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Systems of Observables in Axiomatic Quantum Mechanics

Stanley Gudder

Systems of observables are considered in an axiomatic framework for quantum mechanics which generalizes the usual Hilbert space formulation. Familiar concepts such as complete systems of observables and superselection rules are generalized and it is shown that many of the Hilbert space theorems carry over to this abstract formalism. Also functions of observables are considered and some theorems due to John von Neumann are generalized.

Journal of Mathematical PhysicsOct 1, 196712 citations

John von Neumann's mathematical “Utopia” in quantum theory

Giovanni Valente

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsAug 20, 20085 citations

Unary Classical Mechanics

Peter Morgan

Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving an operator algebraic variant of the Koopman--von Neumann approach. A classically natural approach to the Poisson bracket in a Koopman--von Neumann Hilbert space formalism gives a classical logic that is the same as quantum logic: in both cases the logic of projections on Hilbert space. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical approach that can also be successfully applied to the measurement problem of quantum mechanics.Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving an operator algebraic variant of the Koopman--von Neumann approach. A classically natural approach to the Poisson bracket in a Koopman--von Neumann Hilbert space formalism gives a classical logic that is the same as quantum logic: in both cases the logic of projections on Hilbert space. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical approach that can also be successfully applied to the measurement problem of quantum mechanics.

Jan 2, 20191 citation

A note on the Hilbert space formalism of quantum mechanics

Bedřich Pondělíček

Reports on Mathematical PhysicsAug 1, 19750 citations

Von Neumann’s Hilbert Space Formalism

Anthony Duncan A, Michel Janssen B

Abstract This chapter covers the completion of the formalism of quantum mechanics and its statistical interpretation in the first two installments of a remarkable trilogy of work by von Neumann in 1927. In the first, von Neumann introduces the modern Hilbert space formalism and demonstrates the equivalence of matrix and wave mechanics by showing that they are just two different instantiations of Hilbert space, one being the space of square-summable sequences, the other being the space of square integrable functions. In the second, drawing on work by von Mises on the foundations of probability theory, he provides a careful formulation of the statistical interpretation of this formalism. In doing so, he introduces such staples of modern quantum mechanics as density operators characterizing quantum statistical ensembles and the distinction between pure and mixed states.Abstract This chapter covers the completion of the formalism of quantum mechanics and its statistical interpretation in the first two installments of a remarkable trilogy of work by von Neumann in 1927. In the first, von Neumann introduces the modern Hilbert space formalism and demonstrates the equivalence of matrix and wave mechanics by showing that they are just two different instantiations of Hilbert space, one being the space of square-summable sequences, the other being the space of square integrable functions. In the second, drawing on work by von Mises on the foundations of probability theory, he provides a careful formulation of the statistical interpretation of this formalism. In doing so, he introduces such staples of modern quantum mechanics as density operators characterizing quantum statistical ensembles and the distinction between pure and mixed states.

Aug 17, 20231 citation

John von Neumann and the Foundations of Quantum Physics

Miklós Rédei, Michael Stöltzner

Springer eBooksJan 1, 200184 citations

Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)

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