Magnetic monopole - Knowledge

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energy of the colliding particles can be produced. Beyond this, very little is known theoretically about the creation of magnetic monopoles in high-energy particle collisions. This is due to their large magnetic charge, which invalidates all the usual calculational techniques. As a consequence, collider-based searches for magnetic monopoles cannot, as yet, provide lower bounds on the mass of magnetic monopoles. They can however provide upper bounds on the probability (or cross section) of pair production, as a function of energy.

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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 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.'

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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.

3989:{\displaystyle {\begin{aligned}\mathbf {E} =&-\nabla \phi _{\mathrm {e} }-{\frac {\partial \mathbf {A} _{\mathrm {e} }}{\partial t}}-{\frac {1}{\varepsilon _{0}}}\nabla \times \mathbf {A} _{\mathrm {m} }\\\mathbf {B} =&-\mu _{0}\nabla \phi _{\mathrm {m} }-\mu _{0}{\frac {\partial \mathbf {A} _{\mathrm {m} }}{\partial t}}+\nabla \times \mathbf {A} _{\mathrm {e} }\\\end{aligned}}} 2428: 3748:{\displaystyle {\begin{aligned}\mathbf {E} =&-\nabla \phi _{\mathrm {e} }-{\frac {1}{c}}{\frac {\partial \mathbf {A} _{\mathrm {e} }}{\partial t}}-\nabla \times \mathbf {A} _{\mathrm {m} }\\\mathbf {B} =&-\nabla \phi _{\mathrm {m} }-{\frac {1}{c}}{\frac {\partial \mathbf {A} _{\mathrm {m} }}{\partial t}}+\nabla \times \mathbf {A} _{\mathrm {e} }\\\end{aligned}}} 8635: 8293: 7915: 2938:{\displaystyle {\begin{aligned}\Box \phi _{\mathrm {e} }=&-{\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}\\\Box \mathbf {A} _{\mathrm {e} }=&-\mu _{0}\mathbf {j} _{\mathrm {e} }\\\Box \phi _{\mathrm {m} }=&-{\frac {\rho _{\mathrm {m} }}{\mu _{0}}}\\\Box \mathbf {A} _{\mathrm {m} }=&-\varepsilon _{0}\mathbf {j} _{\mathrm {m} }\\\end{aligned}}} 3352: 7593: 7747: 3188: 1344: 3170:{\displaystyle {\begin{aligned}\Box \phi _{\mathrm {e} }=&-{\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}\\\Box \mathbf {A} _{\mathrm {e} }=&-\mu _{0}\mathbf {j} _{\mathrm {e} }\\\Box \phi _{\mathrm {m} }=&-\rho _{\mathrm {m} }\\\Box \mathbf {A} _{\mathrm {m} }=&-{\frac {\mathbf {j} _{\mathrm {m} }}{c^{2}}}\\\end{aligned}}} 8301: 7444: 2696:{\displaystyle {\begin{aligned}\Box \phi _{\mathrm {e} }=&-4\pi \rho _{\mathrm {e} }\\\Box \mathbf {A} _{\mathrm {e} }=&-{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {e} }\\\Box \phi _{\mathrm {m} }=&-4\pi \rho _{\mathrm {m} }\\\Box \mathbf {A} _{\mathrm {m} }=&-{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {m} }\\\end{aligned}}} 7013: 6756: 7138: 2128: 6881: 9792:, 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. 7967: 2288: 10622:

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

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detector. The particles it is looking for damage the plastic sheets that comprise the nuclear track detectors along their path, with various identifying features. Further, the aluminum bars can trap sufficiently slowly moving magnetic monopoles. The bars can then be analyzed by passing them through a

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Passing a magnetic monopole through a coil of wire induces a net current in the coil. This is not the case for a magnetic dipole or higher order magnetic pole, for which the net induced current is zero, and hence the effect can be used as an unambiguous test for the presence of magnetic monopoles. In

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If you imagine a big sphere in space, you can deform an infinitesimal loop that starts and ends at the north pole as follows: stretch out the loop over the western hemisphere until it becomes a great circle (which still starts and ends at the north pole) then let it shrink back to a little loop while

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

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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

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High-energy particle colliders have been used to try to create magnetic monopoles. Due to the conservation of magnetic charge, magnetic monopoles must be created in pairs, one north and one south. Due to conservation of energy, only magnetic monopoles with masses less than half of the center of mass

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According to standard inflationary cosmology, magnetic monopoles produced before inflation would have been diluted to an extremely low density today. Magnetic monopoles may also have been produced thermally after inflation, during the period of reheating. However, the current bounds on the reheating

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simply connected, doesn't have quantized charges and does not admit Dirac monopoles. The mathematical definition is equivalent to the physics definition provided that—following Dirac—gauge fields are allowed that are defined only patch-wise, and the gauge field on different patches are glued after a

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searches for these particles based on theories that define them as long lived (they do not quickly decay), as well as being highly ionizing (their interaction with matter is predominantly ionizing). In 2019 the search for magnetic monopoles in the ATLAS detector reported its first results from data

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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

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Magnetic monopoles have also inspired condensed-matter physicists to discover analogous states and excitations in systems such as spin ices and Bose–Einstein condensates. However, despite the importance of those developments in their own fields, they do not resolve the question of the existence of

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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

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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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Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975 and 1982 produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive. Therefore, whether monopoles exist remains an open question. Further advances in

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In the universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black

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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

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transformation, it would be the other way around. The key empirical fact is that all particles ever observed have the same ratio of magnetic charge to electric charge. Duality transformations can change the ratio to any arbitrary numerical value, but cannot change that all particles have the same

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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

242:. Since 2009, numerous news reports from the popular media have incorrectly described these systems as the long-awaited discovery of the magnetic monopoles, but the two phenomena are only superficially related to one another. These condensed-matter systems remain an area of active research. (See 9700:

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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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

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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). To do this with as little energy as possible, you should leave only the U(1) gauge group in the

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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

7753: 1131: 860: 3519:{\displaystyle {\begin{aligned}&{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\phi _{\mathrm {e} }+\nabla \cdot \mathbf {A} _{\mathrm {e} }=0\\&{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\phi _{\mathrm {m} }+\nabla \cdot \mathbf {A} _{\mathrm {m} }=0\\\end{aligned}}} 1684: 8487: 8145: 7450: 7600: 3341:{\displaystyle {\begin{aligned}&{\frac {1}{c}}{\frac {\partial }{\partial t}}\phi _{\mathrm {e} }+\nabla \cdot \mathbf {A} _{\mathrm {e} }=0\\&{\frac {1}{c}}{\frac {\partial }{\partial t}}\phi _{\mathrm {m} }+\nabla \cdot \mathbf {A} _{\mathrm {m} }=0\\\end{aligned}}} 8687:

Consider a system consisting of a single stationary electric monopole (an electron, say) and a single stationary magnetic monopole, which would not exert any forces on each other. Classically, the electromagnetic field surrounding them has a momentum density given by the

8476:{\displaystyle {\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}}} 1221: 8679:

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

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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 & R.S. Perry (September 3, 2009) . "Dirac Strings and Magnetic Monopoles in Spin Ice

7293: 5781: 1964: 1215: 9692:(GUT) 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. 6889: 6632: 2277:{\displaystyle {\begin{aligned}\mathbf {F} ={}&q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)+\\&{\frac {q_{\mathrm {m} }}{\mu _{0}}}\left(\mathbf {B} -\mathbf {v} \times {\frac {\mathbf {E} }{c^{2}}}\right)\end{aligned}}} 470:

relate the electric and magnetic fields to each other and to the distribution of electric charge and current. The standard equations provide for electric charge, but they posit zero magnetic charge and current. Except for this constraint, the equations are

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Zhong, Fang; Nagosa, Naoto; Takahashi, Mei S.; Asamitsu, Atsushi; Mathieu, Roland; Ogasawara, Takeshi; Yamada, Hiroyuki; Kawasaki, Masashi; Tokura, Yoshinori; Terakura, Kiyoyuki (2003). "The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space".

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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

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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

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If magnetic charge does not exist – or if it exists but is absent 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

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make predictions about what the horizon volume was, which lead to predictions about present-day monopole density. Early models predicted an enormous density of monopoles, in clear contradiction to the experimental evidence. This was called the

8134:{\displaystyle {\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}}} 5648: 5883: 10108:, 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 9724:

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

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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.

2423:{\displaystyle {\begin{aligned}\mathbf {F} ={}&q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)+\\&q_{\mathrm {m} }\left(\mathbf {B} -\mathbf {v} \times {\frac {\mathbf {E} }{c^{2}}}\right)\end{aligned}}} 9808:

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 spacetime, the monopoles have a finite mass. Monopoles occur in

4084: 442:, 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 1581: 1890: 207:, a string theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen". These theories are not necessarily inconsistent with the experimental evidence. In some theoretical 8642:

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.

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ratio. Since this is the case, a duality transformation can be made that sets this ratio at zero, so that all particles have no magnetic charge. This choice underlies the "conventional" definitions of electricity and magnetism.

9159: 4357: 10769:. 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 4247: 7910:{\displaystyle {\tilde {F}}^{\alpha \beta }=\mu _{0}c(\partial ^{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }A_{\mathrm {m} }^{\alpha })+\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {e} }\nu }} 9659:

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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drastically reduce the predicted number of magnetic monopoles, to a density small enough to make it unsurprising that humans have never seen one. This resolution of the "monopole problem" was regarded as a success of

10285:". 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 1038: 770: 11319:, September 3, 2009. "Oleg Tchernyshyov at Johns Hopkins University 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." 7198: 5466: 5228: 1822: 8630:{\displaystyle {\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}}} 8288:{\displaystyle {\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}}} 4985: 4805: 1592: 952: 756: 7588:{\displaystyle {\tilde {F}}^{\alpha \beta }=\partial ^{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }A_{\mathrm {m} }^{\alpha }+\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {e} }\nu }} 5404: 5166: 9728:

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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The Dirac monopole is a singular solution of Maxwell's equation (because it requires removing the worldline from spacetime); in more sophisticated theories, it is superseded by a smooth solution such as the

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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

7742:{\displaystyle F^{\alpha \beta }=\partial ^{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }A_{\mathrm {e} }^{\alpha }-\mu _{0}c\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {m} }\nu }} 10317:

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

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Maxwell's equations can also be written in a fully symmetric form if one allows for "magnetic charge" analogous to electric charge. With the inclusion of a variable for the density of magnetic charge, say

1339:{\displaystyle \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)} 33:

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

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Monopole"), there has never been reproducible evidence for the existence of magnetic monopoles. The lack of such events places an upper limit on the number of monopoles of about one monopole per 10

7439:{\displaystyle F^{\alpha \beta }=\partial ^{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }A_{\mathrm {e} }^{\alpha }-\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {m} }\nu }} 5236: 4998: 4813: 4633: 10240:

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

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materials. While these should not be confused with hypothetical elementary monopoles existing in the vacuum, they nonetheless have similar properties and can be probed using similar techniques.

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field were created and studied in a spinor Bose–Einstein condensate. This constitutes the first example of a quasi-magnetic monopole observed within a system governed by quantum field theory.

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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

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with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from

4620: 4291: 4571: 7211: 7008:{\displaystyle \partial ^{\alpha }\partial _{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {m} }^{\alpha }={\frac {4\pi }{c}}J_{\mathrm {m} }^{\beta }} 6751:{\displaystyle \partial ^{\alpha }\partial _{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {e} }^{\alpha }={\frac {4\pi }{c}}J_{\mathrm {e} }^{\beta }} 5683: 6600: 6478: 1724: 902: 7133:{\displaystyle \partial ^{\alpha }\partial _{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {m} }^{\alpha }=\varepsilon _{0}J_{\mathrm {m} }^{\beta }} 1896: 1145: 9950:, 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 12646:

Giblin, S. R.; Bramwell, S. T.; Holdsworth, P. C. W.; Prabhakaran, D.; Terry, I. (February 13, 2011). "Creation and measurement of long-lived magnetic monopole currents in spin ice".

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Bramwell, S. T.; Giblin, S. R.; Calder, S.; Aldus, R.; Prabhakaran, D.; Fennell, T. (October 15, 2009). "Measurement of the charge and current of magnetic monopoles in spin ice".

5965: 10411:, installed at the Large Hadron Collider, is currently searching for magnetic monopoles and large supersymmetric particles using nuclear track detectors and aluminum bars around 9004: 6876:{\displaystyle \partial ^{\alpha }\partial _{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {e} }^{\alpha }=\mu _{0}J_{\mathrm {e} }^{\beta }} 411:

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

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is the only known example, there are always finite-mass monopoles. For ordinary electromagnetism, the upper mass bound is not very useful because it is about same size as the

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of the principal bundle, and depends only upon the choice of the principal bundle, and not the specific connection over it. In other words, it is a topological invariant.

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Updates to the theoretical and experimental searches in matter can be found in the reports by G. Giacomelli (2000) and by S. Balestra (2011) in the Bibliography section.

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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

10214:(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 10229:. 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 12209:

T. Ollikainen; K. Tiurev; A. Blinova; W. Lee; D. S. Hall; M. Möttönen (2017). "Experimental Realization of a Dirac Monopole through the Decay of an Isolated Monopole".

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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

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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

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of the system from an effective gas of these quasiparticles was also described. This research went on to win the 2012 Europhysics Prize for condensed matter physics.

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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

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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

9490: 10364:. In his paper it was demonstrated that the path of the cosmic ray event that was claimed due to a magnetic monopole could be reproduced by the path followed by a 4007: 9774:. Lie groups are homogeneous, 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 9751:. This is the Dirac quantization condition, and it is a topological condition that demands that the long distance U(1) gauge field configurations be consistent. 9711:

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.

1529: 1833: 10077:. So a topological classification of the possible connections is reduced to classifying the transition functions. The transition function maps the strip to 8928: 5889: 4525:{\displaystyle {\tilde {F}}^{\alpha \beta }=({\tilde {F}}^{01},{\tilde {F}}^{02},{\tilde {F}}^{03},\;{\tilde {F}}^{23},{\tilde {F}}^{31},{\tilde {F}}^{12})} 12154:, updated August 2015 by D. Milstead and E.J. Weinberg. "To date there have been no confirmed observations of exotic particles possessing magnetic charge." 9098: 128:

in the nineteenth century showed that the magnetism of lodestones was properly explained not by magnetic monopole fluids, but rather by a combination of

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above the particle), and another set of functions for the "southern hemisphere". These two vector potentials are matched at the "equator" (the plane

4137: 385:. This is an expression of the field as the sum of component fields with specific mathematical forms. The first term in the expansion is called the 1126:{\displaystyle -\nabla \times \mathbf {E} -{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}={\frac {4\pi }{c}}\mathbf {j} _{\mathrm {m} }} 855:{\displaystyle \nabla \times \mathbf {B} -{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}={\frac {4\pi }{c}}\mathbf {j} _{\mathrm {e} }} 10404:

collected from the LHC Run 2 collisions at center of mass energy of 13 TeV, which at 34.4 fb is the largest dataset analyzed to date.

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Arkani-Hamed, Nima; Motl, Luboš; Nicolis, Alberto; Vafa, Cumrun (2007). "The string landscape, black holes and gravity as the weakest force".

11293: 10273:. According to that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetry breaking took place. 7144: 5410: 5172: 1774: 1679:{\displaystyle \nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {j} _{\mathrm {e} }} 12262:

Yakaboylu, E.; Deuchert, A.; Lemeshko, M. (December 6, 2017). "Emergence of Non-Abelian Magnetic Monopoles in a Quantum Impurity Problem".

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where collective behavior leads to emergent phenomena that resemble magnetic monopoles in certain respects, including most prominently the

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Ray, M. W.; Ruokokoski, E.; Kandel, S.; Möttönen, M.; Hall, D. S. (2014). "Observation of Dirac monopoles in a synthetic magnetic field".

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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

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would allow, therefore the correlation length for making magnetic monopoles must be at least as big as the horizon size determined by the

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is something whose magnetic field is predominantly or exactly described by the magnetic dipole term of the multipole expansion. The term

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is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist.

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temperature span 18 orders of magnitude and as a consequence the density of magnetic monopoles today is not well constrained by theory.

9826:, the lightest charged particles cannot be too heavy. The lightest monopole should have a mass less than or comparable to its charge in 5357: 5119: 14257: 12052:"Search for magnetic monopoles and stable high-electric-charge objects in 13 TeV proton-proton collisions with the ATLAS detector" 8812:

However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the

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Heras, J. A.; Baez, G. (2009). "The covariant formulation of Maxwell's equations expressed in a form independent of specific units".

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Price, P. B.; Shirk, E. K.; Osborne, W. Z.; Pinsky, L. S. (August 25, 1975). "Evidence for Detection of a Moving Magnetic Monopole".

9813:, and there the core size is the lattice size. In general, they are expected to occur whenever there is a short-distance regulator. 1730: 14139: 1411: 12020: 9256:

If we maximally extend the definition of the vector potential for the southern hemisphere, it is defined everywhere except for a

12115:

If the structures of the magnetic fields appear to be magnetic monopoles, that are macroscopic in size, then this is a wormhole.

8838:

of an electrically charged particle (a "probe charge") that orbits the "equator" generally changes by a phase, much like in the

5346:{\displaystyle A_{\mathrm {m} }^{\alpha }=(A_{\mathrm {m} }^{0},A_{\mathrm {m} }^{1},A_{\mathrm {m} }^{2},A_{\mathrm {m} }^{3})} 5108:{\displaystyle A_{\mathrm {e} }^{\alpha }=(A_{\mathrm {e} }^{0},A_{\mathrm {e} }^{1},A_{\mathrm {e} }^{2},A_{\mathrm {e} }^{3})} 4923:{\displaystyle J_{\mathrm {m} }^{\alpha }=(J_{\mathrm {m} }^{0},J_{\mathrm {m} }^{1},J_{\mathrm {m} }^{2},J_{\mathrm {m} }^{3})} 4743:{\displaystyle J_{\mathrm {e} }^{\alpha }=(J_{\mathrm {e} }^{0},J_{\mathrm {e} }^{1},J_{\mathrm {e} }^{2},J_{\mathrm {e} }^{3})} 10163:

in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension

6228: 14277: 13057: 12991: 12972: 12953: 12934: 12861: 11743: 11405: 11376: 10337:

There have been many searches for preexisting magnetic monopoles. Although there has been one tantalizing event recorded, by

10233:

of one and an electric charge of one, and there are no lighter particles that conserve these values. On the other hand, the

9970:

is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over

11430: 5479: 2043:{\displaystyle -\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=\mu _{0}\mathbf {j} _{\mathrm {m} }} 9613:

for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.

9249:—provides a natural explanation of charge quantization, without invoking the need for magnetic monopoles; but only if the 13181: 8816:

at the origin. We must define one set of functions for the vector potential on the "northern hemisphere" (the half-space

472: 10395:

currently has the most stringent cross section limits for magnetic monopoles of 1 and 2 Dirac charges, produced through

12027:. International symposium on lepton and photon interactions at high energies, Aug 21, 1975. p. 967. Archived from 10549:—except insofar as studies of analogous situations can help confirm that the mathematical analyses involved are sound. 9876: 4297: 1429: 289: 4577: 14037: 12824: 11950: 10808: 10735: 9902: 9668: 9282: 7288:{\displaystyle \partial _{\alpha }A_{\mathrm {e} }^{\alpha }=0,\quad \partial _{\alpha }A_{\mathrm {m} }^{\alpha }=0} 5776:{\displaystyle F^{\alpha \beta }=-F^{\beta \alpha },\quad {\tilde {F}}^{\alpha \beta }=-{\tilde {F}}^{\beta \alpha }} 4253: 315: 11105: 10218:, of which the most basic state was a monopole. The charge on magnetic monopoles predicted by GUTs is either 1 or 2 9884: 8735:

must also be quantized. This means that if even a single magnetic monopole existed in the universe, and the form of

4536: 297: 14118: 10326:

loop the induced current is long-lived. By using a highly sensitive "superconducting quantum interference device" (

10237:, essentially a heavy electron, can decay into the electron plus two quanta of energy, and hence it is not stable. 1959:{\displaystyle -\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {j} _{\mathrm {m} }} 1210:{\displaystyle \mathbf {F} =q_{\mathrm {e} }\left(\mathbf {E} +{\frac {\mathbf {v} }{c}}\times \mathbf {B} \right)} 12419:

Qi, X.-L.; Li, R.; Zang, J.; Zhang, S.-C. (2009). "Inducing a Magnetic Monopole with Topological Surface States".

9974:

connection over it. Note that a connection over a trivial bundle can never give us a nontrivial principal bundle.

