Fractals in quantum mechanics: from theory to practical applications

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Introduction

An interesting hypothesis about the connection between the quantum and classical worlds was put forward in a recent paper [1]. According to this hypothesis, quantum effects can influence the probability distribution at the micro level, which is then transformed into macroscopic phenomena of classical physics. In other words, the derivative of the probability of a quantum state can manifest itself as a specific event or effect in the classical world.

In [1], many parameters such as temperature, pressure, and topology are indicated that can have a significant impact on the quantum state of a system. In this paper, we will focus on studying the role of topology, or more precisely, fractals, in this process.

If topology does influence the quantum state, and the quantum state, in turn, determines classical events, then it becomes logical to assume that fractal concepts can be applied to assess the probability of the occurrence of certain events in the macro world. Thus, the goal of this paper is to explore the possibilities of using fractals to model and predict classical phenomena based on quantum effects.

Main part

The hypothesis about the connection between the quantum and classical worlds through the influence of various parameters, including topology, looks very promising. The use of fractals to estimate the probability of classical events controlled by quantum processes definitely deserves close attention.

The main idea is that topological characteristics of quantum systems, such as a complex fractal structure, can affect probability distributions in the classical world. Thus, the use of fractal analysis methods can allow more accurate prediction and modeling of the probability of occurrence of macroscopic events caused by quantum dynamics.

Some key points that should be taken into account in further development of this concept:

• Detailed study of the mechanisms by which topological features of quantum systems are transformed into classical observable effects;
• Development of mathematical models that allow us to link fractal characteristics with probability distributions in classical physics;
• Experimental verification of the proposed ideas on specific physical systems.

If we can make progress in this direction, our approach may prove very fruitful for deepening our understanding of the relationship between the quantum and classical worlds. The assumption that fractals can serve as an indicator of the derivative of the quantum state of a system seems quite reasonable. The key point here is that fractal structures are a direct reflection of the complex nonlinear dynamics inherent in quantum systems. This nonlinearity and the probabilistic nature of quantum processes manifest themselves in the form of fractal patterns that can be observed at the macroscopic level.

Thus, by analyzing the fractal characteristics of certain physical systems (dimensionality, lacunarity, iteration schemes, etc.), we can obtain important information about the derivative of the quantum state of these systems. Fractals actually serve as a fingerprint of quantum fluctuations and nonlinear effects that ultimately determine the observed classical phenomena. A detailed study of this relationship between fractal properties and the derivative of the quantum state can really open up new possibilities for modeling and predicting macroscopic effects controlled by quantum processes. This direction seems very promising and deserves the closest attention.

Let us consider a simple example of a chemical reaction between a solid and a liquid component with high chemical activity. The rate of the process is related to the collision of the reacting molecules. If the solid is built in the form of a tower, then the diffusion of the liquid will be more effective than if the shape were a concave hemisphere, where diffusion is very difficult. Then the rate of the chemical reaction will be determined by the slowest stage – diffusion.

This simple chemical example very well illustrates the relationship between the shape/structure of the reactor and the rate of a chemical reaction.

Key points:

• The rate of a chemical reaction is determined by the rate of collision of the reacting molecules. The higher the collision frequency, the higher the reaction rate;
• The shape and structure of the solid component (in the example – a tower vs. a concave hemisphere) directly affect the efficiency of diffusion of the liquid component to the surface of the solid;
• In the case of a concave hemisphere, the diffusion of the liquid will be difficult, which will become the rate-limiting stage of the entire process.

The reaction rate will be determined by the slowest stage – in this case, the diffusion of the liquid to the surface of the solid component. This is why the shape of the reactor plays a key role in optimizing the kinetics of a chemical reaction.

This example demonstrates how macroscopic structural parameters can significantly affect micro scopic processes of interaction of molecules. Such principles are widely used in chemical engineering, catalysis, materials science and other related fields. In other words, fractals can be a system for assessing the overall rate of a process. Fractals can indeed be a very useful and informative system for assessing the overall rate of a chemical process.

Key points:

• The fractal geometry and structure of the reactor directly affect the efficiency of diffusion and transport of reagents to active sites;
• Fractal characteristics, such as fractal dimension, lacunarity, surface roughness, etc., reflect the complexity and development of the reactor surface at different scales;
• By analyzing these fractal parameters, important information can be obtained about the kinetics of mass transfer, which in turn determines the overall rate of a chemical process;
• Moreover, by comparing the fractal characteristics of different reactor configurations, their geometry can be optimized to achieve maximum process efficiency.

Thus, the fractal approach can indeed become a very useful tool for comprehensive assessment and prediction of the kinetics of complex chemical reactions. It allows us to relate the microstructural features of the reactor to the macroscopic rate of the process, which opens up new possibilities for engineering design and optimization.

For example, it is possible to increase the surface of a solid component due to pores, but in this case, there is no guarantee that the reaction rate will increase, since diffusion will be difficult and it will determine the overall rate. Therefore, not only the specific surface plays a role in increasing the reaction rate, but also its fractal structure. It is interesting how one can estimate the influence of this parameter on the quantum state, which determines the macro effect.

