[Paper Review] Extended probability theory and quantum mechanics I: non-classical events, partitions, contexts, quadratic probability spaces

Jiří Souček|arXiv (Cornell University)|Aug 2, 2010

Probability and Statistical Research3 citations

TL;DR

This paper introduces Extended Probability Theory (EPT) by redefining events as partitions of a sample space Ω, where elements within the same partition part are indistinguishable. Incompatibility arises when events conflict in their partition structure, and contexts—maximal sets of compatible events—restore classical probability. The key contribution is the formulation of quadratic probability spaces, which generalize Kolmogorov’s framework and provide a foundation for modeling quantum mechanics as Markov processes, realizing Einstein’s vision of QM as a stochastic theory beyond classical probability.

ABSTRACT

In the paper the basic concepts of extended probability theory are introduced. The basic idea: the concept of an event as a subset of Ωis replaced with the concept of an event as a partition. The partition is any set of disjoint non-empty subsets of Ω(i.e. partition=subset+its decomposition). Interpretation: elements inside certain part are indistinguishable, while elements from different parts are distinguishable. There are incompatible events, e.g {{e1},{e2}} and {e1,e2}. This is logical incompatibility analogical to the impossibility to have and simultaneously not to have the which-way information in the given experiment. The context is the maximal set of mutually compatible events. Each experiment has associated its context. In each context the extended probability is reduced to classical probability. Then the quadratic representation of events, partitions and probability measures is developed. At the end the central concept of quadratic probability spaces (which extend Kolmogorov probability spaces) is defined and studied. In the next paper it will be shown that quantum mechanics can be represented as the theory of Markov processes in the extended probability theory (Einstein's vision of QM).

Motivation & Objective

• To develop a foundational framework for Extended Probability Theory (EPT) that generalizes classical probability by redefining events as partitions rather than subsets.
• To introduce the concepts of incompatibility and context as essential for interpreting quantum phenomena, particularly the which-way information in quantum experiments.
• To construct quadratic probability spaces that extend Kolmogorov’s probability model, enabling a representation of quantum mechanics within a stochastic framework.

Proposed method

• Events are redefined as partitions of the sample space Ω, where each part contains indistinguishable elementary events.
• Incompatible events are defined as those that cannot coexist in the same context due to conflicting partition structures, e.g., distinguishing vs. indistinguishable elements.
• A context is defined as a maximal set of mutually compatible partitions, within which classical probability theory holds.
• Quadratic structures are introduced via operations (+) on partitions, leading to a quadratic representation of events and probability measures.
• The probability measure remains additive, but the event space is endowed with a quadratic algebraic structure derived from partition decomposition.
• The framework is formalized into quadratic probability spaces, which generalize Kolmogorov’s probability spaces and support the representation of quantum mechanics as Markov processes in subsequent work.

Experimental results

Research questions

• RQ1 How can events in probability theory be redefined to capture non-classical features such as incompatibility and context dependence?
• RQ2 What is the role of partition structure in modeling quantum mechanical phenomena like the which-way information?
• RQ3 How can a probability theory be constructed that supports quadratic event structures while preserving additivity of the probability measure?
• RQ4 In what way do contexts restore classical probability within an extended framework, and how do they relate to experimental realizability?
• RQ5 Can quantum mechanics be represented as a Markov process within this extended probability framework, thereby realizing Einstein’s vision of QM as a stochastic theory?

Key findings

• Events are redefined as partitions of the sample space Ω, where elements within the same part are indistinguishable and elements from different parts are distinguishable.
• Incompatible events arise when two partitions conflict in their partitioning of the same elements, such as distinguishing e16 and e20 in one partition but treating them as indistinguishable in another.
• A context is defined as a maximal set of mutually compatible partitions, and within each context, the extended probability theory reduces to classical probability theory.
• The framework introduces quadratic probability spaces that generalize Kolmogorov’s probability spaces by endowing events with a quadratic algebraic structure derived from partition decomposition.
• The probability measure remains additive, but the event algebra is non-linear due to the quadratic structure of partitions, distinguishing EPT from quantum measure theory (QMT), which uses additive event structures.
• The construction provides a foundation for modeling quantum mechanics as Markov processes in extended probability theory, fulfilling Einstein’s vision of QM as a stochastic theory.