Magnetic Monopoles| Magnetic Monopoles |
Abelian / Dirac Monopoles
> s.a. QED [flux quantization, curved spacetime].
* 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 H2(\(\mathbb R^3\) \ {0};\(\mathbb R\))
= H2(S2;\(\mathbb R\))
= \(\mathbb R\).
* Classification: The gauge group
is U(1), so monopoles are classified by U(1)-bundles P over \(\mathbb R\)
\ {0}, homotopic to S2, i.e., by elements of
π1(S1)
= \(\mathbb Z\), the integer n (magnetic charge) being evaluated by calculating
the characteristic class c1(P),
and integrating it over S2:
C1 = −\(1\over2\pi\)tr ∫ S2 F = n .
* Coupling strength:
Since ge/\(\hbar\)c = n/2, from
e2/\(\hbar\)c = 1/137
we get g2/\(\hbar\)c
= (137/4) n2, 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; solutions and phenomenology.
@ General: Dirac PRS(31);
Ramsey PR(58) [and discrete symmetries];
Ferrell & Hopfield Physics(64);
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 in(05)ht/04;
Aloisi & Nali Ulisse-a1608-RG [Dirac];
news sn(18)jan [brief search update].
@ Reviews: Carrigan NC(65);
Sandars CP(66);
Amaldi in(68);
Goldhaber & Trower AJP(90)may [RL],
91;
Shnir 05;
Rajantie CP(12)-a1204;
Mavromatos & Mitsou IJMPA(20)-a2005.
@ 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];
Dimock a2005 [quantum charged particle];
> 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 the loop representation];
Goldhaber & Heras a1710 [with non-zero photon mass];
Heras CP(18)-a1810 [rev].
@ 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 PLB(16)-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 = \(1\over2\)∫ d3x [B2 + (Dφ)2 + λ(φ2 − C2)] ,
where B:= ∇ × A + [A, A]
and Dφ:= ∇φ + [A, φ];
One way to obtain solutions is to minimize E with the constraint
φ2 = C2,
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;
Evslin JHEP(18)-a1801 [spiked, with two charged scalar Higgs fields].
@ 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 proc(04)ht [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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