Hermann Kolanoski, "Magnetic Monopoles" 18.2.2005
Magnetic Monopoles
• 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
Hermann Kolanoski, AMANDA Literature Discussion 8.+15.Feb.2005
Hermann Kolanoski, "Magnetic Monopoles" 28.2.2005
E-B-Symmetry of Maxwell Equations
In vacuum:
00
00
t
EB
t
BE
BE
Symmetric for ),(),( EBBE
more general:
B
E
B
E
cos
sin
sin
cos
'
'
Measurable effects are independent of a rotation by
Hermann Kolanoski, "Magnetic Monopoles" 38.2.2005
With charges and currents
em
me
jt
EBj
t
BE
BE
cos
sin
sin
cos
e
e
jt
EB
t
BE
BE
0
0
Can only be reconciled with our known form if
emconst
(ratio of electric and magnetic charge is the same for all particles)
0 B
Simultaneous rotation of
m
e
m
e
j
j
B
E
, ,
by
Hermann Kolanoski, "Magnetic Monopoles" 48.2.2005
Dirac Monopole
rm e
r
qrB
2
1
4)(
sphere
m SdBq
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: ABB
0
Solutions:
e
r
qrA m
sin
cos1
4)(
“+”: singular for negative z axis
“-”: singular for 0 positive z axis
z
x
yA
Hermann Kolanoski, "Magnetic Monopoles" 58.2.2005
More about monopole solutions
Except for z axis:
AA
AA
:
0)(
Not simply connected region
2sin
1
2mm q
er
qAA
discontinuous function
Flux through a sphere around monopole:
)0()2(
)( )(
2
0
2
0
2
0
ldldAldA
SdASdASdBqm
Discontinuity of necessary for flux 0
+
-
z
equator
Hermann Kolanoski, "Magnetic Monopoles" 68.2.2005
Quantisation of the Dirac Monopole
)/( with 2
2
qeAiet
im
Schrödinger equation for particle with charge q:
Invariance under gauge transformation:
)exp( , ieAA
Must be single valued function
neqm 2
If only one monopole in the world e quantized
Hermann Kolanoski, "Magnetic Monopoles" 78.2.2005
Dirac Monopoles Summarized:
rr
qB m
3
eqm 2
137
Dirac monopoles exhibit the basic features which define a monopole
and help you detecting it:
ee
ne
cnqm 2
137
22
(strong-weak duality)
(monopole with
“standard electrodynamics”)
pointlike
But not in
“spontaneous symmetry breaking”
(SSB) scenarios like GUT monopoles
- quantized charge
- large charge
- B-field:
- localisation 0r
4’s wrong
Hermann Kolanoski, "Magnetic Monopoles" 88.2.2005
GUT monopoles and such
YLC USUSUnSU )1()2()3()(
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
Hermann Kolanoski, "Magnetic Monopoles" 98.2.2005
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
Example: Consider a Higgs potential in 1-dim
V() = (2-m2/ )2 = (2-)2
Classification of stable solutions:
+ -
+ +
- +
+ -
- -
kink solutions stable
+-
V()
Hermann Kolanoski, "Magnetic Monopoles" 108.2.2005
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: 0 0 jxdQj
space
n
Analogously for topological charges:
Example kink solution:
2 ,0 ,2)()(1
and
0 1
0
jdxQ
jj
Hermann Kolanoski, "Magnetic Monopoles" 118.2.2005
Topological index etc http://ww
w.m
athematik.ch/m
athematiker/E
uler.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
Q = #corners - #edges + # planes = 2 “conserved number”
torus:
bretzel: Q = -1
Q = 0
or . . .
Hermann Kolanoski, "Magnetic Monopoles" 128.2.2005
Topology
A Topologist is someone who can't tell thedifference between a doughnut and a coffee cup.
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.
How To Catch A Lion
Hermann Kolanoski, "Magnetic Monopoles" 138.2.2005
Deformations and Homotopy Classes
Simple example:
circle circle
: S1S1
0() = 0
0’() =t 0t(2-)
trivial (b)
(c)
for t 0 0’ 0 same homotopy class
1() = •
•
n() = n
continuous mapping mod 2 (d)
prototype mapping of Q=n class
homotopy class defined by
“winding number” Q
dd
dQ
2
02
1
nQ
Q
Q
n
:
1 :
0 :
1
0
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
Set of homotopy classes is a group
which is isomorphic to Z
Hermann Kolanoski, "Magnetic Monopoles" 148.2.2005
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 Sn
phys 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
Hermann Kolanoski, "Magnetic Monopoles" 158.2.2005
Homotopy Classes (examples)
internal “vectors” mapped
onto the real space
Going around S2phys
maps out a path in S2int
Going around S2phys
maps out a path in S2int
8
7
6
5
4
3
2
1
S2phys
7
2
1
3
4
5
6
8
S2int
8
7
6
5
4
3
2
1
S2phys
2
1
3
4
56
8
7S2
int
Q=0
Q=1
Hermann Kolanoski, "Magnetic Monopoles" 168.2.2005
Homotopy Classes (more examples)
internal “vectors” mapped
onto the real space
Going around S2phys
maps out a path in S2int
Going around S2phys
maps out a path in S2int
8
7
6
5
4
3
2
1
S2phys
1- 8
S2int
8
7
6
5
4
32
1
S2phys
2
1
3
4
56
8
7S2
int
Q=0
Q=2
109
15
14
13
12
11
16
9
10
11
12
13
14
15
16
Hermann Kolanoski, "Magnetic Monopoles" 178.2.2005
Topological DefectsKnown from: Crystal growing, self-organizing structures, wine glass left/right of plate ….
