An algebraic representation of continuous superselection rules
Renzo Cirelli, Franco Gallone, Bella Gubbay · Journal of Mathematical Physics · 1975
From the logic approach to quantum and classical mechanics, the W*−algebraic approach is deduced in dependence of a suitable ’’prestate.’’ An algebraic representation of the logic description is in fact constructed in a framework in which continuous superselection rules can be present. Logic propositions, observables, and states are represented by decomposable projections, decomposable self−adjoint operators, and normal states in a direct integral of Hilbert spaces. In this representation each algebraic term becomes the representative of a homologous logic one and the expectation values as well as the superselection rules are conserved. When a principle of ’’undistinguishability’’ is taken into account, the representation is faithful. In the classical case, the representation results in Koopman’s formalism.