Solitons and boundaries

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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

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Transcript of Solitons and boundaries

Page 1: Solitons and boundaries

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