Many-Hilbert-spaces theory of quantum measurements

Mikio Namiki

Abner Shimony, Howard Stein

Masatoshi Namiki, Saverio Pascazio

John F. Cyranski

It is shown how quantum theory (QT) can be expressed as a probability theory on Hilbert space, treated as a measure space. The approach generalizes the work of Bach and clarifies the ’’generalized trace’’ of Langerholc. It permits the description of both bounded and unbounded observables as measurable functions. Dynamical evolution can be described in terms of stochastic processes.

E. Celeghini

Abstract The Paradigms introduced in philosophy of science one century ago are shown to be quite more satisfactory of that introduced by Galileo. This is particularly evident in the physics based on Hilbert Spaces and related mathematical structures that we apply in this paper to Quantum Mechanics and to Theory of Images. An exhaustive discussion, that include the algebraic analysis of the operators acting on them, exhibits that the Hilbert Spaces-that have fixed dimension-must be generalized to the Rigged Hilbert Spaces that contain spaces with continuous and discrete dimensions. This is the property of Rigged Hilbert Spaces that allows a consistent formal description of the physics we are considering. Theory of Quantum Mechanics and of Images are similar and the fundamental difference between them comes from the definition of measure that is outside the theory of the spaces: while in Quantum Mechanics the measure is a probabilistic action, in Images it is a classical functional.

Terence William Barrett

A method of defining structural information in Hilbert space is described. Instead of commencing an information theory analysis with a simple signal and progressing to an n-dimensional information space, the reverse procedure can be followed: the signal is defined completely in both real and imaginary spaces, and is then represented as a dimensionless number in Hilbert space. Thus, a signal can be analyzed into a Hilbert space measure of information. This measure has similarities to the relativity of states of Everett's (1957) reformulation of quantum mechanics involving observer-observed interaction. Whereas Everett's theory deals with the totality of all possible ways in which a state function can be decomposed into the sum of products of state functions for subsystems of the overall system, the theory presented here deals with the totality of possible ways in which a Hilbert space measure of signal state can be decomposed by signal definition into reciprocally related measures of the state for subsystems of the overall system. In the overall system presented here, a signal is given a concise representation either as a quadratic form on a Hilbert space, or as a Fourier coefficients in isomorphic function space. The total information content of a signal qualified as a dimensionless number in Hilbert space may be given a representation in familiar units by the use of such methods. A primary postulate of quantum mechanics is obeyed in that all physically relevant information about a system is derivable from the knowledge of the state function of the system. Thus, a signal is defined firstly as an information measure, and secondly in the familiar units of frequency, bandwidth, midperiod, and duration.