Solitons and boundaries


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  • At this point I use the pen to change the sign of the mass term and delete the word “no” in the red sentence.
  • Solitons and boundaries

    1. 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. 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. 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. 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. 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. 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. 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. 8. Localized finite-energy solutions An example: where Two vacuum solutions: Look for kink solution with
    9. 9. Finding the static kink solution with Mechanical analogue: Time, Position.
    10. 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. 11. Boosting the kink Applying Lorentz transformation gives moving kink:
    12. 12. Scattering kink and anti-kink These kinks and anti-kinks are not solitons. They are “solitary waves” or “lumps”.
    13. 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. 14. Exact two-solitons solutions
    15. 15. Scattering soliton and antisoliton These really are solitons!
    16. 16. Scattering soliton and antisoliton Before scattering: After scattering: Same shape
    17. 17. Time advance during scattering The solitons experience a time advance while scattering through each other.
    18. 18. Breather solution A breather is formed from a soliton and an antisoliton oscillating around each other.
    19. 19. Affine Toda potential <ul><li>Generalize the sine-Gordon potential </li></ul>For example sl(3): Simple roots of affine Lie algebra
    20. 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. 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. 22. Method of images Place an oppositely moving and inverted mirror particle behind the boundary Dirichlet boundary condition is automatically satisfied
    23. 23. Neumann boundary Impose Neumann Boundary condition
    24. 24. Method of images Neumann boundary condition is automatically satisfied Place an oppositely moving mirror particle behind the boundary
    25. 25. Kink in  4 theory What will happen to our kink when it hits the boundary with Dirichlet boundary condition It comes back!
    26. 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. 27. Sine-Gordon Soliton reflection
    28. 28. Sine-Gordon Soliton reflection Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.
    29. 29. Soliton reflection Center of mass of soliton-mirror soliton pair
    30. 30. Time advance during reflection For an attractive boundary condition the soliton experiences a time advance during reflection.
    31. 31. Time delay during reflection For a repulsive boundary condition the soliton experiences a time delay during reflection.
    32. 32. Boundary bound states A soliton can bind to its mirror antisoliton to form a boundary bound state, the boundary breather.
    33. 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. 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. 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. 36. Classical soliton scattering
    37. 37. Quantum soliton scattering Scattering amplitude
    38. 38. Soliton S-matrix Possible processes in sine-Gordon: Identical particles Transmission Reflection (does not happen classically)
    39. 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. 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. 41. Bound states breather Poles in the amplitudes corresponding to bound states
    42. 42. Generalization to boundary: Scattering amplitude Reflection amplitude
    43. 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. 44. Bound states breather Boundary breather Poles in the amplitudes corresponding to bound states
    45. 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. 46. Sine-Gordon as perturbed CFT Free field two-point functions: Perturbing operator Euclidean space-time
    47. 47. Topological charge Action on soliton states:
    48. 48. Non-local charges Non-local currents:
    49. 49. Commutation relations
    50. 50. S-matrix from symmetry Determines S-matrix uniquely up to scaling. Much simpler than Yang-Baxter equation.
    51. 51. Representation and Coproduct
    52. 52. Result for sine-Gordon S-matrix
    53. 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. 54. Boundary quantum groups Derived using boundary conformal perturbation theory Delius, MacKay, hep- th /0112023
    55. 55. Reflection matrix from symmetry Determines reflection matrix uniquely up to scaling. Much simpler than the reflection equation.
    56. 56. Action on tensor products
    57. 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. 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. 59. Sine-Gordon model coupled to boundary oscillator