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arXiv:1705.06987 · Physics of Particles and Nuclei Letters

Fundamental quantal paradox and its resolution

Felix M. Lev

Published May 1, 2017• 2 citations
arXivHTMLPDF

Abstract

The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory. As a consequence, coordinate wave functions of photons emitted by stars have cosmic sizes. This results in a paradox because predictions of the theory contradict observations. The reason of the paradox and its resolution are discussed.

DOI
10.1134/s1547477117030128

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The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory. As a consequence, coordinate wave functions of photons emitted by stars have cosmic sizes. This results in a paradox because predictions of the theory contradict observations. The reason of the paradox and its resolution are discussed.
The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. As a consequence, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum. This leads to several paradoxes. The most striking of them is that coordinate wave functions of photons emitted by stars have cosmic sizes and strong arguments indicate that this contradicts observational data. We argue that the above postulate is based neither on strong theoretical arguments nor on experimental data and propose a new consistent definition of the position operator. Then WPS in directions perpendicular to the particle momentum is absent and the paradoxes are resolved. Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. Implications of the results for entanglement, quantum locality and the problem of time in quantum theory are discussed.
The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. As a consequence, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum. This leads to the following paradoxes: if the major part of photons emitted by stars are in wave packet states (what is the most probable scenario) then we should see not separate stars but only an almost continuous background from all stars; no anisotropy of the CMB radiation should be observable; data on gamma-ray bursts, signals from directional radio antennas (in particular, in experiments on Shapiro delay) and signals from pulsars show no signs of WPS. In addition, a problem arises why there are no signs of WPS for protons in the LHC ring. We argue that the above postulate is based neither on strong theoretical arguments nor on experimental data and propose a new consistent definition of the position operator. Then WPS in directions perpendicular to the particle momentum is absent and the paradoxes are resolved. Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. Implications of the results for entanglement, quantum locality and the problem of time in quantum theory are discussed.
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.
Quantum physics is a successful field of science with many relevant applications in everyday life. An exciting question is still, what is the fundamental reason for the postulates about quanta? An answer would provide a deeper insight into nature. Such an insight can sometimes be achieved with help of a paradox. How can we derive the quantum postulates with help of a paradox? In present-day science, space is usually modelled as a single entity. However, special relativity shows that this view is paradoxical. Instead, space consists of several volume portions that move at the velocity of light. According to the observed isotropy of space at a universal scale, space is a statistical average of rapidly moving volume portions. This solves the paradox. Therefrom, the deeper insight is achieved as follows: The dynamics of these volume portions is derived. Therefrom, the Schrödinger equation and the quantum postulates are derived. In this manner, the dynamics of space provides the fundamental reason for the postulates about quanta. In the learning process, the students achieve an intensive awareness, as they experience a cognitive conflict, the paradox. With it, they realize that the basic concept of a single entity of space is insufficient. They realize that the new concept requires rapidly moving volume portions and provides the quantum postulates. A concept for teaching is presented and experiences are reported.
From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.
A simple conceptual experiment which seems to violate the general principles of quantum mechanics is reported. The reasoning leading to this contradiction contains a trick which makes the paradox possible. By discussing the failure of the argument the author wishes to draw attention to a fundamental feature of the wave function which is frequently overlooked. Finally, in light of this discussion, a kind of delayed-choice experiment is described, and an analysis of the related issue of complementarity is made.