9243:

other widely accepted explanation of charge quantization has appeared. (The concept of local gauge invariance—see

14272: 9644:. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of 9260:

line stretched from the origin in the direction towards the northern pole. This semi-infinite line is called the

6484: 6350: 1695: 873: 371: 6212:{\displaystyle \partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {\mu _{0}}{c}}J_{\mathrm {m} }^{\beta }} 13153: 9880: 9468:{\displaystyle \prod _{s}\left(1+ieA_{\mu }{dx^{\mu } \over ds}\,ds\right)=\exp \left(ie\int A\cdot dx\right).} 4991: 293: 10650:

is called a "synthetic magnetic field". In January 2014, it was reported that monopole quasiparticles for the

7951:, and simultaneously change the fields and charges everywhere in the universe as follows (in Gaussian units): 13144: 11159:

Cabrera, Blas (May 17, 1982). "First Results from a Superconductive Detector for Moving Magnetic Monopoles".

10149:

This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to

6041:{\displaystyle \partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {4\pi }{c}}J_{\mathrm {m} }^{\beta }} 958: 9080:{\displaystyle {\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{2\pi \varepsilon _{0}\hbar c^{2}}}\in \mathbb {Z} } 238:, but since they move independently, they can be treated for many purposes as independent magnetic monopole 14134: 12759:

Pietilä, Ville; Möttönen, Mikko (2009). "Creation of Dirac Monopoles in Spinor Bose–Einstein Condensates".

10695: 10459: 866: 141: 10352:

Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in

14252: 13139: 11615: 11043: 9696: 6123:{\displaystyle \partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {1}{c}}J_{\mathrm {m} }^{\beta }} 12147: 11488:

Rindler, Wolfgang (November 1989). "Relativity and electromagnetism: The force on a magnetic monopole".

10451:

groups have used the term "magnetic monopole" to describe a different and largely unrelated phenomenon.

13134: 12109: 11824:

Zel'dovich, Ya. B.; Khlopov, M. Yu. (1978). "On the concentration of relic monopoles in the universe".

10700: 10226: 10191:

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

1027:{\displaystyle -\nabla \times \mathbf {E} -{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}=0} 11312: 9980: 2117:{\displaystyle \mathbf {F} =q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)} 1390:

of a test particle, all defined analogously to the related quantities of electric charge and current;

14210: 10720: 10436: 10424: 8747: 11681: 11651: 13622: 10553: 10448: 10400: 10322:

a wire with finite resistance, the induced current quickly dissipates its energy as heat, but in a

10170:. Another way is to examine the type of topological singularity at a point with the homotopy group 10131: 9865: 9610: 9269: 8839: 278: 227: 133: 12025:

Proceedings of the 1975 international symposium on lepton and photon interactions at high energies

11014:"The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, Literature and General Information" 10111: 9212: 5660: 5643:{\displaystyle {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}(\mathbf {F} \cdot \mathbf {v} ,\;-\mathbf {F} )} 14267: 14178: 13813: 13440: 13174: 12562: 12480: 12127: 11265: 10875:

Castelnovo, C.; Moessner, R.; Sondhi, S. L. (January 3, 2008). "Magnetic monopoles in spin ice".

10605: 9869: 9196: 8871:

Because the electron returns to the same point after the full trip around the equator, the phase

282: 10574:

One example of the work on magnetic monopole quasiparticles is a paper published in the journal

5878:{\displaystyle \partial _{\alpha }F^{\alpha \beta }={\frac {4\pi }{c}}J_{\mathrm {e} }^{\beta }} 187:, in fact, quantized, which is consistent with (but does not prove) the existence of monopoles. 14113: 13450: 11676: 11659: 10542: 9655:, when the flux leaked out from one of its ends it would be indistinguishable from a monopole. 9273: 8736: 8681: 7204: 4130: 4103: 3181: 1407: 633: 603: 463: 196: 145: 11028: 11013: 10798: 9716: 9592:{\displaystyle e\oint _{\partial D}A\cdot dx=e\int _{D}(\nabla \times A)\,dS=e\int _{D}B\,dS.} 14200: 13115: 10730: 10396: 10392: 10291: 10286: 4079:{\displaystyle \Box =\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{{\partial t}^{2}}}} 532: 11065: 10071:. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle 211:, magnetic monopoles are unlikely to be observed, because they are too massive to create in 13086: 13020: 12884: 12778: 12717: 12657: 12520: 12438: 12377: 12281: 12228: 12177: 12073: 11975: 11914: 11875: 11833: 11790: 11668: 11624: 11595: 11542: 11497: 11470: 11228: 11168: 11130: 11077: 10962: 10894: 10740: 10338: 10306: 10211: 10207: 10195: 9963: 9810: 9689: 9316:, unit complex numbers under multiplication. For infinitesimal paths, the group element is 8831: 8813: 5675: 689:

Maxwell's equations and Lorentz force equation with magnetic monopoles: CGS-Gaussian units

360: 212: 208: 179:

magnetic monopoles exist in the universe, then all electric charge in the universe must be

70: 11860: 8776:

and is directed in the radial direction, located at the origin. Because the divergence of

1576:{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}} 8: 14215: 13198: 12151: 10670: 10589: 10546: 10481: 10470: 10455: 10361: 10360:. Price later retracted his claim, and a possible alternative explanation was offered by 10262: 9178: 8740: 8714:

Quantum mechanics dictates, however, that angular momentum is quantized as a multiple of

1885:{\displaystyle -\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=0} 382: 180: 66: 58: 42: 13090: 13024: 12888: 12782: 12721: 12661: 12524: 12442: 12381: 12285: 12232: 12181: 12077: 11979: 11927: 11918: 11902: 11879: 11837: 11802: 11794: 11672: 11628: 11599: 11546: 11501: 11474: 11232: 11172: 11134: 11081: 10966: 10898: 14262: 14158: 13167: 13149: 13102: 13076: 13036: 13010: 12946:

Classical Electrodynamics: From Image Charges to the Photon Mass and Magnetic Monopoles

12900: 12839: 12768: 12741: 12707: 12625: 12599: 12544: 12510: 12462: 12428: 12401: 12367: 12339: 12305: 12271: 12244: 12218: 12063: 12001: 11806: 11780: 11694: 11585: 11558: 11532: 11460: 11394: 11345: 11244: 11218: 11044:"Sur la possibilité d'existence de la conductibilité magnétique et du magnétisme libre" 10994: 10952: 10918: 10884: 10844:

Wen, Xiao-Gang; Witten, Edward, "Electric and magnetic charges in superstring models",

10770: 10600:, the magnetic moments were shown to align into interwoven tubelike bundles resembling 10597: 10342: 10210:

led many theorists to move on to attempt to combine them in a single theory known as a

10203: 10026: 9967: 9805: 8984:{\displaystyle {\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{2\pi \hbar }}\in \mathbb {Z} } 8794: 8676: 7930: 4099: 74: 12028: 5949:{\displaystyle \partial _{\alpha }F^{\alpha \beta }=\mu _{0}J_{\mathrm {e} }^{\beta }} 762: 356: 14227: 14195: 14108: 13332: 13259: 13053: 13040: 13032: 12987: 12968: 12949: 12930: 12904: 12857: 12820: 12794: 12745: 12733: 12698: 12617: 12536: 12466: 12454: 12393: 12343: 12309: 12297: 12089: 11993: 11946: 11845: 11739: 11734: 11554: 11426: 11401: 11372: 11349: 11297: 10986: 10978: 10910: 10804: 10637:, related to superfluid vorticity, which is mathematically analogous to the magnetic 10576: 10408: 10241: 9823: 9755: 8668: 395: 220: 204: 165: 13106: 12629: 12405: 12248: 11810: 11698: 11562: 9154:{\displaystyle 2{\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{\hbar c}}\in \mathbb {Z} } 148:, is the mathematical statement that magnetic monopoles do not exist. Nevertheless, 14205: 13976: 13751: 13612: 13597: 13279: 13190: 13094: 13028: 13001:

Milton, K. A. (2006). "Theoretical and experimental status of magnetic monopoles".

12892: 12790: 12786: 12725: 12665: 12609: 12548: 12528: 12446: 12385: 12331: 12293: 12289: 12236: 12185: 12085: 12081: 12005: 11983: 11922: 11883: 11841: 11798: 11686: 11632: 11550: 11505: 11337: 11332:

Gibney, Elizabeth (January 29, 2014). "Quantum cloud simulates magnetic monopole".

11236: 11176: 11138: 11085: 10970: 10943: 10922: 10902: 10745: 10690: 10388: 10282: 10257:

The length scale over which this special vacuum configuration exists is called the

10105: 10054: 9920: 9633:

have no interference fringes, because the phase factor for any charged particle is

8790: 8693: 1494:

Maxwell's equations then take the following forms (using the same notation above):

493: 476: 467: 348: 192: 129: 125: 96: 50: 12872: 12326:

Elizabeth Gibney (January 29, 2014). "Quantum cloud simulates magnetic monopole".

11248: 10998: 203:, have led to more compelling arguments (detailed below) that monopoles do exist. 77:

theories, which predict their existence. The known elementary particles that have

14173: 14098: 14082: 14022: 13432: 13357: 13347: 13337: 13249: 11451: 11046:[On the possible existence of magnetic conductivity and free magnetism]. 10725: 10710: 10685: 10357: 10323: 10011: 9916: 8689: 1498:

Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units

667: 618: 553: 439: 419: 381:

Mathematically, the magnetic field of an object is often described in terms of a

367: 235: 200: 137: 121: 78: 11887: 11209:

Polchinski, Joseph (February 1, 2004). "Monopoles, Duality, and String Theory".

11180: 403:, and so on. Any of these terms can be present in the multipole expansion of an 103:, or contain phenomena that are mathematically analogous to magnetic monopoles. 14232: 14146: 14103: 13839: 13627: 13400: 13322: 13317: 13239: 12648: 11729: 11142: 10618: 10541:

vorticity), or various other quantum fields. They are not directly relevant to

10522: 10505: 10501: 10428: 10302: 10086: 9800:

of the monopole. Outside the core, the monopole has only magnetic field energy.

9759: 9206: 9091: 8865: 7302: 4242:{\displaystyle F^{\alpha \beta }=(F^{01},F^{02},F^{03},\;F^{23},F^{31},F^{12})} 1403: 705: 682: 660: 653: 404: 375: 352: 333: 329: 13098: 12682: 12335: 12240: 11341: 11240: 14246: 14190: 14042: 14009: 13801: 13771: 13703: 13562: 13342: 13269: 13254: 12567: 11269: 10982: 10705: 10610: 10581: 10369: 10266: 10230: 10061:

to the open 2-ball such that their intersection is homeomorphic to the strip

9834: 9827: 9685: 9673:

In a U(1) gauge group with quantized charge, the group is a circle of radius

9602: 9257: 8835: 8782:

is equal to zero everywhere except for the locus of the magnetic monopole at

2440:

Maxwell's equations can also be expressed in terms of potentials as follows:

1137: 239: 230:

propose a structure superficially similar to a magnetic monopole, known as a

100: 89: 12729: 12613: 12450: 12389: 10824: 14168: 13718: 13708: 13698: 13459: 13410: 13352: 13274: 13229: 12913: 12798: 12737: 12621: 12540: 12458: 12397: 12301: 12093: 11997: 11089: 10990: 10914: 10675: 10601: 10295: 10199: 10143: 10058: 9780:, which is a lift of the identity. Going around the loop twice gets you to 9771: 9305: 9299: 9261: 9245: 8920: 4626: 2463: 1484: 1446: 149: 12590:

M.J.P. Gingras (2009). "Observing Monopoles in a Magnetic Analog of Ice".

11029:"Principles of Physics: Designed for Use as a Textbook of General Physics" 10774: 9253:

gauge group is compact, in which case we have magnetic monopoles anyway.)

455:

opposite strength, and the two poles cannot be separated from each other.

336:, has zero magnetic monopole charge. Therefore, the ordinary phenomena of 14183: 13951: 13854: 13849: 13766: 13761: 13691: 13645: 13602: 13567: 13526: 13418: 13382: 13244: 12372: 9838: 9767: 9478: 7941:

The generalized Maxwell's equations possess a certain symmetry, called a

4107: 12532: 11613:

Shanmugadhasan, S (1952). "The Dynamical Theory of Magnetic Monopoles".

11590: 11465: 10974: 10906: 9927:

is the gauge group, and it acts on each fiber of the bundle separately.

7193:{\displaystyle {\text{...}}={\frac {1}{c^{2}}}J_{\mathrm {m} }^{\beta }} 13925: 13819: 13809: 13791: 13681: 13582: 13517: 13234: 13159: 13081: 13015: 12896: 12131: 12051: 11785: 11690: 11223: 11061: 10627: 10538: 10474: 10432: 10353: 10341:

on the night of February 14, 1982 (thus, sometimes referred to as the "

8672: 5671: 5472: 4349: 1488: 522: 172: 85: 26: 12669: 12208: 12190: 12165: 9601:

So that the phase a charged particle gets when going in a loop is the

9264:

and its effect on the wave function is analogous to the effect of the

5461:{\displaystyle (\phi _{\mathrm {m} }/c,\;\mathbf {A} _{\mathrm {m} })} 5223:{\displaystyle (\phi _{\mathrm {e} }/c,\;\mathbf {A} _{\mathrm {e} })} 1817:{\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\rho _{\mathrm {m} }} 430:, corresponding to the fact that a dipole magnet typically contains a 244: 14067: 14057: 14027: 13920: 13886: 13879: 13756: 13746: 13741: 13713: 13481: 13264: 10715: 10580:

in September 2009, in which researchers described the observation of

10018: 9763: 4980:{\displaystyle (c\rho _{\mathrm {m} },\;\mathbf {j} _{\mathrm {m} })} 4800:{\displaystyle (c\rho _{\mathrm {e} },\;\mathbf {j} _{\mathrm {e} })} 645:(There should not be red and blue circles in the bottom right image.) 337: 257: 231: 169: 117: 11988: 11963: 11636: 11509: 10765:

For the convention where magnetic charge has the weber as unit, see

10567:

to describe the manipulation of magnetic monopole quasiparticles in

10442: 10244:. In this scenario, the dyons arise due to the configuration of the 10100:-bundle formulation, a gauge theory admits Dirac monopoles provided 9854: 347:

Instead, magnetism in ordinary matter is due to two sources. First,

267: 21: 14163: 13991: 13946: 13930: 13891: 13864: 13577: 13572: 13552: 13522: 13512: 13507: 13327: 13302: 13297: 13224: 12645: 12276: 12223: 12068: 10585: 10568: 10557: 10493: 10377: 10365: 10277: 10270: 10250: 10022: 10015: 9695:

The case of the U(1) gauge group is a special case because all its

9606: 9482: 9265: 8916: 8861: 947:{\displaystyle \nabla \cdot \mathbf {B} =4\pi \rho _{\mathrm {m} }} 751:{\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho _{\mathrm {e} }} 562: 473:

symmetric under the interchange of the electric and magnetic fields

447: 38: 12844: 12773: 12712: 12604: 12515: 12433: 11537: 10957: 10889: 10469:. A monopole of this kind, which would help to explain the law of 10435:

are potential magnetic monopoles, representing the entrance to an

9938:-bundle tells you how to glue fibers together at nearby points of 5399:{\displaystyle (\phi _{\mathrm {m} },\mathbf {A} _{\mathrm {m} })} 5161:{\displaystyle (\phi _{\mathrm {e} },\mathbf {A} _{\mathrm {e} })} 14072: 14062: 14032: 13986: 13981: 13956: 13874: 13859: 13786: 13781: 13686: 13671: 13617: 13592: 13557: 13486: 13468: 13207: 11266:"Magnetic Monopoles Detected in a Real Magnet for the First Time" 10857:

S. Coleman, "The Magnetic Monopole 50 years Later", reprinted in

10489: 10346: 10330:) one can, in principle, detect even a single magnetic monopole. 9232: 11735:

The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius

10800:

Dark Cosmos: In Search of Our Universe's Missing Mass and Energy

8647:

electrons have electric charge but not magnetic charge; after a

14052: 14047: 13676: 13663: 13654: 13476: 13390: 13289: 10593: 10521:); instead, they are sources for other fields, for example the 10497: 10485: 10373: 10245: 8996: 4095: 1450: 543: 443: 341: 62: 10616:

In another example, a paper in the February 11, 2011 issue of

10545:

or other aspects of particle physics, and do not help explain

1763:{\displaystyle \nabla \cdot \mathbf {B} =\rho _{\mathrm {m} }} 14077: 14017: 13869: 13728: 13607: 13547: 13502: 13395: 13373: 13216: 10327: 590: 12356: 13844: 13776: 13736: 13312: 13307: 11770: 11576:

Moulin, F. (2002). "Magnetic monopoles and Lorentz force".

11449:

Moulin, F. (2001). "Magnetic monopoles and Lorentz force".

10940: 10680: 10416: 10412: 10234: 10215: 9313: 9250: 99:

systems contain effective (non-isolated) magnetic monopole

34: 11861:"Cosmological production of superheavy magnetic monopoles" 10261:

of the system. A correlation length cannot be larger than

9915:

In mathematics, a (classical) gauge field is defined as a

12500: 12261: 12197:

real magnetic monopoles. Therefore, the search continues.

10825:"Particle Data Group summary of magnetic monopole search" 1436:, there are two conventions for defining magnetic charge 328:

All matter isolated to date, including every atom on the

251: 11294:

Making magnetic monopoles, and other exotica, in the lab

10874: 8900:. In various units, this condition can be expressed as: 6339:{\displaystyle f_{\alpha }=\left{\frac {v^{\beta }}{c}}} 475:. Maxwell's equations are symmetric when the charge and 407:, for example. However, in the multipole expansion of a 156:

conceivably exist, despite not having been seen so far.

11420: 11120: 10584:

resembling magnetic monopoles. A single crystal of the

1433: 12984:

The Leptonic Magnetic Monopole: Theory and Experiments

11066:"Quantised Singularities in the Electromagnetic Field" 8592: 8529: 8496: 8424: 8361: 8310: 8250: 8187: 8154: 8086: 8023: 7976: 168:

theory of magnetic charge started with a paper by the

111: 12563:"'Magnetricity' Observed And Measured For First Time" 11823: 10142:

The total magnetic flux is none other than the first

10114: 9983: 9616:

But if all particle charges are integer multiples of

9493: 9345: 9215: 9101: 9007: 8931: 8490: 8304: 8148: 7970: 7756: 7603: 7453: 7313: 7214: 7147: 7022: 6892: 6765: 6635: 6487: 6353: 6231: 6137: 6055: 5968: 5892: 5816: 5686: 5568: 5554:{\displaystyle f_{\alpha }=(f_{0},f_{1},f_{2},f_{3})} 5482: 5413: 5360: 5239: 5175: 5122: 5001: 4937: 4816: 4757: 4636: 4580: 4539: 4360: 4300: 4256: 4140: 4010: 3762: 3538: 3355: 3191: 2952: 2710: 2474: 2291: 2131: 2062: 1973: 1899: 1836: 1777: 1733: 1698: 1595: 1532: 1224: 1148: 1041: 968: 911: 876: 773: 715: 219:

below), and also too rare in the Universe to enter a

12128:

Quantised Singularities in the Electromagnetic Field

11308: 11306: 9331:

which implies that for finite paths parametrized by

8842:. This phase is proportional to the electric charge 681:

