The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d- dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities, is violated by quantum probabilities in
the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2), to the subspaces H1,H2. As an application, it is shown that the proof of CHSH inequalities for a system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it is not valid for Dempster- Shafer probabilities. The violation of these inequalities in experiments, supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.
The set of subsystems of a finite quantum system (with variables in Z(n)) together with logical connectives, is a distributive lattice. With regard to this lattice, the (where P(m) is the projector to) obeys a supermodularity inequality, and it is in
terpreted as a lower probability in the sense of the Dempster-Shafer theory, and not as a Kolmogorov probability. It is shown that the basic concepts of the Dempster-Shafer theory (lower and upper probabilities and the Dempster multivaluedness) are pertinent to the quantum formalism of finite systems.
Generalized Bargmann representations which are based on generalized coherent states are considered. The growth of the corresponding analytic functions in the complex plane is studied. Results about the overcompleteness or undercompleteness of discret
e sets of these generalized coherent states are given. Several examples are discussed in detail.
