Magnetic
Monopoles |

**Abelian / Dirac Monopoles** > s.a. QED [flux
quantization, curved spacetime]; solutions and phenomenology.

* __History__: 1931, First
proposed by Dirac; 14 feb 1982, Event with monopole signature seen in Blas
Cabrera's SQUID; 2003, Team from Japan, China
and Switzerland claim they have found indirect evidence,
based on the anomalous Hall effect; 2010, Still no confirmation.

* __Idea__: Particles that
carry isolated N or S magnetic poles; Solutions of ∇ · **B** =
0, with **B** not of the form ∇ × **A**;
They are characterized by H^{2}(\(\mathbb R\)^{3} \
{0};\(\mathbb R\))
= H^{2}(S^{2};\(\mathbb R\))
=\(\mathbb R\).

* __Classification__: The gauge
group is U(1), so monopoles are classified by U(1)-bundles *P* over \(\mathbb R\) \
{0}, homotopic to S^{2}, i.e., by elements
of π_{1}(S^{1})
=\(\mathbb Z\), the integer *n* (magnetic charge) being evaluated by
calculating the characteristic class *c*_{1}(*P*), and integrating
it over S^{2}:

*C*_{1} = –\(1\over2\pi\)tr ∫_{ S2} *F*
= *n* .

* __Coupling strength__:
Since *ge*/\(\hbar\)*c* = *n*/2,
from *e*^{2}/\(\hbar\)*c* =
1/137 we get *g*^{2}/\(\hbar\)*c* =
(137/4) *n*^{2}, an enormous value.

* __Particle trajectories__:
A charged particle will spiral inward to a minimum distance, then outward; There are no bound states.

**References** > s.a. gravitational collapse;
electromagnetism.

@ __General__: Dirac PRS(31);
Ramsey PR(58) [and discrete symmetries]; Schwinger Sci(69)aug;
Wu & Yang PRD(75);
Dirac IJTP(78); Yang pr(79); Von Baeyer ThSc(90)jul;
Staruszkiewicz in(92)ht/98;
Bakker et al PRL(98)
[in SU(2) gauge theory];
Lynden-Bell & Nouri-Zonoz RMP(98)gq/96 [interactions];
Kalogeropoulos IJGMP(04)mp/05 [and
differential
characters]; Weinberg 12.

@ __History__: Bais ht/04-in; Aloisi & Nali Ulisse-a1608-RG [Dirac].

@ __Reviews__: Carrigan NC(65); Sandars CP(66);
Amaldi in(68); Goldhaber
& Trower AJP(90)may [RL], 91;
Shnir 05; Rajantie CP(12)-a1204.

@ __Particle dynamics__: Rodrigues Sobreira & Bezerra
de Mello G&C(99)ht/98;
Banerjee & Ghosh IJMPA(00);
Pitelli & Letelier PRD(09)-a0911 [massive
scalar quantum particle]; Ushakov IJTP(11)-a1004 [charged particle in the field of a magnetic
monopole, phase space]; Vaz IJTP(13) [Clifford algebra approach]; > s.a. quantum
particles; spin-1 particles.

@ __And charge quantization__: Dirac PR(48);
Jackiw IJMPA(04)ht/02-conf;
Nesterov & de
la Cruz PLA(04)ht,
JMP(08)ht/04 [and representations of rotation group]; Leal & López JMP(06)ht/04 [in
loop representation].

@ __And general relativity / cosmology__: Gibbons LNP(91)-a1109 [gravitating, and black holes]; Borde et al PRD(99)gq/98 [collisions
and baby universes];
Arreaga et al PRD(00)gq [stability]; Marunović & Prokopec a1411 [global monopoles and topology change].

**Non-Abelian Monopoles** > s.a. symplectic
structures.

* __Idea__: Classical soliton
solutions in gauge theories, with non-abelian (e.g., color) magnetic charge;
Most known exact solutions are static.

* __In Yang-Mills-Higgs theory__:
SU(2)-valued pairs (*A*, *φ*), with *A* a connection and *φ* a
scalar field, with energy

*E* = \(1\over2\)∫ d^{3}*x* [*B*^{2} +(*Dφ*)^{2}
+ *λ*(*φ*^{2}–*C*^{2})] ,

where *B*:= ∇ × *A* +
[*A*,* A*] and *Dφ*:= ∇*φ* + [*A*, *φ*];
One way to obtain solutions is to minimize *E* with
the constraint *φ*^{2} = *C*^{2},
which gives the Bogomolny Equation, with the BPS solutions.

@ __General references__: 't Hooft NPB(74);
Polyakov JETPL(74); Goddard & Olive RPP(78);
Freund IJTP(78); Hitchin
CMP(82);
Díaz & Lázaro-Camí a0811.

@ __Books, reviews__: Marciano IJTP(78); Atiyah & Hitchin 88; Murray mp/01;
Tyurin RMS(02)
[mathematical]; Ritter mp/03;
Konishi LNP(08)ht/07.

@ __And sigma models__: Witten PRD(79); Forgács et al PRL(80).

@ __Quantization__: Auzzi et al NPB(04)ht,
Konishi ht/04-conf
[quantum
and topological aspects]; Qandalji IJTP(06)ht/05 [Dirac,
axial gauge].

@ __Related topics__: Bais & Primack NPB(77)
[spherical]; Mazur & Richter
PLA(85)
[uniqueness]; Labastida & Mariño
PLB(95)ht [Lagrangian],
NPB(95)ht;
Houghton JHEP(00)ht/99 [and
Legendre transform]; van der Bij & Radu
IJMPA(03)
[no Yang-Mills-Higgs rotating in Minkowski]; Bonati et al PRD(10)-a1009 [on the lattice].

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2016