Non-commutative probability and non-commutative processes

TL;DR

This paper explores various non-commutative probability spaces, including those linked to non-commutative space-time, expanding the understanding beyond classical and quantum probability frameworks.

Contribution

It introduces and discusses non-commutative probability spaces associated with non-commutative space-time, broadening the scope of algebraic structures studied in probability theory.

Findings

Analysis of non-commutative probability spaces beyond classical and quantum cases

Identification of non-commutative space-time as a relevant algebraic structure

Discussion of implications for non-commutative probability theory

Abstract

A probability space is a pair ($\mathcal{A},\phi $) where $\mathcal{A}$ is an algebra and $\phi $ a state on the algebra. In classical probability $\mathcal{A}$ is the algebra of linear combinations of indicator functions on the sample space and in quantum probability $\mathcal{A}$ is the Heisenberg or Clifford algebra. However, other algebras are of interest in non-commutative probability. Here one discusses some other non-commutative probability spaces, in particular those associated to non-commutative space-time.