  The partial trace operation is usually considered in composite quantum
systems, to reduce the state on a single subsystem. This operation has a key
role in the decoherence effect and quantum measurements. However, partial trace
operations can be defined in more generic situations. In particular, it can be
used to restrict a quantum state (for a single or several quantum entities) on
a specific region of space, the rest of the universe being treated as an
environment. The reduced state is then interpreted as the state that can be
detected by an ideal probe with a limited spatial extent. In this paper, such
an operation is investigated for systems defined on a Fock Hilbert space. A
generic expression of the reduced density matrix is computed, and it is applied
to several case studies: eigenstates of the number operator, coherent states,
and thermal states. These states admit very different behaviors. In particular,
(i) a decoherence effect happens on eigenstates of the number operator (ii)
coherent or thermal states remain coherent or thermal, but with an
amplitude/temperature reduced non-trivially by the overlap between the field
and the region of interest.
