Our first postulate reads: the coupling to a "yes-no"
monitoring device is described by an operator
in the Hilbert space 
.
In general 
may explicitly
depend on time but here, for simplicity, we will assume that
this is not the case. That means: to any real experimental
device there corresponds a 
. In practice it may be
difficult to produce the 
that describes exactly
a given device. As it is difficult to find the Hamiltonian
that takes into account exactly all the forces that
act in a system. Nevertheless we believe that an exact
Hamiltonian exists, even if it is hard to find or impractical
to apply. Similarly our postulate states that an exact 
exists, although it may be hard to find or impractical
to apply. Then we use an approximate one, a model one.
It should be noticed that we do not assume that 
is an orthogonal projection. This reflects the fact that our
device - although giving definite "yes-no" answers,
gives them acting upon possibly fuzzy criteria. In the
limit when the criteria become sharp one should think
of 
as 
, where
is a coupling constant of physical dimension
and E a projection operator. In the general case
it is usually convenient to write 
where 
is dimensionless.
It is also important to notice that the property that is being
monitored by the device
need not be an elementary one. Using the concepts
of quantum logic (cf. [18,19]) the property need not
be atomic -
it can be a composite property. In such a case,
when thinking about physical implementation of the procedure
determining whether the property holds or no, there are two
extreme cases. Roughly speaking the situation here is similar to
that occurring in discussion of superpositions of state preparation
procedures. Some procedures lead to a coherent superpositions,
some other lead to mixtures. Similarly with composite detectors:
one possibility is that we have
a distributed array of detectors that can act independently of each
other, and our event consists
on activating one of them. Another possibility is
that we have a coherent distributed detector like a
solid state lattice that acts as one detector.
In the first case (called incoherent) 
will
be of the form
while in the second coherent case:
where 
are operators associated to individual
constituents of the detector's array. More can be said
about this important topic, but we will not need to
analyze it in more details for the present purpose.