Solvable System with non-trivial Boundary Conditions : Quantum Group and Excahnge Algebra

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In this project, we chose "quantum field theory with non-trivial boundary condition" as an interesting and promising generalisation and extension of integrable quantum field theory, solvable lattice models and two dimensional conformal field theory in the whole space, which have been rather well understood. At first, for the general question : given an exactly solvable theory in the whole space, " is it solvable in a half space by imposing appropriate boundary conditions? " we gave a positive answer for general affine Toda field theories. The next question was : "are all the classically integrable field theories on a half space quantum integrable? " We answered that only a very limited part was allowed by the stability. The criterion for stability was applied to the "solitons" in the affine Toda field theory with "pure imaginary coupling constant". Together with the hermiticity we concluded that "quantum corrections to the soliton masses" was not justifiable. The integrability of the half-space non-linear sigma models was addressed and it was shown the infinite set of non-local charges was not conserved for the free boundary condition in half space, in sharp contrast to the affine Toda case. The reduction of the equation of motion of affine Toda field theory was investigated systematically and comprehensively. Many new reduction relations were found. We investigated relatively simple physical systems with finite degrees of freedom and discussed the effects of the quantum group (algebra), which was another big element of the current research. The representation of the minimum uncertainty states in quantum optics, like the coherent and the squeezed states were given in terms of various quantum algebras.