s.
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| | |
| 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.
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| | |
| 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.)
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| | |
| 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.
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| | |
| 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]].
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| | |
| ==Topological interpretation==
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| | |
| ===Dirac string===
| |
| {{Main|Dirac string}}
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| 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.
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| | |
| In electrodynamics, the group is [[U(1)]], unit complex numbers under multiplication. For infinitesimal paths, the group element is 1 + ''iA''<sub>''μ''</sub>''dx''<sup>''μ''</sup> which implies that for finite paths parametrized by ''s'', the group element is:
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| | |
| :<math>\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} ds \right) = \exp \left( ie\int A\cdot dx \right) . </math>
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| | |
| 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:
| |
| | |
| :<math>e \oint_{\partial D} A\cdot dx = e \int_D (\nabla \times A) dS = e \int_D B \, dS.</math>
| |
| | |
| 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 [[Aharonov–Bohm effect|interference fringes]] for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.
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| | |
| 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 ''e''<sup>2π''i''</sup> = 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.
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| | |
| 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.
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| | |
| ===Grand unified theories===
| |
| {{main|'t Hooft–Polyakov monopole}}
| |
| 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 space|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.
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| | |
| 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.
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| | |
| 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.
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| | |
| The argument is topological:
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| | |
| # 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 [[Poincaré conjecture|''lassoing the sphere'']].
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| # 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.
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| # 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.
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| # 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 2π''N''/''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.
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| # 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 ''P''<sup>2</sup>, three times to ''P''<sup>3</sup>, 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.
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| # 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.
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| | |
| 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 gauge theory|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.
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| | |
| ===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.<ref>Nima Arkani-Hamed, Lubos Motl, Alberto Nicolis, Cumrun Vafa: [http://arxiv.org/abs/hep-th/0601001 The String Landscape, Black Holes and Gravity as the Weakest Force](arXiv:hep-th/0601001, JHEP 0706:060,2007)</ref> The lightest monopole should have a mass less than or comparable to its charge in [[natural units]].
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| | |
| 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]].
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| | |
| ===Mathematical formulation===
| |
| In mathematics, a (classical) gauge field is defined as a [[connection form|connection]] over a [[principal bundle|principal G-bundle]] over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately.
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| | |
| 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.
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| 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 class]]es 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.
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| | |
| If space time is '''R'''<sup>4</sup> the space of all possible connections of the ''G''-bundle is [[connected space|connected]]. But consider what happens when we remove a [[timelike]] [[worldline]] from spacetime. The resulting spacetime is [[homotopy|homotopically equivalent]] to the [[topological sphere]] ''S''<sup>2</sup>.
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| A principal ''G''-bundle over ''S''<sup>2</sup> is defined by covering ''S''<sup>2</sup> by two [[chart (topology)|charts]], each [[homeomorphic]] to the open 2-ball such that their intersection is homeomorphic to the strip ''S''<sup>1</sup>×''I''. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle ''S''<sup>1</sup>. 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''.
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| 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.
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| 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.
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| | |
| 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 π<sub>''d'' − 2</sub>(''G'').
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| ==Grand unified theories==
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| In more recent years, a new class of theories has also suggested the existence of magnetic monopoles.
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| | |
| 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 [[dyon]]s, 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.
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| 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 law]]s. 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.
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| 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 ''[[topology|topological]]'' state into which they can decay.
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| 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 (physics)|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 tensor|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.