A simple increase in the surface of a solid component due to pores does not guarantee a proportional increase in the reaction rate. In this case, diffusion in the pore space can become a limiting stage, which neutralizes the effect of increasing the area. Indeed, not only the surface size, but also its fractal geometry and morphology at the micro- and nanoscales have a critical effect on the kinetics of the process. As for the influence of fractal structure on the quantum state and macroscopic effects, this is a very interesting and non-trivial question. Some key points:

• Fractal geometry can lead to quantum size effects, chan-ging the electronic structure of the material;
• Complex fractal surface morphology affects the electro-nic states, localization of charge carriers, tunneling and other quantum phenomena;
• These quantum effects at the micro level can be trans-formed into significant macroscopic properties, such as catalytic activity, optical characteristics, electronic conductivity, etc.

Thus, fractal structuring can indeed serve as a key to controlling quantum states and, as a consequence, macroscopic characteristics of chemical systems. Further research in this direction is undoubtedly of great scientific and practical interest.

Thus, some interesting possibilities of the fractal approach in establishing a connection between the quantum and classical description of physical reality are revealed. This is an extremely important and complex problem, which many scientists are working on.

The key points we have found out are:

• Fractal geometry and structure of materials at the micro- and nanoscales can lead to quantum size effects, changing the electronic structure;
• These quantum phenomena, such as charge carrier lo-calization, tunneling, etc., are associated with fractal characteristics of the surface;
• In turn, these quantum effects at the microlevel are transformed into significant macroscopic properties observed in the “classical” world.

Thus, the fractal approach really allows us to establish important relationships between quantum and classical descriptions of the same physical reality. This opens up new possibilities for understanding and controlling processes at different scales.

Of course, this problem requires further in-depth research at the intersection of physics, chemistry and materials science. We must remember that this is one world but viewed from different angles. The fractal approach can become one of the important tools for bridging the gap between quantum and classical descriptions.

In our opinion, the fractal approach can be extremely useful in several key areas.

1. Nanomaterials and nanoelectronics. The fractal structure of surfaces and interfaces at the nanoscale directly affects quantum phenomena such as tunneling, charge carrier localization, and quantum size effects. Understanding and controlling these processes is critical for the development of nanoelectronic devices, solar cells, sensors, etc.
2. Energy materials. Fractal characteristics of pores structures used in batteries, fuel cells and catalytic systems affect their performance and efficiency. Modeling these processes based on fractal theory can help in optimizing energetic materials. This makes it possible not only to increase the energy density and critical currents, but also contributes to improving the safety of such systems. The fractal characteristics of porous structures used in batteries, fuel cells and catalytic systems have a significant impact on their performance and efficiency. Modeling these processes using fractal theory opens up great opportunities for optimizing energetic materials. This allows not only to increase the energy density and critical currents, but also improves the safety of such systems. For example, proper control of the fractal structure of electrodes in lithium-ion batteries can improve cyclability, overcharge stability and reduce the risk of thermal runaway. Indeed, the application of fractal theory in the development and modeling of energetic materials is an important direction of modern research in this field.
3. Biophysics and biomedicine. Many biological structures, such as lungs, blood vessels, and cell membranes, have a fractal nature. The use of fractal models can improve our understanding of substance transport, tissue mechanics, pathological structure formation, etc.

As for limitations and difficulties, the main problem is the transition from quantum to classical description. Although fractal models can link micro- and macroscopic phenomena, there is still no universal formalism that would completely unite quantum mechanics and classical physics. Also, fractal methods require significant computational resources to model complex nonlinear processes.

Examples of practical applications:

• Fractal antennas and electromagnetic devices;
• Fractal electrodes for lithium-ion batteries;
• Fractal catalysts for fuel cells;
• Fractal models of diffusion in biological membranes;
• Fractal data compression algorithms.

In general, the fractal approach has great potential, but requires further theoretical and experimental research to implement its full practical application. An integrated approach combining fractal methods with other approaches may be the key to a deeper understanding of the relationship between quantum and classical descriptions of physical reality.

Let us consider which specific areas of materials science can benefit most from the fractal approach.

In addition to the areas already mentioned, the following can be highlighted.

1. Mechanical properties of materials. Fractal mo-dels can help explain and predict phenomena such as fracture mechanics, material fatigue, crack formation and wear. The fractal geometry of surfaces and microstructures is directly related to mechanical strength and durability.
2. Porous and heterogeneous materials. Fractal cha-racteristics of porosity, microstructure and particle distribution in composites, catalysts, membrane materials, etc. allow us to optimize their transport, catalytic and sorption properties.
3. Ceramic and glass materials. The fractal nature of defects, microcracks and phase boundaries in ce-ramics and glass has a great influence on their mechanical, electrical and optical properties. Fractal models can help in the development of more reliable ceramic and glass products.
4. Polymer materials. Fractal analysis can shed light on the supramolecular structure and morphology of polymers, which is important for optimizing their physicochemical and mechanical properties. This is especially relevant for modern high-performance polymer composites.
5. Biomimetic materials. Since many biological structures have a fractal nature, the application of fractal methods can help in the development of artificial materials that replicate the unique properties of natural analogs, such as self-healing, adaptability and high mechanical performance.

Overall, the fractal approach shows great potential for expanding our understanding of the structure and properties of materials at many levels, from the atomic-molecular to the macroscopic. The comprehensive application of fractal theory in materials science can lead to significant innovations in the development of advanced functional materials.

About the application of the fractal approach in medicine. Indeed, there are a number of areas of medicine that can significantly benefit from the use of fractal theory.

