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# Magnetic Monopoles - PowerPoint PPT Presentation

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

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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

Hermann Kolanoski, "Magnetic Monopoles"

In vacuum:

Symmetric for

more general:

Measurable effects are independent of a rotationby

Hermann Kolanoski, "Magnetic Monopoles"

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)

Hermann Kolanoski, "Magnetic Monopoles"

r

A

y

x

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

Hermann Kolanoski, "Magnetic Monopoles"

+

equator

-

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

Hermann Kolanoski, "Magnetic Monopoles"

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

Hermann Kolanoski, "Magnetic Monopoles"

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

Hermann Kolanoski, "Magnetic Monopoles"

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"

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:

kink solutions  stable

Hermann Kolanoski, "Magnetic Monopoles"

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:

Hermann Kolanoski, "Magnetic Monopoles"

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

http://www.mathematik.ch/mathematiker/Euler.jpg

sphere:

or . . .

or

Q = #corners - #edges + # planes = 2“conserved number”

torus:

Q = 0

bretzel:

Q = -1

Hermann Kolanoski, "Magnetic Monopoles"

A Topologist is someone who can't tell thedifference between a doughnut and a coffee cup.

Topology

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.

Hermann Kolanoski, "Magnetic Monopoles"

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 Q=n class

homotopy class defined by

“winding number” Q

Set of homotopy classes is a group

which is isomorphic to Z

Hermann Kolanoski, "Magnetic Monopoles"

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

Hermann Kolanoski, "Magnetic Monopoles"

1

6

1

4

8

8

2

2

Going around S2phys

maps out a path in S2int

S2phys

7

3

7

S2int

3

4

6

5

1

8

2

1

Going around S2phys

maps out a path in S2int

8

2

7

S2phys

3

S2int

7

3

6

4

5

4

6

5

Homotopy Classes (examples)

Q=0

internal “vectors” mapped

onto the real space

Q=1

Hermann Kolanoski, "Magnetic Monopoles"

Going around S2phys

maps out a path in S2int

Going around S2phys

maps out a path in S2int

Homotopy Classes (more examples)

1- 8

1

8

2

S2phys

Q=0

7

3

S2int

4

6

5

internal “vectors” mapped

onto the real space

9

10

16

1

1

8

16

15

2

2

3

14

4

7

Q=2

S2phys

11

13

15

S2int

5

3

12

6

4

11

7

6

10

8

14

5

12

9

13

Hermann Kolanoski, "Magnetic Monopoles"

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

Hermann Kolanoski, "Magnetic Monopoles"

Hermann Kolanoski, "Magnetic Monopoles"

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 )

Hermann Kolanoski, "Magnetic Monopoles"

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

Hermann Kolanoski, "Magnetic Monopoles"

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 (3+1) dim!

• t’Hooft 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:

Hermann Kolanoski, "Magnetic Monopoles"

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 = F2sphere in the internal space to the

r =  sphere in real space

Hermann Kolanoski, "Magnetic Monopoles"

A topological current can be defined by:

And yields the topological charge or winding number:

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

Hermann Kolanoski, "Magnetic Monopoles"

Lorentz covariant Maxwell Equations

Hermann Kolanoski, "Magnetic Monopoles"

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:

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

Hermann Kolanoski, "Magnetic Monopoles"

Follows:

Q = topological charge

= 0, 1, 2, …

Magnetic monopole charge:

Quantisation as for Dirac

Hermann Kolanoski, "Magnetic Monopoles"

• 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"

TC = 1027 K

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

Hermann Kolanoski, "Magnetic Monopoles"

• 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

Hermann Kolanoski, "Magnetic Monopoles"