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| 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.<ref>{{Cite journal|title=On the concentration of relic monopoles in the universe|first=Ya.|last=Zel'dovich|coauthors=M. Yu. Khlopov|year=1978|journal=Phys. Lett.|volume=B79|pages=239–41|bibcode=1978PhLB...79..239Z|last2=Khlopov|doi=10.1016/0370-2693(78)90232-0|issue=3}}</ref><ref>{{Cite journal|title=Cosmological production of superheavy magnetic monopoles | doi = 10.1103/PhysRevLett.43.1365|year=1979|journal=Phys. Rev. Lett.|volume=43|issue=19|pages=1365|first=John|last=Preskill|bibcode = 1979PhRvL..43.1365P }}</ref> 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 [[Inflation (cosmology)|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.<ref>{{Cite journal|title=Magnetic Monopoles | doi = 10.1146/annurev.ns.34.120184.002333|year=1984|journal=Ann. Rev. Nucl. Part. Sci.|volume=34|pages=461|first=John|last=Preskill|bibcode = 1984ARNPS..34..461P }}</ref> This resolution of the "monopole problem" was regarded as a success of [[Inflation (cosmology)|cosmic inflation theory]]. (However, of course, it is only a noteworthy success if the particle-physics monopole prediction is correct.<ref>Rees, Martin. (1998). ''Before the Beginning'' (New York: Basic Books) p. 185 ISBN 0-201-15142-1</ref>) 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]].
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| 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.
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| | |
| == 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"<ref>http://www.nature.com/nature/journal/v429/n6987/full/429010a.html</ref>), there has never been reproducible evidence for the existence of magnetic monopoles.<ref name="PRL-48-1378" /> The lack of such events places a limit on the number of monopoles of about one monopole per 10<sup>29</sup> [[nucleon]]s.
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| Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in [[cosmic ray]]s by the team led by [[P. Buford Price]].<ref name="PRL-35-487"/> Price later retracted his claim, and a possible alternative explanation was offered by Alvarez.<ref>{{cite conference|first=Luis W|last=Alvarez|title=Analysis of a Reported Magnetic Monopole|editor=ed. Kirk, W. T.|conference=International symposium on lepton and photon interactions at high energies, Aug 21, 1975|booktitle=Proceedings of the 1975 international symposium on lepton and photon interactions at high energies|pages=967|url=http://usparc.ihep.su/spires/find/hep/www?key=93726}}</ref> 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 [[nuclear decay|decaying]] first to [[osmium]], and then to [[tantalum]].
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| Other experiments rely on the strong coupling of monopoles with [[photon]]s, 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 {{val|600|u=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 10<sup>17</sup> {{val/units|GeV/c2}}.
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| 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.
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| == "Monopoles" in condensed-matter systems ==
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| Since around 2003, various [[condensed-matter physics]] groups have used the term "magnetic monopole" to describe a different and largely unrelated phenomenon.<ref name=TchernyshyovQuote/><ref name=GibneyQuote/>
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| A true magnetic monopole would be a new [[elementary particle]], and would violate [[Gauss's law for magnetism|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,<ref>"[http://users.physik.fu-berlin.de/~kleinert/files/dirac1931.pdf Quantised Singularities in the Electromagnetic Field]" [[Paul Dirac]], ''Proceedings of the Royal Society'', May 29, 1931. Retrieved February 1, 2014.</ref> has never been observed in experiments.
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| 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 ([[proton]]s, [[neutron]]s, [[electron]]s, [[photon]]s); in other words, they are [[quasi-particle]]s. They are not sources for the [[magnetic field|'''B'''-field]] (i.e., they do not violate [[Gauss's law for magnetism|the law ∇⋅'''B''' = 0]]); instead, they are sources for other fields, for example the [[magnetic field|'''H'''-field]],<ref name=Castelnovo/> or the "B*-field" (related to [[superfluid]] vorticity)<ref name=Ray/> 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.<ref name=Gibney/>
