Boundary Exchange Algebras and Scattering on the Half Line

Abstract

Some algebraic aspects of field quantization in space-time with boundaries are discussed. We introduce an associative algebra B R , whose exchange properties are inferred from the scattering processes in integrable models with reflecting boundary conditions on the half line. The basic properties of B R are established and the Fock representations associated with certain involutions in B R are derived. We apply these results for the construction of quantum fields and for the study of scattering on the half line.

Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

“…In the same way, the Hamiltonian of [9] is the counterpart of the NLS model on the half-line whose symmetry is given by the reflection algebra [15], showing the consistency of the approach of [10]. In [15], the concept of boundary algebra [16] was crucial to establish all the properties of NLS on the half-line as an integrable system.…”

“…Then, in [9], the role of the so-called reflection group was emphasized and in [16], the Weyl group W N associated to the Lie algebra B N replaced the permutation group in the construction of a Fock space for systems on the half-line. Let us note that the same group proved to be fundamental in the constructions of [25] corresponding to an interacting gas on the half-line where the usual δ interaction was replaced by another contact interaction, the so-called δ ′ interaction.…”

Junctions of anyonic Luttinger wires

“…For all these reasons, we investigate in this paper the Tomonaga-Luttinger (TL) model on a junction with an arbitrary number n of arms as depicted in Fig. 1 (a junction with two wires n = 2 can be seen as a defect on the line, a problem that has been largely investigated [29][30][31][32][33][34][35][36][37][38][39][40] in the past). To solve this problem, at the junction we impose conditions that are probably not obvious for an electronic problem, but they show the advantage to be exactly solvable.…”

Solving the quantum nonlinear Schrödinger equation with δ-type impurity

“…An intuitive explanation for this breaking is that before such particles scatter, one of them must necessarily cross the impurity. The nontrivial transmission is therefore the origin of the symmetry breaking in S. This conclusion agrees with the observation that in systems which allow only reflection ͑e.g., models on the half-line͒, one can have [14][15][16][17][18][19] both Galilean ͑Lorentz͒ invariant and nonconstant bulk scattering matrices.…”

“…The study [12][13][14][15][16][17][18][19] of the special case of purely reflecting impurities ͑bound-aries͒ indicates factorized scattering theory [20][21][22][23][24] as the most efficient method for dealing with this kind of problem. The method provides on-shell information about the system and allows to derive the exact scattering matrix.…”