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Modern but not entirely coordinated foundations of quantum physics are described in the book “The Quantum Challenge”. The difficulties and philosophical problems of this area of science are discussed. Discussions of many great scientists who paved the foundations of the physics of micro-world are described. These discussions are still urgent. The diversity of interpretations of the wave function, light interference, uncertainty principle, complementarity and completeness of micro-world description are stressed in this book. Difficulties and problems of quantum mechanics described in this book allowed the author of the present communication to propose a new approach based on the infinitely small metrics The difference of infinitesimals in two geometries allows one to explain W. Heisenberg’s uncertainty principle. Interconnection of the images in these geometries is possible with the help of Weierstrass integral transform. This approach allows one to describe the interference of light behind the screen with slits as a sum of the corpuscular component (Weierstrass transform) and the wave component (Fourier transform).

The authors of the books describing the foundations of quantum mechanics usually try to avoid the difficulties and to make no mention of the unsolved problems of this area of science. An exception to the publications of this kind is “The Quantum Challenge” by G. Greenstein and A. Zajonc [

This book [

Uncertainly principle and complementarity are discussed in [

It should be indicated that the author of the present communication is educated in Chemistry, so the notions of quantum mechanics and E. Schrödinger’s equation are rather shallow. However, the author has always been interested in the philosophical problems of science, its history and development [2 - 4]. In this connection, book [

At the beginning of the XX century physicists came across the problem of the quantitative description of the world. Results of experiments did not allow one to make a complete and noncontradictory description of processes at the micro level [

However, to describe micro-particles possessing discrete properties, it was necessary to develop new physics, which would operate with discrete transitions. In other words, physics needed a new discrete tool. Mathematical approaches of this kind, in which discreteness had been already designed-in, had been proposed about a hundred years ago [

It should be noted that the creation of quantum mechanics proceeded using the methods of analytical mathematics. Thirty years later, the first computers appeared. After the next twenty years, it became possible to carry out numerical modeling, to built up plots, and finally to visualize calculations completely with the help of computers. These advances promoted the broad application of computers to solve numerically E. Schrödinger’s equation and other problems of quantum mechanics. In addition, with the development of computers, the possibilities of the numerical modeling of integral Fourier, Laplace, Weierstrass etc. transforms were opened. Some of them may be used to describe the phenomena of microworld [9 , 10].

Only the idea of the wave nature of light is used in physics manuals to describe the interference of light [9 - 12]. The transition of light through one or two slits in a non-transparent screen is usually considered. Diffraction at a semi-plane is also considered. No manualsd include the examples of interference at three, five or more slits in a screen, except for diffraction at a grating. In addition, the manuals consider only the case of rather remote recording of interference (from 2 m. Fig.6.2 [

The obtained result is in good agreement with the experiment. It is necessary to give D. Bohm’s genius his due, for having found a classical constituent in the wave function. However, E. Schrödinger developing his equation did not expect the presence of a corpuscular component in the wave function. So, according to D. Bohm, to provide a descritpion of light interference at any distance from the screen with the slits, it is necessary to take into account both the corpuscular and the wave components of light. The approach proposed by D. Bohm was further developed in [5 - 7], but the classical scatting behind a screen with holes was described by the integral Weierstrass’ transformation, with its core being varied depending on the distance to the recording site. The contribution from classical scattering decreased with an increase in the distance, for example, as Exp(−αR^{2}), where α is a constant, and R is the distance from the screen. The wave function was described by the squared Fourier screen transformation, the contribution from which increased as {1 − Exp(−αR^{2})}. The total value was normalized over the energy that passe4d through the slits in the screen. The results of numerical calculation of diffraction were published in [5 , 6 , 8]. This approach allows one to describe the distance and interference of light for any number of slits in the screen and at any distance from the screen. Theoretical considerations substantiating the application of Weierstrass and Fourier transforms to the description of light interference will be presented below.

