Magnetic Monopoles

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


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


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


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


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


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


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


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


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()


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


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 . . .


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


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


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


Homotopy Classes (examples)

internal “vectors” mapped

onto the real space

Going around S2phys

maps out a path in S2int

Going around S2phys


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


Homotopy Classes (more examples)

internal “vectors” mapped

onto the real space

Going around S2phys


Going around S2phys


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


Topological DefectsKnown from: Crystal growing, self-organizing structures, wine glass left/right of plate ….


Defects and Anti-Defects


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) ?


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


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


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


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


Lorentz covariant Maxwell Equations

AAF

kF

jF

monopole) with ( 0

4

2

1

ijk

ijkii BFEF

2

10

Reminder:


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


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


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


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


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

Documents

Magnetic Monopoles