There have been many discussions about whether all observational propositions in nonrelativistic quantum mechanics can be taken as simultaneously definite without contradiction. Von Neumann, Jauch and Piron, and Bell have shown that not all observational propositions in nonrelativistic quantum mechanics can be interpreted in this way. In the present paper, I examine these theorems from an operator algebraic point view in order to apply these theorems to algebraic quantum field theory. Then I point out that not all observational propositions in local algebras can be taken as definite, in addition to nonrelativistic quantum mechanics.
In this paper, using quantum groups, noncommutative geometry, and the
q-sequence theory, a complete algebraization scheme of quantum mechanics in view of Bohm's implicate order idea is given. It is pointed out that, in fact, there is some
q-implicate order, and concrete expressions are given.