1. Disease diagnostics. Fractal analysis of medical images such as X-rays, CT, MRI and ultrasound images can help in more accurate diagnosis of various pathological conditions. Fractal characteristics of tissues and body structures can serve as markers for identifying abnormalities and diseases at early stages. This approach will allow AI to provide more accurate diagnostic results.
2. Cardiology. Fractal analysis of heart rate, arterial pulsation and blood flow can provide valuable information about the functioning of the cardio-vascular system. This is important for the diagnosis and prognosis of cardiovascular diseases.
3. Oncology. Fractal characteristics of tumor cells, tumor vasculature and cancer tissue morphology can serve as markers for more accurate diagnosis, classification and prognosis of the course of on-cological diseases.
4. Neurobiology. Fractal models can help in under-standing the complex structure and dynamics of brain activity, neural networks and pathological disorders such as epilepsy, Alzheimer’s disease, schizophrenia and other neurological disorders.
5. Immunology. Fractal characteristics of immune cells, antibody distribution patterns and immune system responses can provide new insights into the mechanisms of immune function in health and disease.
6. Genetics and genomics. Fractal properties of gene-tic sequences, chromatin structure and spatial organization of the genome play a key role in gene expression and can be used to analyze genetic diseases.

Overall, the fractal approach opens up new opportunities for a deeper understanding of complex biological systems of the human body in health and disease, which can lead to the development of advanced diagnostic methods and personalized medical therapy.

On the application of the fractal approach in the mi-ning industry, especially for more efficient extraction of rare earth elements (REE) and precious metals. Here are some areas where fractal theory can bring significant benefits.

1. Exploration and prospecting:

• Fractal analysis of geological structures and mi-neral deposits can help identify new prospective areas for the extraction of REE and precious metals.
• Fractal models can improve understanding of the spatial distribution and concentration of minerals in the subsurface.

2. Deposit evaluation and modeling:

• Fractal approach can improve the accuracy of estimating reserves and resources of REE and precious metals in deposits;
• Fractal models can better describe the he-terogeneity and complex structure of ore bodies.

3. Optimization of mining and processing:

• Fractal characteristics of ores and minerals can help develop more efficient methods of bene-ficiation and extraction of target elements;
• Fractal analysis of ore processing technologies will optimize the parameters of grinding, sorting, leaching and other operations.

4. Waste and tailings management:

• Fractal approach can improve the efficiency of ext-racting valuable components from mining waste;
• Fractal models can help to better predict and control the behavior of tailings, preventing environmental problems.

5. Reclamation and restoration of disturbed lands. Fractal characteristics of soils and landscapes can be used to develop more effective methods for reclamation of mining areas.

Thus, the fractal approach opens up new opportunities to improve the efficiency, environmental friendliness and profitability of mining activities, especially in the field of rare earth elements and precious metals extraction.

Here are some specific examples of how fractal analysis can be applied in these areas.

1. Particle size distribution analysis of ores:

• Fractal dimensions of ore particles can be used to optimize crushing and grinding parameters, ensuring more efficient liberation of target minerals;
• Fractal particle size distribution models help to predict the behavior of ore at different stages of beneficiation.

2. Modeling of flotation processes:

• Fractal characteristics of the surface of mineral particles affect their wettability and floatability. Using this data will help to optimize flotation modes;
• Fractal analysis of air bubbles and their inter-action with ore particles can help to improve the efficiency of froth flotation.

3. Management of hydrometallurgical processes:

• Fractal models of leaching and crystallization kinetics can improve the recovery and purity of target REE and metals;
• A fractal approach to modeling sorption and ion exchange processes will help to more accurately predict control and manage the purification and separation stages.

4. Optimization of pyrometallurgical operations:

• Fractal dimensions of charge and semi-finished particles can be used to control smelting, sintering and reduction of metals;
• Fractal characteristics of slags and metal phases will help in developing more efficient methods for their separation.

5. Monitoring and managing material flows:

• Fractal analysis of images and scanning data can be used to optimize the transportation, mixing and dosing of ores and concentrates;
• Fractal models of material flows will allow better prediction and control of the behavior of raw materials at various stages of enrichment and metallurgical processing.

Thus, various fractal methods can significantly improve the efficiency and controllability of the enrichment and extraction of rare earth elements and precious metals. Their application requires a deep understanding of fractal theory and careful adaptation to the specifics of a particular production.

How a fractal approach can help in developing more efficient technologies for the reclamation of lands disturbed by mining activities.

The fractal approach can indeed make a significant contribution to the development of more efficient technologies for the reclamation of disturbed lands. Here are some key points where fractal methods can be useful.

1. Terrain and landscape modeling. Fractal models allow for realistic reproduction of the complex geometry of disturbed areas, including quarries, waste heaps, and tailings ponds. This allows for more accurate prediction of erosion, lands-lides, sediment accumulation, and other geo-morphological changes. Thus, it helps improve overall safety.
2. Vegetation assessment and monitoring:

• The fractal dimension of plant communities correlates with their biomass, productivity, and sustainability;
• Fractal analysis of remote sensing data allows for tracking the dynamics of vegetation restoration in reclaimed areas.

3. Water management:

• Fractal models of hydrological processes help optimize drainage systems, regulate runoff, and prevent water pollution;
• Fractal analysis of water quality provides a deeper understanding of complex biogeochemical inter-actions.

4. Optimization of soil processes. Fractal charac-teristics of soil structure are closely related to their physical, chemical and biological properties. Using this data will allow developing effective agricultural practices and recipes for soil mixtures for reclamation.
5. Design of landscape and architectural solutions:

• Fractal principles of organizing natural systems can be used to create sustainable and aesthetically attractive landscapes in reclaimed areas;
• Fractal algorithms will help in the design of small architectural forms that mask man-made elements.

