Time-dependent integrable quantum field theories with dynamical boundaries

Abstract

We present a general method to construct and solve time-dependent integrable chiral quantum field theories on finite intervals from autonomous scattering and reflection data. Spatial homogeneity forces the spectral labels of right- and left-moving particles to be affine with a common slope, so bulk and boundary couplings sample one spectral trajectory. A complete round trip shifts the spectral labels and gives boundary quantum Knizhnik--Zamolodchikov equations. The Yang--Baxter and reflection equations provide local consistency, while boundary-qKZ flatness provides global consistency. We reconstruct the local Hamiltonian by Cayley inversion of the bulk and boundary matching matrices. We apply the construction to rational $SU(N)$ scattering with fundamental boundary impurities and to an $O(3)$ model with a spin-$\tfrac12$ impurity. For the $SU(2)$ impurity, we construct explicit boundary-qKZ solutions as lattice sums of off-shell boundary Bethe vectors and recover the static open-chain Bethe equations when the drive rate vanishes