The use of the Fourier integral transform for the description of wave phenomena is well known. For instance, this transform was used by R. Ditchburn for the description of light interference [

It should be stressed that similar transformations but with infinitely large values are already known in physics. Ancient people believed that temperature could take any value from −∞ to +∞. However, Lord Kelvin proposed to transfer the minimal temperature to a finite value (−275.15˚C) and accept this value to be equal to zero. In this case, many thermodynamic expressions are written in a simpler form. In fact, this approach is the transfer of an infinitely remote value (−∞ for temperature) to the zero of Kelvin’s scale. This approach also automatically transfers unattainability of the infinitely far point into the zero of Kelvin’s scale, and this value becomes unattainable. Another finite unattainable value for any object with non-zero mass is the velocity of light. Ancient philosophers (with rare exception) thought that the velocity of light is infinite. However, in 1905, A. Einstein developed the Special Theory of Relativity (SR) to adjust the laws of classical mechanics and electrodynamics. According to this theory, the velocity of light measured in any inertial reference system is the same and independent of the motion of the system and irradiator. According to SR, the velocity of light is the maximal velocity with unattainability attribute.

These two examples show that in modern physics it is sometimes useful to make unusual deformations (approximations) of infinitely large values to finite ones. The assumption concerning an increased value of infinitesimal has some similarity with these two examples. It is still unclear whether it would be easier to explain the properties of micro-particles if we assume an exaggeration (increase) of infinitesimal.

The authors of book [

In the approach proposed by the author of the present communication, the size of infinitesimal in the geometry of the moving photon is larger than the distance between the slits on the screen. Because of this, the two slits are perceived as a single one in this geometry. However, interference is recorded in the geometry of the experimenter. Because of this, it is possible to describe interference as a sum of the corpuscular and wave components. At present, mathematical and topological approaches to the comparison of two geometries with different infinitesimal metrics have not been elaborated yet. However, the author of the present communication hopes that the assumption concerning exaggeration of infinitesimal will provide a better explanation of the features of micro-particle motion. Maybe, this will allow one to combine classical mechanics with quantum mechanics, similarly to how A. Einstein and other scientists succeeded in uniting classical mechanics with electrodynamics.

The author is grateful to S.P. Babailov (professor, physicist by training) for the discussion and valuable comments.

The author declares no conflicts of interest regarding the publication of this paper.

Paragraph from the book Stabnikov P.A. “Ramki, v kotorykh razvivayetsya materiya (Frames in which matter develops)”, published in 2018 by Palmarium Academic Publishing.

ISBN: 978-620-2-38223-6

Let us consider a task. There are two pupils standing before the blackboard. Each pupil holds several circles of the same size, but the circles in the hands of the first pupil are larger than those in the hands of the second pupil. Let every circle be an infinitely small point specified for each pupil. How will these pupils measure an increasing size of a segment? If the segment is shorter than the diameter of the smallest circle, both pupils will say that the length of the segment does not exceed the infinitely small value (a point). If the length of the segment is longer than the size of the smallest circle but shorter than the length of the larger circle, one pupil will say that the segment is small but its length may be estimated, while the other pupil will say that the segment still does not exceed the infinitesimal. When the segment length becomes larger than the sizes of both kinds of circles, both pupils will be able to estimate the length of the segment relying on the sizes of the circles identifiable as infinitely small points. They also may estimate the length of longer segments packing their circles along the segment to cover it completely. Of course, their results will differ from each other but they will be represented by jogged lines (

It follows from

This approach declaring the differences by the metrics of infinitesimals for the geometry of the motion of micro particles potentially allows us to explain the discreteness with an increase in the distance. It should also be noted that the images in the geometry with the larger value of infinitesimal will be more blurred or fuzzy. This approach was developed as an alternative with the help of which one might explain W. Heisenberg’s uncertainty by geometric statements. The most important item is that this geometric approach is more fundamental because it is based on clear geometric statements, unlike for the wave-corpuscle dualism relying on two antagonistic notions: a wave and a particle.

This addition is an attempt of the topological extension of the geometric principles of an infinitely small value. According to this approach, the geometry of microworld does not differ from macro geometry except for the size of an infinitely small point. In the opinion of the author, this approach allows us to explain the discreteness of microworld. In addition, this will allow us to extent Galilean relativity principle to the micro level.