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| There are a number of examples in [[condensed-matter physics]] where collective behavior leads to emergent phenomena that resemble magnetic monopoles in certain respects,<ref name=symmetrymagazine>[http://www.symmetrymagazine.org/breaking/2009/01/29/making-magnetic-monopoles-and-other-exotica-in-the-lab/ Making magnetic monopoles, and other exotica, in the lab], [[Symmetry Breaking]], January 29, 2009. Retrieved January 31, 2009.</ref><ref>Zhong, Fang; Naoto Nagosa, Mei S. Takahashi, Atsushi Asamitsu, Roland Mathieu, Takeshi Ogasawara, Hiroyuki Yamada, Masashi Kawasaki, Yoshinori Tokura, Kiyoyuki Terakura (October 3, 2003). "The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space". ''Science'' 302 (5642) 92–95. {{doi|10.1126/science.1089408}}. ISSN 1095-9203. http://www.sciencemag.org/cgi/content/abstract/302/5642/92. Retrieved August 2, 2007.</ref><ref>[http://www.sciencemag.org/cgi/content/abstract/1167747 Inducing a Magnetic Monopole with Topological Surface States], [[American Association for the Advancement of Science]] (AAAS) ''Science Express'' magazine, Xiao-Liang Qi, Rundong Li, Jiadong Zang, Shou-Cheng Zhang, January 29, 2009. Retrieved January 31, 2009.</ref><ref>[http://www.sciencedaily.com/releases/2013/05/130531103910.htm '' Artificial Magnetic Monopoles Discovered '']</ref> including most prominently the [[spin ice]] materials.<ref name=Castelnovo>{{cite journal |doi=10.1038/nature06433 |title=Magnetic monopoles in spin ice |authors=C. Castelnovo, R. Moessner and S. L. Sondhi |journal=Nature |volume=451 |pages=42–45 |date=January 3, 2008}}</ref><ref>[http://www.nature.com/nature/journal/v461/n7266/abs/nature08500.html Nature 461, 956–959 (15 October 2009); ] {{doi|10.1038/nature08500}} PMID 19829376, Steven Bramwell et al</ref> 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.
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| One example of the work on magnetic monopole quasiparticles is a paper published in the journal ''[[Science (journal)|Science]]'' in September 2009, in which researchers Jonathan Morris and Alan Tennant from the [[Helmholtz-Zentrum Berlin|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 [[quasiparticle]]s 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 string]]s. At the [[crystallographic defect|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.<ref name=sciencedaily>
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| {{cite web
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| |url=http://www.sciencedaily.com/releases/2009/09/090903163725.htm
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| |title=Magnetic Monopoles Detected in a Real Magnet for the First Time
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| |publisher=[[Science Daily]]
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| |date=September 4, 2009
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| |accessdate=September 4, 2009
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| }}</ref><ref>
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| {{cite journal
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| |doi=10.1126/science.1178868
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| |title=Dirac Strings and Magnetic Monopoles in Spin Ice Dy<sub>2</sub>Ti<sub>2</sub>O<sub>7</sub>
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| |author=D.J.P. Morris, D.A. Tennant, S.A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czter-nasty, M. Meissner, K.C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R.S. Perry
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| |journal=[[Science (journal)|Science]]
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| |submitted=2009-07-09
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| |date=September 3, 2009
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| |bibcode = 2009Sci...326..411M
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| |pmid=19729617|arxiv = 1011.1174
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| |volume=326
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| |issue=5951
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| |pages=411–4}}</ref>
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| This research went on to win the 2012 Europhysics Prize for condensed matter physics
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| 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.<ref>
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| {{cite journal
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| |doi=10.1038/nphys1896
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| |url=http://www.nature.com/nphys/journal/v7/n3/full/nphys1896.html
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| |title=Creation and measurement of long-lived magnetic monopole currents in spin ice
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| |author=S. R. Giblin, S. T. Bramwell, P. C. W. Holdsworth, D. Prabhakaran & I. Terry
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| |publisher=[[Nature Physics]]
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| |date=February 13, 2011
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| |accessdate=February 28, 2011
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| |bibcode = 2011NatPh...7..252G
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| |volume=7
