Magnetic Monopoles
Hermann Kolanoski, AMANDA Literature Discussion 8.+15.Feb.2005
•
•
•
•
•
How large is a monopole?
Is a monopole a particle?
How do monopoles interact?
What are topological charges?
What is a homotopy class?
Content:
• Dirac monopoles
• Topological charges
• A model with spontaneous symmetry breaking by a Higgs field
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Hermann Kolanoski, "Magnetic Monopoles"
1
E-B-Symmetry of Maxwell Equations
In vacuum:
Symmetric for
more general:
 
E  0

  B
 E 
0
t
 
B  0

  E
 B 
0
t
 
 
( E , B )  ( B,  E )

 E '   cos 
  
 B'    sin 
 

sin   E 
  
cos   B 
Measurable effects are independent of a rotation by 
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Hermann Kolanoski, "Magnetic Monopoles"
2
With charges and currents
 
  E  e

  B
 E 
0
t
 
B  0

  E 
 B 
 je
t
Simultaneous rotation of

E
  ,
 B
 
 
  E  e

  

B
 E 
  jm
t

 je 
  ,
j 
 m
 e 
 
 m 
by
 cos 

  sin 
 
  B  m

  E 
 B 
 je
t
sin  

cos  
 
Can only be reconciled with our known   B  0 form if
e/m  const
(ratio of electric and magnetic charge is the same for all particles)
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Dirac Monopole
Assume that a magnetic monopole
with charge qm exists (at the origin):
 
q 1 
B(r )  m 2 er
4 r
 
qm   BdS
In these units qm is also the flux:
sphere
Except for the origin it still holds:
z

A
Solutions:
y
x
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
  
B  0  B    A
 
qm 1  cos  
A (r )  
e
4 r sin 
“+”: singular for     negative z axis
“-”: singular for   0  positive z axis
Hermann Kolanoski, "Magnetic Monopoles"
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More about monopole solutions

 
  ( A  A )  0



   : A  A  
Except for z axis:
Not simply connected region
discontinuous function


  qm  
qm 1 
A  A 
e  
   
2 r sin 
 2 
Flux through a sphere around monopole:

 
 
qm   BdS   (  A ) dS 



 
 (  A ) dS

2
  2   2  
  A dl   A dl    dl   (2 )   (0)
0
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0
0
Hermann Kolanoski, "Magnetic Monopoles"
z
+
equator
Discontinuity of 
necessary for flux  0
5
Quantisation of the Dirac Monopole
Schrödinger equation for particle with charge q:
 2

   i
2m
t

 
with     ieA
(e  q /  )
Invariance under gauge transformation:

 
A  A   ,
  exp( ie ) 
Must be single valued function
 eqm  2 n
If only one monopole in the world  e quantized
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Dirac Monopoles Summarized:
Dirac monopoles exhibit the basic features which define a monopole
and help you detecting it:
4’s wrong
- quantized charge
- large charge
- B-field:
- localisation
qm  n
c
e 137
n

e
2e
2
2
qm 
137
e
2
 q 
B  m3 r
r

r 0
(strong-weak duality)
(monopole with
“standard electrodynamics”)
pointlike
But not in
“spontaneous symmetry breaking”
(SSB) scenarios like GUT monopoles
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GUT monopoles and such
Grand Unification: our know Gauge Groups are embedded in a larger group:
e.g.
SU (n)  SU (3)C  SU (2) L U (1)Y
Monopole construction:
• Take a gauge group which spontaneously breaks down into U(1)em
• Determine the fields and the equations of motion
• Search for
• stable,
• non-dissipative,
• finite energy
solutions of the field equations (solitons)
• Identify solution with magnetic monopole
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Finite energy solutions
For a solution to have finite energy it has to approach the
vacuum solution(s) at , i.e. minimal energy density
 boundary conditions at 
V()
Example: Consider a Higgs potential in 1-dim
V() = (2-m2/ )2 = (2-s2)2
-s
+s

Classification of stable solutions:
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+
-
+s
+s
-s
+s
+s
-s
-s
-s

Hermann Kolanoski, "Magnetic Monopoles"
kink solutions  stable
9
Conserved topological charges
A kink is stable: classically no “hopping” from one vacuum into the other
like a knot in a rope fixed at both sides by “boundary conditions”
How is the fact that the node cannot be removed expressed mathematically?
 “conserved topological charges”
Noether charges:
 j  0  Q 
n
0
d
x
j

space
Analogously for topological charges:
Example kink solution:
j 
1
s
       j   0
and

