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Journal of Hechi University

Probability in Physics

WU Xin-zhong

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

According to the development of probability concept in physics histrory,this paper discusses the roles of probability concept in sciencific philosophy,molecules kinematics theory,statisticle mechanics,quantum mechanics,reveals logical relations among various probability concepts,and gives a philosophical demonstration that probabiliy concept embodies dialectical unity between determinacy and randomicity in physics process.

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According to the development of probability concept in physics histrory,this paper discusses the roles of probability concept in sciencific philosophy,molecules kinematics theory,statisticle mechanics,quantum mechanics,reveals logical relations among various probability concepts,and gives a philosophical demonstration that probabiliy concept embodies dialectical unity between determinacy and randomicity in physics process.
The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.
Probability concept in physics entered into statistical physics and quantum physics by molecules kinematics; and curvature concept in physics as applying differential geometry to physics, entered into analytical mechanics long ago. Along with introducing space-time curvature concept into general relativity, curvature concept became more important; gauge field theory regards field intensity as curvature of fibre bundles. Curvature concept in quantum mechanics germinated from original derivation of Schrodinger equation; catastrophe scientist Rene Thom advanced curvature interpretations of ψ function and entropy according to differential geometry. Guoqiu Zhao advanced curvature interpretation of quantum mechanics; this new interpretation made relativity theory and quantum mechanics more harmonious, and regarded ψ function as a curvature function. So far Zhao’s quantum curvature interpretation is nearest to Schrodinger’s scientific thought and Einstein’s physics ideal.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
A physical theory is a construction of thought which is founded on experience so as to constitute knowledge of the natural world. Propositions in physics are constituted of concepts which express the properties and processes of the physical world. For purposes of record and communication concepts are designated by the terms of a language, such as mathematical symbols, and philosophical discussion may be based on linguistic forms. In this essay, however, the element of discussion will be the concept as a mode of thought through which are grasped the attributes of physical things and the relations exemplified in physical processes. The present philosophical analysis of a physical theory, quantum mechanics, is to be carried out by critical analysis of its basic concepts. In particular, the program is to show that in quantum mechanics the concept of physical object exhibits the characters of discontinuity, complementarity, and relativity.
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 paper reviewed the construction and development of quantum mechanics,and analyzed the corresponding philosophical debate of some paradox and interpretation.It also investigated the dialectic determinism relationship between philosophy and the development of quantum mechanics from an epistemology and ontological point of view,and discussed the attribution and the essence of determinism from the achievements of quantum mechanics.It concluded that the determinism is recovered both in theoretical framework and substance validation.