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We study a coupled system that describes the interacting dynamics between abulk field, confined to a finite region with timelike boundary, and a boundaryobservable. In our system the dynamics of the boundary observable prescribesdynamical boundary conditions for the bulk field. We cast our classical systemin the form of an abstract linear Klein-Gordon equation, in an enlarged Hilbertspace for the bulk field and the boundary observable. This makes it possible toapply to our coupled system the general methods of quantization. In particular,we implement the Fock quantization in full detail. Using this quantization westudy the Casimir effect in our coupled system. Specifically, we compute therenormalized local state polarization and the local Casimir energy, which wecan define for both the bulk field and the boundary observable of our system.Numerical examples in which the integrated Casimir energy is positive ornegative are presented.

Quantum field theory with dynamical boundary conditions and the Casimir effect