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
Famous quotes containing the words quantum, mechanical and/or system:
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Industry has operated against the artisan in favor of the idler, and also in favor of capital and against labor. Any mechanical invention whatsoever has been more harmful to humanity than a century of war.”
—Rémy De Gourmont (1858–1915)
“In the course of the actual attainment of selfish ends—an attainment conditioned in this way by universality—there is formed a system of complete interdependence, wherein the livelihood, happiness, and legal status of one man is interwoven with the livelihood, happiness, and rights of all. On this system, individual happiness, etc. depend, and only in this connected system are they actualized and secured.”
—Georg Wilhelm Friedrich Hegel (1770–1831)