Quantum Logic - Projections As Propositions

Projections As Propositions

The so-called Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R3, the state space is the position-momentum space R6. We will merely note here that an observable is some real-valued function f on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value f(x), that is the value of f for some particular system state x, is obtained by a process of measurement of f. The propositions concerning a classical system are generated from basic statements of the form

• Measurement of f yields a value in the interval for some real numbers a, b.

It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to that of some Boolean algebra of subsets of the state space. By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's laws. These are analogous to the rules relating boolean conjunctives and material implication in classical propositional logic. For technical reasons, we will also assume that the algebra of subsets of the state space is that of all Borel sets. The set of propositions is ordered by the natural ordering of sets and has a complementation operation. In terms of observables, the complement of the proposition {f ≥ a} is {f < a}.

We summarize these remarks as follows:

• The proposition system of a classical system is a lattice with a distinguished orthocomplementation operation: The lattice operations of meet and join are respectively set intersection and set union. The orthocomplementation operation is set complement. Moreover this lattice is sequentially complete, in the sense that any sequence {E_i}_i of elements of the lattice has a least upper bound, specifically the set-theoretic union:

In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely-defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f, the following equation holds:

In case f is the indicator function of an interval, the operator f(A) is a self-adjoint projection, and can be interpreted as the quantum analogue of the classical proposition

• Measurement of A yields a value in the interval .