Magnetic Monopoles

1
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

2
E-B-Symmetry of Maxwell Equations
In vacuum
Symmetric for
more general
Measurable effects are independent of a rotation
by ?
3
With charges and currents
?
Simultaneous rotation of
by
Can only be reconciled with our known
form if re/rm const (ratio of electric and
magnetic charge is the same for all particles)
4
Dirac Monopole
Assume that a magnetic monopole with charge qm
exists (at the origin)
In these units qm is also the flux
Except for the origin it still holds
Solutions
singular for ? ? ? ? negative z
axis - singular for ? ? 0 ? positive z
axis
5
More about monopole solutions
Except for z axis
Not simply connected region
discontinuous function
Flux through a sphere around monopole
Discontinuity of ? necessary for flux ? 0
6
Quantisation of the Dirac Monopole
Schrödinger equation for particle with charge q
Invariance under gauge transformation
Must be single valued function
If only one monopole in the world ? e quantized
7
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
(strong-weak duality) (monopole with standard
electrodynamics) pointlike But not in
spontaneous symmetry breaking (SSB) scenarios
like GUT monopoles
8
GUT monopoles and such
Grand Unification our know Gauge Groups are
embedded in a larger group
e.g.

• 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

9
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
? - ?
s s
-s s
s -s
-s -s
?
kink solutions ? stable
10
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
Analogously for topological charges Example
kink solution
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Topological index etc
Do you know Eulers polyeder theorem?
Consider the class of rubber-like continuous
deformations of a body to any polyeder ?
classes of mappings with conserved topological
index
http//www.mathematik.ch/mathematiker/Euler.jpg
sphere
?
or . . .
or
Q corners - edges planes 2
conserved number
torus
?
Q 0
bretzel
Q -1
?
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Topology

• A Topologist is someone who can't tell
  thedifference 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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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 circle ? circle ? S1?S1
?0(?) 0 ?0(?)
trivial (b)
t? 0???? t(2?-?) ??????
(c)
for t ? 0 ?0 ? ?0 ? same homotopy class

• ?1(?) ?
• ?n(?) n?

continuous mapping mod 2? (d)
prototype mapping of Qn class
homotopy class defined by winding number Q
Set of homotopy classes is a group which is
isomorphic to Z
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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)
Q0
internal vectors mapped onto the real space
Q1
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Homotopy Classes (more examples)
1
1- 8
8
2
S2phys
Q0
7
3
S2int
4
6
5
internal vectors mapped onto the real space
9
10
1
16
1
8
16
2
15
2
3
14
4
7
S2phys
Q2
11
S2int
13
15
5
3
6
12
4
7
11
6
8
5
10
12
14
9
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
The bosonic sector has
3 gauge fields W?a 3-component Higgs field
?(?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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Lagrangian of Georgi-Glashow Model
Higgs potential VEV ? 0 and not unique free
phase of ?
Field tensor
Covariant derivative
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

• This is a system of 15 coupled non-linear
  differential equations in (31) dim!
• tHooft and Polyakov searched for soliton
  solutions with the restriction to
• 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
relatively uninteresting solution with no gauge
fields and constant Higgs field in the whole
space
The energy vanishes for
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Finite energy solutions of the equations of motion
Solutions for
Important is that here the covariant derivative
has to vanish at ?.
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

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Lorentz covariant Maxwell Equations
Reminder
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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).
tHooft found a gauge invariant definition of the
em field tensor
For the special case ? (0, 0, 1) one gets
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
26
B-Field in GG Model
Follows
Q topological charge 0, 1, 2,
Magnetic monopole charge
Quantisation as for Dirac
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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.
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
29
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-Kleingrotha
  us/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