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.
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.
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.
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.
According to the Kolmogorovian Censorship Hypothesis, everything that quantum theory says about the world in the language of the quantum mechanical Hilbert space formalism is actually about relationships between ordinary relative frequencies expressible in operational terms using classical Kolmogorovian probability theory. In other words, a quantum theoretical description of a system should in principle be translatable into a purely operational-probabilistic description. However, our goal in this paper is different; we do not want to deal with the problem how to translate the known theory of quantum mechanics into operational terms, or to reconstruct the theory from postulates which can be interpreted in operational terms. Our aim is somewhat broader and points in the opposite direction. We start with a general scheme for the operational description of an arbitrary physical system. The description is based solely on the notion of observable events (measurement operations and measurement results) and on general, empirically established simple laws concerning their relative frequency. These laws are so simple and fundamental that they apply equally to any physical system. In the first part of the paper, we outline the basic elements of such an operational-probabilistic theory. In the second part, we discuss how this operational-probabilistic description compares to the quantum mechanical description and to what extent the standard Hilbert space quantum mechanics can be regarded as a reformulation of the general operational-probabilistic theory.
Introduction Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ , P ) is replaced by the pair ( L ∞ (Ω), E (.) = ∫ Ω . d P ) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space ( N ,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form ( L ∞ (Ω), E ), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S ( H )), if we take N = B ( H ) and Ф the state defined by Ф( X ) = 〈 ψ, X ψ 〉 (or Ф ( X ) = tr(ρ X )) for X ∊ B ( H ). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces. A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments. However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.
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.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.