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| |issue=3}}</ref>
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| In [[superfluid]]s, 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<ref>[http://prl.aps.org/abstract/PRL/v103/i3/e030401 Ville Pietilä, Mikko Möttönen, ''Creation of Dirac Monopoles in Spinor Bose–Einstein Condensates'', Phys. Rev. Lett. 103, 030401 (2009)]</ref> for the '''B'''* field were created and studied in a spinor Bose–Einstein condensate.<ref name=Ray/> This constitutes the first example of a magnetic monopole observed within a system governed by quantum field theory.<ref name=Gibney>{{cite journal |doi=10.1038/nature.2014.14612 |author=Elizabeth Gibney |journal=Nature (news section) |date=29 January 2014 |title=Quantum cloud simulates magnetic monopole}}</ref>
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| == Further descriptions in particle physics ==
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| In physics the phrase "magnetic monopole" usually denoted a [[Yang–Mills theory|Yang–Mills potential]] ''A'' and [[Higgs field]] ''ϕ'' whose equations of motion are determined by the Yang–Mills [[action (physics)|action]]
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| :<math>\int (F_A,F_A)+(D_A \phi,D_A \phi) - \lambda(1 - \| \phi \|^2 )^2.</math>
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| In mathematics, the phrase customarily refers to a static solution to these equations in the [[Bogomol'nyi–Prasad Sommerfield bound|Bogomolny–Parasad–Sommerfeld limit]] ''λ'' → ''ϕ'' which realizes, within topological class, the absolutes minimum of the functional
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| :<math>\int_{R^3} (F_A,F_A)+(D_A \phi,D_A \phi).</math>
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| This means that it in a [[connection (vector bundle)|connection]] ''A'' on a [[Principal bundle|principal ''G''-bundle]] over ''R''<sup>3</sup> (c.f. also Connections on a manifold; principal ''G''-object) and a section ''ϕ'' of the associated [[adjoint bundle]] of [[Lie algebra]]s such that the [[curvature]] ''F''<sub>''A''</sub> and [[covariant derivative]] ''D''<sub>''A''</sub> ''ϕ'' satisfy the Bogomolny equations
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| :<math>F_A = * D_A \phi</math>
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| and the [[boundary condition]]s.
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| :<math>\| \phi \| = 1- \frac{m}{r} + \theta(r^2), \quad \| D_A \phi \| = \mathcal{O} (r^2)</math>
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| Pure mathematical advances in the theory of monopoles from the 1980s onwards have often proceeded on the basis of physically motived questions.
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| The equations themselves are invariant under [[Gauge theory|gauge transformation]] and orientation-preserving symmetries. When γ is large, ''ϕ''/||''ϕ''|| defines a mapping from a 2-sphere of radius γ in '''R'''<sup>3</sup> 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 8π''k'' and the coefficient ''m'' in the [[asymptotic expansion]] of ϕ/||ϕ|| is ''k''/2.
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| 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
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| :<math>A = \left( \frac{1}{\sinh{\gamma}} - \frac{1}{\gamma} \right) \ \epsilon_{ijk} \frac{x_j}{\gamma} \sigma_k \, dx_i, </math>
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| :<math>\phi = \left( \frac{1}{\tanh{\gamma}} - \frac{1}{\gamma} \right) \ \frac{x_j}{\gamma} \sigma_i </math>
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| In 1980, C.H.Taubes<ref>{{cite book |author=A.Jaffe, C.H.Taubes| title=Vortices and monopoles| publisher= |edition= | year=1980| isbn=}}</ref> 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]].
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| There are two ways of solving the [[Bogomolny equations]]. The first is by [[twistor]] methods. In the formulation of N.J.Hitchin,<ref>{{cite book |author=N.J.Hitchin | title=Monopoles and geodesics |publisher= | year=1982| isbn=}}</ref> an arbitrary solution corresponds to a [[holomorphic vector bundle]] over the [[complex surface]] ''TP''<sup>1</sup>, the [[tangent bundle]] of the projective line. This is naturally isomorphic to the space of oriented straight lines in '''R'''<sup>3</sup>.
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| The boundary condition show that the holomorphic bundle is an extension of line bundles determined by a compact [[algebraic curve]] of genus (''k'' − 1)<sup>2</sup> (the spectral curve) in ''TP''<sup>1</sup>, satisfying certain constraints.
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| The second method, due to W.Nahm,<ref>{{cite book |author=W.Nahm| title=The construction of all self-dual monopoles by the ADHM mothod| publisher= |edition= | year=1982 |isbn= }}</ref> 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]].