The extended Maxwell's equations are as follows, in

216: 183:(Dirac quantization condition). The electric charge 11652:"On Generalized Electromagnetism and Dirac Algebra" 11289: 11287: 9737:, the magnetic flux through the sphere is equal to 9477:The map from paths to group elements is called the 8860:of the source. Dirac was originally considering an 8711:, and is independent of the distance between them. 1502: 25:It is impossible to make magnetic monopoles from a 11393: 10312: 10122: 9998: 9715:going over the eastern hemisphere. This is called 9591: 9467: 9223: 9153: 9079: 8983: 8629: 8475: 8287: 8133: 7909: 7741: 7587: 7438: 7287: 7192: 7132: 7007: 6875: 6750: 6594: 6472: 6338: 6211: 6122: 6040: 5948: 5877: 5775: 5642: 5553: 5460: 5398: 5345: 5222: 5160: 5107: 4979: 4922: 4799: 4742: 4614: 4565: 4524: 4337: 4285: 4241: 4078: 3988: 3747: 3518: 3340: 3169: 2937: 2695: 2422: 2276: 2116: 2042: 1958: 1884: 1816: 1762: 1718: 1678: 1575: 1338: 1209: 1125: 1026: 946: 896: 854: 750: 116:Many early scientists attributed the magnetism of 11371:(Fourth ed.). Boston: Pearson. p. 339. 11303: 10477:in 1931, has never been observed in experiments. 10081:, and the different ways of mapping a strip into 8889:added to the wave function must be a multiple of 4338:{\displaystyle (-\mathbf {E} /c,\;-\mathbf {B} )} 14244: 13150:Creative Commons Attribution/Share-Alike License 12325: 11284: 10276:Cosmological models of the events following the 9833:So in a consistent holographic theory, of which 9754:When the U(1) gauge group comes from breaking a 9239:proportional to the elementary electric charge. 8883:must be unchanged, which implies that the phase 8851:of the probe, as well as to the magnetic charge 4615:{\displaystyle (-\mathbf {B} ,\;\mathbf {E} /c)} 12965:Classical Electrodynamics: A Modern Perspective 12817:The Geometry and Dynamics of Magnetic Monopoles 12758: 11300:, January 29, 2009. Retrieved January 31, 2009. 11260: 11258: 9796:neighborhood of one point, which is called the 4286:{\displaystyle (-\mathbf {E} ,\;-\mathbf {B} )} 12589: 12583: 12163: 11612: 11444: 11442: 11360: 11327: 11325: 11154: 11152: 9954:element associated to a path act on the fiber 4566:{\displaystyle (-\mathbf {B} ,\;\mathbf {E} )} 624:formed by two hypothetical magnetic monopoles. 245:§ "Monopoles" in condensed-matter systems 13175: 12870: 12814: 12418: 11712: 11710: 11708: 11204: 11202: 11116: 11114: 11108:, University of California, Berkeley, 2007–08 11101: 11099: 11016:. University Press – via Google Books. 10643:-field. Because of the similarity, the field 10014:. But consider what happens when we remove a 10006:the space of all possible connections of the 9942:. It starts with a continuous symmetry group 9288: 8739:is valid, all electric charges would then be 1453:-meter (A⋅m). The conversion between them is 1406:. For all other definitions and details, see 670:(black/white) and magnetic charge (red/blue). 175:in 1931. In this paper, Dirac showed that if 12981: 12854:Modern Problems in Classical Electrodynamics 11255: 11041: 10870: 10868: 10866: 8830:through the particle), and they differ by a 152:pointed out in 1894 that magnetic monopoles 16:Hypothetical particle with one magnetic pole 12496: 12494: 12138:, May 29, 1931. Retrieved February 1, 2014. 11738:. London: Faber and Faber. pp. 185–9. 11439: 11322: 11149: 11048:Séances de la Société Française de Physique 11031:. Century Company – via Google Books. 10186: 10021:from spacetime. The resulting spacetime is 9883:. Unsourced material may be challenged and 9844: 6595:{\displaystyle f_{\alpha }=\leftv^{\beta }} 6473:{\displaystyle f_{\alpha }=\leftv^{\beta }} 1719:{\displaystyle \nabla \cdot \mathbf {B} =0} 897:{\displaystyle \nabla \cdot \mathbf {B} =0} 479:density are zero everywhere, as in vacuum. 438:on the other side. This is analogous to an 296:. Unsourced material may be challenged and 13182: 13168: 13132: 13067:Sutcliffe, P. M. (1997). "BPS monopoles". 12911: 12481:"Artificial magnetic monopoles discovered" 12021:"Analysis of a Reported Magnetic Monopole" 11705: 11208: 11199: 11111: 11096: 5628: 5440: 5202: 4959: 4779: 4595: 4554: 4455: 4323: 4271: 4199: 13113: 13080: 13066: 13014: 12843: 12772: 12711: 12641: 12639: 12603: 12514: 12432: 12371: 12321: 12319: 12275: 12222: 12189: 12067: 11987: 11926: 11784: 11680: 11649: 11589: 11536: 11522: 11464: 11366: 11313:"Magnetic monopoles spotted in spin ices" 11222: 11211:International Journal of Modern Physics A 10956: 10888: 10863: 10194:During the early 1970s, the successes of 10116: 9986: 9903:Learn how and when to remove this message 9662: 9579: 9553: 9410: 9217: 9147: 9073: 8977: 7936: 316:Learn how and when to remove this message 13189: 13131:This article incorporates material from 12833: 12555: 12491: 11961: 11900: 11858: 11761:, section 6.11, equation (6.153), p. 275 11027:Magie, William Francis (June 26, 2018). 11011: 10592:was cooled to a temperature between 0.6 10571:, in analogy to the word "electricity". 10454:A true magnetic monopole would be a new 8864:whose wave function is described by the 8746:Although it would be possible simply to 2435: 1587:Ampère's law (with Maxwell's extension) 676: 124:. However, an improved understanding of 106: 29:. If a bar magnet is cut in half, it is 20: 12962: 12924: 11728: 11487: 11425:(4th ed.). John Wiley & Sons. 11158: 10443:"Monopoles" in condensed-matter systems 8696:, which is proportional to the product 8662: 4094:Maxwell's equations in the language of 693: 344:do not derive from magnetic monopoles. 14245: 13653: 13000: 12943: 12871:Hitchin, N. J.; Murray, M. K. (1988). 12636: 12316: 11575: 11448: 11421:Mansfield, M.; O'Sullivan, C. (2011). 11391: 11331: 10796: 10596:and 2.0 kelvin. Using observations of 458: 252:Poles and magnetism in ordinary matter 217:§ Searches for magnetic monopoles 13163: 13047: 12873:"Spectral curves and the ADHM method" 11643: 11060: 11026: 10936: 10934: 10932: 10790: 4089: 370:, the most important of which is the 12851: 11396:McGraw-Hill Encyclopaedia of Physics 9881:adding citations to reliable sources 9848: 9684:. Such a U(1) gauge group is called 496:density" variable in the equations, 294:adding citations to reliable sources 261: 159: 12815:Atiyah, M. F.; Hitchin, N. (1988). 12166:"The search for magnetic monopoles" 12049: 12018: 11928:10.1146/annurev.ns.34.120184.002333 11606: 5670:The electromagnetic tensor and its 446:and the negative charge is made of 112:Early science and classical physics 13: 11650:Fryberger, David (February 1989). 11106:Lecture notes by Robert Littlejohn 11070:Proceedings of the Royal Society A 10929: 10552:There are a number of examples in 10484:in systems of everyday particles ( 9541: 9502: 9126: 9114: 9029: 9017: 8953: 8941: 8456: 8435: 8339: 8321: 8114: 8095: 8001: 7985: 7897: 7881: 7844: 7829: 7814: 7799: 7729: 7713: 7666: 7651: 7636: 7621: 7575: 7559: 7525: 7510: 7495: 7480: 7426: 7410: 7376: 7361: 7346: 7331: 7305:–Ferrari-Shanmugadhasan relation) 7268: 7253: 7231: 7216: 7179: 7119: 7089: 7074: 7064: 7049: 7034: 7024: 6994: 6959: 6944: 6934: 6919: 6904: 6894: 6862: 6832: 6817: 6807: 6792: 6777: 6767: 6737: 6702: 6687: 6677: 6662: 6647: 6637: 6542: 6512: 6408: 6378: 6284: 6256: 6198: 6139: 6109: 6057: 6027: 5970: 5935: 5894: 5864: 5818: 5449: 5423: 5387: 5370: 5329: 5309: 5289: 5269: 5246: 5211: 5185: 5149: 5132: 5091: 5071: 5051: 5031: 5008: 4968: 4950: 4906: 4886: 4866: 4846: 4823: 4788: 4770: 4726: 4706: 4686: 4666: 4643: 4061: 4050: 4018: 3976: 3962: 3950: 3942: 3931: 3909: 3900: 3865: 3851: 3822: 3814: 3803: 3791: 3782: 3735: 3721: 3709: 3701: 3690: 3668: 3659: 3634: 3620: 3608: 3600: 3589: 3567: 3558: 3500: 3486: 3477: 3462: 3458: 3422: 3408: 3399: 3384: 3380: 3322: 3308: 3299: 3284: 3280: 3251: 3237: 3228: 3213: 3209: 3145: 3119: 3098: 3076: 3057: 3023: 2990: 2966: 2925: 2891: 2858: 2834: 2815: 2781: 2748: 2724: 2683: 2644: 2623: 2595: 2576: 2537: 2516: 2488: 2365: 2314: 2207: 2154: 2077: 2034: 2004: 1994: 1977: 1950: 1930: 1920: 1903: 1867: 1857: 1840: 1808: 1778: 1754: 1734: 1699: 1670: 1640: 1630: 1596: 1555: 1533: 1430:International System of Quantities 1292: 1239: 1163: 1117: 1082: 1072: 1045: 1009: 999: 972: 938: 912: 877: 846: 811: 801: 774: 742: 716: 14: 14289: 14258:Hypothetical elementary particles 13125: 13117:Magnetism of Elementary Particles 12836:Magnetic Monopole Bibliography-II 12107:"Could wormholes really exist?". 10254:state into which they can decay. 9770:, but where all closed loops are 9312:In electrodynamics, the group is 9053: 8967: 8684:and still have magnetic charges. 1414:form, remove the factors of 234:. The ends of a flux tube form a 14226: 14119:Timeline of particle discoveries 12136:Proceedings of the Royal Society 11012:Chisholm, Hugh (June 26, 2018). 9999:{\displaystyle \mathbb {R} ^{4}} 9853: 9816: 8667:One of the defining advances in 8611: 8597: 8509: 8500: 8450: 8429: 8333: 8315: 8269: 8255: 8167: 8158: 7945:. One can choose any real angle 5633: 5621: 5613: 5443: 5381: 5205: 5143: 4962: 4782: 4597: 4588: 4556: 4547: 4328: 4308: 4276: 4264: 3970: 3936: 3876: 3859: 3808: 3768: 3729: 3695: 3645: 3628: 3594: 3544: 3494: 3416: 3316: 3245: 3139: 3113: 3051: 3017: 2919: 2885: 2809: 2775: 2677: 2638: 2570: 2531: 2395: 2385: 2377: 2342: 2334: 2326: 2297: 2249: 2239: 2231: 2182: 2174: 2166: 2137: 2105: 2097: 2089: 2064: 2028: 1998: 1984: 1944: 1924: 1910: 1861: 1847: 1785: 1741: 1706: 1664: 1634: 1603: 1540: 1327: 1314: 1304: 1274: 1261: 1251: 1226: 1198: 1185: 1175: 1150: 1111: 1076: 1052: 1003: 979: 919: 884: 840: 805: 781: 723: 589: 542: 266: 12982:Lochak, G.; Stumpf, H. (2015). 12808: 12752: 12676: 12473: 12412: 12350: 12255: 12202: 12157: 12141: 12120: 12100: 12043: 12012: 11962:Brumfiel, Geoff (May 6, 2004). 11955: 11945:(New York: Basic Books) p. 185 11935: 11894: 11852: 11817: 11764: 11758: 11752: 11722: 11716: 11569: 11516: 11481: 11414: 11385: 11369:Introduction to electrodynamics 11193: 11187: 11054: 10797:Hooper, Dan (October 6, 2009). 10766: 10759: 10504:. They are not sources for the 10399:pair production. A team led by 10313:Searches for magnetic monopoles 9804:Hence, the Dirac monopole is a 9609:has a magnetic flux, there are 9605:through the loop. When a small 9293: 9283:'t Hooft–Polyakov monopole 7251: 5788:The generalized equations are: 5722: 1396:is the particle's velocity and 372:electron magnetic dipole moment 195:, particularly developments in 13154:GNU Free Documentation License 13148:, which is licensed under the 13003:Reports on Progress in Physics 12915:Magnetic Monopole Bibliography 12819:. Princeton University Press. 12791:10.1103/physrevlett.103.030401 12294:10.1103/PhysRevLett.119.235301 12086:10.1103/PhysRevLett.124.031802 11773:Journal of High Energy Physics 11035: 11020: 11005: 10851: 10838: 10817: 10563:Some researchers use the term 9550: 9538: 7855: 7795: 7764: 7461: 6561: 6439: 6298: 6156: 6074: 5987: 5758: 5730: 5637: 5609: 5548: 5496: 5455: 5414: 5393: 5361: 5340: 5260: 5217: 5176: 5155: 5123: 5102: 5022: 4974: 4938: 4917: 4837: 4794: 4758: 4737: 4657: 4609: 4581: 4560: 4540: 4519: 4507: 4485: 4463: 4440: 4418: 4396: 4386: 4368: 4332: 4301: 4280: 4257: 4236: 4157: 1423: 1: 11803:10.1088/1126-6708/2007/06/060 11400:(2nd ed.). McGraw-Hill. 10784: 9822:hole can decay completely by 9707:The argument is topological: 9688:. Any U(1) that comes from a 8789:, one can locally define the 1481:1 Wb = 1 H⋅A = (1 H⋅m)(1 A⋅m) 1445:, each with different units: 450:, but a magnetic dipole does 41:, but would instead be a new 14278:Unsolved problems in physics 14135:History of subatomic physics 11846:10.1016/0370-2693(78)90232-0 11367:Griffiths, David J. (2013). 10500:); in other words, they are 10123:{\displaystyle \mathbb {R} } 9224:{\displaystyle \mathbb {Z} } 8898:Dirac quantization condition 4110:in this article as follows: 7: 13140:Encyclopedia of Mathematics 12856:. Oxford University Press. 12113:. No. 24. April 2014. 12050:Aad, Georges el al (2020). 11964:"Physics: The waiting game" 11888:10.1103/PhysRevLett.43.1365 11616:Canadian Journal of Physics 11525:European Journal of Physics 11490:American Journal of Physics 11181:10.1103/PhysRevLett.48.1378 10663: 10513:(i.e., they do not violate 10447:Since around 2003, various 10222:, depending on the theory. 10206:and the mathematics of the 9697:irreducible representations 9622:, solenoids with a flux of 8720:, so therefore the product 1828:Faraday's law of induction 765:(with Maxwell's extension) 697:Without magnetic monopoles 389:term, the second is called 10: 14294: 13033:10.1088/0034-4885/69/6/R02 12948:(2nd ed.). Springer. 11907:Annu. Rev. Nucl. Part. Sci 11555:10.1088/0143-0807/30/1/003 11143:10.1103/PhysRevLett.35.487 11050:(in French). Paris: 76–77. 10736:'t Hooft–Polyakov monopole 9669:'t Hooft–Polyakov monopole 9666: 9297: 9289:Topological interpretation 8803:equals the magnetic field 8692:, and it also has a total 1690:Gauss's law for magnetism 959:Faraday's law of induction 643:from magnetic monopoles). 639:found in ordinary matter ( 374:, which is related to its 332:and every particle in the 255: 14224: 14127: 14091: 14008: 13969: 13939: 13913: 13909: 13900: 13832: 13800: 13727: 13662: 13644: 13540: 13495: 13467: 13458: 13449: 13431: 13409: 13381: 13372: 13288: 13215: 13206: 13197: 13114:Vonsovsky, S. V. (1975). 13099:10.1142/S0217751X97002504 12927:Classical Electrodynamics 12336:10.1038/nature.2014.14612 12241:10.1103/PhysRevX.7.021023 11342:10.1038/nature.2014.14612 11241:10.1142/S0217751X0401866X 10848:, Volume 261, pp. 651–677 10813:– via Google Books. 10721:Magnetic monopole problem 10696:Gauss's law for magnetism 10460:Gauss's law for magnetism 7597: 7208: 6759: 6626: 5886: 5562: 4990: 4931: 4751: 4625: 3756: 3349: 1589: 1526: 1510: 1505: 867:Gauss's law for magnetism 767: 709: 552:Fields due to stationary 142:Gauss's law for magnetism 14152:mathematical formulation 13747:Eta and eta prime mesons 12023:. In Kirk, W. T. (ed.). 10752: 10554:condensed-matter physics 10449:condensed-matter physics 10132:universal covering group 10023:homotopically equivalent 9845:Mathematical formulation 9337:, the group element is: 8797:of the vector potential 8675:'s work on developing a 1519:Ampere-meter convention 1511:With magnetic monopoles 1375:magnetic current density 700:With magnetic monopoles 228:condensed matter systems 134:electron magnetic moment 81:are electric monopoles. 13814:Double-charm tetraquark 12929:(3rd ed.). Wiley. 12925:Jackson, J. D. (1999). 