Editing Quantum logic (section)

=== Propositional lattice of a quantum mechanical system ===
In the [[Hilbert space]] formulation of quantum mechanics as presented by von&nbsp;Neumann, a physical observable is represented by some (possibly [[bounded operator|unbounded]]) densely defined [[self-adjoint operator]] ''A'' on a Hilbert space ''H''.  ''A'' has a [[Spectral theorem|spectral decomposition]], which is a [[projection-valued measure]] E defined on the Borel subsets of '''R'''.  In particular, for any bounded [[Borel function]] ''f'' on '''R''', the following extension of ''f'' to operators can be made: <math display=block> f(A) = \int_{\mathbb{R}} f(\lambda) \, d \operatorname{E}(\lambda).</math>

In case ''f'' is the indicator function of an interval [''a'', ''b''], the operator ''f''(''A'') is a self-adjoint projection onto the subspace of [[generalized eigenvector]]s of ''A'' with eigenvalue in {{closed-closed|''a'',''b''}}.  That subspace can be interpreted as the quantum analogue of the classical proposition
* Measurement of ''A'' yields a value in the interval [''a'', ''b''].

This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's ''Axiom VII'':
* The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of ''H''; the negation of a proposition ''V'' is the orthogonal complement ''V''<sup>⊥</sup>.

The space ''Q'' of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {''V''<sub>''i''</sub>}<sub>''i''</sub> of elements of ''Q'' has a least upper bound.  Here disjointness of ''W''<sub>1</sub> and ''W''<sub>2</sub> means ''W''<sub>2</sub> is a subspace of ''W''<sub>1</sub><sup>⊥</sup>. The least upper bound of {''V''<sub>''i''</sub>}<sub>''i''</sub> is the closed internal [[direct sum]].