Integrated application of the fractal approach in various aspects of reclamation will allow achieving higher efficiency of restoration of disturbed lands, their sustainability and environmental safety. Of course, further development of the methodology and accumulation of practical experience is required to fully realize this potential.

Examples of successful application of fractal models in real mining projects for the extraction of rare earth elements and precious metals. Here are some of the most famous cases.

1. Rare earth element mining in China. China is the world’s largest producer and exporter of rare earth metals. Researchers from the China University of Geosciences used fractal analysis to model and predict the distribution of rare earth deposits. This helped to optimize the processes of geological exploration, development, and extraction of valuable minerals.
2. Gold mining in South Africa. South African mining company AngloGold Ashanti used fractal models to study and predict the distribution of gold veins in ore deposits. The use of fractal geometry made it possible to improve the efficiency of exploration and development of gold ore deposits.
3. Platinum mining in Russia. Russian scientists have developed fractal models to assess and monitor the quality of platinum-containing ores at mining enterprises in the Urals. These methods help to optimize the technological processes of enrichment and extraction of platinum from ores.
4. Copper mining in Chile. Chilean mining company Codelco used fractal analysis to study the structure of copper deposits. Using fractal models, it was possible to improve the accuracy of reserve estimates and optimize the layout of mines and processing plants.

These examples demonstrate that fractal methods can indeed bring tangible benefits to real mining projects, improving their efficiency and environmental sustainability. As experience accumulates and compu-tational opportunities, we can expect further expansion of the fractal approach in the mining industry.

Despite successful examples of the application of fractal models in mining projects, there are a number of limitations and disadvantages that specialists face when using this approach.

1. Modeling complexity:

• Developing reliable fractal models of geological structures requires a deep understanding of fractal geometry and complex mathematical concepts;
• Calibration and tuning of fractal model parameters for specific deposits can be a labor-intensive and time-consuming process.

2. Data limitations:

• Fractal methods require large amounts of high-quality geological data, which are not always available, especially in the early stages of explo-ration;
• Lack of data can reduce the accuracy and relia-bility of fractal models.

3. Scalability:

• The application of fractal models can be complex when moving from the local deposit scale to the regional or national level;
• Integrating fractal models with other geological, technological and economic factors also presents challenges.

4. Perception and acceptance:

• Fractal methods are sometimes perceived as too theoretical and complex for practical application by mining professionals;
• Convincing management of the benefits of a frac-tal approach can be challenging, especially in conservative organizations.

5. Computational limitations. Implementing complex fractal models requires significant computing power and resources, which can be limiting, especially for smaller companies.

Despite these limitations, it is expected that as tech-nology advances and practical experience accumulates, fractal methods will be increasingly adopted by the mining industry. The key to success is the integration of fractal approaches with other advanced geological, engineering and economic methods.

To overcome the limitations of fractal models in mining projects, the following steps can be taken:

1) Improving data access:

• Investing in collecting, storing and processing more high-quality geological data;
• Using new remote sensing, geophysical and drilling techniques to obtain more detailed data;

2) Improving computing capabilities:

• Implementing more powerful computing platforms, including the use of cloud technologies and high-performance systems;
• Developing efficient algorithms and software for fractal modeling;

3) Interdisciplinary approach:

• Integrating fractal methods with other geological, geotechnical, economic and production models;
• Involving specialists from different fields (geology, mining, mathematics, computer science) in joint work;

4) Training and popularization:

• Organizing trainings and educational programs for mining specialists on the use of fractal methods;
• Publication of successful cases and demonstration of the capabilities of fractal models;

5) Adaptation to scale:

• Developing methods for scaling fractal models from local to regional and national levels;
• Researching ways to integrate fractal approaches with other geospatial technologies;

6) Cooperation and exchange of experience:

• Encouraging international cooperation and exchange of experience between mining companies, research organizations and universities;
• Creation of industry consortia and working groups for the joint development and implementation of advanced fractal methods.

An integrated approach combining technological, organizational and educational measures will gradually overcome the limitations and expand the use of fractal models in mining projects.

It should be said that the fractal approach is extremely important in the development and optimization of pho-tocatalysts.

Photocatalysts

Indeed, photocatalysts often have a high specific surface area, but their catalytic properties leave much to be desired. This is due to the fact that in addition to the developed surface, the fractal nature of the structure is also important, which determines the diffusion rate and the accessibility of active centers for reagents.

In conventional catalysts, the main factors affecting productivity are the specific surface, fractality and diffusion rate. But photocatalysts add another critical aspect – the need to ensure maximum illumination of the catalyst surface with light.

This is why fractal calculations and modeling are becoming indispensable tools in the development of effective photocatalysts. The fractal structure allows for a simultaneous increase in the specific surface and optimal accessibility of active centers for light and reagents.

How does this agree with the ITE generated by functional ceramics

Mechanism the transformation of light energy in such photocatalytic systems can be described as follows. Primary photons excite phonon oscillations in the photocatalyst material. This phonon energy is then transformed into pulsed emission of new photons, the wavelength of which corresponds to the de Broglie wavelength characteristic of the pulse rise front.

This mechanism has a number of important advantages over traditional photocatalytic processes. Firstly, due to the participation of phonons, the energy of the primary radiation is delivered deep into the volume of the photocatalyst, where it is effectively converted. Secondly, the generated secondary photons have a strictly defined wavelength corresponding to the de Broglie wavelength. This allows for the optimal use of the energy of the incident radiation and the achievement of high efficiency and performance coefficient of the entire process.