Hermann Kolanoski, "Magnetic Monopoles" 198.2.2005
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 Wa
3-component Higgs field =(1,2,3)
internal SU(2) index
(in SU(2) x U(1) we have in addition a U(1) field B )
W3 = A (em field) ?
Hermann Kolanoski, "Magnetic Monopoles" 208.2.2005
Lagrangian of Georgi-Glashow Model
22
4
1
2
1
4
1),( FDDGGtxL aaaaaa
Higgs potential: VEV 0
and not unique: free phase of
Field tensor
Covariant derivative
cbaaa WWgWWG abc
cbabcaa WgD
This Lagrangian has been constructed to be invariant under
local SU(2) gauge transformations
Remark: Mass spectrum of the G-G model
Hermann Kolanoski, "Magnetic Monopoles" 218.2.2005
Equations of Motion of G-G ModelBy the Euler-Lagrange variational principle one finds “as usual” the
equations of motion: cbabca DgGD
aabba FDD
2
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:
2233
4
1
2
1
4
1)( FDDGGxdxxdE aaaa
iaija
iji
The energy vanishes for:
00 )(
)()( )(
0)( )(
)(
2
ai
iai
aa
ai
Diii
Fxxii
xWi
relatively uninteresting
solution with no gauge fields
and constant Higgs field in the
whole space
Hermann Kolanoski, "Magnetic Monopoles" 228.2.2005
Finite energy solutions of the equations of motion
Solutions for
2
3/2 a 0r :for followsit
for 0)(but 0
F
Dr
rxE
aa
ai
Important is that here the covariant derivative has to vanish at .
cbi
abcai
cbi
abcai
ai
Wg
WgD
0
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 transformationof the a a = F2 sphere in the internal space to the r = sphere in real space
Hermann Kolanoski, "Magnetic Monopoles" 238.2.2005
Identification as monopole
0 :conserved iswhich
ˆˆˆ8
1
k
k cbaabc
A topological current can be defined by:
And yields the topological charge or winding number:
ckbjaabcijki
S phys
dkxdQ
ˆˆˆ8
1
2
20
3
‘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
Hermann Kolanoski, "Magnetic Monopoles" 248.2.2005
Lorentz covariant Maxwell Equations
AAF
kF
jF
monopole) with ( 0
4
2
1
ijk
ijkii BFEF
2
10
Reminder:
Hermann Kolanoski, "Magnetic Monopoles" 258.2.2005
Elm.Field in G-G Model
Association of vector potential A with the gauge field W3 does not work
because it is not gauge invariant (the Wa mix under gauge trafo).
cbaabcg
aa DDGF ˆˆˆˆ 1
For the special case = (0, 0, 1) one gets:
33 WWF
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
t’Hooft found a gauge invariant definition of the em field tensor:
breaks SU(2) symmetry
cannot hold in the whole space
for solutions with Q 0
Hermann Kolanoski, "Magnetic Monopoles" 268.2.2005
B-Field in GG Model
kgg
F cbaabc
14ˆˆˆ
2
12
1
ijk
ijk BFw ith 2
1
Follows: gkB / 4 0
Magnetic monopole charge:g
Q
g
kxdqm 03 Q = topological charge
= 0, 1, 2, …
Quantisation as for Dirac
eme q
nqgq
Hermann Kolanoski, "Magnetic Monopoles" 278.2.2005
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
Hermann Kolanoski, "Magnetic Monopoles" 288.2.2005
Birth of monopoles
TC = 1027 K
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
Hermann Kolanoski, "Magnetic Monopoles" 298.2.2005
Literature
• "Electromagnetic Duality for Children"
http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/EDC.pdf
• All about the Dirac Monopole: Jackson, Electrodynamics
…. strengthened by the first introduction to homotopy on the
corridor of the Physics Institut by Michael Mueller-Preussker
• 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