  We quantise canonical free-field zero modes $p$, $q$ on a half-plane $p>0$
both, for the Liouville field theory and its reduced Liouville particle
dynamics. We describe the particle dynamics in detail, calculate one-point
functions of particle vertex operators, deduce their zero mode realisation on
the half-plane, and prove that the particle vertex operators act self-adjointly
on a Hilbert space $L^2(\rr_+)$ on account of symmetries generated by the
$S$-matrix. Similarly, self-adjointness of the corresponding vertex operator of
Liouville field theory in the zero mode sector is obtained by applying the
Liouville reflection amplitude, which is derived by the operator method.
