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arXiv:1803.04913

On non-commutativity in quantum theory (I): from classical to quantum probability

Luca Curcuraci

Published Mar 13, 2018• 0 citations
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Abstract

A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.

DOI
10.48550/arxiv.1803.04913

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A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.
The thesis is about the peculiar probabilistic structure of quantum mechanics. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables. A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. A method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability is proposed and applied to a series of model to reobtain the quantum behaviour od a particle. In particular, using ordinary probability theory, the kinematics of a point-like particle jumping at random over a discrete random space is studied. After the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space. Using a suitable random process for the space one is able to recover non-relativistic quantum mechanics in a suitable limit.
In this article, we continue our investigation on the role of non-commutativity in quantum theory. Using the method explained in "On non-commutativity in quantum theory (I): from classical to quantum probability", we analyze two toy models which exhibit non-commutativity between the corresponding position and velocity random variables. In particular, using ordinary probability theory, we study the kinematics of a point-like particle jumping at random over a discrete random space. We show that, after the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space.
This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).
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.