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An Axiomatic Theory of Quaternionic Probability Extending Kolmogorov’s Framework

Carlos Riveros Berger

Published Sep 28, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.20944/preprints202509.2314.v1

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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.
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
Kolmogorov's foundation of probability takes measure spaces, σσσ-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation.
We demonstrate that the contextual approach to Kolmogorov probability model gives the possibility to unify this conventional model of probability with the quantum (Hilbert space) probability model. In fact, the Kolmogorov model can exhibit all distinguishing features of the quantum probability model. In particular, by using the contextual (interference) formula of total probability one can construct complex amplitudes of Kolmogorov probabilities. There exists a natural Hilbert space structure on the space of those complex amplitudes. Classical (Kolmogorovian) random variables are represented by in general noncommutative operators in the Hilbert space of complex amplitudes. The existence of such a contextual representation of the Kolmogorovian model looks very surprising in the view of the orthodox quantum tradition. Supported by EU-network “Quantum Probability and Applications”.
This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called p-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.
Bateman's dual oscillator embeds a damped harmonic oscillator and an independent anti-damped adjoint coordinate in a conservative doubled system. We develop a split-quaternionic formulation that remains consistent from the classical equations through ordinary complex canonical quantization. The doubled dynamics is written as a two-sided hypercomplex evolution whose conserved indefinite quadratic form is the split norm, while the sign of the generator square unifies the underdamped, overdamped, and critical regimes. The Bateman Lagrangian is recovered as a real scalar action on the split-complex configuration subalgebra. After quantization, operator-valued split quaternions belong to the complexified algebra C⊗RHs≃M2(C)\mathbb C\otimes_{\mathbb R}\mathbb H_s\simeq M_2(\mathbb C)C⊗R​Hs​≃M2​(C). Noncommuting mechanical components require a symmetrized quantum split norm, which exactly reproduces the symmetrically ordered canonical Hamiltonian. Null idempotents resolve the crossed canonical structure of the doubled phase space. In particular, damping yields [my^˙,mx^˙]=iℏmγ[m\dot{\widehat y},m\dot{\widehat x}]=\mathrm{i}\hbar mγ[my​˙​,mx˙]=iℏmγ and a Robertson--Schrödinger uncertainty relation. The limit γ→0γ\to0γ→0 retains the doubled positive/negative geometry; the ordinary single-oscillator Heisenberg relation is recovered only after selecting the positive normal mode, or by undoubling before quantization. The scalar split action gives the exact quadratic propagator and Gaussian evolution. A formal trace over the auxiliary coordinate is finally compared with high-temperature Caldeira--Leggett dynamics, clarifying that the doubled model is a coherent zero-noise embedding rather than a microscopic reservoir theory.