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arXiv:1709.03875

Two Approaches to Measurability Concept and Quantum Theory

Alexander Shalyt-Margolin

Published Sep 5, 2017• 0 citations
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Abstract

In the present paper, in terms of the measurability concept introduced in the previous works of the author, a quantum theory is studied. Within the framework of this concept, several examples are considered using the Schrodinger picture; analogs of Fourier transformations from the momentum representation to the coordinate one and vice versa are constructed. It is shown how to produce a measurable analog of the Heisenberg picture. At the end of this paper the obtained results are used to substantiate another (more general) definition of the measurability concept that is not based on the Heisenberg Uncertainty Principle and its generalization, as it has been in the previous works of the author, and may be equally suitable for both non-relativistic and relativistic cases.

DOI
10.48550/arxiv.1709.03875

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In the present paper, in terms of the measurability concept introduced in the previous works of the author, a quantum theory is studied. Within the framework of this concept, several examples are considered using the Schrodinger picture; analogs of Fourier transformations from the momentum representation to the coordinate one and vice versa are constructed. It is shown how to produce a measurable analog of the Heisenberg picture. At the end of this paper the obtained results are used to substantiate another (more general) definition of the measurability concept that is not based on the Heisenberg Uncertainty Principle and its generalization, as it has been in the previous works of the author, and may be equally suitable for both non-relativistic and relativistic cases.
The Heisenberg uncertainty principle is a consequence of the postulate that coordinate and momentum representations are related to each other by the Fourier transform. This postulate has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. We argue that the postulate is based neither on strong theoretical arguments nor on experimental data. A position operator proposed in our recent publication resolves inconsis-tencies of standard approach and sheds a new light on important problems of quantum theory. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.
A new classification of quantum mechanical quantities according to their measurability is discussed. We have three different types of quantities: first, exactly measurable quantities which commute with all the conserved quantities; second, approximately measurable quantities which do not commute with all additive conserved quantities; and finally nonmeasurable quantities which are conjugate to the absolute invariants. The epistemological implications of the measurement process are discussed and it is pointed out that the quantum mechanical description of this process does not furnish an adequate explanation of the acquisition of knowledge.
By the example of a Fourier transform, the possibilities of Hilbert space geometry applications for statistical model construction are analyzed. In accordance with Bohr's complementarity principle, mutually-complementary coordinate and momentum representations are presented. It was demonstrated that the characteristic function of coordinate distribution may be considered as a convolution of the psi-function in momentum representation and vice versa. The naturalness of coordinate and momentum operators introduction is demonstrated. A probabilistic interpretation of Hilbert space geometry is given. Cauchy-Bunyakowsky (Cauchy-Schwartz), Cramer-Rao and uncertainty inequalities are considered in the same framework. The principal postulates of quantum informatics as a natural science are presented. It is demonstrated that quantum informatics serves as a theoretic basis for both probability theory and quantum mechanics.
This paper presents a novel approach to understanding the role of harmonic dynamics and gaining a deeper appreciation for its impact within and outside of quantum mechanics. This includes consequences of harmonic dynamics and the uncertainty principle for anomalous diffusion and for the time-frequency analysis of chirp signals. In this approach, we consider a contact transformation to view a system of canonical variables with coordinate xxx and momentum pxp_xpx​ in the context of a new system of "generalized" coordinates and momentum. This new system is first studied in the context of non-relativistic quantum mechanics. The classical analog is then explored by use of the Poisson bracket equation. From this, new implications are demonstrated in classical phenomena. One is for a new model of Anomalous and Normal Diffusion. In another, we introduce the concept of the "Mixed Fourier Transform" which explores a new Gaussian Fourier Transform kernel in terms of the generalized variables. This has the ultimate objective of "harmonizing" chirp signals or producing a harmonic signal from an otherwise non-harmonic chirp.