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AIP conference proceedings

Fuzzy Quantum Probability Theory

Stanley Gudder

Published Jan 1, 2005• 8 citations
Publisher

Abstract

This paper surveys some of the recent results that have been obtained in fuzzy quantum probability theory. In this theory fuzzy events are represented by quantum effects and fuzzy random variables are represented by quantum measurements. Probabilities, conditional probabilities and sequential products of effects are discussed. The relationship between the law of total probability and compatibility of measurements is treated. Results concerning independent effects are given. Finally, properties of the sequential product, almost sharp effects and nearly sharp effects are discussed.

DOI
10.1063/1.1874565

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This paper surveys some of the recent results that have been obtained in fuzzy quantum probability theory. In this theory fuzzy events are represented by quantum effects and fuzzy random variables are represented by quantum measurements. Probabilities, conditional probabilities and sequential products of effects are discussed. The relationship between the law of total probability and compatibility of measurements is treated. Results concerning independent effects are given. Finally, properties of the sequential product, almost sharp effects and nearly sharp effects are discussed.
The goal of this study is to extend quantum theory to the field of fuzzy phenomena. Using the idea and method of fuzzy mathematics, the quantum probability theory of fuzzy event is proposed, with the form of fuzzy event route integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory changes quantum mechanics, making it a special instance of the former. Under some conditions, this theory breaches unitarity, gauge invariance, probability conservation, and information conservation, rendering these concepts approximate. This new theory, which does not require renormalization and can naturally produce meaningful results that are consistent with the experiments, is the proper theory for describing microscopic high-speed phenomena, whereas quantum mechanics is only a proper theory for describing microscopic low-speed phenomena. Because this theory is not divergent under the conditions that there is no renormalization and an infinite number of offsetting terms, it can constitute the theoretical framework required for gravity quantization.
“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.
A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.
A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator E E on a Hilbert space that satisfies 0 ≤ E ≤ I 0\le E\le I . We define effects E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n to be sequentially independent if the result of any sequential measurement of E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.
Various versions of noncommutative probability theory are surveyed. It is stressed that the main motivation and applications of these noncommutative theories is quantum mechanics. A review of traditional probability theory and its unsharp version are presented. Sharp and unsharp Hilbert space probability theories are considered next. We then present a general discussion of observables and statistical maps. Finally, we consider sequential effect algebras and show that they unify and generalize all of these types of probability theories.