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| :<math>\frac{dT_1}{ds}=[T_2,T_3],\ \ \frac{dT_2}{ds}=[T_3,T_1],\ \ \frac{dT_3}{ds}=[T_1,T_2]</math>
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| where ''T<sub>i</sub>''(''s'') is a ''k''×''k'' -matrix valued function on (0,2).
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| 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 ''R''<sup>4</sup>.
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| The equivalence of the two methods for ''SU''(2) and their general applicability was established in<ref>{{cite book |author=N.J.Hitchin |title=On the construction of monopoles |publisher= |year=1983 |isbn= }}</ref> (see also<ref name="N.J.Hitchin 1999">{{cite book |author=N.J.Hitchin |title=Integrable sustems in Riemannian geometry |publisher= C-L.Terng (ed.) |edition= K.Uhlenbeck |year=1999 |isbn= }}</ref>). Explicit formulas for ''A'' and <math>\phi</math> are difficult to obtain by either method, despite some exact solutions of Nahm's equations in symmetric situations.<ref>{{cite book |author=N.J.Hitchin, N.S.Manton, M.K.Murray |title=Symmetric Monopoles |publisher= |year=1995 |isbn= }}</ref>
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| 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<ref>{{cite book |author=M.K.Murray |title=Monopoles and spectral curves for arbitrary Lie groups |publisher= |edition= |year=1983 |isbn= }}</ref> 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.<ref>{{cite book |author=J.Hurtubise, M.K.Murray |title=On the construction of Monopoles for the classical groups |publisher= |edition= |year=1989 |isbn= }}</ref>
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| The [[moduli space]] (c.f. also Moduli theory) of all ''SU''(2) monopoles of charge ''k'' up to gauge equivalence was shown by Taubes<ref>{{cite book |author=C.H.Taubes| title=Stability in Yang–Mills theories| publisher= |edition= | year=1983| isbn=}}</ref> to be a smooth non-compact manifold of dimension 4''k'' − 1. Restricting to gauge transformations that preserve the connection at infinity gives a 4''k''-dimensional manifold ''M''<sub>''k''</sub>, which is a circle bundle over the true moduli space and carries a natural complete hyperKähler metric<ref name="M.F.Atiyah, N.J.Hitchin 1988">{{cite book |author=M.F.Atiyah, N.J.Hitchin |title=The geometry and dynamics of magnetic monopoles |publisher=Princeton Univ.Press |year=1988 |isbn= }}</ref> (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 ''P''<sub>1</sub> to itself.<ref>{{cite book |author=S.K.Donaldson| title=Nahm’s equations and the classification of monopoles| publisher= |year=1984 |isbn= }}</ref>
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| The metric is known in twistor terms,<ref name="M.F.Atiyah, N.J.Hitchin 1988"/> and its [[Kähler potential]] can be written using the Riemann [[theta functions]] of the spectral curve,<ref name="N.J.Hitchin 1999"/> but only the case ''k'' = 2 is known in a more conventional and usable form<ref name="M.F.Atiyah, N.J.Hitchin 1988"/> (as of 2000). This Atiyah–Hitchin manifold, the Einstein [[Taub-NUT metric]] and '''R'''<sup>4</sup> 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.
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| A cyclic ''k''-fold conering of ''M''<sub>''k''</sub> splits isometrically is a product <math>\tilde{M_k} \times S^1 \times R^3</math>, where <math>\tilde{M_k}</math> is the space of strongly centred monopoles. This space features in an application of [[S-duality]] in theoretical physics, and in<ref>{{cite book |author=G.B.Segal, A.Selby |title=The cohomology of the space of magnetic monopoles |publisher=| edition= |year=1996 |isbn= }}</ref> G.B.Segal and A.Selby studied its topology and the ''L''<sup>2</sup> harmonic forms defined on it, partially confirming the physical prediction.
