Conditional probability in quantum axiomatics

Wawrzyniec Guz

Oľga Nánásiová

Carlos Riveros Berger

We present an axiomatic construction of quaternionic probability, extending Kolmogorov’s classical framework to the noncommutative algebra of quaternions. The theory introduces quaternionic probability spaces, conditional probabilities, Bayes’ rules, independence, random variables, expectations, and transport equations, all formulated in a consistent manner. Classical probability is recovered through scalar projection, while restriction to complex subalgebras reproduces the standard quantum formalism. Uniquely quaternionic structures arise, including noncommutative conditional probabilities, inequivalent forms of independence, and quaternionic transport laws. The framework further develops quaternionic Markov chains, entropy, and divergence measures that separate scalar uncertainty from vectorial coherence. Several illustrative examples are provided to show how quaternionic probability captures order effects, hidden correlations, and orthogonal divergences—features invisible to both classical and complex approaches. These results establish quaternionic probability as a rigorous generalization of Kolmogorov’s axioms and as a potential foundation for future studies in noncommutative probability, integrable structures, and quaternionic extensions of mathematical physics.

Yvon Gauthier

We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality.We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert.Finite probability theory requires finite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space.The conclusion hints at a constructivist physics.

Oľga Nánásiová

Platon C. Deliyannis

An alternative axiomatic system describing the concept of conditioning (or preparation) of states in a quantum logic is proposed and its consequences developed. The main difference between this and the system proposed by Pool is our reliance on Mielnik’s idea of transition probabilities, thereby avoiding some ad hoc hypotheses as well as the theory of Baer *-semigroups.