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- 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
- 2. Content <ul><li>About classical solitons </li></ul><ul><li>What happens at boundaries </li></ul><ul><li>Quantum soliton scattering and reflection </li></ul><ul><li>Quantum group symmetry </li></ul><ul><li>Boundary quantum groups </li></ul>
- 3. What are solitons? <ul><li>Solitons are classical solutions to some field equations. They are localised packets of energy that travel undistorted in shape with some uniform velocity. </li></ul><ul><li>Solitons resemble particles and this is our reason to be interested in them. </li></ul><ul><li>Solitons (as opposed to solitary waves) regain their shape after scattering through each other. </li></ul>
- 4. Free massless field theory <ul><li>Lagrangian density: </li></ul><ul><li>Field equation: Klein-Gordon equation </li></ul><ul><li>General solution: </li></ul><ul><li>Localized solutions </li></ul><ul><li>Move at the speed of light </li></ul>behave like free particles
- 5. Free massless field theory <ul><li>Lagrangian density: </li></ul><ul><li>Field equation: Klein-Gordon equation </li></ul><ul><li>General solution: </li></ul><ul><li>Localized solutions </li></ul><ul><li>Move at the speed of light </li></ul><ul><li>Do not interact </li></ul>behave like free particles
- 6. Free mass ive field theory <ul><li>Lagrangian density: </li></ul><ul><li>Field equation: </li></ul><ul><li>massive Klein-Gordon equation </li></ul><ul><li>Dispersion </li></ul>There are no localized classical solutions that could serve as models for particles in this theory.
- 7. Interacting field theory <ul><li>Lagrangian density: </li></ul><ul><li>Field equation: </li></ul><ul><li>Dispersion </li></ul>There are no localized classical solutions that could serve as models for particles in this theory.
- 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.
- 10. Energy of the kink <ul><li>Energy density </li></ul><ul><li>Energy </li></ul>localized energy Inverse dependence on coupling constant typical of solitary waves
- 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”.
- 13. Sine-Gordon theory <ul><li>Lagrangian: </li></ul><ul><li>Field equation: </li></ul><ul><li>Soliton solution: </li></ul>Cosine potential Soliton
- 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.
- 19. Affine Toda potential <ul><li>Generalize the sine-Gordon potential </li></ul>For example sl(3): Simple roots of affine Lie algebra
- 20. Summary of Part I: About classical solitons <ul><li>We like to look for localized finite-energy solutions which behave like particles. </li></ul><ul><li>Any theory with degenerate vacua has such solitary waves (kinks). </li></ul><ul><li>Energy of solitary waves goes as 1/ </li></ul><ul><li>Usually kinks break up during scattering. </li></ul><ul><li>Solitons however survive scattering (this is due to integrability). </li></ul>Rajaraman: Solitons and Instantons , North Holland 1982
- 21. What happens at boundaries? <ul><li>We now restrict to the half-line or an interval by imposing a boundary condition. </li></ul>What will happen to a free wave when it hits the boundary? Let us impose the Dirichlet boundary condition
- 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!
- 26. Kink in 4 theory <ul><li>Now let us try the same with Neumann boundary condition </li></ul>It does not come 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.
- 34. Summary of Part II: What happens at boundaries <ul><li>If one imposes integrable boundary conditions, solitons will reflect off the boundary. </li></ul><ul><li>In the sine-Gordon model the solutions on the half-line can be obtained from the method of images </li></ul><ul><li>New solutions exist which are localized near the boundary (boundary bound states). </li></ul><ul><li>We also propose the study of dynamical boundaries. </li></ul>
- 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
- 45. Summary of Part III: Quantum scattering and reflection <ul><li>Solitons lead to quantum particle states </li></ul><ul><li>Multi-soliton scattering and reflection amplitudes factorize </li></ul><ul><li>Scattering matrices are solutions of the Yang-Baxter equation </li></ul><ul><li>Reflection matrices are solutions of the reflection equation </li></ul><ul><li>Spectrum of bound states and boundary states follows from the pole structure of the scattering and reflection amplitudes </li></ul>
- 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
- 53. Summary of Part IV: Quantum group symmetry <ul><li>Non-local symmetry charges generate quantum affine algebra </li></ul><ul><li>Solitons form spin ½ multiplet of this symmetry </li></ul><ul><li>Non-local action of multi-soliton states given by non-cocommutative coproduct </li></ul><ul><li>S-matrix given by intertwiner of tensor product representations (R-matrix) </li></ul>
- 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
- 58. Summary of Part V: Boundary quantum groups <ul><li>A boundary breaks the quantum group symmetry to a subalgebra </li></ul><ul><li>This boundary quantum group is not a Hopf algebra </li></ul><ul><li>The reflection matrix is an intertwiner of representations of the boundary quantum group </li></ul>
- 59. Sine-Gordon model coupled to boundary oscillator

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