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| Magnetic monopole on hyperbolic three-space were investigated from the twistor point of view b M.F.Atiyah<ref>{{cite book |author=M.F.Atiyah |title=Magnetic monopoles in hyperbolic space, Vector bundles on algebraic varieties |publisher=Oxford Univ.Press |year=1987 |isbn= }}</ref> (replacing the complex surface ''TP''<sup>1</sup> by the comoplement of the anti-diagonal in ''P''<sup>1</sup> × ''P''<sup>1</sup>) and in terms of discrete Nahm equations by Murray and M.A.Singer.<ref>{{cite book |author=M.K.Murray |title=On the complete integrability of the discrete Nahm equations |publisher= |edition= |year=2000 |isbn= }}</ref>
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| == See also ==
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| {{colbegin|3}}
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| * [[Bogomolny equations]]
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| * [[Dirac string]]
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| * [[Dyon]]
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| * [[Felix Ehrenhaft]]
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| * [[Gauss's law for magnetism]]
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| * [[Halbach array]]
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| * [[Instanton]]
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| * [[Meron (physics)|Meron]]
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| * [[Soliton (topological)|Soliton]]
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| * [['t Hooft–Polyakov monopole]]
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| * [[Wu–Yang monopole]]
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| {{colend}}
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| ==Notes==
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| {{reflist|2}}
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| <!--NOTE: The numbers in square brackets [4] and [15] have been left for the editor to identify where they belong in the text.-->
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| *[4] {{cite book |author=N.J.Hitchin, M.K.Murray |title=Spectral curves and the ADHM method |publisher= |year=1988 |isbn= }}
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| *[15] {{cite book |author=P.M.Sutcliffe |title=BPS monopoles |publisher= |edition= |year=1997 |isbn= }}
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| ==References==
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| *{{cite book | author=Brau, Charles A. | title=Modern Problems in Classical Electrodynamics | publisher=Oxford University Press | year=2004 | isbn=0-19-514665-4}}
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| *{{cite book | author=Jackson, John David | title=Classical Electrodynamics | edition=3rd | location=New York | publisher=Wiley | year=1999 | isbn=0-471-30932-X}}
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| *{{cite journal |last=Milton |first=Kimball A. |title=Theoretical and experimental status of magnetic monopoles |journal=Reports on Progress in Physics |volume=69 |issue=6 |date=June 2006 |pages=1637–1711 |doi=10.1088/0034-4885/69/6/R02 |arxiv=hep-ex/0602040|bibcode = 2006RPPh...69.1637M }}
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| *{{cite book | author=Shnir, Yakov M. | title=Magnetic Monopoles | publisher=Springer-Verlag | year=2005 | isbn=3-540-25277-0}}
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| ==External links==
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| * [http://arxiv.org/abs/hep-ex/0302011 Magnetic Monopole Searches (lecture notes)]
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| * [http://pdg.lbl.gov/2004/listings/s028.pdf Particle Data Group summary of magnetic monopole search]
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| * [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.
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| * [http://www.drillingsraum.com/magnetic_monopole/magnetic_monopole.html Interview with Jonathan Morris] about magnetic monopoles and magnetic monopole quasiparticles. Drillingsraum, April 16, 2010
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| * [http://www.nature.com/news/2009/090903/full/news.2009.881.html ''Nature'', 2009]
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| * [http://www.sciencedaily.com/releases/2009/09/090903163725.htm ''Sciencedaily'', 2009]
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| * {{cite news|title=Observation of Magnetic Monopoles in Spin Ice
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| |author=H. Kadowaki, N. Doi, Y. Aoki, Y.Tabata, T.J. Sato, J.W. Lynn, K. Matsuhira, Z. Hiroi
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| |year=2009
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| |location=Japan
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| |url=http://arxiv.org/pdf/0908.3568v2.pdf
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| |arxiv=0908.3568v2
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| }}
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| * [http://www.youtube.com/watch?v=QmBng7Y7mgk ''Video of lecture by Paul Dirac on magnetic monopoles'', 1975]
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| {{SpringerEOM attribution |id=magnetic_monopole |title=Magnetic Monopole |name=N. Hitchin }}
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| {{DEFAULTSORT:Magnetic Monopole}}
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| [[Category:Magnetic monopoles| ]]
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| [[Category:Magnetism]]
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| [[Category:Quantum field theory]]
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| [[Category:Hypothetical particles]]
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| [[Category:Unsolved problems in physics]]
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