  We discuss the introduction of boundary Hilbert spaces for a class of
physical systems for which it is not possible to factor their state spaces as
tensor products of Hilbert spaces naturally associated to their boundaries and
bulks respectively. In order to do this we make use of the so called trace
operators that play a relevant role in the analysis of PDE's in bounded
regions. By taking advantage of these operators and some functorial aspects of
the construction of Fock spaces, we will show how to obtain quantum dynamics at
the boundaries defined in appropriate Hilbert spaces associated with them.
