- Algebraic and measure-theoretic properties of classes of subspaces of an inner product space
Review articleOpen access
2007/01/01 Simple chapter DOI: 10.1016/B978-044452870-4/50025-X
Publisher SummaryIn the classical Kolmogorov probability theory, the set of experimentally verifiable events assigned to physical systems can be identified with a measurable space. The order relation ≤ induced by the lattice operations V and Λ is logically interpreted as the implication relation. A probability measure is a countably additive, normalized and nonnegative function μ on Σ. Random variables are the Σ-measurable real-valued functions on X. The algebraic study of quantum logics that generalize Boolean σ-algebras has given rise to the theory of orthomodular posets, and the study of states to non-commutative measure theory. One of the most important quantum logic is the projection lattice of a Hilbert space H. The basic axiom of the Hilbert space model is that the events of a quantum system can be represented by projections on a Hilbert space or, equivalently, the collection L (H) of closed subspaces of a Hilbert space H. The transition from the Boolean σ-algebra Σ to the projection lattice L (H) consists in replacing the disjointedness of sets by a geometric concept of the orthogonality of subspaces.
Request full text