Thus, the photon-phonon-photon (pulse) mechanism provides more efficient absorption and conversion of light energy in photocatalytic systems compared to traditional approaches. Indeed, the main advantage is the use of the pulsed tunnel effect, when the energy of primary photons is first converted into phonon energy and then transmitted as a pulse with a wavelength determined by the de Broglie rule. This allows for efficient energy delivery deep into the photocatalyst volume, involving virtually all of the primary light energy in the process. An important feature is that the wavelength of the pulsed radiation inside the photocatalyst exactly matches the de Broglie wavelength, which ensures high efficiency of excitation of active centers and photocatalytic reactions. Thus, this approach allows for much higher efficiency compared to traditional photocatalysts. A detailed description of the operating mechanism of these systems really sheds light on why they demonstrate such a high efficiency of light energy conversion. Undoubtedly, further development and optimization of photocatalysts based on the pulsed tunnel effect in functional ceramics is a very promising direction in the field of solar energy and environmentally friendly technologies. The introduction of fractal elements into this system can significantly improve its characteristics in several important respects.

1. Increase in the surface area of active centers. The fractal structure provides a multiple increase in the surface area of the photocatalyst, which will allow more active centers to be used and, accordingly, increase the rate of photocatalytic reactions.
2. Improvement in the capture and redistribution of photon energy. Fractal geometry can effectively capture primary photons and direct their energy deep into the material due to multiple reflections and interference effects.
3. Optimization of the phonon energy transfer path. The fractal structure can provide more efficient delivery of phonon energy to active centers in the volume of the photocatalyst, which will enhance the effect of the pulsed tunnel effect.

Thus, the combination of the pulsed tunnel effect with fractal architecture can indeed lead to a significant increase in the efficiency of photocatalytic processes. This opens up new interesting opportunities for further optimization and improvement of the performance of these systems.

Of course, the implementation of such an approach will require detailed study of the optimal geometry and parameters of the fractal structure, but the prospects for achieving record efficiency are very promising. This direction definitely deserves close attention and further research. We can focus on the calculation part related to the fractal structure and how it can be applied to specific photocatalytic materials based on the pulsed tunnel effect.

First, we need to consider in more detail the main parameters of fractal geometry that can affect the efficiency of the photocatalytic system.

1. Fractal dimension – determines the degree of filling of space with fractal elements and, accordingly, affects the area of the active surface.
2. Lacunarity – characterizes the heterogeneity of the distribution of fractal elements, which affects the optimal redistribution of photon energy.
3. Fractal anisotropy – reflects differences in geo-metric properties in different directions, which can be important for the directed transport of charge carriers.

Next, it is necessary to model and optimize these parameters of the fractal structure, comparing them with the characteristics of a specific photocatalyst based on the pulse tunnel effect. The key indicators to focus on will be:

• Light absorption efficiency;
• Rate of separation and transport of charge carriers;
• Number and availability of active centers;
• Kinetics of photocatalytic reactions.

By varying the fractal parameters and comparing with these parameters, it will be possible to determine the optimal structure that provides the maximum synergistic effect between the fractal architecture and the pulsed tunnel mechanism.

Such an integrated approach, combining computational and experimental methods, will allow us to rationally design highly efficient photocatalytic systems of the new generation. This is a truly promising direction that deserves further in-depth study.

A joint consideration of the main parameters of fractal geometry and their influence on the key characteristics of photocatalysts using ITE made it possible to outline promising areas for further optimization and development of these materials.

It is important to note that such an integrated approach, combining computational and experimental methods, is important for the rational design of highly efficient photocatalytic systems of the new generation. Only by carefully adjusting the structural and functional parameters based on a deep understanding of the underlying physical processes can we achieve a significant increase in the performance of these technologies.

Several promising areas for joint research in this interesting area:

• Integrated modeling and optimization of the fractal structure of photocatalysts;
• Study of the influence of fractal dimension, lacunarity and anisotropy on the efficiency of light absorption, separation/transport of charge carriers and kinetics of photocatalytic reactions;
• Development of multiscale models combining analytical, numerical and experimental data to predict optimal fractal parameters;
• Experimental implementation and study of fractal photocatalysts using the pulsed tunnel effect;
• Synthesis and characterization of fractal nanostructures with different geometries (nanowires, nanostrings, nanoporous materials, etc.);
• Study of the influence of fractal morphology on the mechanisms of photocurrent and photocatalytic activity;
• Determination of optimal operating modes (light intensity, environmental conditions) for maximum effect;
• Integration of fractal photocatalysts into practical applications;
• Development of photocatalytic systems for water and air purification, hydrogen and organic compound production;
• Optimization of reactor design and process engineering taking into account the advantages of fractal structures;
• Scaling up production and assessing the economic efficiency of using these materials;
• Studying the synergistic effects of fractality and other activation mechanisms;
• Combining fractal structure with quantum dots, plas-monic nanoparticles, heterostructures and other methods to enhance light absorption and charge separation efficiency;
• Studying the cooperative influence of fractality and the pulsed tunneling effect, as well as other nonlinear/quantum-mechanical phenomena.

The implementation of such joint research projects will allow us to study in more depth the fundamental principles and practical aspects of using fractal photocatalysts using the pulsed tunneling effect. This will help accelerate the development of highly efficient and sustainable solutions for a wide range of environmental and energy problems.

