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Natural Science

Geometric Interpretation of the Uncertainty Principle

Pavel A. Stabnikov

Published Jan 1, 2019• 4 citations
Publisher

Abstract

The problems of unattainable infinity and infinitesimal are discussed. Limitations connected with the absolute zero of temperature and the maximal velocity are considered, as well as the consequences of these limitations. A geometric approach is proposed as an alternative to the wave-particle duality to explain the anomalous motion of micro objects. The basis of the geometric approach is a comparison between two geometries differing from each other in the metric of infinitesimal. The interconnection of these geometries is possible through the direct and inverse Weierstrass transformation. The application of this transformation allows one to explain diffraction effects.

DOI
10.4236/ns.2019.115017

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The problems of unattainable infinity and infinitesimal are discussed. Limitations connected with the absolute zero of temperature and the maximal velocity are considered, as well as the consequences of these limitations. A geometric approach is proposed as an alternative to the wave-particle duality to explain the anomalous motion of micro objects. The basis of the geometric approach is a comparison between two geometries differing from each other in the metric of infinitesimal. The interconnection of these geometries is possible through the direct and inverse Weierstrass transformation. The application of this transformation allows one to explain diffraction effects.
A novel geometric approach is proposed for the development of the wave-particle notions. This approach is based on a comparison of the two geometries with different sizes of an infinitesimal point. It is assumed that the smaller is object mass, the larger is the size of the infinitesimal point in comparison with the point of the geometry of macro world. Within this approach, the smaller is object mass, the larger is the uncertainty of its position from the viewpoint of macro objects (macro geometry). This approach provides a natural ex-planation of Heisenberg’s indeterminancy principle. Formally, this approach appears as an unusual operation with an infinitesimal value (point). However, it should be noted that unusual operations (though with infinitely large values) are already known in physics. These are unattainability of the absolute zero of temperature and unattainability of the maximal velocity of movement. Interconnection of the two geometries with different sizes of infinitesimal values is possible with the help of the direct and inverse Weierstrass transformation. At present, diffraction effects are described using the wave notions about the light and Fourier transform. The diffraction of light is usually registered at a distance not less than 1 - 3 metres between the screens in one of which there is a slit or several slits. This distance is about 106 times longer than the wavelength of the radiation. In the present work, an approach is proposed that allows one to describe the light fluxes at short distances between the screens with the help of Fourier and Weierstrass transforms.
Using Weyl's geometrical interpretation of quantum mechanics the fundamental relations of the transformation theory can be treated without making use of Dirac's $\ensuremath{\delta}$ function. The principle of uncertainty is particularly clear in this system. It is shown that in the discrete representation for a definite axis of the coordinate qqq all Eigenwerte of the conjugate momentum ppp are equally probable. It is also shown how in the limit of very fine subdivision the results of Heisenberg and Bohr are obtained. The whole treatment receives an easy interpretation by applying it to the problem of the diffraction of waves by a grating. Weyl's method of discussing quantities with continuous ranges of values is found to be equivalent to investigating diffraction phenomena in the limiting case of an infinitesimal grating space. The quantum condition $i(PQ\ensuremath{-}QP)=1$ is shown to be correct in a purely operational sense in spite of the fact that as a matrix equation it is untrue.
Wave-particle duality is often considered as the modern answer to the problem of the nature of light after more than 2000 years of questioning. It is also the answer given by quantum physics concerning the nature of matter particles and any other radiations. The main objective of this work is to analyze the relations that are existing between this concept of wave-particle duality, the uncertainty principle and the concepts of phase space and microstates considered in statistical mechanics. It is mainly highlighted that while the concepts of phase space and microstates were already introduced in classical physics before the discovery of the wave-particle duality, a correct understanding of them cannot be achieved without the use of the concept of quantum phase space and phase space representation of quantum mechanics which are directly related to the uncertainty principle. The possibility of using these concepts of quantum phase space and phase space representations of quantum mechanics to help in a deeper description of the wave-particle duality and in the study of some current issues related to foundational problems of quantum mechanics like quantum decoherence and the measurement problem is also discussed.
The principle of uncertainty is one of the essential principles of quantum mechanics as it outlines that there is a reflective limit to the measurement of microscopic particle motion. However it is impossible to measure at once and accurately both the position and momentum data of any microscopic particle no matter what means are used to do so. This paper starts by discussing the notion and mathematical derivation of the uncertainty principle as put forth by Heisenberg explaining the physical relevance of the principle. It introduces the natural uncertainty the principle introduces to the dynamics of particles in the microscopic world, and the significant part it plays in the theory of quantum mechanics. The article overviews the explanations of the principle on phenomena including atomic stability and scanning tunneling microscopes, and the interference of measurement actions on the states of particles and the shortcomings of measurement precision. According to the wave-particle duality of quantum mechanics, this article gives a required theoretical basis of microscopic phenomena. Likewise, the use of the research theory on the uncertainty principle by distinguished scholars in the physics fraternity helps the readers in achieving the knowledge in related subjects more easily.
Wave and particle properties of a quantum object cannot be observed simultaneously. In particular, the fringe visibility in an interferometer is limited by the amount of which-way information which can be obtained. This limit is set by the recently discovered duality relation. So far, all derivations of the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstrate that it is alternatively possible to derive the duality relation in the form of an uncertainty relation for some suitably chosen observables.
This paper presents a mathematical approach, based on the spectral analysis and Fourier transformation to derive the principle of uncertainty. The approach presented in this paper is found to be different from those appeared so far in the literature. It shows that the uncertainty principle is, in fact, an aliasing phenomenon and emerges merely from the quantization effect. The concept of ‘certainty’ is also introduced and discussed. It is shown that observation rates higher than the speed of a particle are the only possible approaches to the absence of uncertainty in the measurements of the observable quantities ofthe entity. This would imply that for particles travelling at the velocity of light, data acquisition rates higher than the speed of light should be required. Therefore, the uncertainty principle is related to the theory of relativity and exists so far as the velocity of light poses as an upperlimit universal constant. Otherwise, it would render to ‘certainty’ if our observation abilities could escalade at rates above the speed of light. This is to say that the uncertainty can be an inherent intrinsic property grounded in the universal speed limitation, which also subdues our observation abilities too; else demanding challenges on Einstein’s definition of simultaneity, and hence its consequences, has to be expected.