1. Hermann Kolanoski, "Magnetic Monopoles"1 8.2.2005 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

2. Hermann Kolanoski, "Magnetic Monopoles"2 8.2.2005 E-B-Symmetry of Maxwell Equations In vacuum: Symmetric for more general: Measurable effects are independent of a rotation by 

3. Hermann Kolanoski, "Magnetic Monopoles"3 8.2.2005 With charges and currents  Can only be reconciled with our known form if  e  m  const (ratio of electric and magnetic charge is the same for all particles) Simultaneous rotation of by

4. Hermann Kolanoski, "Magnetic Monopoles"4 8.2.2005 Dirac Monopole Assume that a magnetic monopole with charge q m exists (at the origin): In these units q m is also the flux: Except for the origin it still holds: Solutions: “+”: singular for     negative z axis “-”: singular for   0  positive z axis z x y A 

5. Hermann Kolanoski, "Magnetic Monopoles"5 8.2.2005 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 ++ -- z equator

6. Hermann Kolanoski, "Magnetic Monopoles"6 8.2.2005 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

7. Hermann Kolanoski, "Magnetic Monopoles"7 8.2.2005 Dirac Monopoles Summarized: Dirac monopoles exhibit the basic features which define a monopole and help you detecting it: (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 4  ’s wrong

8. Hermann Kolanoski, "Magnetic Monopoles"8 8.2.2005 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

9. Hermann Kolanoski, "Magnetic Monopoles"9 8.2.2005 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 -m 2 / ) 2 = (  2 -   ) 2 Classification of stable solutions: +  -  ++ ++ -- ++ ++ -- -- --  kink solutions  stable ++ -- V(  ) 

10. Hermann Kolanoski, "Magnetic Monopoles"10 8.2.2005 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: Analogously for topological charges: Example kink solution:

11. Hermann Kolanoski, "Magnetic Monopoles"11 8.2.2005 Topological index etc http://www.mathematik.ch/mathematiker/Euler.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...  

12. Hermann Kolanoski, "Magnetic Monopoles"12 8.2.2005 Topology A Topologist is someone who can't tell the difference 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

13. Hermann Kolanoski, "Magnetic Monopoles"13 8.2.2005 Deformations and Homotopy Classes Simple example: circle  circle  : S1  S1  0 (  ) = 0  0 ’(  ) = t  0  t(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 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

14. Hermann Kolanoski, "Magnetic Monopoles"14 8.2.2005 Homotopy Group  n (S m ) The topology of our stable, finite energy solutions of field equations (e.g. the Higgs fields later) by mappings of sphere S m int in an internal space  sphere S n phys in real space:  n (S m ) (group of homotopy classes S n  S m ) = Z An example is the mapping of a 3-component Higgs field  =(  1,  2,  3 ) onto a sphere in R 3 If in additon  is normalised, |  |=1, all field configurations  lie on a sphere S 2 int in internal space Internal space

15. Hermann Kolanoski, "Magnetic Monopoles"15 8.2.2005 Homotopy Classes (examples) internal “vectors” mapped onto the real space Going around S 2 phys maps out a path in S 2 int Going around S 2 phys maps out a path in S 2 int 8 7 6 5 4 3 2 1 S 2 phys 7 2 1 3 4 5 6 8 S 2 int 8 7 6 5 4 3 2 1 S 2 phys 2 1 3 4 5 6 8 7 S 2 int Q=0 Q=1

16. Hermann Kolanoski, "Magnetic Monopoles"16 8.2.2005 Homotopy Classes (more examples) internal “vectors” mapped onto the real space Going around S 2 phys maps out a path in S 2 int Going around S 2 phys maps out a path in S 2 int 8 7 6 5 4 3 2 1 S 2 phys 1- 8 S 2 int 8 7 6 5 4 3 2 1 S 2 phys 2 1 3 4 5 6 8 7 S 2 int Q=0 Q=2 10 9 15 14 13 12 11 16 9 10 11 12 13 14 15 16

17. Hermann Kolanoski, "Magnetic Monopoles"17 8.2.2005 Topological Defects Known from: Crystal growing, self-organizing structures, wine glass left/right of plate ….

18. Hermann Kolanoski, "Magnetic Monopoles"18 8.2.2005 Defects and Anti-Defects

19. Hermann Kolanoski, "Magnetic Monopoles"19 8.2.2005 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 W  a 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  ) W  3 = A  (em field) ?

20. Hermann Kolanoski, "Magnetic Monopoles"20 8.2.2005 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

21. Hermann Kolanoski, "Magnetic Monopoles"21 8.2.2005 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 (i)be static and (ii) to satisfy W 0 a (x)=0 for all x,a  only spatial indices in the EM involved Search for solutions which minimize the energy: The energy vanishes for: relatively uninteresting solution with no gauge fields and constant Higgs field in the whole space

22. Hermann Kolanoski, "Magnetic Monopoles"22 8.2.2005 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 = F 2 sphere in the internal space to the r =  sphere in real space

23. Hermann Kolanoski, "Magnetic Monopoles"23 8.2.2005 Identification as monopole A topological current can be defined by: And yields the topological charge or winding number: ‘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

24. Hermann Kolanoski, "Magnetic Monopoles"24 8.2.2005 Lorentz covariant Maxwell Equations Reminder:

25. Hermann Kolanoski, "Magnetic Monopoles"25 8.2.2005 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). For the special case  = (0, 0, 1) one gets: 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

26. Hermann Kolanoski, "Magnetic Monopoles"26 8.2.2005 B-Field in GG Model Follows: Magnetic monopole charge: Q = topological charge = 0, 1, 2, … Quantisation as for Dirac

27. Hermann Kolanoski, "Magnetic Monopoles"27 8.2.2005 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

28. Hermann Kolanoski, "Magnetic Monopoles"28 8.2.2005 Birth of monopoles T C = 10 27 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

29. Hermann Kolanoski, "Magnetic Monopoles"29 8.2.2005 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