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THE INTERPRETATION OF QUANTUM MECHANICS

Wolfgang Weidlich

Published Aug 1, 1960• 0 citations
OpenAlex

Abstract

General conditions are given for the relationship of a possible parameter space, in which a given description of the physical system should occur, with the Hilbert space of quantam mechanics. The validity of quantum theoretical measurement axioms and motion equations is assumed. (tr-auth)

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FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS V: INTERPRETATION, SUMMARY AND DISCUSSION

BOZHIDAR Z. ILIEV

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.

arXiv:quant-ph/9902068•International Journal of Modern Physics A•Jan 20, 2002•12 citations

An Attempt to Understand Relational Quantum Mechanics

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Abstract We search for a possible mathematical formulation of some of the key ideas of the relational interpretation of quantum mechanics and study their consequences. We also briefly overview some proposals of relational quantum mechanics for an axiomatic reconstruction of the Hilbert space formulation of quantum mechanics.

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THE PROBLEM OF IRREVERSIBILITY WITHIN THE FRAMEWORK OF A GENERALIZED ENTROPY CONCEPT

Felix Engelmann, Marc Roy Feix, Ettore Minardi +1 more

In view of the fact that physical measurements represent transmission of information, a general definition based on the measure of lack of knowledge in information theory can be introduced for different entropies each of them corresponding to one particular kind of measurement. By means of this entropy concept the problem of the consistency of microscopic reversibility and maeroscopic irreversibility is studied. Under certain conditions it can be shown that the reversible microscopic equation of motion leads to the result that all entropies corresponding to reduced quantities, i.e., to quantities which do not determine the microscopic state of the system completely, can never decrease. However, it is to be noted that this behavior is symmetric in time, in the classical as well as in the quantum mechanical case. Irreversibility is so related on the one hand to incomplete description of a system, but on the other also to the fact that theoretical statements can be verified by experiments in the future only. The validity of the results obtained within the framework of ensemble theory for a single physical system is dtscussed. As applications of the general ideas a system of free particles and the Boltzmann gas are treated. (auth)

•Nov 1, 1961•1 citation

RANDOM HILBERT SPACE AND WIENER’S INTERPRETATION OF QUANTUM MECHANICS

Yu. P. Rybakov

The special representation of quantum mechanics suggested by Wiener is studied, the wave function being considered as Gaussian random variable, i. e. the vector of the random Hilbert space. The connection between this representation and the well-known Einstein’s program aiming at creating the consistent field formulation of particle physics is revealed, with particles being represented as solitons, clots of some material field satisfying nonlinear equations.

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Quantum mechanics

Peter Szekeres

Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.

Cambridge University Press eBooks•Dec 16, 2004•0 citations
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.
In view of the fact that physical measurements represent transmission of information, a general definition based on the measure of lack of knowledge in information theory can be introduced for different entropies each of them corresponding to one particular kind of measurement. By means of this entropy concept the problem of the consistency of microscopic reversibility and maeroscopic irreversibility is studied. Under certain conditions it can be shown that the reversible microscopic equation of motion leads to the result that all entropies corresponding to reduced quantities, i.e., to quantities which do not determine the microscopic state of the system completely, can never decrease. However, it is to be noted that this behavior is symmetric in time, in the classical as well as in the quantum mechanical case. Irreversibility is so related on the one hand to incomplete description of a system, but on the other also to the fact that theoretical statements can be verified by experiments in the future only. The validity of the results obtained within the framework of ensemble theory for a single physical system is dtscussed. As applications of the general ideas a system of free particles and the Boltzmann gas are treated. (auth)
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.