At this point I use the pen to change the sign of the mass term and delete the word “no” in the red sentence.
Transcript
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Solitons and Boundaries Delivered at CBPF Rio de Janeiro June 28, 2002 Gustav W Delius Department of Mathematics The University of York United Kingdom
Solitons are classical solutions to some field equations. They are localised packets of energy that travel undistorted in shape with some uniform velocity.
Solitons resemble particles and this is our reason to be interested in them.
Solitons (as opposed to solitary waves) regain their shape after scattering through each other.
We now restrict to the half-line or an interval by imposing a boundary condition.
What will happen to a free wave when it hits the boundary? Let us impose the Dirichlet boundary condition
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Method of images Place an oppositely moving and inverted mirror particle behind the boundary Dirichlet boundary condition is automatically satisfied
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Soliton reflection Center of mass of soliton-mirror soliton pair
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Time advance during reflection For an attractive boundary condition the soliton experiences a time advance during reflection.
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Time delay during reflection For a repulsive boundary condition the soliton experiences a time delay during reflection.
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Boundary bound states A soliton can bind to its mirror antisoliton to form a boundary bound state, the boundary breather.
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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.
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Summary of Part II: What happens at boundaries
If one imposes integrable boundary conditions, solitons will reflect off the boundary.
In the sine-Gordon model the solutions on the half-line can be obtained from the method of images
New solutions exist which are localized near the boundary (boundary bound states).
We also propose the study of dynamical boundaries.
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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:
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Soliton S-matrix Possible processes in sine-Gordon: Identical particles Transmission Reflection (does not happen classically)
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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
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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 =
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Bound states breather Poles in the amplitudes corresponding to bound states
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Generalization to boundary: Scattering amplitude Reflection amplitude
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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:
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Bound states breather Boundary breather Poles in the amplitudes corresponding to bound states
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Summary of Part III: Quantum scattering and reflection
Solitons lead to quantum particle states
Multi-soliton scattering and reflection amplitudes factorize
Scattering matrices are solutions of the Yang-Baxter equation
Reflection matrices are solutions of the reflection equation
Spectrum of bound states and boundary states follows from the pole structure of the scattering and reflection amplitudes
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Sine-Gordon as perturbed CFT Free field two-point functions: Perturbing operator Euclidean space-time
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