Q   dx j 0 

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1
s
 ()   ()   2,
Hermann Kolanoski, "Magnetic Monopoles"
0,  2
10
Topological index etc
http://www.mathematik.ch/mathematiker/Euler.jpg
Do you know Euler’s polyeder theorem?
Consider the class of “rubber-like” continuous deformations
of a body to any polyeder
 classes of mappings with conserved topological index
sphere:

or
or . . .
Q = #corners - #edges + # planes = 2
torus:

bretzel:

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“conserved number”
Q=0
Q = -1
Hermann Kolanoski, "Magnetic Monopoles"
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Topology
A Topologist is someone who can't tell the
difference between a doughnut and a coffee cup.
How To Catch A Lion
1.7 A topological method
We observe that the lion possesses the topological gender of a torus. We embed the desert in a four
dimensional space. Then it is possible to apply a deformation [2] of such a kind that the lion when returning to
the three dimensional space is all tied up in itself. It is then completely helpless.
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Hermann Kolanoski, "Magnetic Monopoles"
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Deformations and Homotopy Classes
Consider continuous mappings f, g of a space M into a space N
f, g are called homotope if they can be continuously deformed into each other
Simple example:
0() = 0
circle  circle
: S1S1
0’() =
t
t(2-)
trivial (b)
0
2
for t  0 0’  0
(c)
 same homotopy class
1() = 
continuous mapping mod 2 (d)
n() = n
prototype mapping of Q=n class
•
•
homotopy class defined by
“winding number” Q
1
Q
2
2

0
d
d
d
0 : Q  0
Set of homotopy classes is a group
which is isomorphic to Z
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Hermann Kolanoski, "Magnetic Monopoles"
1 : Q  1
n : Q  n
13
Homotopy Group n(Sm)
The topology of our stable, finite energy solutions of field equations
(e.g. the Higgs fields later) by mappings of
sphere Smint in an internal space  sphere Snphys in real space:
n(Sm) (group of homotopy classes Sn Sm) = Z
An example is the mapping of a
3-component Higgs field =(1, 2, 3)
onto a sphere in R3
If in additon  is normalised, ||=1, all
field configurations  lie on a sphere S2int
in internal space
Internal space
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Homotopy Classes (examples)
5
6
1
8
2
Q=0
4
2
Going around S2phys
maps out a path in S2int
S2phys
7
1
8
3
S2int
7
3
4
6
5
internal “vectors” mapped
onto the real space
1
1
8
2
8
2
Q=1
S2phys
7
Going around S2phys
maps out a path in S2int
S2int
3
4
6
4
6
5
5
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7
3
Hermann Kolanoski, "Magnetic Monopoles"
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Homotopy Classes (more examples)
1- 8
1
8
2
Q=0
Going around S2phys
maps out a path in S2int
S2phys
7
S2int
3
4
6
5
internal “vectors” mapped
onto the real space
9
10
1
16
15
16
2
3
4
Q=2
8
2
14
S2phys
13
1
Going around S2phys
maps out a path in S2int
5
12
7
11
S2int
15
3
6
11
4
7
10
9
6
8
14
5
12
13
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Topological Defects
Known from: Crystal growing, self-organizing structures, wine glass left/right of plate ….
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Defects and Anti-Defects
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The ‘t Hooft – Polyakov Monopole
Georgi – Glashow model:
Early attempt for electro-weak unification using
SU(2) gauge group with SSB to U(1)em
internal SU(2) index
The bosonic sector has
3 gauge fields
3-component Higgs field
Wa
=(1,2,3)
W3 = A (em field) ?
(in SU(2) x U(1) we have in addition a U(1) field B )
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19
Lagrangian of Georgi-Glashow Model





1 a a 1
1
a
a

L( x , t )   G G  D D     a a  F 2
4
2
4

2
Higgs potential: VEV  0
and not unique: free phase of 
Field tensor
a
G
  Wa   Wa  g abcWbWc
Covariant derivative
D a    a  g abcWb c
This Lagrangian has been constructed to be invariant under
local SU(2) gauge transformations
Remark: Mass spectrum of the G-G model
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Equations of Motion of G-G Model
By the Euler-Lagrange variational principle one finds “as usual” the
equations of motion:
a
abc
 b
c
D G
 g
D  


