  We have studied in a previous work the quantization of a mixed bulk-boundary
system describing the coupled dynamics between a bulk quantum field confined to
a spacetime with finite space slice and with timelike boundary, and a boundary
observable defined on the boundary. Our bulk system is a quantum field in a
spacetime with timelike boundary and a dynamical boundary condition-the
boundary observable's equation of motion. Owing to important physical
motivations, in such previous work we have computed the renormalized local
state polarization and local Casimir energy for both the bulk quantum field and
the boundary observable in the ground state and in a Gibbs state at finite,
positive temperature. In this work, we introduce an appropriate notion of
coherent and thermal coherent states for this mixed bulk-boundary system, and
extend our previous study of the renormalized local state polarization and
local Casimir energy to coherent and thermal coherent states. We also present
numerical results for the integrated Casimir energy and for the Casimir force.
