Transcript
Solitons and Boundaries
Delivered at CBPFRio de JaneiroJune 28, 2002
Gustav W DeliusDepartment of MathematicsThe University of YorkUnited Kingdom
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Content
I. About classical solitons II. What happens at boundaries III. Quantum soliton scattering and
reflection IV. Quantum group symmetryV. Boundary quantum groups
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What are solitons? 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.
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Free massless field theory
Lagrangian density:
Field equation: Klein-Gordon equation
General solution:
Localized solutions
• Move at the speed of light
behave like free particles
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Free massless field theory
Lagrangian density:
Field equation: Klein-Gordon equation
General solution:
Localized solutions
• Move at the speed of light
• Do not interact
behave like free particles
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Free massive field theory
Lagrangian density:
Field equation:massive Klein-Gordon
equation
Dispersion
There are no localized classical solutions that could serve as models for particles in this theory.
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Interacting field theory
Lagrangian density:
Field equation:
Dispersion
There are no localized classical solutions that could serve as models for particles in this theory.
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Localized finite-energy solutions
An example:
where
Two vacuum solutions:
Look for kink solution with
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Finding the static kink solution
with
Mechanical analogue: Time, Position.
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Energy of the kink Energy density
Energylocalized energy
Inverse dependence oncoupling constant typicalof solitary waves
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Boosting the kink
Applying Lorentz transformation
gives moving kink:
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Scattering kink and anti-kink
These kinks and anti-kinks are not solitons.They are “solitary waves” or “lumps”.
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Sine-Gordon theory
Lagrangian:
Field equation:
Soliton solution:
Cosine potential
Soliton
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Exact two-solitons solutions
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Scattering soliton and antisoliton
These really are solitons!
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Scattering soliton and antisoliton
Before scattering:
After scattering:
Same shape
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Time advance during scattering
The solitons experiencea time advance while scattering through each other.
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Breather solution
A breather is formed from a soliton and an antisoliton oscillating around each other.
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Summary of Part I:About classical solitons We like to look for localized finite-energy
solutions which behave like particles. Any theory with degenerate vacua has
such solitary waves (kinks). Energy of solitary waves goes as 1/ Usually kinks break up during scattering. Solitons however survive scattering (this is
due to integrability).
Rajaraman: Solitons and Instantons, North Holland 1982
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What happens at boundaries? 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 imagesPlace an oppositely moving and inverted mirror particle behind the boundary
Dirichlet boundary condition
is automatically satisfied
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Neumann boundary
Impose NeumannBoundary condition
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Method of images
Neumann boundary condition
is automatically satisfied
Place an oppositely moving mirror particle behind the boundary
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Kink in 4 theory
What will happen to our kink when it hits the boundary with Dirichlet boundary condition
It comes back!
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Kink in 4 theory
Now let us try the same with Neumann boundary condition
It does not come back.
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Sine-Gordon Soliton reflection
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Sine-Gordon Soliton reflection
Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.
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Soliton reflection
Center of mass of soliton-mirror soliton pair
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Time advance during reflection
For an attractiveboundary conditionthe soliton experiencesa time advanceduring reflection.
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Time delay during reflection
For a repulsiveboundary conditionthe soliton experiencesa time delayduring reflection.
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Boundary bound states
A soliton can bind to its mirror antisoliton to form aboundary bound state, the boundary breather.
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Integrable Boundary Conditions
Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.
Determines locationof mirror solitons
Determines location ofthird 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
Rapidity:
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.
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Classical soliton scattering
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Quantum soliton scattering
Scattering amplitude
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Soliton S-matrix
Possible processesin 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.
Semiclassical phase shift Time delay
Number of bound states
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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 theamplitudescorrespondingto bound states
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Generalization to boundary:
Scattering amplitude Reflection amplitude
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Factorization
= Yang-Baxterequation
= Reflectionequation
Cherednik, Theor.Math.Phys. 61 (1984) 977Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.
One way to obtain amplitudes is to solve:
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Bound states
breatherBoundary breather
Poles in theamplitudescorrespondingto 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-pointfunctions:
Perturbing operator
Euclideanspace-time
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Topological charge
Action on soliton states:
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Non-local charges
Non-local currents:
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Commutation relations
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S-matrix from symmetry
Determines S-matrix uniquely up to scaling.
Much simpler than Yang-Baxter equation.
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Representation and Coproduct
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Result for sine-Gordon S-matrix
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Summary of Part IV:
Quantum group symmetry Non-local symmetry charges generate
quantum affine algebra Solitons form spin ½ multiplet of this
symmetry Non-local action of multi-soliton states
given by non-cocommutative coproduct
S-matrix given by intertwiner of tensor product representations (R-matrix)
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Boundary quantum groups
Derived using boundary conformal perturbation theory
Delius, MacKay, hep-th/0112023
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Reflection matrix from symmetry
Determines reflection matrix uniquely up to scaling.
Much simpler than the reflection equation.
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Action on tensor products
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Result for sine-Gordon K-matrix
We also determined higher-spin reflection matrices,GWD and R. Nepomechie, J.Phys.A 35 (2002) L341
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Summary of Part V:
Boundary quantum groups A boundary breaks the quantum
group symmetry to a subalgebra This boundary quantum group is
not a Hopf algebra The reflection matrix is an
intertwiner of representations of the boundary quantum group
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Sine-Gordon model coupled to boundary oscillator
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