12912:Giacomelli, G. (2000), 12730:10.1126/science.1178868 12614:10.1126/science.1181510 12451:10.1126/science.1167747 12390:10.1126/science.1089408 12264:Physical Review Letters 12164:Arttu Rajantie (2016). 11901:Preskill, John (1984). 11859:Preskill, John (1979). 11161:Physical Review Letters 11123:Physical Review Letters 10777:, eqn (4), for example. 10292:cosmic inflation theory 10085:are given by the first 10047:is defined by covering 9946:that acts on the fiber 9197:reduced Planck constant 8896:. This is known as the 4352:electromagnetic tensor 4104:electromagnetic tensors 2054:Lorentz force equation 1410:. For the equations in 1362:magnetic charge density 632:field due to a natural 556:and magnetic monopoles. 376:quantum-mechanical spin 223:with much probability. 14273:Hypothetical particles 11941:Rees, Martin. (1998). 11660:Foundations of Physics 11392:Parker, C. B. (1994). 11090:10.1098/rspa.1931.0130 10701:Ginzburg–Landau theory 10543:grand unified theories 10202:in the development of 10187:Grand unified theories 10139:gauge transformation. 10124: 10000: 9964:characteristic classes 9663:Grand unified theories 9593: 9469: 9274:quantization condition 9225: 9155: 9081: 8985: 8631: 8477: 8289: 8135: 7943:duality transformation 7937:Duality transformation 7911: 7743: 7589: 7440: 7289: 7205:Lorenz gauge condition 7194: 7134: 7009: 6877: 6752: 6596: 6474: 6340: 6213: 6124: 6042: 5950: 5879: 5777: 5644: 5555: 5462: 5400: 5347: 5224: 5162: 5109: 4981: 4924: 4801: 4744: 4616: 4567: 4526: 4339: 4287: 4243: 4131:Electromagnetic tensor 4080: 3990: 3749: 3520: 3342: 3182:Lorenz gauge condition 3171: 2939: 2697: 2424: 2278: 2118: 2044: 1960: 1886: 1818: 1764: 1720: 1680: 1577: 1479:, since the units are 1340: 1211: 1127: 1028: 948: 898: 856: 752: 634:magnetic dipole moment 604:electric dipole moment 197:grand unified theories 46: 14211:Wave–particle duality 14201:Relativistic particle 13338:Electron antineutrino 13048:Shnir, Y. M. (2005). 12834:Balestra, S. (2011), 11423:Understanding Physics 11042:Pierre Curie (1894). 10437:Einstein–Rosen bridge 10393:Large Hadron Collider 10125: 10001: 9594: 9470: 9226: 9156: 9082: 8986: 8877:of its wave function 8632: 8478: 8290: 8136: 7957:Charges and currents 7912: 7744: 7590: 7441: 7290: 7195: 7135: 7010: 6878: 6753: 6597: 6475: 6341: 6214: 6125: 6043: 5951: 5880: 5778: 5676:antisymmetric tensors 5659:The signature of the 5645: 5556: 5463: 5401: 5348: 5225: 5163: 5110: 4982: 4925: 4802: 4745: 4617: 4568: 4527: 4340: 4288: 4244: 4125:SI units (Wb or A⋅m) 4081: 3991: 3750: 3521: 3343: 3172: 2940: 2698: 2436:Potential formulation 2425: 2279: 2119: 2045: 1961: 1887: 1819: 1765: 1721: 1681: 1578: 1341: 1212: 1128: 1029: 949: 899: 857: 753: 677:In Gaussian cgs units 533:magnetic flux density 213:particle accelerators 107:Historical background 24: 13441:Faddeev–Popov ghosts 13191:Particles in physics 13133:N. Hitchin (2001) , 13069:Int. J. Mod. Phys. A 12963:Lechner, K. (2018). 12852:Brau, C. A. (2004). 11943:Before the Beginning 11903:"Magnetic Monopoles" 10537:-field" (related to 10458:, and would violate 10339:Blas Cabrera Navarro 10307:particle accelerator 10212:Grand Unified Theory 10208:strong nuclear force 10196:quantum field theory 10112: 9981: 9877:improve this section 9690:grand unified theory 9611:interference fringes 9491: 9343: 9270:Aharonov–Bohm effect 9213: 9099: 9005: 8929: 8840:Aharonov–Bohm effect 8832:gauge transformation 8814:Dirac delta function 8663:Dirac's quantization 8488: 8302: 8146: 7968: 7754: 7601: 7451: 7311: 7212: 7145: 7020: 6890: 6763: 6633: 6627:Maxwell's equations 6485: 6351: 6229: 6135: 6053: 5966: 5890: 5814: 5684: 5566: 5480: 5411: 5358: 5237: 5173: 5120: 4999: 4935: 4814: 4755: 4634: 4578: 4537: 4358: 4298: 4254: 4138: 4102:clear. We introduce 4008: 3760: 3536: 3353: 3189: 2950: 2708: 2472: 2460:Maxwell's equations 2289: 2129: 2060: 1971: 1897: 1834: 1775: 1731: 1696: 1593: 1530: 1222: 1146: 1039: 966: 909: 874: 771: 713: 361:elementary particles 290:improve this section 140:of other particles. 61:that is an isolated 14216:Particle chauvinism 14159:Subatomic particles 13135:"Magnetic Monopole" 13091:1997IJMPA..12.4663S 13025:2006RPPh...69.1637M 12944:Lacava, F. (2022). 12889:1988CMaPh.114..463H 12783:2009PhRvL.103c0401P 12722:2009Sci...326..411M 12662:2011NatPh...7..252G 12533:10.1038/nature08500 12525:2009Natur.461..956B 12443:2009Sci...323.1184Q 12427:(5918): 1184–1187. 12382:2003Sci...302...92F 12286:2017PhRvL.119w5301Y 12233:2017PhRvX...7b1023O 12182:2016PhT....69j..40R 12152:Particle data group 12078:2020PhRvL.124c1802A 12031:on February 4, 2009 11980:2004Natur.429...10B 11919:1984ARNPS..34..461P 11880:1979PhRvL..43.1365P 11838:1978PhLB...79..239Z 11795:2007JHEP...06..060A 11673:1989FoPh...19..125F 11629:1952CaJPh..30..218S 11600:2001NCimB.116..869M 11547:2009EJPh...30...23H 11502:1989AmJPh..57..993R 11475:2001NCimB.116..869M 11272:. September 4, 2009 11233:2004IJMPA..19S.145P 11217:(supp01): 145–154. 11173:1982PhRvL..48.1378C 11135:1975PhRvL..35..487P 11082:1931RSPSA.133...60D 11076:(821). London: 60. 10975:10.1038/nature12954 10967:2014Natur.505..657R 10907:10.1038/nature06433 10899:2008Natur.451...42C 10859:Aspects of Symmetry 10671:Bogomolny equations 10630:, there is a field 10590:dysprosium titanate 10547:charge quantization 10482:emergent phenomenon 10471:charge quantization 10456:elementary particle 10423:The astrophysicist 10362:Luis Walter Alvarez 10356:by the team led by 9718:lassoing the sphere 9179:vacuum permittivity 8999:-meter convention) 8737:Maxwell's equations 8682:Maxwell's equations 8464: 8443: 8122: 8103: 7854: 7824: 7676: 7646: 7535: 7505: 7386: 7356: 7278: 7241: 7189: 7129: 7099: 7059: 7004: 6969: 6929: 6872: 6842: 6802: 6747: 6712: 6672: 6208: 6119: 6037: 5945: 5874: 5339: 5319: 5299: 5279: 5256: 5101: 5081: 5061: 5041: 5018: 4916: 4896: 4876: 4856: 4833: 4736: 4716: 4696: 4676: 4653: 3530:Relation to fields 1499: 1408:Maxwell's equations 1351:In these equations 690: 492:, there is also a " 464:Maxwell's equations 459:Maxwell's equations 383:multipole expansion 146:Maxwell's equations 59:elementary particle 43:elementary particle 14253:Magnetic monopoles 13050:Magnetic Monopoles 12897:10.1007/BF01242139 12571:. October 15, 2009 12148:Magnetic Monopoles 11691:10.1007/bf00734522 10803:. Harper Collins. 10598:neutron scattering 10259:correlation length 10204:electroweak theory 10120: 10027:topological sphere 9996: 9968:algebraic topology 9921:principal G-bundle 9806:topological defect 9589: 9465: 9355: 9221: 9151: 9092:Gaussian-cgs units 9077: 8981: 8627: 8621: 8581: 8515: 8473: 8467: 8448: 8427: 8413: 8347: 8285: 8279: 8239: 8173: 8131: 8125: 8108: 8089: 8075: 8009: 7931:Levi-Civita symbol 7907: 7838: 7808: 7739: 7660: 7630: 7585: 7519: 7489: 7436: 7370: 7340: 7299:Relation to fields 7285: 7262: 7225: 7190: 7173: 7130: 7113: 7083: 7043: 7005: 6988: 6953: 6913: 6873: 6856: 6826: 6786: 6748: 6731: 6696: 6656: 6592: 6470: 6336: 6223:Lorentz force law 6209: 6192: 6120: 6103: 6038: 6021: 5960:Faraday–Gauss law 5946: 5929: 5875: 5858: 5794:Maxwell equations 5773: 5640: 5551: 5458: 5396: 5343: 5323: 5303: 5283: 5263: 5240: 5220: 5158: 5105: 5085: 5065: 5045: 5025: 5002: 4977: 4920: 4900: 4880: 4860: 4840: 4817: 4797: 4740: 4720: 4700: 4680: 4660: 4637: 4612: 4563: 4522: 4335: 4283: 4239: 4100:Lorentz covariance 4090:Tensor formulation 4076: 3986: 3984: 3745: 3743: 3516: 3514: 3338: 3336: 3167: 3165: 2935: 2933: 2693: 2691: 2420: 2418: 2274: 2272: 2114: 2040: 1956: 1882: 1814: 1760: 1716: 1676: 1573: 1497: 1412:nondimensionalized 1336: 1207: 1123: 1024: 944: 894: 852: 748: 688: 580:charge induces an 434:on one side and a 415:term is non-zero. 57:is a hypothetical 47: 14240: 14239: 14196:Massless particle 14004: 14003: 14000: 13999: 13965: 13964: 13828: 13827: 13640: 13639: 13636: 13635: 13588:Magnetic monopole 13536: 13535: 13427: 13426: 13368: 13367: 13348:Muon antineutrino 13333:Electron neutrino 13120:. Mir Publishers. 13075:(26): 4663–4706. 13059:978-3-540-25277-1 12993:978-0-12-802463-8 12974:978-3-319-91808-2 12955:978-3-031-05098-5 12936:978-0-471-30932-1 12863:978-0-19-514665-3 12670:10.1038/nphys1896 12598:(5951): 375–376. 12509:(7266): 956–959. 12191:10.1063/PT.3.3328 12019:Alvarez, Luis W. 11874:(19): 1365–1368. 11745:978-0-571-22278-0 11407:978-0-07-051400-3 11378:978-0-321-85656-2 11298:Symmetry Breaking 11167:(20): 1378–1381. 10951:(7485): 657–660. 10846:Nuclear Physics B 10775:physics/0508099v1 10473:as formulated by 10409:MoEDAL experiment 10269:of the expanding 10242:symmetry breaking 10227:conservation laws 9913: 9912: 9905: 9824:Hawking radiation 9786:, three times to 9756:compact Lie group 9408: 9346: 9164: 9163: 9141: 9067: 8971: 8640: 8639: 7921: 7920: 7767: 7464: 7171: 7151: 6986: 6729: 6605: 6604: 6564: 6551: 6442: 6429: 6334: 6301: 6190: 6159: 6101: 6077: 6019: 5990: 5856: 5808:Ampère–Gauss law 5761: 5733: 5653: 5652: 5607: 5606: 4510: 4488: 4466: 4443: 4421: 4399: 4371: 4074: 4045: 3999: 3998: 3957: 3849: 3829: 3716: 3685: 3615: 3584: 3469: 3454: 3391: 3376: 3291: 3276: 3220: 3205: 3161: 3006: 2874: 2764: 2673: 2566: 2433: 2432: 2409: 2263: 2223: 2011: 1937: 1874: 1647: 1625: 1571: 1516:Weber convention 1506:Without magnetic 1487:– the SI unit of 1483:, where H is the 1349: 1348: 1321: 1268: 1192: 1107: 1089: 1067: 1016: 994: 836: 818: 796: 572:charge induces a 349:electric currents 326: 325: 318: 221:particle detector 205:Joseph Polchinski 160:Quantum mechanics 130:electric currents 67:particle theories 55:magnetic monopole 14285: 14230: 14206:Virtual particle 13977:Mesonic molecule 13911: 13910: 13907: 13906: 13752:Bottom eta meson 13660: 13659: 13651: 13650: 13623:W′ and Z′ bosons 13613:Sterile neutrino 13598:Majorana fermion 13465: 13464: 13456: 13455: 13379: 13378: 13358:Tau antineutrino 13213: 13212: 13204: 13203: 13184: 13177: 13170: 13161: 13160: 13147: 13121: 13110: 13084: 13063: 13044: 13018: 13009:(6): 1637–1711. 12997: 12978: 12959: 12940: 12921: 12920: 12908: 12877:Comm. Math. Phys 12867: 12848: 12847: 12830: 12803: 12802: 12776: 12756: 12750: 12749: 12715: 12680: 12674: 12673: 12643: 12634: 12633: 12607: 12587: 12581: 12580: 12578: 12576: 12559: 12553: 12552: 12518: 12498: 12489: 12488: 12485:sciencedaily.com 12477: 12471: 12470: 12436: 12416: 12410: 12409: 12375: 12373:cond-mat/0310232 12354: 12348: 12347: 12323: 12314: 12313: 12279: 12259: 12253: 12252: 12226: 12206: 12200: 12199: 12193: 12161: 12155: 12145: 12139: 12124: 12118: 12117: 12104: 12098: 12097: 12071: 12047: 12041: 12040: 12038: 12036: 12016: 12010: 12009: 11991: 11959: 11953: 11939: 11933: 11932: 11930: 11898: 11892: 11891: 11865: 11856: 11850: 11849: 11821: 11815: 11814: 11788: 11768: 11762: 11756: 11750: 11749: 11726: 11720: 11714: 11703: 11702: 11684: 11656: 11647: 11641: 11640: 11610: 11604: 11603: 11593: 11573: 11567: 11566: 11540: 11520: 11514: 11513: 11485: 11479: 11478: 11468: 11446: 11437: 11436: 11432:978-0-47-0746370 11418: 11412: 11411: 11399: 11389: 11383: 11382: 11364: 11358: 11357: 11329: 11320: 11310: 11301: 11291: 11282: 11281: 11279: 11277: 11262: 11253: 11252: 11226: 11206: 11197: 11191: 11185: 11184: 11156: 11147: 11146: 11118: 11109: 11103: 11094: 11093: 11058: 11052: 11051: 11039: 11033: 11032: 11024: 11018: 11017: 11009: 11003: 11002: 10960: 10938: 10927: 10926: 10892: 10872: 10861: 10855: 10849: 10842: 10836: 10835: 10829: 10821: 10815: 10814: 10794: 10778: 10763: 10746:Magnetic current 10741:Wu–Yang monopole 10691:Flatness problem 10656: 10649: 10642: 10636: 10536: 10527: 10520: 10510: 10468: 10427:argues that the 10389:ATLAS experiment 10287:cosmic inflation 10283:monopole problem 10182: 10174: 10169: 10162: 10156:dimensions with 10155: 10129: 10127: 10126: 10121: 10119: 10106:simply connected 10103: 10099: 10092: 10084: 10080: 10076: 10070: 10052: 10046: 10040: 10033: 10009: 10005: 10003: 10002: 9997: 9995: 9994: 9989: 9977:If spacetime is 9957: 9953: 9949: 9945: 9941: 9937: 9926: 9923:over spacetime. 9908: 9901: 9897: 9894: 9888: 9857: 9849: 9791: 9785: 9779: 9750: 9742: 9736: 9683: 9678: 9654: 9649: 9643: 9638: 9632: 9627: 9621: 9598: 9596: 9595: 9590: 9575: 9574: 9537: 9536: 9509: 9508: 9474: 9472: 9471: 9466: 9461: 9457: 9421: 9417: 9409: 9407: 9399: 9398: 9397: 9384: 9382: 9381: 9354: 9336: 9330: 9230: 9228: 9227: 9222: 9220: 9204: 9194: 9193: 9176: 9160: 9158: 9157: 9152: 9150: 9142: 9140: 9132: 9131: 9130: 9129: 9119: 9118: 9117: 9106: 9086: 9084: 9083: 9078: 9076: 9068: 9066: 9065: 9064: 9052: 9051: 9035: 9034: 9033: 9032: 9022: 9021: 9020: 9009: 8990: 8988: 8987: 8982: 8980: 8972: 8970: 8959: 8958: 8957: 8956: 8946: 8945: 8944: 8933: 8905: 8904: 8895: 8894: 8888: 8882: 8876: 8859: 8850: 8829: 8822: 8808: 8802: 8791:vector potential 8788: 8781: 8775: 8774: 8772: 8771: 8766: 8763: 8734: 8719: 8710: 8694:angular momentum 8657: 8655: 8636: 8634: 8633: 8628: 8626: 8625: 8618: 8617: 8604: 8603: 8586: 8585: 8520: 8519: 8512: 8503: 8482: 8480: 8479: 8474: 8472: 8471: 8460: 8459: 8453: 8439: 8438: 8432: 8418: 8417: 8352: 8351: 8344: 8343: 8342: 8336: 8326: 8325: 8324: 8318: 8294: 8292: 8291: 8286: 8284: 8283: 8276: 8275: 8262: 8261: 8244: 8243: 8178: 8177: 8170: 8161: 8140: 8138: 8137: 8132: 8130: 8129: 8118: 8117: 8099: 8098: 8080: 8079: 8014: 8013: 8006: 8005: 8004: 7990: 7989: 7988: 7954: 7953: 7950: 7928: 7916: 7914: 7913: 7908: 7906: 7905: 7901: 7900: 7889: 7888: 7879: 7878: 7853: 7848: 7847: 7837: 7836: 7823: 7818: 7817: 7807: 7806: 7791: 7790: 7778: 7777: 7769: 7768: 7760: 7748: 7746: 7745: 7740: 7738: 7737: 7733: 7732: 7721: 7720: 7711: 7710: 7689: 7688: 7675: 7670: 7669: 7659: 7658: 7645: 7640: 7639: 7629: 7628: 7616: 7615: 7594: 7592: 7591: 7586: 7584: 7583: 7579: 7578: 7567: 7566: 7557: 7556: 7534: 7529: 7528: 7518: 7517: 7504: 7499: 7498: 7488: 7487: 7475: 7474: 7466: 7465: 7457: 7445: 7443: 7442: 7437: 7435: 7434: 7430: 7429: 7418: 7417: 7408: 7407: 7385: 7380: 7379: 7369: 7368: 7355: 7350: 7349: 7339: 7338: 7326: 7325: 7294: 7292: 7291: 7286: 7277: 7272: 7271: 7261: 7260: 7240: 7235: 7234: 7224: 7223: 7199: 7197: 7196: 7191: 7188: 7183: 7182: 7172: 7170: 7169: 7157: 7152: 7149: 7139: 7137: 7136: 7131: 7128: 7123: 7122: 7112: 7111: 7098: 7093: 7092: 7082: 7081: 7072: 7071: 7058: 7053: 7052: 7042: 7041: 7032: 7031: 7014: 7012: 7011: 7006: 7003: 6998: 6997: 6987: 6982: 6974: 6968: 6963: 6962: 6952: 6951: 6942: 6941: 6928: 6923: 6922: 6912: 6911: 6902: 6901: 6882: 6880: 6879: 6874: 6871: 6866: 6865: 6855: 6854: 6841: 6836: 6835: 6825: 6824: 6815: 6814: 6801: 6796: 6795: 6785: 6784: 6775: 6774: 6757: 6755: 6754: 6749: 6746: 6741: 6740: 6730: 6725: 6717: 6711: 6706: 6705: 6695: 6694: 