  An L operator is presented related to an infinite dimensional limit of the
fusion R matrices for U_q(A^{(1)}_{n-1}) and U_q(D^{(1)}_n). It is factorized
into the local propagation operators which quantize the deterministic dynamics
of particles and antiparticles in the soliton cellular automata known as the
box-ball systems and their generalizations. Some properties of the dynamical
amplitudes are also investigated.
