A Basis for Statistical Theory and Quantum Theory

The purpose of the present paper is to give a general theory of the quantum mechanical representation of particle populations.The first part of the paper, Sections 1 to 5, is devoted to a review of mathematical principles of quantum theory, with particular emphasis on the role played by probability concepts, using an approach adapted to the subsequent development of the theory of particle populations. This approach, which goes back in its essentials to von Neumann [20], leans heavily on the subsequent work of Wigner, Mackey, Jauch, Segal, Wightman and many others (see e.g., Mackey [15], Jauch [11], Streater and Wightman [26]). Sections 6 to 9 deal with the representation of finite particle populations: i.e., quantum systems where the total number of particles is an observable. In Section 10 a brief sketch is given of the generalization of the theory to infinite populations where the total number of particles is not an observable, as e.g., in the statistical theory of an infinitely extended gas (see Ruelle [22]). Finally, Section 11 treats some simple examples.

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Bridge from Classical Statistical Theory of Open Systems to Quantum Theory

Statistical Theory of Open Systems ◽

10.1007/978-94-011-0175-2_23 ◽

1995 ◽

pp. 525-554

Author(s):

Yu. L. Klimontovich

Keyword(s):

Statistical Theory ◽

Quantum Theory ◽

Open Systems

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Particle populations and number operators in quantum theory

Advances in Applied Probability ◽

10.2307/1425806 ◽

1972 ◽

Vol 4 (1) ◽

pp. 39-80 ◽

Cited By ~ 1

Author(s):

J. E. Moyal

Keyword(s):

Statistical Theory ◽

Quantum Theory ◽

Subsequent Development ◽

Quantum Systems ◽

Quantum Mechanical ◽

Subsequent Work ◽

Von Neumann ◽

Quantum Mechanical Representation ◽

Finite Particle ◽

Number Of Particles

The purpose of the present paper is to give a general theory of the quantum mechanical representation of particle populations.The first part of the paper, Sections 1 to 5, is devoted to a review of mathematical principles of quantum theory, with particular emphasis on the role played by probability concepts, using an approach adapted to the subsequent development of the theory of particle populations. This approach, which goes back in its essentials to von Neumann [20], leans heavily on the subsequent work of Wigner, Mackey, Jauch, Segal, Wightman and many others (see e.g., Mackey [15], Jauch [11], Streater and Wightman [26]). Sections 6 to 9 deal with the representation of finite particle populations: i.e., quantum systems where the total number of particles is an observable. In Section 10 a brief sketch is given of the generalization of the theory to infinite populations where the total number of particles is not an observable, as e.g., in the statistical theory of an infinitely extended gas (see Ruelle [22]). Finally, Section 11 treats some simple examples.

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