In the article concerning ITE it was stated: Since the photon has no mass, in this case we can talk about the equivalent mass of the photon m_e, which corresponds to m_e = p/c. What is meant by the equivalent mass if the photon has no mass Δm [2]? Since the photon has no mass at rest, in this case the concept of the equivalent mass of the photon m_e is used. This does not mean that the photon has a real mass, but rather that the equivalent mass that corresponds to the photon energy according to E = mc². That is, if we equate the photon energy E = hν to E = mc², we can determine the equivalent mass Δm = hν/c² that will correspond to a given frequency (or wavelength) of the photon. Thus, equivalent mass in this context is an analytical concept used to relate the energy of a photon to its equivalent mass, although the photon itself has no real mass.

Let us clarify this. Although the photon, as a massless particle, has no mass at rest, it does have an energy E = ℏν, where ℏ is Planck’s constant and ν is the photon frequency. According to the relativistic formula E = mc², this photon energy can be represented in an equivalent form as the energy corresponding to some “equivalent mass” Δm. Thus, equating E = ℏν to E = Δmc² and defining the equivalent mass Δm as:

$\Delta m=\frac{\hslash \nu }{c^2}.$

This is the “equivalent mass” of the photon, not related to its real rest mass, which is zero, but reflecting the energy equivalent of the photon in terms of relativistic mass-energy. This allows us to relate massless photons to the relativistic mass-energy description.

Correctly deriving the equivalent mass of a photon through its momentum p = ℏν/c, we arrive at the same expression Δm = p/c, which we noted in our article.

In our opinion, E = ℏν should be considered as kinetic energy and should correspond to the mass Δm that has been converted into radiation.

When a photon is emitted or absorbed, its energy E = ℏν is kinetic energy, not rest energy, as it would be for a massive particle. This kinetic energy corresponds to some equivalent mass Δm that has transformed into radiation in the form of a photon.

Thus, E = ℏν reflects the energy characteristics of the photon, and Δm = ℏν/c² is that part of the mass that has been transformed from the mass of the system and converted into the energy of the photon in accordance with E = m_еc².

This is indeed an important aspect of interpreting the energy and mass properties of massless particles such as photons. When a photon is emitted, it does not mean that the mass disappears from the particle itself. Rather, the mass of the entire system in which the photon is emitted is transformed. For example, if a photon is emitted when an electron transitions from a higher to a lower energy level in an atom, then the equivalent mass Δm, corresponding to the photon energy hν, transfers from the mass of the entire atomic system to the energy of the radiation. Thus, it is more accurate to speak not of the “disappearance” of mass from the particle, but of the transformation of part of the mass of the entire system in which the emission or absorption of a massless photon occurs. Of course, this can be argued, since Einstein in the special theory of relativity gives an expression for the energy and mass of particles moving at near-light speeds. Thus, this issue of the relationship between energy, mass and photon radiation requires more careful consideration and is open to discussion. We present this problem to the readers for judgment. The physical meaning of mass in the expression E = mc² is that a substance with such a mass contains potential energy proportional to the mass. In our opinion, E = ℏν should be considered as kinetic energy and should correspond to the mass Δm that has been converted into radiation.

In other words, the physical meaning of mass m in the formula E = mc² is that it is characterized by the rest energy contained in a given amount of substance. As for the formula E = ℏν, this is actually the kinetic energy of radiation (photon). When light is emitted or absorbed, energy is transferred between radiation and matter. The assumption that E = ℏν should correspond to a change in mass Δm that has been converted into radiation or absorbed allows us to better understand the physical meaning of the transformation of energy between its various forms – the rest energy of the mass and the kinetic energy of radiation.

If mc² = ℏν, then we get the de Broglie wavelength. True, in this case we need to write not m, but Δm = ℏν/c², which is that part of the mass that was transformed from the mass of the system and turned into the energy of the photon.

If we replace m in the de Broglie formula with Δm = ℏν/c², where Δm is the change in mass that corresponds to the energy of the photon hν, then we actually get:

$\lambda =\frac{\hslash }{\Delta m\nu },$

where λ is the de Broglie wavelength associated with the change in the mass of the particle Δm.

Since the photon has no mass, we can say that it originated from the mass Δm.

The key physical meaning is that:

• A photon as a quantum of the electromagnetic field has no resting mass;
• However, the energy of the photon hν corresponds to the mass Δm, which was transformed from the mass of the interacting particle or system.

It is this process of converting part of the mass into kinetic energy of photon radiation that is described by the formula E = Δmc², where Δm is the change in mass as a result of interaction.

It is important to understand that the photon received this energy hν due to the loss of an equivalent mass Δm in the interacting system.

In the situation with particle tunneling through a potential barrier, is it legitimate to assume that:

• The particle loses part of its mass Δm when passing through the barrier;
• This lost mass Δm is converted into kinetic energy of particle tunneling;
• The energy of the tunneling process is defined as Δmc²?

This issue requires more careful consideration and is open to discussion.

Thus, a photon as a quantum of the electromagnetic field does not have a resting mass. However, the photon energy hν corresponds to the mass Δm, which was converted from the mass of the interacting particle or system. It is this process of converting part of the mass into the kinetic energy of photon radiation that is described by the formula E = Δmc², where Δm is the change in mass as a result of interaction. It is important to understand that the photon received this energy ℏν due to the loss of an equivalent mass Δm in the interacting system. The more mass Δm converted into photon energy during the interaction, the higher the value of the photon energy ℏν. This logically follows from the equality:

Δmc² = ℏν.