6685: 6684: 6671: 6666: 6665: 6655: 6654: 6645: 6644: 6610: 6609: 6601: 6599: 6598: 6593: 6591: 6590: 6581: 6577: 6576: 6575: 6574: 6566: 6565: 6557: 6552: 6547: 6546: 6545: 6535: 6530: 6529: 6517: 6516: 6515: 6497: 6496: 6479: 6477: 6476: 6471: 6469: 6468: 6459: 6455: 6454: 6453: 6452: 6444: 6443: 6435: 6430: 6428: 6424: 6423: 6413: 6412: 6411: 6401: 6396: 6395: 6383: 6382: 6381: 6363: 6362: 6345: 6343: 6342: 6337: 6335: 6330: 6329: 6320: 6318: 6314: 6313: 6312: 6311: 6303: 6302: 6294: 6289: 6288: 6287: 6274: 6273: 6261: 6260: 6259: 6241: 6240: 6218: 6216: 6215: 6210: 6207: 6202: 6201: 6191: 6186: 6185: 6176: 6171: 6170: 6169: 6161: 6160: 6152: 6147: 6146: 6129: 6127: 6126: 6121: 6118: 6113: 6112: 6102: 6094: 6089: 6088: 6087: 6079: 6078: 6070: 6065: 6064: 6047: 6045: 6044: 6039: 6036: 6031: 6030: 6020: 6015: 6007: 6002: 6001: 6000: 5992: 5991: 5983: 5978: 5977: 5955: 5953: 5952: 5947: 5944: 5939: 5938: 5928: 5927: 5915: 5914: 5902: 5901: 5884: 5882: 5881: 5876: 5873: 5868: 5867: 5857: 5852: 5844: 5839: 5838: 5826: 5825: 5791: 5790: 5782: 5780: 5779: 5774: 5772: 5771: 5763: 5762: 5754: 5744: 5743: 5735: 5734: 5726: 5718: 5717: 5699: 5698: 5666: 5661:Minkowski metric 5649: 5647: 5646: 5641: 5636: 5624: 5616: 5608: 5605: 5604: 5595: 5590: 5589: 5574: 5570: 5560: 5558: 5557: 5552: 5547: 5546: 5534: 5533: 5521: 5520: 5508: 5507: 5492: 5491: 5467: 5465: 5464: 5459: 5454: 5453: 5452: 5446: 5433: 5428: 5427: 5426: 5405: 5403: 5402: 5397: 5392: 5391: 5390: 5384: 5375: 5374: 5373: 5352: 5350: 5349: 5344: 5338: 5333: 5332: 5318: 5313: 5312: 5298: 5293: 5292: 5278: 5273: 5272: 5255: 5250: 5249: 5229: 5227: 5226: 5221: 5216: 5215: 5214: 5208: 5195: 5190: 5189: 5188: 5167: 5165: 5164: 5159: 5154: 5153: 5152: 5146: 5137: 5136: 5135: 5114: 5112: 5111: 5106: 5100: 5095: 5094: 5080: 5075: 5074: 5060: 5055: 5054: 5040: 5035: 5034: 5017: 5012: 5011: 4986: 4984: 4983: 4978: 4973: 4972: 4971: 4965: 4955: 4954: 4953: 4929: 4927: 4926: 4921: 4915: 4910: 4909: 4895: 4890: 4889: 4875: 4870: 4869: 4855: 4850: 4849: 4832: 4827: 4826: 4806: 4804: 4803: 4798: 4793: 4792: 4791: 4785: 4775: 4774: 4773: 4749: 4747: 4746: 4741: 4735: 4730: 4729: 4715: 4710: 4709: 4695: 4690: 4689: 4675: 4670: 4669: 4652: 4647: 4646: 4621: 4619: 4618: 4613: 4605: 4600: 4591: 4572: 4570: 4569: 4564: 4559: 4550: 4531: 4529: 4528: 4523: 4518: 4517: 4512: 4511: 4503: 4496: 4495: 4490: 4489: 4481: 4474: 4473: 4468: 4467: 4459: 4451: 4450: 4445: 4444: 4436: 4429: 4428: 4423: 4422: 4414: 4407: 4406: 4401: 4400: 4392: 4382: 4381: 4373: 4372: 4364: 4344: 4342: 4341: 4336: 4331: 4316: 4311: 4292: 4290: 4289: 4284: 4279: 4267: 4248: 4246: 4245: 4240: 4235: 4234: 4222: 4221: 4209: 4208: 4195: 4194: 4182: 4181: 4169: 4168: 4153: 4152: 4113: 4112: 4106:and preliminary 4085: 4083: 4082: 4077: 4075: 4073: 4072: 4067: 4058: 4057: 4048: 4046: 4044: 4043: 4031: 4026: 4025: 3995: 3993: 3992: 3987: 3985: 3981: 3980: 3979: 3973: 3958: 3956: 3948: 3947: 3946: 3945: 3939: 3929: 3927: 3926: 3914: 3913: 3912: 3899: 3898: 3879: 3870: 3869: 3868: 3862: 3850: 3848: 3847: 3835: 3830: 3828: 3820: 3819: 3818: 3817: 3811: 3801: 3796: 3795: 3794: 3771: 3754: 3752: 3751: 3746: 3744: 3740: 3739: 3738: 3732: 3717: 3715: 3707: 3706: 3705: 3704: 3698: 3688: 3686: 3678: 3673: 3672: 3671: 3648: 3639: 3638: 3637: 3631: 3616: 3614: 3606: 3605: 3604: 3603: 3597: 3587: 3585: 3577: 3572: 3571: 3570: 3547: 3525: 3523: 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1677: 1675: 1674: 1673: 1667: 1661: 1660: 1648: 1646: 1638: 1637: 1628: 1626: 1624: 1623: 1611: 1606: 1582: 1580: 1579: 1574: 1572: 1570: 1569: 1560: 1559: 1558: 1548: 1543: 1500: 1496: 1482: 1478: 1477: 1462: 1444: 1419: 1401: 1395: 1385: 1372: 1359: 1345: 1343: 1342: 1337: 1335: 1331: 1330: 1322: 1317: 1312: 1307: 1297: 1296: 1295: 1282: 1278: 1277: 1269: 1264: 1259: 1254: 1244: 1243: 1242: 1229: 1216: 1214: 1213: 1208: 1206: 1202: 1201: 1193: 1188: 1183: 1178: 1168: 1167: 1166: 1153: 1132: 1130: 1129: 1124: 1122: 1121: 1120: 1114: 1108: 1103: 1095: 1090: 1088: 1080: 1079: 1070: 1068: 1060: 1055: 1033: 1031: 1030: 1025: 1017: 1015: 1007: 1006: 997: 995: 987: 982: 953: 951: 950: 945: 943: 942: 941: 922: 903: 901: 900: 895: 887: 861: 859: 858: 853: 851: 850: 849: 843: 837: 832: 824: 819: 817: 809: 808: 799: 797: 789: 784: 757: 755: 754: 749: 747: 746: 745: 726: 691: 687: 602:field due to an 593: 546: 530: 520: 516: 504: 494:magnetic current 491: 477:electric current 468:electromagnetism 321: 314: 310: 307: 301: 270: 262: 193:particle physics 138:magnetic moments 126:electromagnetism 97:condensed matter 51:particle physics 14293: 14292: 14288: 14287: 14286: 14284: 14283: 14282: 14243: 14242: 14241: 14236: 14220: 14174:Nuclear physics 14123: 14087: 14023:Davydov soliton 13996: 13961: 13935: 13896: 13824: 13796: 13723: 13632: 13532: 13491: 13445: 13423: 13405: 13364: 13284: 13193: 13188: 13128: 13060: 12994: 12975: 12956: 12937: 12918: 12864: 12827: 12811: 12806: 12761:Phys. Rev. Lett 12757: 12753: 12706:(5951): 411–4. 12695: 12691: 12687: 12681: 12677: 12644: 12637: 12588: 12584: 12574: 12572: 12561: 12560: 12556: 12499: 12492: 12479: 12478: 12474: 12417: 12413: 12366:(5642): 92–95. 12355: 12351: 12324: 12317: 12260: 12256: 12207: 12203: 12162: 12158: 12146: 12142: 12125: 12121: 12110:All About Space 12106: 12105: 12101: 12056:Phys. Rev. Lett 12048: 12044: 12034: 12032: 12017: 12013: 11989:10.1038/429010a 11974:(6987): 10–11. 11960: 11956: 11940: 11936: 11899: 11895: 11868:Phys. Rev. Lett 11863: 11857: 11853: 11822: 11818: 11769: 11765: 11757: 11753: 11746: 11730:Farmelo, Graham 11727: 11723: 11719:, section 6.11. 11715: 11706: 11682:10.1.1.382.3733 11654: 11648: 11644: 11637:10.1139/p52-021 11611: 11607: 11591:math-ph/0203043 11578:Nuovo Cimento B 11574: 11570: 11521: 11517: 11510:10.1119/1.15782 11496:(11): 993–994. 11486: 11482: 11466:math-ph/0203043 11452:Nuovo Cimento B 11447: 11440: 11433: 11419: 11415: 11408: 11390: 11386: 11379: 11365: 11361: 11330: 11323: 11311: 11304: 11292: 11285: 11275: 11273: 11264: 11263: 11256: 11207: 11200: 11192: 11188: 11157: 11150: 11119: 11112: 11104: 11097: 11059: 11055: 11040: 11036: 11025: 11021: 11010: 11006: 10939: 10930: 10883:(7174): 42–45. 10873: 10864: 10856: 10852: 10843: 10839: 10827: 10823: 10822: 10818: 10811: 10795: 10791: 10787: 10782: 10781: 10764: 10760: 10755: 10750: 10711:Horizon problem 10686:Felix Ehrenhaft 10666: 10651: 10644: 10638: 10631: 10531: 10523: 10514: 10506: 10502:quasi-particles 10462: 10445: 10431:of macroscopic 10358:P. Buford Price 10343:Valentine's Day 10324:superconducting 10315: 10189: 10180: 10172: 10171: 10164: 10157: 10150: 10115: 10113: 10110: 10109: 10101: 10097: 10090: 10082: 10078: 10072: 10062: 10048: 10042: 10038: 10029: 10007: 9990: 9985: 9984: 9982: 9979: 9978: 9955: 9951: 9947: 9943: 9939: 9935: 9924: 9909: 9898: 9892: 9889: 9874: 9858: 9847: 9819: 9787: 9781: 9775: 9740: 9738: 9732: 9676: 9674: 9671: 9665: 9647: 9645: 9636: 9634: 9625: 9623: 9617: 9599: 9570: 9566: 9532: 9528: 9501: 9497: 9492: 9489: 9488: 9475: 9435: 9431: 9400: 9393: 9389: 9385: 9383: 9377: 9373: 9360: 9356: 9350: 9344: 9341: 9340: 9332: 9326: 9317: 9302: 9296: 9291: 9216: 9214: 9211: 9210: 9200: 9191: 9182: 9175: 9169: 9146: 9133: 9125: 9124: 9120: 9113: 9112: 9108: 9107: 9105: 9100: 9097: 9096: 9072: 9060: 9056: 9047: 9043: 9036: 9028: 9027: 9023: 9016: 9015: 9011: 9010: 9008: 9006: 9003: 9002: 8976: 8960: 8952: 8951: 8947: 8940: 8939: 8935: 8934: 8932: 8930: 8927: 8926: 8892: 8890: 8884: 8878: 8872: 8858: 8852: 8849: 8843: 8824: 8817: 8804: 8798: 8783: 8777: 8767: 8764: 8762: 8756: 8755: 8753: 8752: 8733: 8727: 8721: 8715: 8709: 8703: 8697: 8690:Poynting vector 8665: 8653: 8648: 8620: 8619: 8610: 8609: 8606: 8605: 8596: 8595: 8588: 8587: 8580: 8579: 8568: 8556: 8555: 8541: 8525: 8524: 8514: 8513: 8508: 8505: 8504: 8499: 8492: 8491: 8489: 8486: 8485: 8466: 8465: 8455: 8454: 8449: 8445: 8444: 8434: 8433: 8428: 8420: 8419: 8412: 8411: 8400: 8388: 8387: 8373: 8357: 8356: 8346: 8345: 8338: 8337: 8332: 8331: 8328: 8327: 8320: 8319: 8314: 8313: 8306: 8305: 8303: 8300: 8299: 8278: 8277: 8268: 8267: 8264: 8263: 8254: 8253: 8246: 8245: 8238: 8237: 8226: 8214: 8213: 8199: 8183: 8182: 8172: 8171: 8166: 8163: 8162: 8157: 8150: 8149: 8147: 8144: 8143: 8124: 8123: 8113: 8112: 8105: 8104: 8094: 8093: 8082: 8081: 8074: 8073: 8062: 8050: 8049: 8035: 8019: 8018: 8008: 8007: 8000: 7999: 7995: 7992: 7991: 7984: 7983: 7979: 7972: 7971: 7969: 7966: 7965: 7946: 7939: 7924: 7896: 7895: 7894: 7890: 7884: 7880: 7865: 7861: 7849: 7843: 7842: 7832: 7828: 7819: 7813: 7812: 7802: 7798: 7786: 7782: 7770: 7759: 7758: 7757: 7755: 7752: 7751: 7749: 7728: 7727: 7726: 7722: 7716: 7712: 7697: 7693: 7684: 7680: 7671: 7665: 7664: 7654: 7650: 7641: 7635: 7634: 7624: 7620: 7608: 7604: 7602: 7599: 7598: 7574: 7573: 7572: 7568: 7562: 7558: 7543: 7539: 7530: 7524: 7523: 7513: 7509: 7500: 7494: 7493: 7483: 7479: 7467: 7456: 7455: 7454: 7452: 7449: 7448: 7446: 7425: 7424: 7423: 7419: 7413: 7409: 7394: 7390: 7381: 7375: 7374: 7364: 7360: 7351: 7345: 7344: 7334: 7330: 7318: 7314: 7312: 7309: 7308: 7300: 7273: 7267: 7266: 7256: 7252: 7236: 7230: 7229: 7219: 7215: 7213: 7210: 7209: 7184: 7178: 7177: 7165: 7161: 7156: 7148: 7146: 7143: 7142: 7124: 7118: 7117: 7107: 7103: 7094: 7088: 7087: 7077: 7073: 7067: 7063: 7054: 7048: 7047: 7037: 7033: 7027: 7023: 7021: 7018: 7017: 6999: 6993: 6992: 6975: 6973: 6964: 6958: 6957: 6947: 6943: 6937: 6933: 6924: 6918: 6917: 6907: 6903: 6897: 6893: 6891: 6888: 6887: 6867: 6861: 6860: 6850: 6846: 6837: 6831: 6830: 6820: 6816: 6810: 6806: 6797: 6791: 6790: 6780: 6776: 6770: 6766: 6764: 6761: 6760: 6742: 6736: 6735: 6718: 6716: 6707: 6701: 6700: 6690: 6686: 6680: 6676: 6667: 6661: 6660: 6650: 6646: 6640: 6636: 6634: 6631: 6630: 6622:SI units (A⋅m) 6616:Gaussian units 6607:Alternatively, 6586: 6582: 6567: 6556: 6555: 6554: 6553: 6541: 6540: 6536: 6534: 6522: 6518: 6511: 6510: 6506: 6505: 6501: 6492: 6488: 6486: 6483: 6482: 6464: 6460: 6445: 6434: 6433: 6432: 6431: 6419: 6415: 6414: 6407: 6406: 6402: 6400: 6388: 6384: 6377: 6376: 6372: 6371: 6367: 6358: 6354: 6352: 6349: 6348: 6325: 6321: 6319: 6304: 6293: 6292: 6291: 6290: 6283: 6282: 6278: 6266: 6262: 6255: 6254: 6250: 6249: 6245: 6236: 6232: 6230: 6227: 6226: 6203: 6197: 6196: 6181: 6177: 6175: 6162: 6151: 6150: 6149: 6148: 6142: 6138: 6136: 6133: 6132: 6114: 6108: 6107: 6093: 6080: 6069: 6068: 6067: 6066: 6060: 6056: 6054: 6051: 6050: 6032: 6026: 6025: 6008: 6006: 5993: 5982: 5981: 5980: 5979: 5973: 5969: 5967: 5964: 5963: 5940: 5934: 5933: 5923: 5919: 5907: 5903: 5897: 5893: 5891: 5888: 5887: 5869: 5863: 5862: 5845: 5843: 5831: 5827: 5821: 5817: 5815: 5812: 5811: 5803:SI units (A⋅m) 5797:Gaussian units 5764: 5753: 5752: 5751: 5736: 5725: 5724: 5723: 5710: 5706: 5691: 5687: 5685: 5682: 5681: 5664: 5632: 5620: 5612: 5600: 5596: 5591: 5585: 5581: 5569: 5567: 5564: 5563: 5542: 5538: 5529: 5525: 5516: 5512: 5503: 5499: 5487: 5483: 5481: 5478: 5477: 5448: 5447: 5442: 5441: 5429: 5422: 5421: 5417: 5412: 5409: 5408: 5386: 5385: 5380: 5379: 5369: 5368: 5364: 5359: 5356: 5355: 5334: 5328: 5327: 5314: 5308: 5307: 5294: 5288: 5287: 5274: 5268: 5267: 5251: 5245: 5244: 5238: 5235: 5234: 5210: 5209: 5204: 5203: 5191: 5184: 5183: 5179: 5174: 5171: 5170: 5148: 5147: 5142: 5141: 5131: 5130: 5126: 5121: 5118: 5117: 5096: 5090: 5089: 5076: 5070: 5069: 5056: 5050: 5049: 5036: 5030: 5029: 5013: 5007: 5006: 5000: 4997: 4996: 4967: 4966: 4961: 4960: 4949: 4948: 4944: 4936: 4933: 4932: 4911: 4905: 4904: 4891: 4885: 4884: 4871: 4865: 4864: 4851: 4845: 4844: 4828: 4822: 4821: 4815: 4812: 4811: 4787: 4786: 4781: 4780: 4769: 4768: 4764: 4756: 4753: 4752: 4731: 4725: 4724: 4711: 4705: 4704: 4691: 4685: 4684: 4671: 4665: 4664: 4648: 4642: 4641: 4635: 4632: 4631: 4601: 4596: 4587: 4579: 4576: 4575: 4555: 4546: 4538: 4535: 4534: 4513: 4502: 4501: 4500: 4491: 4480: 4479: 4478: 4469: 4458: 4457: 4456: 4446: 4435: 4434: 4433: 4424: 4413: 4412: 4411: 4402: 4391: 4390: 4389: 4374: 4363: 4362: 4361: 4359: 4356: 4355: 4327: 4312: 4307: 4299: 4296: 4295: 4275: 4263: 4255: 4252: 4251: 4230: 4226: 4217: 4213: 4204: 4200: 4190: 4186: 4177: 4173: 4164: 4160: 4145: 4141: 4139: 4136: 4135: 4122:Gaussian units 4092: 4068: 4060: 4059: 4053: 4049: 4047: 4039: 4035: 4030: 4021: 4017: 4009: 4006: 4005: 3983: 3982: 3975: 3974: 3969: 3968: 3949: 3941: 3940: 3935: 3934: 3930: 3928: 3922: 3918: 3908: 3907: 3903: 3894: 3890: 3883: 3875: 3872: 3871: 3864: 3863: 3858: 3857: 3843: 3839: 3834: 3821: 3813: 3812: 3807: 3806: 3802: 3800: 3790: 3789: 3785: 3775: 3767: 3763: 3761: 3758: 3757: 3742: 3741: 3734: 3733: 3728: 3727: 3708: 3700: 3699: 3694: 3693: 3689: 3687: 3677: 3667: 3666: 3662: 3652: 3644: 3641: 3640: 3633: 3632: 3627: 3626: 3607: 3599: 3598: 3593: 3592: 3588: 3586: 3576: 3566: 3565: 3561: 3551: 3543: 3539: 3537: 3534: 3533: 3513: 3512: 3499: 3498: 3493: 3492: 3476: 3475: 3471: 3461: 3456: 3448: 3444: 3439: 3435: 3434: 3421: 3420: 3415: 3414: 3398: 3397: 3393: 3383: 3378: 3370: 3366: 3361: 3356: 3354: 3351: 3350: 3335: 3334: 3321: 3320: 3315: 3314: 3298: 3297: 3293: 3283: 3278: 3268: 3264: 3263: 3250: 3249: 3244: 3243: 3227: 3226: 3222: 3212: 3207: 3197: 3192: 3190: 3187: 3186: 3164: 3163: 3155: 3151: 3144: 3143: 3138: 3137: 3135: 3128: 3118: 3117: 3112: 3111: 3105: 3104: 3097: 3096: 3092: 3085: 3075: 3074: 3070: 3064: 3063: 3056: 3055: 3050: 3049: 3043: 3039: 3032: 3022: 3021: 3016: 3015: 3009: 3008: 3000: 2996: 2989: 2988: 2984: 2982: 2975: 2965: 2964: 2960: 2953: 2951: 2948: 2947: 2932: 2931: 2924: 2923: 2918: 2917: 2911: 2907: 2900: 2890: 2889: 2884: 2883: 2877: 2876: 2868: 2864: 2857: 2856: 2852: 2850: 2843: 2833: 2832: 2828: 2822: 2821: 2814: 2813: 2808: 2807: 2801: 2797: 2790: 2780: 2779: 2774: 2773: 2767: 2766: 2758: 2754: 2747: 2746: 2742: 2740: 2733: 2723: 2722: 2718: 2711: 2709: 2706: 2705: 2690: 2689: 2682: 2681: 2676: 2675: 2662: 2660: 2653: 2643: 2642: 2637: 2636: 2630: 2629: 2622: 2621: 2617: 2604: 2594: 2593: 2589: 2583: 2582: 2575: 2574: 2569: 2568: 2555: 2553: 2546: 2536: 2535: 2530: 2529: 2523: 2522: 2515: 2514: 2510: 2497: 2487: 2486: 2482: 2475: 2473: 2470: 2469: 2461: 2455:SI units (A⋅m) 2449:Gaussian units 2438: 2417: 2416: 2403: 2399: 2394: 2392: 2384: 2376: 2375: 2371: 2364: 2363: 2359: 2355: 2354: 2341: 2333: 2325: 2324: 2320: 2313: 2312: 2308: 2306: 2304: 2296: 2292: 2290: 2287: 2286: 2271: 2270: 2257: 2253: 2248: 2246: 2238: 2230: 2229: 2225: 2217: 2213: 2206: 2205: 2201: 2199: 2195: 2194: 2181: 2173: 2165: 2164: 2160: 2153: 2152: 2148: 2146: 2144: 2136: 2132: 2130: 2127: 2126: 2104: 2096: 2088: 