Coupling The Sine-Gordon Field Theory to a Mechanical System at the Boundary

Abstract

We describe an integrable system consisting of the sine-Gordon field, restricted to the half line, and coupled to a non-linear oscillator at the boundary. By extension of the coupling constant to imaginary values we also outline the equivalent system for the sinh-Gordon field. We show how Sklyanin's formalism can be applied to situations with dynamic boundary conditions, and illustrate the method with the derivation of our example system.

The massive Klein–Gordon field coupled to a harmonic oscillator at the boundary

“…Relation (38) holds as ρ is always positive, (39) requires that ̺ is a root of (26) and (40) uses the relations between roots of a cubic equation 6 . By substituting the general solutions (36) and (37) into the Hamiltonian (14) and applying the othoganality relations (38)-(40) we can rewrite the Hamiltonian as an infinite sum of independent harmonic oscillators each corresponding to one mode of oscillation of the field 7 ,…”

“…By using the definitions (28)-(35) we have described the general classical solutions for the field as the linear superposition of independent modes of oscillation. The Hamiltonian in this basis can be written as a sum over an infinite number of harmonic oscillators each corresponding to one of the modes6 Let the roots of the cubic z 3 + c 2 z 2 + c 1 z + c 0 be z 1 , z 2 and z 3 . These roots satisfy the relations z 1 z 2 z 3 = −c 0 , z 1 z 2 + z 1 z 3 + z 2 z 3 = c 1 and z 1 + z 2 + z 3 = −c 2 .…”

“…As well as the 'bulk' solutions there can exist square integrable boundary bound state solutions. Such solutions can found by allowing the momentum, ρ, of the bulk solutions, given in this case by (6), to take imaginary values, ρ = i̺. Because the bulk solutions satisfy the boundary condition (4) and the equation of motion of the bulk field (5) the imaginary momentum solutions will also satisfy these equations.…”

“…the boundary is repulsive and there exists no boundary bound state. Taking each term of the Hamiltonian in turn at t = 0, substituting the expressions for the bulk solutions (6) we find that…”

“…The third term on the right of (12) arises from any boundary bound states of the system, such contributions will fall off to zero as x 1 → −∞ and x 2 → −∞ due to the localization of the boundary bound state particles close to the boundary. Using these creation and annihilation operators, the definition of the quantum reflection matrix (12) and the expressions for the general bulk solutions of the field (6) it is simple to calculate the R-matrix for the Robin boundary to be…”