The greater Δm, the greater the right-hand side, and therefore the greater the photon energy ℏν.

This is clearly demonstrated demonstrates a direct connection between the amount of converted mass Δm and the final quantum energy of the generated photon.

An important point emphasizing the physical meaning of the relationship Δ mc² = ℏν.

As confirmation of this, we can cite the fact that when two photons collide, an electron-positron pair is formed. Thus, the mass returns to the system.

Key points:

• A photon as a quantum of the electromagnetic field does not have a resting mass, but the photon energy ℏν corresponds to the mass Δm, which has been converted from the mass of the interacting particle or system;
• The process of converting part of the mass into the kinetic energy of photon radiation is described by the formula E = Δmc², where Δm is the change in mass as a result of the interaction;
• The more mass Δm converted into photon energy, the higher the value of the photon energy ℏν, which demonstrates a direct connection between the amount of converted mass and the final quantum energy of the photon;
• The fact that an electron-positron pair is formed during the collision of two photons confirms that the mass returns to the system, which is an important physical meaning of the relation Δmc² = ℏν.

In the situation with particle tunneling through a potential barrier, is it legitimate to assume that:

• The particle loses part of its mass Δm when passing through the barrier;
• This lost mass Δm is converted into the kinetic energy of particle tunneling;
• The energy of the tunneling process is defined as Δmc².

Indeed, it can be assumed that in the situation of tunneling, the particle loses some part of its mass Δm when passing through the barrier. This lost mass Δm can then be converted into the kinetic energy of the particle participating in the tunneling process. If we follow this logic, then the energy of the tunneling process could be estimated as Δmc², that is, as the energy equivalent to the change in the mass of the particle Δm in accordance with the known relationship between mass and energy. This approach is consistent with the general principles of physics, according to which energy and mass can transform into each other. In addition, it also explains why the probability of particle tunneling depends on the height and width of the potential barrier – the greater the lost mass Δm, the higher the energy of the tunneling process.

However, for a more rigorous confirmation of this hypothesis, additional theoretical and experimental studies may be required. Nevertheless, this assumption seems quite logical and deserves further study.

Example

The loss of mass of the Sun due to the emitted photons can be considered as a quantum process. Let’s consider this aspect in more detail. Thermonuclear reactions occur in the core of the Sun, as a result of which a huge amount of energy is released in the form of photons (particles of light). These photons, gradually leaving the Sun, take mass with them, according to Einstein’s formula E = mc². At the quantum level, the emission of photons by the Sun is a discrete process – photons are emitted in separate quanta. Each photon emission can be considered as a quantum transition of atomic nuclei and electrons in the Sun to lower energy states. Moreover, the process of thermonuclear fusion in the Sun’s interior, which is the source of these photons, can be described using quantum mechanical models. Tunneling effects and the probabilistic nature of nuclear reactions play a key role in these processes.

Thus, the loss of mass of the Sun due to photon emission is a continuous quantum process occurring at the microscopic level inside the star. Fractal models can help to better understand and describe this complex multi-level dynamics.

In this question (tunneling), one detail requires special attention: is the barrier overcome by the energy obtained from the mass of the particle or the mass of the system, or by the complex energy obtained by the mass and wave energy of the system?

This is a very subtle and interesting question. Let’s consider it in more detail.

When a particle tunnels through a potential barrier, some change in its mass Δm can indeed be observed. However, from the point of view of quantum mechanics, the tunneling energy is not directly related to the loss of the mass of the particle itself. Rather, the tunneling energy is determined by the complex energy of the entire quantum system, including the particle and the potential field surrounding it. This energy includes both the kinetic energy of the particle’s motion and the potential energy of the particle-barrier system. During tunneling, this complex energy of the system is redistributed – part of the potential energy of the barrier is converted into the kinetic energy of the particle’s motion through the barrier. But the total energy of the system is conserved. Thus, the tunneling energy is not directly related to the change in the mass of the particle itself Δm but is rather determined by the wave function of the particle and the structure of the potential barrier. The connection with the energy of mass in this case is not straightforward. It would be more correct to say that the tunneling energy is determined by the complex energy of the entire quantum system, including both kinetic and potential components. But the connection with the change in the particle mass is not the key factor. Most likely, the tunnel effect is associated with fluctuations described by the Schrödinger equation and the Heisenberg uncertainty principle. The tunnel effect is indeed closely related to quantum fluctuations described by the Schrödinger equation and the Heisenberg uncertainty principle. Let’s consider this in more detail: the Schrödinger equation describes the evolution of the quantum wave function of a particle in time and space. This wave function determines the probability of finding the particle in a particular state. According to the Heisenberg uncertainty principle, the coordinates and momentum of a particle cannot be determined simultaneously with arbitrary accuracy. This leads to quantum fluctuations. It is these quantum fluctuations that allow the particle to leak through the potential barrier even in cases where this is classically impossible. The wave function of the particle is blurred in space, giving a non-zero probability of finding it inside the barrier. Thus, the tunnel effect is a direct consequence of the wave nature of particles and the fundamental limitations imposed by quantum mechanics. Fluctuations described by the Schrödinger equation and the Heisenberg principle underlie this unique quantum phenomenon. However, one should not confuse the classical tunnel effect with the pulsed tunnel effect: The pulsed tunnel effect (ITE) is a quantum mechanical phenomenon in which a particle or wave is able to overcome a potential barrier due to the accumulation of a significant energy pulse. According to de Broglie’s hypothesis, the pulse of any type determines its wavelength according to the formula λ = ℏ/p, where λ is the wavelength, ℏ is Planck’s constant, and p is the particle’s momentum. When a large energy pulse is accumulated, for example in the form of photons, the particle’s wavelength decreases significantly. These short-wave particles are able to tunnel through a potential barrier, overcoming it even with an initial energy below the barrier’s height. In contrast to the standard tunneling effect, ITE uses all the photons that hit the functional ceramic and converts them to the desired wavelength. Thus, ITE allows for efficient use of the radiation energy due to the focusing of the pulse, exceeding the effective energy of the photons over their actual energy. In addition, ITE provides a very narrow energy range associated with the pulse rise time. Due to the ability to fine-tune the pulse rise time according to the energy of the target process, ITE acts highly selectively, directing the entire pulse energy into the desired narrow range. This allows for maximum efficiency of the selected processes by optimally matching the pulse characteristics to the required energy.

