Download Magnetic Monopoles and Group Theory

Magnetic Monopoles and the
Homotopy Groups
Physics 251
Jonathan Cornell
6/8/2011
Solitons



Finite-energy, non-dissipative solutions of the
classical wave equations
Cannot arise in linear theories such as
electrodynamics, because there is dispersion
Non linear theories allow cancelation of
dispersive effects
Solitons in Field Theory

M is the set of vacuum field configurations:

Define a topological quantum number πd2(G/H), where G maps one vacuum to
another and H is the isotopy group (G/H is
the manifold of the vacuum)
Solitons in 1+1 Dimensions
1
L  (  ) 2  V ( )
2
1 2 2
V ( )   m    4
2
Euler—Lagrange Equations give us the
equation of motion
 2  2
2
3


m



2
2
t
x
This has static solutions at zero and the
potential minimum

  0,

Kinky Soliton Solutions

Equation of motion can also be satisfied by
the soliton kink solution:
1.0
0.5
4
2
2
0.5
1.0
4
Topological Charge

The scalar fields in 2D have a conserved
current, leading to a topological charge:

For the kink solution:
Charge
Topological Quantum
Number
Q
2

 Z2 
 0    Z2
 e 
Dirac Monopoles

To construct a magnetic monopole, you can
have A of the form

g (1  cos( )) ˆ
AN 

4r sin(  )

g (1  cos( )) ˆ
AS 

4r sin(  )


g
B   A 
rˆ
2
4r
Magnetic Monopole with charge g
Dirac Quantization Condition



Singularities at θ=0 and θ=π correspond to
Dirac string
Moving in a circle around string, particle wave
function picks up a phase (e-ieg)
This phase factor must be equal to 1 for
string to be undetectable
e-ieg = 1
eg = 2πn
Dirac Quantization Condition




From previous calculation we see that charge
is quantized
Monopoles require a compact U(1) gauge
group
Can be embedded in a larger compact gauge
group i.e. SU(2)
Also can arise in Kaluza-Klein theories
Soliton in 2+1 Dimensions (Nielsen-Olsen
Model)



Here Dμ=∂μ – ieAμ and Φ is a complex valued
field



Vacuum is identified with Fμν=0, Dμϕ=0,

The vacuum fields have the values  ei ( )

Boundary Conditions


As r→∞, we need to have vacuum states.
Working in gauge where A0=0 and
considering the requirements on vacuum
solution:
Defining χ(θ)=nθ where n   refers to a
homotopy class we find that magnetic field
through (x,y) plane is quantized.
‘t Hooft-Polyakov Model (3+1
dimensions)

Φa are scalar fields in the adjoint representation of
SU(2) (a=1,2,3).

The vacuum state manifold is a 2 sphere with
radius

 2 (SO(3) / SO(2))  1 (SO(2))  Z
Boundary Conditions

r→∞, need vacuum state, Dμϕa=0

Shown by ‘t Hooft that Wμa and Fμν are
related by (Φ={ϕa}):

Working out electric and magnetic fields
using the r→∞ limit value of Wμa
Ei   F0i  0
1
1
H i    ijk Fjk  3 ri
2
er
Quantization of Magnetic Charge



This is same as Dirac quantization condition
(eg = 2πn) with n = 2!
Considering all possible solutions of the
equations of motion for this theory, it can be
shown we get the Dirac quantization
condition with n even
Agrees with result for possible homotopy
classes
Mass of Monopoles
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

M mon 
4M V
e 2 , where MV is gauge
t’ Hooft showed
boson mass
Monopoles arise in GUTs, with mass of the scale of
the symmetry breaking of the theory
SU(5):
GUT monopoles are too massive to be produced in
accelerators, but could have been produced in early
universe
Monopole searches focus on accelerators, cosmic
rays, and on monopoles possibly bound in matter
To date, nothing found
References
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Boya L. et al. “Homotopy and Solitons.” Fortschritte der
Physik 26, 175-214 (1978).
Coleman, S. “Magnetic Monopole Fifty Years Later.” from The
Unity of the Fundamental Interactions, A. Zichichi, ed.
European Physical Society, 1981.
Dine, Michael. Supersymmetry and String Theory. Cambridge
University Press, 2007.
Goddard P. and D.I. Olive, “Magnetic monopoles in gauge
field theories.” Rep. Prog. Phys., 41, 1357-1437 (1978).
Harvey, Jeffrey A. “Magnetic Monopoles, Duality, and
Supersymmetry.” arXiv: hep-th/9603086
Milstead, D. and E.J. Weinberg. “Magnetic Monopoles.” from
K. Nakamura et al. (Particle Data Group), JPG 37, 075021
(2010).