The logic of a physical system

[Fredrik]

In both classical and quantum mechanics, there's a lattice called the logic of the physical system. In CM, it's the set of all subsets of phase space. In QM, it's the set of all closed subspaces of a Hilbert space. Sloppy presentations of this subject say that the members of the logic represent statements of the form

"A measurement using measuring device A will, with probability 1, have a result in the set E".

But the members of the logic can't correspond to such statements, at least not bijectively. The set of statements of this form are in bijective correspondence with pairs (A,E), where A is a measuring device and E is a set of real numbers. The subset of the classical phase space that corresponds to a given pair (A,E) is the set of all states that would make the corresponding statement true. This set doesn't uniquely determine the pair (A,E). For example, the set of states that make (A,ℝ) true is the same as the set of states that make (B,ℝ) true. It's the set of all states, i.e. the phase space itself.

So the members of the logic can at best correspond to some sort of equivalence classes of such pairs. My question is, is there a theory-independent definition (i.e. one that works with both CM and QM) of an equivalence relation ~ on the set S of (A,E) pairs, such that the set of equivalence classes S/~ can be mapped bijectively onto the logic?

 

[Fredrik]

After some additional thought, I think the answer is that we first define the terms "state" and "observable" in a theory-independent way. (Araki does this on the first few pages of "Mathematical theory of quantum fields"). Then we let the A in the pair (A,E) be an observable (an equivalence class of measuring devices), not a single measuring device. Finally, we define an equivalence relation on the set of (A,E) pairs, by saying that (A,E)~(B,F) if, for each state, the statement corresponding to (A,E) is true if and only if the statement corresponding to (B,F) is true.