2087: 2083: 2076: 2075: 2071: 2063: 2061: 2058: 2057: 2033: 2032: 2027: 2026: 2020: 2016: 2003: 1997: 1993: 1991: 1983: 1972: 1969: 1968: 1949: 1948: 1943: 1942: 1929: 1923: 1919: 1917: 1909: 1898: 1895: 1894: 1866: 1860: 1856: 1854: 1846: 1835: 1832: 1831: 1807: 1806: 1802: 1796: 1792: 1784: 1776: 1773: 1772: 1753: 1752: 1748: 1740: 1732: 1729: 1728: 1705: 1697: 1694: 1693: 1669: 1668: 1663: 1662: 1656: 1652: 1639: 1633: 1629: 1627: 1619: 1615: 1610: 1602: 1594: 1591: 1590: 1565: 1561: 1554: 1553: 1549: 1547: 1539: 1531: 1528: 1527: 1507: 1480: 1476: 1470: 1464: 1461: 1455: 1454: 1443: 1437: 1426: 1415: 1397: 1391: 1388:magnetic charge 1384: 1378: 1371: 1365: 1358: 1352: 1326: 1313: 1311: 1303: 1302: 1298: 1291: 1290: 1286: 1273: 1260: 1258: 1250: 1249: 1245: 1238: 1237: 1233: 1225: 1223: 1220: 1219: 1197: 1184: 1182: 1174: 1173: 1169: 1162: 1161: 1157: 1149: 1147: 1144: 1143: 1116: 1115: 1110: 1109: 1096: 1094: 1081: 1075: 1071: 1069: 1059: 1051: 1040: 1037: 1036: 1008: 1002: 998: 996: 986: 978: 967: 964: 963: 937: 936: 932: 918: 910: 907: 906: 883: 875: 872: 871: 845: 844: 839: 838: 825: 823: 810: 804: 800: 798: 788: 780: 772: 769: 768: 741: 740: 736: 722: 714: 711: 710: 679: 674: 673: 672: 671: 668:electric charge 649: 648: 647: 625: 619:magnetic dipole 617:field due to a 610: 594: 586: 585: 557: 547: 526: 518: 510: 503: 497: 490: 484: 461: 440:electric dipole 420:magnetic dipole 368:magnetic moment 359:. Second, many 353:magnetic fields 322: 311: 305: 302: 287: 271: 260: 254: 236:magnetic dipole 201:quantum gravity 162: 122:electric charge 114: 109: 101:quasi-particles 79:electric charge 17: 12: 11: 5: 14291: 14281: 14280: 14275: 14270: 14268:Gauge theories 14265: 14260: 14255: 14238: 14237: 14233:Physics portal 14225: 14222: 14221: 14219: 14218: 14213: 14208: 14203: 14198: 14193: 14188: 14187: 14186: 14176: 14171: 14166: 14161: 14156: 14155: 14154: 14147:Standard Model 14144: 14143: 14142: 14131: 14129: 14125: 14124: 14122: 14121: 14116: 14114:Quasiparticles 14111: 14106: 14101: 14095: 14093: 14089: 14088: 14086: 14085: 14080: 14075: 14070: 14065: 14060: 14055: 14050: 14045: 14040: 14035: 14030: 14025: 14020: 14014: 14012: 14010:Quasiparticles 14006: 14005: 14002: 14001: 13998: 13997: 13995: 13994: 13989: 13984: 13979: 13973: 13971: 13967: 13966: 13963: 13962: 13960: 13959: 13954: 13949: 13943: 13941: 13937: 13936: 13934: 13933: 13928: 13923: 13917: 13915: 13904: 13898: 13897: 13895: 13894: 13889: 13884: 13883: 13882: 13877: 13872: 13867: 13862: 13857: 13847: 13842: 13836: 13834: 13830: 13829: 13826: 13825: 13823: 13822: 13817: 13806: 13804: 13802:Exotic hadrons 13798: 13797: 13795: 13794: 13789: 13784: 13779: 13774: 13769: 13764: 13759: 13754: 13749: 13744: 13739: 13733: 13731: 13725: 13724: 13722: 13721: 13716: 13711: 13706: 13701: 13696: 13695: 13694: 13689: 13684: 13679: 13668: 13666: 13657: 13648: 13642: 13641: 13638: 13637: 13634: 13633: 13631: 13630: 13628:X and Y bosons 13625: 13620: 13615: 13610: 13605: 13600: 13595: 13590: 13585: 13580: 13575: 13570: 13565: 13560: 13555: 13550: 13544: 13542: 13538: 13537: 13534: 13533: 13531: 13530: 13520: 13515: 13510: 13505: 13499: 13497: 13493: 13492: 13490: 13489: 13484: 13479: 13473: 13471: 13462: 13453: 13447: 13446: 13444: 13443: 13437: 13435: 13429: 13428: 13425: 13424: 13422: 13421: 13415: 13413: 13407: 13406: 13404: 13403: 13401:W and Z bosons 13398: 13393: 13387: 13385: 13376: 13370: 13369: 13366: 13365: 13363: 13362: 13361: 13360: 13355: 13350: 13345: 13340: 13335: 13325: 13320: 13315: 13310: 13305: 13300: 13294: 13292: 13286: 13285: 13283: 13282: 13277: 13272: 13267: 13262: 13257: 13255:Strange (quark 13252: 13247: 13242: 13237: 13232: 13227: 13221: 13219: 13210: 13201: 13195: 13194: 13187: 13186: 13179: 13172: 13164: 13127: 13126:External links 13124: 13123: 13122: 13111: 13082:hep-th/9707009 13064: 13058: 13045: 13016:hep-ex/0602040 12998: 12992: 12979: 12973: 12960: 12954: 12941: 12935: 12922: 12909: 12883:(3): 463–474. 12868: 12862: 12849: 12831: 12825: 12810: 12807: 12805: 12804: 12751: 12693: 12689: 12685: 12675: 12656:(3): 252–258. 12649:Nature Physics 12635: 12582: 12554: 12490: 12472: 12411: 12349: 12315: 12270:(23): 235301. 12254: 12201: 12156: 12150:, report from 12140: 12119: 12099: 12042: 12011: 11954: 11934: 11913:(1): 461–530. 11893: 11851: 11816: 11786:hep-th/0601001 11763: 11751: 11744: 11721: 11704: 11667:(2): 125–159. 11642: 11623:(3): 218–225. 11605: 11584:(8): 869–877. 11568: 11515: 11480: 11459:(8): 869–877. 11438: 11431: 11413: 11406: 11384: 11377: 11359: 11321: 11302: 11283: 11254: 11224:hep-th/0304042 11198: 11186: 11148: 11129:(8): 487–490. 11110: 11095: 11053: 11034: 11019: 11004: 10928: 10862: 10850: 10837: 10816: 10809: 10788: 10786: 10783: 10780: 10779: 10757: 10756: 10754: 10751: 10749: 10748: 10743: 10738: 10733: 10728: 10723: 10718: 10713: 10708: 10703: 10698: 10693: 10688: 10683: 10678: 10673: 10667: 10665: 10662: 10619:Nature Physics 10582:quasiparticles 10444: 10441: 10376:, and then to 10314: 10311: 10303:X and Y bosons 10188: 10185: 10175: 10118: 10087:homotopy group 9993: 9988: 9911: 9910: 9861: 9859: 9852: 9846: 9843: 9818: 9815: 9802: 9801: 9793: 9766:with the same 9760:covering space 9752: 9729: 9726: 9722: 9712: 9667:Main article: 9664: 9661: 9588: 9585: 9582: 9578: 9573: 9569: 9565: 9562: 9559: 9556: 9552: 9549: 9546: 9543: 9540: 9535: 9531: 9527: 9524: 9521: 9518: 9515: 9512: 9507: 9504: 9500: 9496: 9487: 9464: 9460: 9456: 9453: 9450: 9447: 9444: 9441: 9438: 9434: 9430: 9427: 9424: 9420: 9416: 9413: 9406: 9403: 9396: 9392: 9388: 9380: 9376: 9372: 9369: 9366: 9363: 9359: 9353: 9349: 9339: 9322: 9298:Main article: 9295: 9292: 9290: 9287: 9231:is the set of 9219: 9207:speed of light 9173: 9166: 9165: 9162: 9161: 9149: 9145: 9139: 9136: 9128: 9123: 9116: 9111: 9104: 9094: 9088: 9087: 9075: 9071: 9063: 9059: 9055: 9050: 9046: 9042: 9039: 9031: 9026: 9019: 9014: 9000: 8992: 8991: 8979: 8975: 8969: 8966: 8963: 8955: 8950: 8943: 8938: 8924: 8913: 8912: 8909: 8866:Dirac equation 8856: 8847: 8793:such that the 8760: 8731: 8725: 8707: 8701: 8669:quantum theory 8664: 8661: 8638: 8637: 8624: 8616: 8613: 8608: 8607: 8602: 8599: 8594: 8593: 8591: 8584: 8578: 8575: 8572: 8569: 8567: 8564: 8561: 8558: 8557: 8554: 8551: 8548: 8545: 8542: 8540: 8537: 8534: 8531: 8530: 8528: 8523: 8518: 8511: 8507: 8506: 8502: 8498: 8497: 8495: 8483: 8470: 8463: 8458: 8452: 8447: 8446: 8442: 8437: 8431: 8426: 8425: 8423: 8416: 8410: 8407: 8404: 8401: 8399: 8396: 8393: 8390: 8389: 8386: 8383: 8380: 8377: 8374: 8372: 8369: 8366: 8363: 8362: 8360: 8355: 8350: 8341: 8335: 8330: 8329: 8323: 8317: 8312: 8311: 8309: 8296: 8295: 8282: 8274: 8271: 8266: 8265: 8260: 8257: 8252: 8251: 8249: 8242: 8236: 8233: 8230: 8227: 8225: 8222: 8219: 8216: 8215: 8212: 8209: 8206: 8203: 8200: 8198: 8195: 8192: 8189: 8188: 8186: 8181: 8176: 8169: 8165: 8164: 8160: 8156: 8155: 8153: 8141: 8128: 8121: 8116: 8111: 8107: 8106: 8102: 8097: 8092: 8088: 8087: 8085: 8078: 8072: 8069: 8066: 8063: 8061: 8058: 8055: 8052: 8051: 8048: 8045: 8042: 8039: 8036: 8034: 8031: 8028: 8025: 8024: 8022: 8017: 8012: 8003: 7998: 7994: 7993: 7987: 7982: 7978: 7977: 7975: 7962: 7961: 7958: 7938: 7935: 7919: 7918: 7904: 7899: 7893: 7887: 7883: 7877: 7874: 7871: 7868: 7864: 7860: 7857: 7852: 7846: 7841: 7835: 7831: 7827: 7822: 7816: 7811: 7805: 7801: 7797: 7794: 7789: 7785: 7781: 7776: 7773: 7766: 7763: 7736: 7731: 7725: 7719: 7715: 7709: 7706: 7703: 7700: 7696: 7692: 7687: 7683: 7679: 7674: 7668: 7663: 7657: 7653: 7649: 7644: 7638: 7633: 7627: 7623: 7619: 7614: 7611: 7607: 7596: 7582: 7577: 7571: 7565: 7561: 7555: 7552: 7549: 7546: 7542: 7538: 7533: 7527: 7522: 7516: 7512: 7508: 7503: 7497: 7492: 7486: 7482: 7478: 7473: 7470: 7463: 7460: 7433: 7428: 7422: 7416: 7412: 7406: 7403: 7400: 7397: 7393: 7389: 7384: 7378: 7373: 7367: 7363: 7359: 7354: 7348: 7343: 7337: 7333: 7329: 7324: 7321: 7317: 7306: 7296: 7295: 7284: 7281: 7276: 7270: 7265: 7259: 7255: 7250: 7247: 7244: 7239: 7233: 7228: 7222: 7218: 7207: 7201: 7200: 7187: 7181: 7176: 7168: 7164: 7160: 7155: 7140: 7127: 7121: 7116: 7110: 7106: 7102: 7097: 7091: 7086: 7080: 7076: 7070: 7066: 7062: 7057: 7051: 7046: 7040: 7036: 7030: 7026: 7015: 7002: 6996: 6991: 6985: 6981: 6978: 6972: 6967: 6961: 6956: 6950: 6946: 6940: 6936: 6932: 6927: 6921: 6916: 6910: 6906: 6900: 6896: 6884: 6883: 6870: 6864: 6859: 6853: 6849: 6845: 6840: 6834: 6829: 6823: 6819: 6813: 6809: 6805: 6800: 6794: 6789: 6783: 6779: 6773: 6769: 6758: 6745: 6739: 6734: 6728: 6724: 6721: 6715: 6710: 6704: 6699: 6693: 6689: 6683: 6679: 6675: 6670: 6664: 6659: 6653: 6649: 6643: 6639: 6628: 6624: 6623: 6620: 6619:SI units (Wb) 6617: 6614: 6603: 6602: 6589: 6585: 6580: 6573: 6570: 6563: 6560: 6550: 6544: 6539: 6533: 6528: 6525: 6521: 6514: 6509: 6504: 6500: 6495: 6491: 6480: 6467: 6463: 6458: 6451: 6448: 6441: 6438: 6427: 6422: 6418: 6410: 6405: 6399: 6394: 6391: 6387: 6380: 6375: 6370: 6366: 6361: 6357: 6346: 6333: 6328: 6324: 6317: 6310: 6307: 6300: 6297: 6286: 6281: 6277: 6272: 6269: 6265: 6258: 6253: 6248: 6244: 6239: 6235: 6224: 6220: 6219: 6206: 6200: 6195: 6189: 6184: 6180: 6174: 6168: 6165: 6158: 6155: 6145: 6141: 6130: 6117: 6111: 6106: 6100: 6097: 6092: 6086: 6083: 6076: 6073: 6063: 6059: 6048: 6035: 6029: 6024: 6018: 6014: 6011: 6005: 5999: 5996: 5989: 5986: 5976: 5972: 5961: 5957: 5956: 5943: 5937: 5932: 5926: 5922: 5918: 5913: 5910: 5906: 5900: 5896: 5885: 5872: 5866: 5861: 5855: 5851: 5848: 5842: 5837: 5834: 5830: 5824: 5820: 5809: 5805: 5804: 5801: 5800:SI units (Wb) 5798: 5795: 5786: 5785: 5784: 5783: 5770: 5767: 5760: 5757: 5750: 5747: 5742: 5739: 5732: 5729: 5721: 5716: 5713: 5709: 5705: 5702: 5697: 5694: 5690: 5668: 5651: 5650: 5639: 5635: 5631: 5627: 5623: 5619: 5615: 5611: 5603: 5599: 5594: 5588: 5584: 5580: 5577: 5573: 5561: 5550: 5545: 5541: 5537: 5532: 5528: 5524: 5519: 5515: 5511: 5506: 5502: 5498: 5495: 5490: 5486: 5475: 5469: 5468: 5457: 5451: 5445: 5439: 5436: 5432: 5425: 5420: 5416: 5406: 5395: 5389: 5383: 5378: 5372: 5367: 5363: 5353: 5342: 5337: 5331: 5326: 5322: 5317: 5311: 5306: 5302: 5297: 5291: 5286: 5282: 5277: 5271: 5266: 5262: 5259: 5254: 5248: 5243: 5231: 5230: 5219: 5213: 5207: 5201: 5198: 5194: 5187: 5182: 5178: 5168: 5157: 5151: 5145: 5140: 5134: 5129: 5125: 5115: 5104: 5099: 5093: 5088: 5084: 5079: 5073: 5068: 5064: 5059: 5053: 5048: 5044: 5039: 5033: 5028: 5024: 5021: 5016: 5010: 5005: 4994: 4992:Four-potential 4988: 4987: 4976: 4970: 4964: 4958: 4952: 4947: 4943: 4940: 4930: 4919: 4914: 4908: 4903: 4899: 4894: 4888: 4883: 4879: 4874: 4868: 4863: 4859: 4854: 4848: 4843: 4839: 4836: 4831: 4825: 4820: 4808: 4807: 4796: 4790: 4784: 4778: 4772: 4767: 4763: 4760: 4750: 4739: 4734: 4728: 4723: 4719: 4714: 4708: 4703: 4699: 4694: 4688: 4683: 4679: 4674: 4668: 4663: 4659: 4656: 4651: 4645: 4640: 4629: 4623: 4622: 4611: 4608: 4604: 4599: 4594: 4590: 4586: 4583: 4573: 4562: 4558: 4553: 4549: 4545: 4542: 4532: 4521: 4516: 4509: 4506: 4499: 4494: 4487: 4484: 4477: 4472: 4465: 4462: 4454: 4449: 4442: 4439: 4432: 4427: 4420: 4417: 4410: 4405: 4398: 4395: 4388: 4385: 4380: 4377: 4370: 4367: 4353: 4346: 4345: 4334: 4330: 4326: 4322: 4319: 4315: 4310: 4306: 4303: 4293: 4282: 4278: 4274: 4270: 4266: 4262: 4259: 4249: 4238: 4233: 4229: 4225: 4220: 4216: 4212: 4207: 4203: 4198: 4193: 4189: 4185: 4180: 4176: 4172: 4167: 4163: 4159: 4156: 4151: 4148: 4144: 4133: 4127: 4126: 4123: 4120: 4117: 4091: 4088: 4087: 4086: 4071: 4066: 4063: 4056: 4052: 4042: 4038: 4034: 4029: 4024: 4020: 4016: 4013: 3997: 3996: 3978: 3972: 3967: 3964: 3961: 3955: 3952: 3944: 3938: 3933: 3925: 3921: 3917: 3911: 3906: 3902: 3897: 3893: 3889: 3886: 3884: 3882: 3878: 3874: 3873: 3867: 3861: 3856: 3853: 3846: 3842: 3838: 3833: 3827: 3824: 3816: 3810: 3805: 3799: 3793: 3788: 3784: 3781: 3778: 3776: 3774: 3770: 3766: 3765: 3755: 3737: 3731: 3726: 3723: 3720: 3714: 3711: 3703: 3697: 3692: 3684: 3681: 3676: 3670: 3665: 3661: 3658: 3655: 3653: 3651: 3647: 3643: 3642: 3636: 3630: 3625: 3622: 3619: 3613: 3610: 3602: 3596: 3591: 3583: 3580: 3575: 3569: 3564: 3560: 3557: 3554: 3552: 3550: 3546: 3542: 3541: 3531: 3527: 3526: 3511: 3508: 3502: 3496: 3491: 3488: 3485: 3479: 3474: 3467: 3464: 3460: 3451: 3447: 3443: 3438: 3436: 3433: 3430: 3424: 3418: 3413: 3410: 3407: 3401: 3396: 3389: 3386: 3382: 3373: 3369: 3365: 3360: 3358: 3348: 3333: 3330: 3324: 3318: 3313: 3310: 3307: 3301: 3296: 3289: 3286: 3282: 3275: 3272: 3267: 3265: 3262: 3259: 3253: 3247: 3242: 3239: 3236: 3230: 3225: 3218: 3215: 3211: 3204: 3201: 3196: 3194: 3184: 3178: 3177: 3158: 3154: 3147: 3141: 3134: 3131: 3129: 3127: 3121: 3115: 3110: 3107: 3106: 3100: 3095: 3091: 3088: 3086: 3084: 3078: 3073: 3069: 3066: 3065: 3059: 3053: 3046: 3042: 3038: 3035: 3033: 3031: 3025: 3019: 3014: 3011: 3010: 3003: 2999: 2992: 2987: 2981: 2978: 2976: 2974: 2968: 2963: 2959: 2956: 2955: 2945: 2927: 2921: 2914: 2910: 2906: 2903: 2901: 2899: 2893: 2887: 2882: 2879: 2878: 2871: 2867: 2860: 2855: 2849: 2846: 2844: 2842: 2836: 2831: 2827: 2824: 2823: 2817: 2811: 2804: 2800: 2796: 2793: 2791: 2789: 2783: 2777: 2772: 2769: 2768: 2761: 2757: 2750: 2745: 2739: 2736: 2734: 2732: 2726: 2721: 2717: 2714: 2713: 2703: 2685: 2679: 2672: 2668: 2665: 2659: 2656: 2654: 2652: 2646: 2640: 2635: 2632: 2631: 2625: 2620: 2616: 2613: 2610: 2607: 2605: 2603: 2597: 2592: 2588: 2585: 2584: 2578: 2572: 2565: 2561: 2558: 2552: 2549: 2547: 2545: 2539: 2533: 2528: 2525: 2524: 2518: 2513: 2509: 2506: 2503: 2500: 2498: 2496: 2490: 2485: 2481: 2478: 2477: 2467: 2457: 2456: 2453: 2452:SI units (Wb) 2450: 2447: 2437: 2434: 2431: 2430: 2414: 2406: 2402: 2397: 2391: 2387: 2383: 2379: 2374: 2367: 2362: 2358: 2356: 2353: 2349: 2344: 2340: 2336: 2332: 2328: 2323: 2316: 2311: 2307: 2303: 2299: 2295: 2294: 2284: 2268: 2260: 2256: 2251: 2245: 2241: 2237: 2233: 2228: 2220: 2216: 2209: 2204: 2198: 2196: 2193: 2189: 2184: 2180: 2176: 2172: 2168: 2163: 2156: 2151: 2147: 2143: 2139: 2135: 2134: 2124: 2112: 2107: 2103: 2099: 2095: 2091: 2086: 2079: 2074: 2070: 2066: 2055: 2051: 2050: 2036: 2030: 2023: 2019: 2015: 2009: 2006: 2000: 1996: 1990: 1986: 1982: 1979: 1976: 1966: 1952: 1946: 1941: 1935: 1932: 1926: 1922: 1916: 1912: 1908: 1905: 1902: 1892: 1881: 1878: 1872: 1869: 1863: 1859: 1853: 1849: 1845: 1842: 1839: 1829: 1825: 1824: 1810: 1805: 1799: 1795: 1791: 1787: 1783: 1780: 1770: 1756: 1751: 1747: 1743: 1739: 1736: 1726: 1715: 1712: 1708: 1704: 1701: 1691: 1687: 1686: 1672: 1666: 1659: 1655: 1651: 1645: 1642: 1636: 1632: 1622: 1618: 1614: 1609: 1605: 1601: 1598: 1588: 1584: 1583: 1568: 1564: 1557: 1552: 1546: 1542: 1538: 1535: 