An order for quantum observables
Stanley Gudder · Czech digital mathematics library · 2006
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators 5(H) on a complex Hilbert space H.The usual order A < B on S(H) is determined by assuming that the expectation of A is not greater than the expectation of B for every state of the system.We may think of < as a numerical order on S(H).In this article we introduce a new order _< on S(H) that may be interpreted as a logical order.This new order is determined by assuming that A _< B if the proposition that A has a value in A implies the proposition that B has a value in A for every Borel set A not containing 0. We give various characterizations of this order and show that it is generated by an orthosum 0 that endows S(H) with the structure of a generalized orthoalgebra.We also show that the usual order < cannot be generated by an orthosum.We demonstrate that if we restrict 0 to an interval [0, A] C 5(H), then we obtain a structure that is isomorphic to an orthomodular lattice of projections on H.The lattice structure of S(H) is investigated and unlike (5(H), <) it is shown that (5(H), _<) is a near-lattice in the sense that if A, B _< C, then A A B and A\/ B exist.Moreover, we show that if dim(H) < oo, then A A B always exists.We also consider the commutative case in which observables are represented by fuzzy random variables.