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An-Najah University Journal for Research - A (Natural Sciences)

The Intersection Operation in Light of the Concept of Joint Observables in Fuzzy (Operational) Probability Theory

Taghreed Z. Nasr

Published Aug 1, 2011• 0 citations
Publisher

Abstract

“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrains than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. We will try to discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition. ظهرت نظرية الاحتمالات الفازية العملية عام 1995 كتوسعة سَلِسة لنظرية الاحتمالات التقليدية، و كان من المتوقع أن يكون للنظرية أهمية كبيرة في تطور ميكانيكا الكم. و لكن النظرية لم تتطور كما كان متوقعا، لذا كانت الحاجة ماسة لمراجعة الأسس الأولية للنظرية. في هذا البحث نجد أن الترابط الكبير بين عملية التقاطع و مفهوم المتغيرات العشوائية المرتبطة يتطلب منا توسعة مفهوم عملية التقاطع، لذا سوف ندرس مدى فاعلية تعريف عملية التقاطع في ميكانيكا الكم، و مدى اتساق هذا التعريف مع النظرية، و مناقشة إمكانية وجود حلول بديلة.

DOI
10.35552/anujr.a.25.1.139

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“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrains than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. We will try to discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition. ظهرت نظرية الاحتمالات الفازية العملية عام 1995 كتوسعة سَلِسة لنظرية الاحتمالات التقليدية، و كان من المتوقع أن يكون للنظرية أهمية كبيرة في تطور ميكانيكا الكم. و لكن النظرية لم تتطور كما كان متوقعا، لذا كانت الحاجة ماسة لمراجعة الأسس الأولية للنظرية. في هذا البحث نجد أن الترابط الكبير بين عملية التقاطع و مفهوم المتغيرات العشوائية المرتبطة يتطلب منا توسعة مفهوم عملية التقاطع، لذا سوف ندرس مدى فاعلية تعريف عملية التقاطع في ميكانيكا الكم، و مدى اتساق هذا التعريف مع النظرية، و مناقشة إمكانية وجود حلول بديلة.
“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrained than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. In this paper, discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition.
Quantum theory does not provide a unique definition for the joint probability of two noncommuting observables, which is the next important question after the Born's probability for a single observable. Instead, various definitions were suggested, e.g., via quasiprobabilities or via hidden-variable theories. After reviewing open issues of the joint probability, we relate it to quantum imprecise probabilities, which are noncontextual and are consistent with all constraints expected from a quantum probability. We study two noncommuting observables in a two-dimensional Hilbert space and show that there is no precise joint probability that applies for any quantum state and is consistent with imprecise probabilities. This contrasts with theorems by Bell and Kochen-Specker that exclude joint probabilities for more than two noncommuting observables, in Hilbert space with dimension larger than two. If measurement contexts are included into the definition, joint probabilities are not excluded anymore, but they are still constrained by imprecise probabilities.
New scientific theories have been and will always be required for the progress of science. Two examples of new theories that were advanced in the 20th century are quantum mechanics and fuzzy set theory. Both deal with the concept of uncertainty and both use mathematical tools that are essentially different from those used in classical probability theory. In both, the introduction of new concepts (theoretical terms) was necessary because phenomena and processes observed in the real world could not be explained with existing mathematical concepts. This paper examines the possibilities of (re)interpreting uncertainty in quantum mechanics with the aid of fuzzy sets and systems
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
The aim of this paper is to present and discuss a probabilistic framework that is adequate for the formulation of quantum theory and faithful to its applications. Contrary to claims, which are examined and rebutted, that quantum theory employs a nonclassical probability theory based on a nonclassical “logic,” the probabilistic framework set out here is entirely classical and the “logic” used is Boolean. The framework consists of a set of states and a set of quantities that are interrelated in a specified manner. Each state induces a classical probability space on the values of each quantity. The quantities, so considered, become statistical variables (not random variables). Such variables need not have a “joint distribution.” For the quantum theoretic application, there is a uniform procedure that defines and determines the existence of such “joint distributions” for statistical variables. A general rule is provided and it is shown to lead to the usual compatibility-commutivity requirements of quantum theory. The paper concludes with a brief discussion of interference and the misunderstandings that are involved in the false move from interference to nonclassical probability.