Convex and Sequential Effect Algebras. (arXiv:1802.01265v1 [quant-ph])
We present a mathematical framework for quantum mechanics in which the basic
entities and operations have physical significance. In this framework the
primitive concepts are states and effects and the resulting mathematical
structure is a convex effect algebra. We characterize the convex effect
algebras that are classical and those that are quantum mechanical. The quantum
mechanical ones are those that can be represented on a complex Hilbert space.
We next introduce the sequential product of effects to form a convex sequential
effect algebra. This product makes it possible to study conditional
probabilities and expectations.