The statistical interpretation is a later add-on to quantum mechanics and is neither obvious in
the quantum mechanical Hilbert space formalism nor in line with Kolmogorov's measuretheoretical axioms of probability theory. With Kolmogorov's axioms, it is assumed that the
events form a Boolean lattice while von Neumann[2]
already pointed out that this does not hold
for the quantum-mechanical events and, moreover, Kolmogorov's axioms imply the so-called
Bell inequalities
[1] which do not hold in quantum mechanics and were disproved by several
physical experiments.
The paper presents a new axiomatic model using probabilistic interpretations from the very
beginning, covering quantum mechanics (with a certain exception) as well as Kolmogorov's
model and revealing a new phenomenon unknown in both these existing theories: non-trivial
state-independent conditional probabilities. This phenomenon is called statistical predictability.
In Kolmogorov's model, the conditional probability of an event F under another event E becomes
independent of the underlying probability measure (state) only in the two trivial cases where
either E implies F or E implies the negation of F.
The incompatibility of events, well-known from quantum mechanics (non-commuting
projections), is defined in a very basic probabilistic way. In Kolmogorov's model, all events are
mutually compatible, and the new axiomatic model, indeed, reduces to Kolmogorov's model if it
is assumed that all events are mutually compatible or if it is assumed that the events form a
Boolean lattice. The new model can therefore be considered a non-Boolean extension of
Kolmogorov's probability theory.