Statistical structure of quantum theory

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.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.

D. Reidel Pub. Co. eBooksJan 1, 19764 citations

Degenerate Bose System. I. Quantum Statistical Theory

Franz Mohling

The theory of a degenerate system of interacting bosons is developed using the $x$ -ensemble formulation of quantum statistics, previously introduced by Lee and Yang. Particular attention is devoted to the self-energy structure of the graphs of the theory, and it is shown that the complete analysis of this structure involves an intimate mixing of the effects of particle statistics and particle interactions. Therefore, the effect of statistics, which causes Bose-Einstein condensation, becomes mixed with the dynamics in a real system. A prescription is given for the grand potential in which all self-energy effects are included in the line factors of master graphs. An expression for the pair-distribution function of a degenerate Bose system is also derived.The theory of a degenerate system of interacting bosons is developed using the $x$-ensemble formulation of quantum statistics, previously introduced by Lee and Yang. Particular attention is devoted to the self-energy structure of the graphs of the theory, and it is shown that the complete analysis of this structure involves an intimate mixing of the effects of particle statistics and particle interactions. Therefore, the effect of statistics, which causes Bose-Einstein condensation, becomes mixed with the dynamics in a real system. A prescription is given for the grand potential in which all self-energy effects are included in the line factors of master graphs. An expression for the pair-distribution function of a degenerate Bose system is also derived.

Physical ReviewAug 17, 196417 citations

Statistical Structure of Quantum Mechanics

Bassano Vacchini

Graduate texts in physicsJan 1, 20240 citations

Statistical Mechanics in Observable System Theory: A Quasi-Axiomatic Approach

Shiro Ishikawa

I have proposed OST (Observable System Theory) as a foundational theory of system theory. Roughly speaking, OST is a theory that unifies statistics and quantum mechanics. In comparison with ordinary statistics, its main advantage is that it possesses an explicit axiomatic structure. In this sense, I have claimed the following two points:(i): OST will, in the future, become the standard understanding of statistics.(ii): OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics.Statistical mechanics, however, involves microscopic states, probability distributions, Gibbs states, coarse graining, time evolution, thermodynamic limits, and so on. For this reason, a complete axiomatization of statistical mechanics is generally regarded as difficult. In the present paper, we propose a different strategy: the system-theoretical part of statistical mechanics is formulated by means of the axiomatic structure of OST, while the remaining parts are treated in the conventional way, withoutforcing them into a single axiomatic system.The aim of this paper is to understand statistical mechanics in the language of OST. In other words, the parts of statistical mechanics in which statistics has traditionally been used are here reformulated by means of OST. Thus, statistical mechanics is examined from the viewpoint of the three-term structure: state, observable, measured value. Although this is only a partial axiomatization, it nevertheless allows us to make the following claim: (iii): OST-based statistical mechanics will, in the future, become the standard understanding of statistical mechanics.This should be regarded as a corollary of (i) above. At least from the foundational viewpoint of how statistical mechanics should be read as a descriptive specification, the author believes that the OST-based understanding of statistical mechanics is more natural than the conventional one.I have proposed OST (Observable System Theory) as a foundational theory of system theory. Roughly speaking, OST is a theory that unifies statistics and quantum mechanics. In comparison with ordinary statistics, its main advantage is that it possesses an explicit axiomatic structure. In this sense, I have claimed the following two points:(i): OST will, in the future, become the standard understanding of statistics.(ii): OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics.Statistical mechanics, however, involves microscopic states, probability distributions, Gibbs states, coarse graining, time evolution, thermodynamic limits, and so on. For this reason, a complete axiomatization of statistical mechanics is generally regarded as difficult. In the present paper, we propose a different strategy: the system-theoretical part of statistical mechanics is formulated by means of the axiomatic structure of OST, while the remaining parts are treated in the conventional way, withoutforcing them into a single axiomatic system.The aim of this paper is to understand statistical mechanics in the language of OST. In other words, the parts of statistical mechanics in which statistics has traditionally been used are here reformulated by means of OST. Thus, statistical mechanics is examined from the viewpoint of the three-term structure: state, observable, measured value. Although this is only a partial axiomatization, it nevertheless allows us to make the following claim: (iii): OST-based statistical mechanics will, in the future, become the standard understanding of statistical mechanics.This should be regarded as a corollary of (i) above. At least from the foundational viewpoint of how statistical mechanics should be read as a descriptive specification, the author believes that the OST-based understanding of statistical mechanics is more natural than the conventional one.

May 30, 20260 citations

Statistical Methods in Quantum Theory

R. J. CLARK, William Harold Watson

NatureJul 7, 19280 citations

The theory of a degenerate system of interacting bosons is developed using the

x

-ensemble formulation of quantum statistics, previously introduced by Lee and Yang. Particular attention is devoted to the self-energy structure of the graphs of the theory, and it is shown that the complete analysis of this structure involves an intimate mixing of the effects of particle statistics and particle interactions. Therefore, the effect of statistics, which causes Bose-Einstein condensation, becomes mixed with the dynamics in a real system. A prescription is given for the grand potential in which all self-energy effects are included in the line factors of master graphs. An expression for the pair-distribution function of a degenerate Bose system is also derived.