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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 Hermann Kolanoski, "Magnetic Monopoles"

2. E-B-Symmetry of Maxwell Equations In vacuum: Symmetric for more general: Measurable effects are independent of a rotationby Hermann Kolanoski, "Magnetic Monopoles"

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) Hermann Kolanoski, "Magnetic Monopoles"

4. 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"

5. 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"

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 Hermann Kolanoski, "Magnetic Monopoles"

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 Hermann Kolanoski, "Magnetic 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 Hermann Kolanoski, "Magnetic Monopoles"

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:  kink solutions  stable Hermann Kolanoski, "Magnetic Monopoles"

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: Hermann Kolanoski, "Magnetic Monopoles"

11. 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"

12. 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  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"

13. 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"

14. 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"

15. 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"

16. 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"

17. Topological Defects Known from:Crystal growing, self-organizing structures, wine glass left/right of plate …. Hermann Kolanoski, "Magnetic Monopoles"

18. Defects and Anti-Defects Hermann Kolanoski, "Magnetic Monopoles"

19. 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"

20. 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"

21. 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"

22. 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"

23. 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"

24. Reminder: Lorentz covariant Maxwell Equations Hermann Kolanoski, "Magnetic Monopoles"

25. 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"

26. B-Field in GG Model Follows: Q = topological charge = 0, 1, 2, … Magnetic monopole charge: Quantisation as for Dirac Hermann Kolanoski, "Magnetic Monopoles"

27. 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"

28. 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"

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-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"