1. Solitons and Boundaries Delivered at CBPF Rio de Janeiro June 28, 2002 Gustav W Delius Department of Mathematics The University of York United Kingdom

8. Localized finite-energy solutions An example: where Two vacuum solutions: Look for kink solution with

9. Finding the static kink solution with Mechanical analogue: Time, Position.

11. Boosting the kink Applying Lorentz transformation gives moving kink:

12. Scattering kink and anti-kink These kinks and anti-kinks are not solitons. They are “solitary waves” or “lumps”.

14. Exact two-solitons solutions

15. Scattering soliton and antisoliton These really are solitons!

16. Scattering soliton and antisoliton Before scattering: After scattering: Same shape

17. Time advance during scattering The solitons experience a time advance while scattering through each other.

18. Breather solution A breather is formed from a soliton and an antisoliton oscillating around each other.

22. Method of images Place an oppositely moving and inverted mirror particle behind the boundary Dirichlet boundary condition is automatically satisfied

23. Neumann boundary Impose Neumann Boundary condition

24. Method of images Neumann boundary condition is automatically satisfied Place an oppositely moving mirror particle behind the boundary

25. Kink in  4 theory What will happen to our kink when it hits the boundary with Dirichlet boundary condition It comes back!

27. Sine-Gordon Soliton reflection

28. Sine-Gordon Soliton reflection Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.

29. Soliton reflection Center of mass of soliton-mirror soliton pair

30. Time advance during reflection For an attractive boundary condition the soliton experiences a time advance during reflection.

31. Time delay during reflection For a repulsive boundary condition the soliton experiences a time delay during reflection.

32. Boundary bound states A soliton can bind to its mirror antisoliton to form a boundary bound state, the boundary breather.

33. Integrable Boundary Conditions Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421. Determines location of mirror solitons Determines location of third stationary soliton Ghoshal,Zamolodchikov, Int.Jour.Mod.Phys.A9(1994)3841.

35. Quantum soliton states Classical solution: Quantum state: Vacuum Soliton Antisoliton Coleman, Classical lumps and their quantum descendants, in “New Phenomena in Subnuclear Physics”. Dashen, Hasslacher, Neveu, The particle spectrum in model field theories from semiclassical functional integral techniques , Phys.Rev.D11(1975)3424. Rapidity:

36. Classical soliton scattering

37. Quantum soliton scattering Scattering amplitude

38. Soliton S-matrix Possible processes in sine-Gordon: Identical particles Transmission Reflection (does not happen classically)

39. Semi-classical limit Jackiw and Woo, Semiclassical scattering of quantized nonlinear waves, Phys.Rev.D12(1975)1643. Faddeev and Korepin, Quantum theory of solitons , Phys. Rep. 42 (1976) 1-87. Time delay Number of bound states Semiclassical phase shift

40. Factorization (Yang-Baxter eq.) Zamolodchikov and Zamolodchikov, Factorized S-Matrices in Two Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Theory Models , Ann. Phys. 120 (1979) 253 The exact S-matrix can be obtained by solving =

41. Bound states breather Poles in the amplitudes corresponding to bound states

42. Generalization to boundary: Scattering amplitude Reflection amplitude

43. Factorization = Yang-Baxter equation = Reflection equation Cherednik, Theor.Math.Phys. 61 (1984) 977 Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841. One way to obtain amplitudes is to solve:

44. Bound states breather Boundary breather Poles in the amplitudes corresponding to bound states

46. Sine-Gordon as perturbed CFT Free field two-point functions: Perturbing operator Euclidean space-time

47. Topological charge Action on soliton states:

48. Non-local charges Non-local currents:

49. Commutation relations

50. S-matrix from symmetry Determines S-matrix uniquely up to scaling. Much simpler than Yang-Baxter equation.

51. Representation and Coproduct

52. Result for sine-Gordon S-matrix

54. Boundary quantum groups Derived using boundary conformal perturbation theory Delius, MacKay, hep- th /0112023

55. Reflection matrix from symmetry Determines reflection matrix uniquely up to scaling. Much simpler than the reflection equation.

56. Action on tensor products

57. Result for sine-Gordon K-matrix We also determined higher-spin reflection matrices, GWD and R. Nepomechie, J.Phys.A 35 (2002) L341

59. Sine-Gordon model coupled to boundary oscillator