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.