1525: 1521: 1520: 1517: 1513: 1512: 1509: 1504: 1474: 1468: 1459: 1441: 1432:used with the 1425: 1422: 1404:speed of light 1382: 1369: 1356: 1347: 1346: 1334: 1329: 1325: 1320: 1316: 1310: 1306: 1301: 1294: 1289: 1285: 1281: 1276: 1272: 1267: 1263: 1257: 1253: 1248: 1241: 1236: 1232: 1228: 1217: 1205: 1200: 1196: 1191: 1187: 1181: 1177: 1172: 1165: 1160: 1156: 1152: 1141: 1134: 1133: 1119: 1113: 1106: 1102: 1099: 1093: 1087: 1084: 1078: 1074: 1066: 1063: 1058: 1054: 1050: 1047: 1044: 1034: 1023: 1020: 1014: 1011: 1005: 1001: 993: 990: 985: 981: 977: 974: 971: 961: 955: 954: 940: 935: 931: 928: 925: 921: 917: 914: 904: 893: 890: 886: 882: 879: 869: 863: 862: 848: 842: 835: 831: 828: 822: 816: 813: 807: 803: 795: 792: 787: 783: 779: 776: 766: 759: 758: 744: 739: 735: 732: 729: 725: 721: 718: 708: 702: 701: 698: 695: 678: 675: 651: 650: 595: 588: 587: 576:field while a 548: 541: 540: 539: 538: 537: 501: 488: 460: 457: 405:electric field 334:Standard Model 330:periodic table 324: 323: 274: 272: 265: 256:Main article: 253: 250: 240:quasiparticles 161: 158: 113: 110: 108: 105: 90:electromagnets 69:, notably the 15: 9: 6: 4: 3: 2: 14290: 14279: 14276: 14274: 14271: 14269: 14266: 14264: 14261: 14259: 14256: 14254: 14251: 14250: 14248: 14235: 14234: 14229: 14223: 14217: 14214: 14212: 14209: 14207: 14204: 14202: 14199: 14197: 14194: 14192: 14191:Exotic matter 14189: 14185: 14182: 14181: 14180: 14179:Eightfold way 14177: 14175: 14172: 14170: 14169:Antiparticles 14167: 14165: 14162: 14160: 14157: 14153: 14150: 14149: 14148: 14145: 14141: 14138: 14137: 14136: 14133: 14132: 14130: 14126: 14120: 14117: 14115: 14112: 14110: 14107: 14105: 14102: 14100: 14097: 14096: 14094: 14090: 14084: 14081: 14079: 14076: 14074: 14071: 14069: 14066: 14064: 14061: 14059: 14056: 14054: 14051: 14049: 14046: 14044: 14041: 14039: 14036: 14034: 14031: 14029: 14026: 14024: 14021: 14019: 14016: 14015: 14013: 14011: 14007: 13993: 13990: 13988: 13985: 13983: 13980: 13978: 13975: 13974: 13972: 13968: 13958: 13955: 13953: 13950: 13948: 13945: 13944: 13942: 13938: 13932: 13929: 13927: 13924: 13922: 13919: 13918: 13916: 13912: 13908: 13905: 13903: 13899: 13893: 13890: 13888: 13885: 13881: 13878: 13876: 13873: 13871: 13868: 13866: 13863: 13861: 13858: 13856: 13853: 13852: 13851: 13848: 13846: 13843: 13841: 13840:Atomic nuclei 13838: 13837: 13835: 13831: 13821: 13818: 13815: 13811: 13808: 13807: 13805: 13803: 13799: 13793: 13790: 13788: 13785: 13783: 13780: 13778: 13775: 13773: 13772:Upsilon meson 13770: 13768: 13765: 13763: 13760: 13758: 13755: 13753: 13750: 13748: 13745: 13743: 13740: 13738: 13735: 13734: 13732: 13730: 13726: 13720: 13717: 13715: 13712: 13710: 13707: 13705: 13704:Lambda baryon 13702: 13700: 13697: 13693: 13690: 13688: 13685: 13683: 13680: 13678: 13675: 13674: 13673: 13670: 13669: 13667: 13665: 13661: 13658: 13656: 13652: 13649: 13647: 13643: 13629: 13626: 13624: 13621: 13619: 13616: 13614: 13611: 13609: 13606: 13604: 13601: 13599: 13596: 13594: 13591: 13589: 13586: 13584: 13581: 13579: 13576: 13574: 13571: 13569: 13566: 13564: 13563:Dual graviton 13561: 13559: 13556: 13554: 13551: 13549: 13546: 13545: 13543: 13539: 13528: 13524: 13521: 13519: 13516: 13514: 13511: 13509: 13506: 13504: 13501: 13500: 13498: 13494: 13488: 13485: 13483: 13480: 13478: 13475: 13474: 13472: 13470: 13466: 13463: 13461: 13460:Superpartners 13457: 13454: 13452: 13448: 13442: 13439: 13438: 13436: 13434: 13430: 13420: 13417: 13416: 13414: 13412: 13408: 13402: 13399: 13397: 13394: 13392: 13389: 13388: 13386: 13384: 13380: 13377: 13375: 13371: 13359: 13356: 13354: 13351: 13349: 13346: 13344: 13343:Muon neutrino 13341: 13339: 13336: 13334: 13331: 13330: 13329: 13326: 13324: 13321: 13319: 13316: 13314: 13311: 13309: 13306: 13304: 13301: 13299: 13296: 13295: 13293: 13291: 13287: 13281: 13278: 13276: 13275:Bottom (quark 13273: 13271: 13268: 13266: 13263: 13261: 13258: 13256: 13253: 13251: 13248: 13246: 13243: 13241: 13238: 13236: 13233: 13231: 13228: 13226: 13223: 13222: 13220: 13218: 13214: 13211: 13209: 13205: 13202: 13200: 13196: 13192: 13185: 13180: 13178: 13173: 13171: 13166: 13165: 13162: 13158: 13157: 13155: 13151: 13146: 13142: 13141: 13136: 13119: 13118: 13112: 13108: 13104: 13100: 13096: 13092: 13088: 13083: 13078: 13074: 13070: 13065: 13061: 13055: 13051: 13046: 13042: 13038: 13034: 13030: 13026: 13022: 13017: 13012: 13008: 13004: 12999: 12995: 12989: 12985: 12980: 12976: 12970: 12966: 12961: 12957: 12951: 12947: 12942: 12938: 12932: 12928: 12923: 12917: 12916: 12910: 12906: 12902: 12898: 12894: 12890: 12886: 12882: 12878: 12874: 12869: 12865: 12859: 12855: 12850: 12846: 12841: 12837: 12832: 12828: 12826:0-691-08480-7 12822: 12818: 12813: 12812: 12800: 12796: 12792: 12788: 12784: 12780: 12775: 12770: 12767:(3): 030401. 12766: 12762: 12755: 12747: 12743: 12739: 12735: 12731: 12727: 12723: 12719: 12714: 12709: 12705: 12701: 12700: 12679: 12671: 12667: 12663: 12659: 12655: 12651: 12650: 12642: 12640: 12631: 12627: 12623: 12619: 12615: 12611: 12606: 12601: 12597: 12593: 12586: 12570: 12569: 12568:Science Daily 12564: 12558: 12550: 12546: 12542: 12538: 12534: 12530: 12526: 12522: 12517: 12512: 12508: 12504: 12497: 12495: 12486: 12482: 12476: 12468: 12464: 12460: 12456: 12452: 12448: 12444: 12440: 12435: 12430: 12426: 12422: 12415: 12407: 12403: 12399: 12395: 12391: 12387: 12383: 12379: 12374: 12369: 12365: 12361: 12353: 12345: 12341: 12337: 12333: 12329: 12322: 12320: 12311: 12307: 12303: 12299: 12295: 12291: 12287: 12283: 12278: 12273: 12269: 12265: 12258: 12250: 12246: 12242: 12238: 12234: 12230: 12225: 12220: 12217:(2): 021023. 12216: 12212: 12205: 12198: 12192: 12187: 12183: 12179: 12175: 12171: 12170:Physics Today 12167: 12160: 12153: 12149: 12144: 12137: 12133: 12129: 12123: 12116: 12112: 12111: 12103: 12095: 12091: 12087: 12083: 12079: 12075: 12070: 12065: 12062:(3): 031802. 12061: 12057: 12053: 12046: 12030: 12026: 12022: 12015: 12007: 12003: 11999: 11995: 11990: 11985: 11981: 11977: 11973: 11969: 11965: 11958: 11952: 11951:0-201-15142-1 11948: 11944: 11938: 11929: 11924: 11920: 11916: 11912: 11908: 11904: 11897: 11889: 11885: 11881: 11877: 11873: 11869: 11862: 11855: 11847: 11843: 11839: 11835: 11832:(3): 239–41. 11831: 11827: 11820: 11812: 11808: 11804: 11800: 11796: 11792: 11787: 11782: 11778: 11774: 11767: 11760: 11755: 11747: 11741: 11737: 11736: 11731: 11725: 11718: 11713: 11711: 11709: 11700: 11696: 11692: 11688: 11683: 11678: 11674: 11670: 11666: 11662: 11661: 11653: 11646: 11638: 11634: 11630: 11626: 11622: 11618: 11617: 11609: 11601: 11597: 11592: 11587: 11583: 11579: 11572: 11564: 11560: 11556: 11552: 11548: 11544: 11539: 11534: 11530: 11526: 11519: 11511: 11507: 11503: 11499: 11495: 11491: 11484: 11476: 11472: 11467: 11462: 11458: 11454: 11453: 11445: 11443: 11434: 11428: 11424: 11417: 11409: 11403: 11398: 11397: 11388: 11380: 11374: 11370: 11363: 11356: 11351: 11347: 11343: 11339: 11335: 11328: 11326: 11318: 11317:Physics World 11314: 11309: 11307: 11299: 11295: 11290: 11288: 11271: 11270:Science Daily 11267: 11261: 11259: 11250: 11246: 11242: 11238: 11234: 11230: 11225: 11220: 11216: 11212: 11205: 11203: 11195: 11190: 11182: 11178: 11174: 11170: 11166: 11162: 11155: 11153: 11144: 11140: 11136: 11132: 11128: 11124: 11117: 11115: 11107: 11102: 11100: 11091: 11087: 11083: 11079: 11075: 11071: 11067: 11063: 11057: 11049: 11045: 11038: 11030: 11023: 11015: 11008: 11000: 10996: 10992: 10988: 10984: 10980: 10976: 10972: 10968: 10964: 10959: 10954: 10950: 10946: 10945: 10937: 10935: 10933: 10924: 10920: 10916: 10912: 10908: 10904: 10900: 10896: 10891: 10886: 10882: 10878: 10871: 10869: 10867: 10860: 10854: 10847: 10841: 10833: 10826: 10820: 10812: 10810:9780061976865 10806: 10802: 10801: 10793: 10789: 10776: 10772: 10768: 10762: 10758: 10747: 10744: 10742: 10739: 10737: 10734: 10732: 10729: 10727: 10724: 10722: 10719: 10717: 10714: 10712: 10709: 10707: 10706:Halbach array 10704: 10702: 10699: 10697: 10694: 10692: 10689: 10687: 10684: 10682: 10679: 10677: 10674: 10672: 10669: 10668: 10661: 10658: 10654: 10647: 10641: 10634: 10629: 10624: 10621: 10620: 10614: 10612: 10611:heat capacity 10607: 10603: 10602:Dirac strings 10599: 10595: 10591: 10587: 10583: 10579: 10578: 10572: 10570: 10566: 10561: 10559: 10555: 10550: 10548: 10544: 10540: 10534: 10529: 10526: 10518: 10512: 10509: 10503: 10499: 10495: 10491: 10487: 10483: 10478: 10476: 10472: 10466: 10461: 10457: 10452: 10450: 10440: 10438: 10434: 10430: 10426: 10421: 10418: 10414: 10410: 10405: 10402: 10398: 10394: 10390: 10385: 10381: 10379: 10375: 10371: 10367: 10363: 10359: 10355: 10350: 10348: 10344: 10340: 10335: 10331: 10329: 10325: 10319: 10310: 10308: 10304: 10299: 10297: 10293: 10288: 10284: 10279: 10274: 10272: 10268: 10264: 10260: 10255: 10253: 10252: 10247: 10243: 10238: 10236: 10232: 10231:lepton number 10228: 10223: 10221: 10217: 10213: 10209: 10205: 10201: 10197: 10192: 10184: 10178: 10167: 10160: 10153: 10147: 10145: 10140: 10137: 10133: 10107: 10094: 10088: 10075: 10069: 10065: 10060: 10056: 10051: 10045: 10041:-bundle over 10035: 10032: 10028: 10024: 10020: 10017: 10013: 9991: 9975: 9973: 9969: 9965: 9959: 9933: 9928: 9922: 9918: 9907: 9904: 9896: 9886: 9882: 9878: 9872: 9871: 9867: 9862:This section 9860: 9856: 9851: 9850: 9842: 9840: 9836: 9835:string theory 9831: 9829: 9828:natural units 9825: 9817:String theory 9814: 9812: 9807: 9799: 9794: 9790: 9784: 9778: 9773: 9769: 9765: 9761: 9757: 9753: 9749: 9745: 9735: 9730: 9727: 9723: 9720: 9719: 9713: 9710: 9709: 9708: 9705: 9701: 9698: 9693: 9691: 9687: 9682: 9670: 9660: 9656: 9653: 9641: 9631: 9620: 9614: 9612: 9608: 9604: 9603:magnetic flux 9586: 9583: 9580: 9576: 9571: 9567: 9563: 9560: 9557: 9554: 9547: 9544: 9533: 9529: 9525: 9522: 9519: 9516: 9513: 9510: 9505: 9498: 9494: 9486: 9484: 9480: 9462: 9458: 9454: 9451: 9448: 9445: 9442: 9439: 9436: 9432: 9428: 9425: 9422: 9418: 9414: 9411: 9404: 9401: 9394: 9390: 9386: 9378: 9374: 9370: 9367: 9364: 9361: 9357: 9351: 9347: 9338: 9335: 9329: 9325: 9321: 9315: 9310: 9307: 9301: 9286: 9284: 9278: 9275: 9271: 9267: 9263: 9259: 9258:semi-infinite 9254: 9252: 9248: 9247: 9240: 9236: 9234: 9208: 9203: 9198: 9189: 9185: 9180: 9172: 9143: 9137: 9134: 9121: 9109: 9102: 9095: 9093: 9090: 9089: 9069: 9061: 9057: 9048: 9044: 9040: 9037: 9024: 9012: 9001: 8998: 8994: 8993: 8973: 8964: 8961: 8948: 8936: 8925: 8922: 8918: 8915: 8914: 8910: 8907: 8906: 8903: 8902: 8901: 8899: 8887: 8881: 8875: 8869: 8867: 8863: 8855: 8846: 8841: 8837: 8836:wave function 8833: 8827: 8820: 8815: 8810: 8807: 8801: 8796: 8792: 8786: 8780: 8770: 8759: 8749: 8744: 8742: 8738: 8730: 8724: 8718: 8712: 8706: 8700: 8695: 8691: 8685: 8683: 8678: 8674: 8670: 8660: 8651: 8644: 8622: 8614: 8600: 8589: 8582: 8576: 8573: 8570: 8565: 8562: 8559: 8552: 8549: 8546: 8543: 8538: 8535: 8532: 8526: 8521: 8516: 8493: 8484: 8468: 8461: 8440: 8421: 8414: 8408: 8405: 8402: 8397: 8394: 8391: 8384: 8381: 8378: 8375: 8370: 8367: 8364: 8358: 8353: 8348: 8307: 8298: 8297: 8280: 8272: 8258: 8247: 8240: 8234: 8231: 8228: 8223: 8220: 8217: 8210: 8207: 8204: 8201: 8196: 8193: 8190: 8184: 8179: 8174: 8151: 8142: 8126: 8119: 8109: 8100: 8090: 8083: 8076: 8070: 8067: 8064: 8059: 8056: 8053: 8046: 8043: 8040: 8037: 8032: 8029: 8026: 8020: 8015: 8010: 7996: 7980: 7973: 7964: 7963: 7959: 7956: 7955: 7952: 7949: 7944: 7934: 7932: 7927: 7917: 7902: 7891: 7885: 7875: 7872: 7869: 7866: 7862: 7858: 7850: 7839: 7833: 7825: 7820: 7809: 7803: 7792: 7787: 7783: 7779: 7774: 7771: 7761: 7734: 7723: 7717: 7707: 7704: 7701: 7698: 7694: 7690: 7685: 7681: 7677: 7672: 7661: 7655: 7647: 7642: 7631: 7625: 7617: 7612: 7609: 7605: 7595: 7580: 7569: 7563: 7553: 7550: 7547: 7544: 7540: 7536: 7531: 7520: 7514: 7506: 7501: 7490: 7484: 7476: 7471: 7468: 7458: 7431: 7420: 7414: 7404: 7401: 7398: 7395: 7391: 7387: 7382: 7371: 7365: 7357: 7352: 7341: 7335: 7327: 7322: 7319: 7315: 7307: 7304: 7298: 7297: 7282: 7279: 7274: 7263: 7257: 7248: 7245: 7242: 7237: 7226: 7220: 7206: 7203: 7202: 7185: 7174: 7166: 7162: 7158: 7153: 7141: 7125: 7114: 7108: 7104: 7100: 7095: 7084: 7078: 7068: 7060: 7055: 7044: 7038: 7028: 7016: 7000: 6989: 6983: 6979: 6976: 6970: 6965: 6954: 6948: 6938: 6930: 6925: 6914: 6908: 6898: 6886: 6885: 6868: 6857: 6851: 6847: 6843: 6838: 6827: 6821: 6811: 6803: 6798: 6787: 6781: 6771: 6743: 6732: 6726: 6722: 6719: 6713: 6708: 6697: 6691: 6681: 6673: 6668: 6657: 6651: 6641: 6629: 6625: 6621: 6618: 6615: 6612: 6611: 6608: 6587: 6583: 6578: 6571: 6568: 6558: 6548: 6537: 6531: 6526: 6523: 6519: 6507: 6502: 6498: 6493: 6489: 6481: 6465: 6461: 6456: 6449: 6446: 6436: 6425: 6420: 6416: 6403: 6397: 6392: 6389: 6385: 6373: 6368: 6364: 6359: 6355: 6347: 6331: 6326: 6322: 6315: 6308: 6305: 6295: 6279: 6275: 6270: 6267: 6263: 6251: 6246: 6242: 6237: 6233: 6225: 6222: 6221: 6204: 6193: 6187: 6182: 6178: 6172: 6166: 6163: 6153: 6143: 6131: 6115: 6104: 6098: 6095: 6090: 6084: 6081: 6071: 6061: 6049: 6033: 6022: 6016: 6012: 6009: 6003: 5997: 5994: 5984: 5974: 5962: 5959: 5958: 5941: 5930: 5924: 5920: 5916: 5911: 5908: 5904: 5898: 5870: 5859: 5853: 5849: 5846: 5840: 5835: 5832: 5828: 5822: 5810: 5807: 5806: 5802: 5799: 5796: 5793: 5792: 5789: 5768: 5765: 5755: 5748: 5745: 5740: 5737: 5727: 5719: 5714: 5711: 5707: 5703: 5700: 5695: 5692: 5688: 5680: 5679: 5677: 5673: 5669: 5662: 5658: 5657: 5656: 5629: 5625: 5617: 5601: 5597: 5592: 5586: 5582: 5578: 5575: 5571: 5543: 5539: 5535: 5530: 5526: 5522: 5517: 5513: 5509: 5504: 5500: 5493: 5488: 5484: 5476: 5474: 5471: 5470: 5437: 5434: 5430: 5418: 5407: 5376: 5365: 5354: 5335: 5324: 5320: 5315: 5304: 5300: 5295: 5284: 5280: 5275: 5264: 5257: 5252: 5241: 5233: 5232: 5199: 5196: 5192: 5180: 5169: 5138: 5127: 5116: 5097: 5086: 5082: 5077: 5066: 5062: 5057: 5046: 5042: 5037: 5026: 5019: 5014: 5003: 4995: 4993: 4989: 4956: 4945: 4941: 4912: 4901: 4897: 4892: 4881: 4877: 4872: 4861: 4857: 4852: 4841: 4834: 4829: 4818: 4810: 4809: 4776: 4765: 4761: 4732: 4721: 4717: 4712: 4701: 4697: 4692: 4681: 4677: 4672: 4661: 4654: 4649: 4638: 4630: 4628: 4624: 4606: 4602: 4592: 4584: 4574: 4551: 4543: 4533: 4514: 4504: 4497: 4492: 4482: 4475: 4470: 4460: 4452: 4447: 4437: 4430: 4425: 4415: 4408: 4403: 4393: 4383: 4378: 4375: 4365: 4354: 4351: 4348: 4347: 4324: 4320: 4317: 4313: 4304: 4294: 4272: 4268: 4260: 4250: 4231: 4227: 4223: 4218: 4214: 4210: 4205: 4201: 4196: 4191: 4187: 4183: 4178: 4174: 4170: 4165: 4161: 4154: 4149: 4146: 4142: 4134: 4132: 4129: 4128: 4124: 4121: 4118: 4115: 4114: 4111: 4109: 4105: 4101: 4097: 4069: 4064: 4054: 4040: 4036: 4032: 4027: 4022: 4014: 4011: 4004: 4003: 4002: 3965: 