D D  a     b b  a  F 2 a
This is a system of 15 coupled non-linear differential equations in (3+1) dim!
t’Hooft and Polyakov searched for soliton solutions with the restriction to
(i) be static and (ii) to satisfy W0a(x)=0 for all x,a
 only spatial indices in the EM involved
Search for solutions which minimize the energy:
2
 
1
1
1
E   d 3 x  ( x )   d 3 x  Gija G aij  Di a D i  a     a a  F 2  
2
4
4

a 
relatively uninteresting
The energy vanishes for: (i ) Wi ( x )  0
solution with no gauge fields


(ii )  a ( x ) a ( x )  F 2
(iii ) Di  0   i  0
a
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(i )
a
Hermann Kolanoski, "Magnetic Monopoles"
and constant Higgs field in the
whole space
21
Finite energy solutions of the equations of motion
Solutions for

E  0 but  ( x )  0 for r  
it follows for r   :
r 3/2 Di a  0 a
 a a  F 2
Important is that here the covariant derivative has to vanish at .
Di a   i a  g abcWi b c  0
  i a   g abcWi b c
It follows that the Higgs field can change the “direction” (=phase) at 
because it can be compensated by the gauge fields.
Therefore the field has in general non-trivial topology
as can be found out from a homotopy transformation
of the a a = F2 sphere in the internal space to the
r =  sphere in real space
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Identification as monopole
‘t Hooft and Polyakov have constructed explicite solutions
here we are only interested in some properties of the solutions:
• Topological charge
• Conserved current
• Monopole field
A topological current can be defined by:
1
 s  abc ˆ a  ˆ b s ˆ c
8
which is conserved :   k   0
k 
And yields the topological charge or winding number:

1
Q   d 3 x k0   
8

2
ˆ a  jˆb  kˆ c
d
s



 i ijk abc

S phys
2
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Reminder:
Lorentz covariant Maxwell Equations
  F   4 j
1
2
 s F s  0 ( k  with monopole)
F     A   A 
F 0i   E i
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 F jk  Bi
1
2 ijk
Hermann Kolanoski, "Magnetic Monopoles"
24
Elm.Field in G-G Model
Association of vector potential A with the gauge field W3 does not work
because it is not gauge invariant (the Wa mix under gauge trafo).
t’Hooft found a gauge invariant definition of the em field tensor:
a
F  ˆ a G
 1g  abcˆ a Dˆ b Dˆ c
For the special case  = (0, 0, 1) one gets:
F   W3   W3
breaks SU(2) symmetry
cannot hold in the whole space
for solutions with Q  0
That means: in regions where  points always in the same (internal) direction
the gauge field in this direction can be considered as the electromagnetic field
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B-Field in GG Model
1
2
 s  F s 
1
1
 s  abc ˆ a  ˆb s ˆ c  4 k 
2g
g
with
Follows:
Magnetic monopole charge:
Quantisation as for Dirac
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1
2
 ijk F jk  Bi

B  4 k0 / g
k
Q
qm   d 3 x 0 
g
g
qe  g  qm  n
Hermann Kolanoski, "Magnetic Monopoles"
Q = topological charge
= 0, 1, 2, …

qe
26
What have we done so far ….?
•
•
•
•
•
Take GUT symmetry group
Break spontaneously down to U(1)em
Search for topologically stable solutions of the field equations
Identify the em part
Find out if there are monopoles (charge, B-field, interaction,..)
Monopoles in the earth magnetic field
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Birth of monopoles
In the GUT symmetry breaking phase the Higgs potential
developed the structure allowing for SSB.
TC = 1027 K
The Higgs field took VEVs randomly in
regions which were causally connected
Beyond this “correlation length” the
Higgs phase is in general different
 monopole density
another discussion
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Literature
• All about the Dirac Monopole: Jackson, Electrodynamics
• "Electromagnetic Duality for Children"
http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/EDC.pdf
• For the Astroparticle Physics: Klapdor-Kleingrothaus/Zuber
and Kolb/Turner: “The Early Universe”
• Most of the content of this talk:
R.Rajaraman: "Solitons and Instantons", North-Holland
…. strengthened by the first introduction to homotopy on the
corridor of the Physics Institut by Michael Mueller-Preussker
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