The Propositional Lattice of A Quantum Mechanical System
This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics. This is essentially Mackey's Axiom VII:
- The orthocomplemented lattice Q of propositions of a quantum mechanical system is the lattice of closed subspaces of a complex Hilbert space H where orthocomplementation of V is the orthogonal complement V⊥.
Q is also sequentially complete: any pairwise disjoint sequence{Vi}i of elements of Q has a least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1⊥. The least upper bound of {Vi}i is the closed internal direct sum.
Henceforth we identify elements of Q with self-adjoint projections on the Hilbert space H.
The structure of Q immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition p, the equations
have exactly one solution, namely the set-theoretic complement of p. In these equations I refers to the atomic proposition which is identically true and 0 the atomic proposition which is identically false. In the case of the lattice of projections there are infinitely many solutions to the above equations.
Having made these preliminary remarks, we turn everything around and attempt to define observables within the projection lattice framework and using this definition establish the correspondence between self-adjoint operators and observables: A Mackey observable is a countably additive homomorphism from the orthocomplemented lattice of the Borel subsets of R to Q. To say the mapping φ is a countably additive homomorphism means that for any sequence {Si}i of pairwise disjoint Borel subsets of R, {φ(Si)}i are pairwise orthogonal projections and
Theorem. There is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H.
This is the content of the spectral theorem as stated in terms of spectral measures.
Read more about this topic: Quantum Logic
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