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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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Magnetic monopoles
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

Hermann Kolanoski, "Magnetic Monopoles"


E b symmetry of maxwell equations
E-B-Symmetry of Maxwell Equations

In vacuum:

Symmetric for

more general:

Measurable effects are independent of a rotationby

Hermann Kolanoski, "Magnetic Monopoles"


With charges and currents
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)

Hermann Kolanoski, "Magnetic Monopoles"


Dirac monopole

z

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"


More about monopole solutions

z

+

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"


Quantisation of the dirac monopole
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

Hermann Kolanoski, "Magnetic Monopoles"


Dirac monopoles summarized
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

Hermann Kolanoski, "Magnetic Monopoles"


Gut monopoles and such
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

  • Hermann Kolanoski, "Magnetic Monopoles"


    Finite energy solutions
    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:

    kink solutions  stable

    Hermann Kolanoski, "Magnetic Monopoles"


    Conserved topological charges
    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:

    Hermann Kolanoski, "Magnetic Monopoles"


    Topological index etc
    Topological index etc

    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"


    Topology

    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"


    Deformations and homotopy classes
    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 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 s m
    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"


    Homotopy classes examples

    5

    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"


    Homotopy classes more examples

    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"


    Topological defects
    Topological Defects

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

    Hermann Kolanoski, "Magnetic Monopoles"


    Defects and anti defects
    Defects and Anti-Defects

    Hermann Kolanoski, "Magnetic Monopoles"


    The t hooft polyakov monopole

    internal SU(2) index

    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"


    Lagrangian of georgi glashow model
    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

    Hermann Kolanoski, "Magnetic Monopoles"


    Equations of motion of g g model
    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 (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"


    Finite energy solutions of the equations of motion
    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 = F2sphere in the internal space to the

    r =  sphere in real space

    Hermann Kolanoski, "Magnetic Monopoles"


    Identification as monopole

    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

    Reminder:

    Lorentz covariant Maxwell Equations

    Hermann Kolanoski, "Magnetic Monopoles"


    Elm field in g g model
    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:

    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"


    B field in gg model
    B-Field in GG Model

    Follows:

    Q = topological charge

    = 0, 1, 2, …

    Magnetic monopole charge:

    Quantisation as for Dirac

    Hermann Kolanoski, "Magnetic Monopoles"


    What have we done so far
    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"


    Birth of 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"


    Literature
    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

    Hermann Kolanoski, "Magnetic Monopoles"