3959: 3953: 3923: 3919: 3915: 3904: 3895: 3891: 3887: 3885: 3880: 3854: 3844: 3840: 3836: 3831: 3825: 3797: 3786: 3779: 3777: 3772: 3724: 3718: 3712: 3682: 3679: 3674: 3663: 3656: 3654: 3649: 3623: 3617: 3611: 3581: 3578: 3573: 3562: 3555: 3553: 3548: 3532: 3529: 3528: 3509: 3506: 3489: 3483: 3472: 3465: 3449: 3445: 3441: 3431: 3428: 3411: 3405: 3394: 3387: 3371: 3367: 3363: 3331: 3328: 3311: 3305: 3294: 3287: 3273: 3270: 3260: 3257: 3240: 3234: 3223: 3216: 3202: 3199: 3185: 3183: 3180: 3179: 3156: 3152: 3132: 3130: 3125: 3108: 3093: 3089: 3087: 3082: 3071: 3067: 3044: 3040: 3036: 3034: 3029: 3012: 3001: 2997: 2985: 2979: 2977: 2972: 2961: 2957: 2946: 2912: 2908: 2904: 2902: 2897: 2880: 2869: 2865: 2853: 2847: 2845: 2840: 2829: 2825: 2802: 2798: 2794: 2792: 2787: 2770: 2759: 2755: 2743: 2737: 2735: 2730: 2719: 2715: 2704: 2670: 2666: 2663: 2657: 2655: 2650: 2633: 2618: 2614: 2611: 2608: 2606: 2601: 2590: 2586: 2563: 2559: 2556: 2550: 2548: 2543: 2526: 2511: 2507: 2504: 2501: 2499: 2494: 2483: 2479: 2468: 2465: 2459: 2458: 2454: 2451: 2448: 2445: 2444: 2441: 2412: 2404: 2400: 2389: 2381: 2372: 2360: 2351: 2347: 2338: 2330: 2321: 2309: 2301: 2285: 2266: 2258: 2254: 2243: 2235: 2226: 2218: 2214: 2202: 2191: 2187: 2178: 2170: 2161: 2149: 2141: 2125: 2110: 2101: 2093: 2084: 2072: 2068: 2056: 2053: 2052: 2021: 2017: 2013: 2007: 1988: 1980: 1974: 1967: 1939: 1933: 1914: 1906: 1900: 1893: 1879: 1876: 1870: 1851: 1843: 1837: 1830: 1827: 1826: 1803: 1797: 1793: 1789: 1781: 1771: 1749: 1745: 1737: 1727: 1713: 1710: 1702: 1692: 1689: 1688: 1657: 1653: 1649: 1643: 1620: 1616: 1612: 1607: 1599: 1586: 1585: 1566: 1562: 1550: 1544: 1536: 1523: 1522: 1518: 1515: 1514: 1501: 1495: 1492: 1490: 1486: 1473: 1467: 1458: 1452: 1448: 1440: 1435: 1431: 1421: 1418: 1413: 1409: 1405: 1400: 1394: 1389: 1381: 1376: 1368: 1363: 1355: 1332: 1323: 1318: 1308: 1299: 1287: 1283: 1279: 1270: 1265: 1255: 1246: 1234: 1230: 1218: 1203: 1194: 1189: 1179: 1170: 1158: 1154: 1142: 1139: 1138:Lorentz force 1136: 1135: 1104: 1100: 1097: 1091: 1085: 1064: 1061: 1056: 1048: 1042: 1035: 1021: 1018: 1012: 991: 988: 983: 975: 969: 962: 960: 957: 956: 933: 929: 926: 923: 915: 905: 891: 888: 880: 870: 868: 865: 864: 833: 829: 826: 820: 814: 793: 790: 785: 777: 764: 761: 760: 737: 733: 730: 727: 719: 707: 704: 703: 699: 696: 692: 686: 684: 669: 665: 663: 658: 656: 646: 642: 638: 635: 631: 628: 627:Bottom right: 623: 620: 616: 613: 608: 605: 601: 598: 592: 583: 579: 575: 571: 567: 564: 560: 555: 551: 545: 536: 534: 529: 525:operator and 524: 514: 506: 500: 495: 487: 480: 478: 474: 469: 465: 456: 453: 449: 445: 441: 437: 433: 429: 425: 421: 416: 414: 410: 406: 402: 398: 397: 392: 388: 384: 379: 377: 373: 369: 366: 362: 358: 355:according to 354: 350: 345: 343: 339: 335: 331: 320: 317: 309: 299: 295: 291: 285: 284: 280: 275:This section 273: 269: 264: 263: 259: 249: 247: 246: 241: 237: 233: 229: 224: 222: 218: 214: 210: 206: 202: 198: 194: 188: 186: 182: 178: 174: 171: 167: 157: 155: 151: 147: 143: 139: 135: 131: 127: 123: 119: 104: 102: 98: 93: 91: 87: 84:Magnetism in 82: 80: 76: 72: 71:grand unified 68: 64: 60: 56: 52: 44: 40: 36: 32: 28: 23: 19: 14231: 13902:Hypothetical 13850:Exotic atoms 13719:Omega baryon 13709:Sigma baryon 13699:Delta baryon 13587: 13451:Hypothetical 13433:Ghost fields 13419:Higgs boson 13353:Tau neutrino 13245:Charm (quark 13138: 13130: 13129: 13116: 13072: 13068: 13052:. Springer. 13049: 13006: 13002: 12986:. Elsevier. 12983: 12967:. Springer. 12964: 12945: 12926: 12914: 12880: 12876: 12853: 12835: 12816: 12809:Bibliography 12764: 12760: 12754: 12703: 12697: 12678: 12653: 12647: 12595: 12591: 12585: 12573:. Retrieved 12566: 12557: 12506: 12502: 12484: 12475: 12424: 12420: 12414: 12363: 12359: 12352: 12327: 12267: 12263: 12257: 12214: 12211:Phys. Rev. X 12210: 12204: 12195: 12173: 12169: 12159: 12143: 12135: 12122: 12114: 12108: 12102: 12059: 12055: 12045: 12033:. Retrieved 12029:the original 12024: 12014: 11971: 11967: 11957: 11942: 11937: 11910: 11906: 11896: 11871: 11867: 11854: 11829: 11825: 11819: 11776: 11772: 11766: 11759:Jackson 1999 11754: 11733: 11724: 11717:Jackson 1999 11664: 11658: 11645: 11620: 11614: 11608: 11581: 11577: 11571: 11531:(1): 23–33. 11528: 11524: 11518: 11493: 11489: 11483: 11456: 11450: 11422: 11416: 11395: 11387: 11368: 11362: 11353: 11333: 11316: 11276:September 4, 11274:. Retrieved 11214: 11210: 11189: 11164: 11160: 11126: 11122: 11073: 11069: 11056: 11047: 11037: 11022: 11007: 10948: 10942: 10880: 10876: 10858: 10853: 10845: 10840: 10831: 10819: 10799: 10792: 10767:Jackson 1999 10761: 10676:Dirac string 10659: 10652: 10645: 10639: 10632: 10625: 10617: 10615: 10575: 10573: 10565:magnetricity 10564: 10562: 10551: 10532: 10524: 10516: 10507: 10479: 10464: 10453: 10446: 10425:Igor Novikov 10422: 10406: 10401:Wendy Taylor 10386: 10382: 10351: 10336: 10332: 10320: 10316: 10300: 10296:proton decay 10275: 10258: 10256: 10249: 10239: 10224: 10219: 10200:gauge theory 10193: 10190: 10176: 10165: 10158: 10151: 10148: 10144:Chern number 10141: 10135: 10095: 10073: 10067: 10063: 10059:homeomorphic 10049: 10043: 10037:A principal 10036: 10030: 9976: 9971: 9960: 9931: 9929: 9914: 9899: 9893:January 2021 9890: 9875:Please help 9863: 9832: 9820: 9811:lattice U(1) 9803: 9797: 9788: 9782: 9776: 9772:contractible 9747: 9743: 9733: 9717: 9706: 9702: 9694: 9680: 9672: 9657: 9651: 9639: 9629: 9618: 9615: 9600: 9476: 9333: 9327: 9323: 9319: 9311: 9306:gauge theory 9303: 9300:Dirac string 9294:Dirac string 9279: 9262:Dirac string 9255: 9246:Gauge theory 9244: 9241: 9237: 9201: 9187: 9183: 9170: 9167: 8923:convention) 8897: 8885: 8879: 8873: 8870: 8853: 8844: 8825: 8818: 8811: 8805: 8799: 8784: 8778: 8768: 8757: 8745: 8728: 8722: 8716: 8713: 8704: 8698: 8686: 8677:relativistic 8666: 8649: 8645: 8641: 7947: 7942: 7940: 7925: 7922: 7750: 7447: 6606: 5787: 5654: 4627:Four-current 4108:four-vectors 4093: 4000: 2464:Lorenz gauge 2439: 1524:Gauss's law 1493: 1471: 1465: 1456: 1438: 1427: 1416: 1398: 1392: 1387: 1379: 1374: 1366: 1361: 1353: 1350: 763:Ampère's law 683:CGS-Gaussian 680: 661: 654: 644: 640: 636: 629: 626: 621: 614: 612:Bottom left: 611: 606: 599: 596: 581: 577: 573: 569: 565: 558: 549: 527: 512: 507: 498: 485: 481: 462: 451: 435: 431: 427: 423: 417: 412: 408: 400: 394: 390: 386: 380: 364: 357:Ampère's law 346: 327: 312: 306:January 2023 303: 288:Please help 276: 243: 225: 191:theoretical 189: 184: 176: 163: 153: 150:Pierre Curie 115: 94: 83: 54: 48: 30: 18: 14184:Quark model 13952:Theta meson 13855:Positronium 13767:Omega meson 13762:J/psi meson 13692:Antineutron 13603:Dark photon 13568:Graviphoton 13527:Stop squark 13235:Down (quark 11062:Dirac, Paul 10628:superfluids 10433:black holes 10354:cosmic rays 10309:to create. 10251:topological 10010:-bundle is 9839:Planck mass 9768:Lie algebra 9479:Wilson loop 1424:In SI units 706:Gauss's law 666:are due to 561:In motion ( 86:bar magnets 75:superstring 14247:Categories 13926:Heptaquark 13887:Superatoms 13820:Pentaquark 13810:Tetraquark 13792:Quarkonium 13682:Antiproton 13583:Leptoquark 13518:Neutralino 13280:antiquark) 13270:antiquark) 13265:Top (quark 13260:antiquark) 13250:antiquark) 13240:antiquark) 13230:antiquark) 13199:Elementary 12277:1705.05162 12224:1611.07766 12176:(10): 40. 12132:Paul Dirac 12069:1905.10130 11826:Phys. Lett 11779:(6): 060. 10785:References 10539:superfluid 10475:Paul Dirac 10096:So in the 9932:connection 9917:connection 8995:SI units ( 8911:Condition 8673:Paul Dirac 7923:where the 5672:Hodge dual 5473:Four-force 2462:(assuming 1508:monopoles 1489:inductance 1447:weber (Wb) 523:divergence 436:south pole 432:north pole 396:quadrupole 173:Paul Dirac 136:, and the 118:lodestones 27:bar magnet 14263:Magnetism 14164:Particles 14109:Particles 14068:Polariton 14058:Plasmaron 14028:Dropleton 13921:Hexaquark 13892:Molecules 13880:Protonium 13757:Phi meson 13742:Rho meson 13714:Xi baryon 13646:Composite 13482:Gravitino 13225:Up (quark 13145:EMS Press 13041:119061150 12905:123573860 12845:1105.5587 12774:0903.4732 12746:206522398 12713:1011.1174 12605:1005.3557 12516:0907.0956 12467:206517194 12434:0811.1303 12344:124109501 12310:206304158 11677:CiteSeerX 11538:0901.0194 11350:124109501 10983:0028-0836 10958:1408.3133 10890:0710.5515 10716:Instanton 10604:. At the 10588:material 10494:electrons 10397:Drell–Yan 10372:first to 10263:causality 10019:worldline 10012:connected 9864:does not 9764:Lie group 9568:∫ 9545:× 9542:∇ 9530:∫ 9514:⋅ 9503:∂ 9499:∮ 9449:⋅ 9443:∫ 9429:⁡ 9395:μ 9379:μ 9348:∏ 9144:∈ 9135:ℏ 9070:∈ 9054:ℏ 9045:ε 9041:π 8974:∈ 8968:ℏ 8965:π 8748:integrate 8741:quantized 8577:ξ 8574:⁡ 8566:ξ 8563:⁡ 8553:ξ 8550:⁡ 8544:− 8539:ξ 8536:⁡ 8409:ξ 8406:⁡ 8398:ξ 8395:⁡ 8385:ξ 8382:⁡ 8376:− 8371:ξ 8368:⁡ 8235:ξ 8232:⁡ 8224:ξ 8221:⁡ 8211:ξ 8208:⁡ 8202:− 8197:ξ 8194:⁡ 8110:ρ 8091:ρ 8071:ξ 8068:⁡ 8060:ξ 8057:⁡ 8047:ξ 8044:⁡ 8038:− 8033:ξ 8030:⁡ 7997:ρ 7981:ρ 7903:ν 7886:μ 7882:∂ 7876:ν 7873:μ 7870:β 7867:α 7863:ε 7851:α 7834:β 7830:∂ 7826:− 7821:β 7804:α 7800:∂ 7784:μ 7775:β 7772:α 7765:~ 7735:ν 7718:μ 7714:∂ 7708:ν 7705:μ 7702:β 7699:α 7695:ε 7682:μ 7678:− 7673:α 7656:β 7652:∂ 7648:− 7643:β 7626:α 7622:∂ 7613:β 7610:α 7581:ν 7564:μ 7560:∂ 7554:ν 7551:μ 7548:β 7545:α 7541:ε 7532:α 7515:β 7511:∂ 7507:− 7502:β 7485:α 7481:∂ 7472:β 7469:α 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8180:= 8175:) 8168:H 8159:E 8152:( 8127:) 8115:m 8096:e 8084:( 8077:) 8021:( 8016:= 8011:) 8002:m 7986:e 7974:( 7948:ξ 7898:e 7892:A 7859:+ 7856:) 7845:m 7840:A 7815:m 7810:A 7796:( 7793:c 7788:0 7780:= 7762:F 7730:m 7724:A 7691:c 7686:0 7667:e 7662:A 7637:e 7632:A 7618:= 7606:F 7576:e 7570:A 7537:+ 7526:m 7521:A 7496:m 7491:A 7477:= 7459:F 7427:m 7421:A 7377:e 7372:A 7347:e 7342:A 7328:= 7316:F 7301:( 7283:0 7280:= 7269:m 7264:A 7249:, 7246:0 7243:= 7232:e 7227:A 7180:m 7175:J 7167:2 7163:c 7159:1 7154:= 7120:m 7115:J 7109:0 7101:= 7090:m 7085:A 7050:m 7045:A 6995:m 6990:J 6984:c 6977:4 6971:= 6960:m 6955:A 6920:m 6915:A 6863:e 6858:J 6852:0 6844:= 6833:e 6828:A 6793:e 6788:A 6738:e 6733:J 6727:c 6720:4 6714:= 6703:e 6698:A 6663:e 6658:A 6584:v 6579:] 6559:F 6549:c 6543:m 6538:q 6532:+ 6520:F 6513:e 6508:q 6503:[ 6499:= 6490:f 6462:v 6457:] 6437:F 6426:c 6421:0 6409:m 6404:q 6398:+ 6386:F 6379:e 6374:q 6369:[ 6365:= 6356:f 6332:c 6323:v 6316:] 6296:F 6285:m 6280:q 6276:+ 6264:F 6257:e 6252:q 6247:[ 6243:= 6234:f 6199:m 6194:J 6188:c 6183:0 6173:= 6154:F 6110:m 6105:J 6099:c 6096:1 6091:= 6072:F 6028:m 6023:J 6017:c 6010:4 6004:= 5985:F 5936:e 5931:J 5925:0 5917:= 5905:F 5865:e 5860:J 5854:c 5847:4 5841:= 5829:F 5756:F 5746:= 5728:F 5720:, 5708:F 5701:= 5689:F 5667:. 5638:) 5634:F 5626:, 5622:v 5614:F 5610:( 5602:2 5598:c 5593:/ 5587:2 5583:v 5576:1 5572:1 5549:) 5544:3 5540:f 5536:, 5531:2 5527:f 5523:, 5518:1 5514:f 5510:, 5505:0 5501:f 5497:( 5494:= 5485:f 5456:) 5450:m 5444:A 5438:, 5435:c 5431:/ 5424:m 5415:( 5394:) 5388:m 5382:A 5377:, 5371:m 5362:( 5341:) 5336:3 5330:m 5325:A 5321:, 5316:2 5310:m 5305:A 5301:, 5296:1 5290:m 5285:A 5281:, 5276:0 5270:m 5265:A 5261:( 5258:= 5247:m 5242:A 5218:) 5212:e 5206:A 5200:, 5197:c 5193:/ 5186:e 5177:( 5156:) 5150:e 5144:A 5139:, 5133:e 5124:( 5103:) 5098:3 5092:e 5087:A 5083:, 5078:2 5072:e 5067:A 5063:, 5058:1 5052:e 5047:A 5043:, 5038:0 5032:e 5027:A 5023:( 5020:= 5009:e 5004:A 4975:) 4969:m 4963:j 4957:, 4951:m 4942:c 4939:( 4918:) 4913:3 4907:m 4902:J 4898:, 4893:2 4887:m 4882:J 4878:, 4873:1 4867:m 4862:J 4858:, 4853:0 4847:m 4842:J 4838:( 4835:= 4824:m 4819:J 4795:) 4789:e 4783:j 4777:, 4771:e 4762:c 4759:( 4738:) 4733:3 4727:e 4722:J 4718:, 4713:2 4707:e 4702:J 4698:, 4693:1 4687:e 4682:J 4678:, 4673:0 4667:e 4662:J 4658:( 4655:= 4644:e 4639:J 4610:) 4607:c 4603:/ 4598:E 4593:, 4589:B 4582:( 4561:) 4557:E 4552:, 4548:B 4541:( 4520:) 4505:F 4498:, 4483:F 4476:, 4461:F 4453:, 4438:F 4431:, 4416:F 4409:, 4394:F 4387:( 4384:= 4366:F 4333:) 4329:B 4321:, 4318:c 4314:/ 4309:E 4302:( 4281:) 4277:B 4269:, 4265:E 4258:( 4237:) 4228:F 4224:, 4215:F 4211:, 4202:F 4197:, 4188:F 4184:, 4175:F 4171:, 4162:F 4158:( 4155:= 4143:F 4070:2 4065:t 4055:2 4041:2 4037:c 4033:1 4023:2 4015:= 3977:e 3971:A 3960:+ 3954:t 3943:m 3937:A 3924:0 3910:m 3896:0 3881:= 3877:B 3866:m 3860:A 3845:0 3837:1 3826:t 3815:e 3809:A 3792:e 3773:= 3769:E 3736:e 3730:A 3719:+ 3713:t 3702:m 3696:A 3683:c 3680:1 3669:m 3650:= 3646:B 3635:m 3629:A 3612:t 3601:e 3595:A 3582:c 3579:1 3568:e 3549:= 3545:E 3510:0 3507:= 3501:m 3495:A 3484:+ 3478:m 3466:t 3450:2 3446:c 3442:1 3432:0 3429:= 3423:e 3417:A 3406:+ 3400:e 3388:t 3372:2 3368:c 3364:1 3332:0 3329:= 3323:m 3317:A 3306:+ 3300:m 3288:t 3274:c 3271:1 3261:0 3258:= 3252:e 3246:A 3235:+ 3229:e 3217:t 3203:c 3200:1 3157:2 3153:c 3146:m 3140:j 3126:= 3120:m 3114:A 3099:m 3083:= 3077:m 3058:e 3052:j 3045:0 3030:= 3024:e 3018:A 3002:0 2991:e 2973:= 2967:e 2926:m 2920:j 2913:0 2898:= 2892:m 2886:A 2870:0 2859:m 2841:= 2835:m 2816:e 2810:j 2803:0 2788:= 2782:e 2776:A 2760:0 2749:e 2731:= 2725:e 2684:m 2678:j 2671:c 2664:4 2651:= 2645:m 2639:A 2624:m 2612:4 2602:= 2596:m 2577:e 2571:j 2564:c 2557:4 2544:= 2538:e 2532:A 2517:e 2505:4 2495:= 2489:e 2413:) 2405:2 2401:c 2396:E 2386:v 2378:B 2373:( 2366:m 2361:q 2352:+ 2348:) 2343:B 2335:v 2331:+ 2327:E 2322:( 2315:e 2310:q 2302:= 2298:F 2267:) 2259:2 2255:c 2250:E 2240:v 2232:B 2227:( 2219:0 2208:m 2203:q 2192:+ 2188:) 2183:B 2175:v 2171:+ 2167:E 2162:( 2155:e 2150:q 2142:= 2138:F 2111:) 2106:B 2098:v 2094:+ 2090:E 2085:( 2078:e 2073:q 2069:= 2065:F 2035:m 2029:j 2022:0 2014:= 2008:t 1999:B 1985:E 1951:m 1945:j 1940:= 1934:t 1925:B 1911:E 1880:0 1877:= 1871:t 1862:B 1848:E 1809:m 1798:0 1790:= 1786:B 1755:m 1746:= 1742:B 1714:0 1711:= 1707:B 1671:e 1665:j 1658:0 1650:= 1644:t 1635:E 1621:2 1617:c 1613:1 1604:B 1567:0 1556:e 1545:= 1541:E 1475:m 1472:q 1469:0 1466:μ 1460:m 1457:q 1442:m 1439:q 1417:c 1399:c 1393:v 1383:m 1380:q 1370:m 1367:j 1357:m 1354:ρ 1333:) 1328:E 1319:c 1315:v 1305:B 1300:( 1293:m 1288:q 1284:+ 1280:) 1275:B 1266:c 1262:v 1256:+ 1252:E 1247:( 1240:e 1235:q 1231:= 1227:F 1204:) 1199:B 1190:c 1186:v 1180:+ 1176:E 1171:( 1164:e 1159:q 1155:= 1151:F 1118:m 1112:j 1105:c 1098:4 1092:= 1086:t 1077:B 1065:c 1062:1 1053:E 1022:0 1019:= 1013:t 1004:B 992:c 989:1 980:E 939:m 927:4 924:= 920:B 892:0 889:= 885:B 847:e 841:j 834:c 827:4 821:= 815:t 806:E 794:c 791:1 782:B 743:e 731:4 728:= 724:E 662:B 655:E 637:m 630:B 622:m 615:B 609:. 607:d 600:E 582:E 574:B 566:v 528:B 513:B 502:m 499:j 489:m 486:ρ 319:) 313:( 308:) 304:( 300:. 286:. 45:.

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Index