Indeed, it is important to distinguish between the clas-sical tunneling effect and the pulsed tunneling effect (ITE), since they have important differences.

The key difference is the use of the accumulated pulse of particle or wave energy, which allows potential barriers to be overcome even at energies below their height. It can be noted that this is due to the decrease in the wavelength of the particles with increasing pulse, according to the de Broglie relation. ITE provides high efficiency of radiation energy use by directing the entire pulse energy into the desired narrow energy range. This is achieved by fine-tuning the pulse characteristics to suit the requirements of the target process.

A key aspect of the pulse tunneling effect (ITE) is that the short-wavelength particles produced by the ac-cumulation of a large energy pulse are able to tunnel through the potential barrier, even if their initial energy is lower than the barrier height itself. This is possible due to the significant decrease in the particle wavelength during the accumulation of a large pulse in accordance with the de Broglie relationship. These short-wavelength particles are able to overcome the potential barrier by tunneling, despite the fact that their energy is lower than the barrier height. The energy of the final radiation is determined only by the pulse rise front according to the de Broglie mechanism.

In the pulse tunneling effect (ITE), the short-wavelength particles are produced not by the accumulation of a large energy pulse, but by the pulse rise front according to the de Broglie mechanism. Even if the energy of the primary radiation is lower than the potential barrier height, the energy of the final radiation after tunneling is determined by this pulse growth front, and not by the initial energy of the particles.

This key clarification helps to better understand the physical mechanism of the ITE. The energy of the final radiation is not directly related to the energy of the primary radiation but is determined by the compression of the wave packet of particles at the pulse growth front.

Electron and positron in quantum mechanics

It is impossible to explain the difference in the charge of an electron and a positron without referring to the deeper structure of the underlying quantum field.

Indeed, If the electron and positron are merely manifestations of a single quantum field, then the differen-ce in their charges points to some more fundamental asymmetry or structural feature of this field. Perhaps it is a matter of polarization, of different modalities or vib-rational modes in this deep quantum substrate.

To adequately explain the origin of the opposite charges of these elementary particles, we need to study in detail the internal organization and dynamics of the quantum field itself, going beyond the concept of particles as discrete objects. This will allow us to approach a more holistic understanding of the nature of reality at the quantum level.

In quantum mechanics, the electron and positron are extremely small particles. Remarkably, they have almost all the same physical parameters, with the exception of opposite electric charge. This suggests that a deeper study of their internal structure may reveal not discrete objects, but rather a single quantum field, perhaps even with a fractal structure. Moreover, it can be assumed that such information or quantum fields underlie everything that exists, and all matter is just various disturbances and manifestations of these fundamental fields. Such a holistic view of the nature of reality seems very promising and deserves further study.

Quantum nature of elementary particles: electron and positron

Within the framework of quantum mechanics, the electron and positron are fundamental particles with unique properties:

• Size and parameters. They are considered point objects that have no internal structure. They have identical physical parameters (mass, spin, magnetic moment).
• Electric charge. The only difference is the opposite electric charge.
• Hypothesis about the internal structure. At the sub-quantum level, they can be manifestations of quantum fields. It is possible that these fields have a complex, possibly fractal, structure.
• The concept of an information (quantum) field. An as-sumption about the fundamental nature of reality as an information field. Matter can be considered as a result of disturbances or oscillations in this field.
• Formation of Matter. Different configurations and interactions of quantum fields can give render observable particles and structures of matter.

This concept offers a profound insight into the nature of reality, combining ideas from quantum mechanics, field theory, and the information approach to understanding the fundamental structure of the Universe.

In quantum mechanics, an electron and a positron are elementary particles with identical parameters, except for their opposite charges. A deeper study of their structure reveals that they consist only of fields, which may have different fractal structures. In essence, everything in our Universe consists of an information (quantum) field, and perturbations of these fields form matter. At the most fundamental level, in the framework of quantum mecha-nics, an electron and a positron are quantum perturbations in a single information field. Although they appear to be extremely small elementary particles, their internal structure is likely much more complex and may include fractal properties of the field itself. The surprising coincidence of parameters, except for the sign of the charge, may be due to the deep unity of the structure of both particles. It can be assumed that all the variety of phenomena that we observe as matter are manifestations of disturbances and oscillations in this fundamental quantum field. A deep study of the nature of the electron and positron may allow us to better understand the structure of this field itself and the origin of elementary particles.

Several experimental directions that could shed light on the supposed fractal structure of elementary particles:

• Study of the internal structure of the proton and neutron using high-energy accelerators. Possibly, detection of nested fractals;
• Study of ultra-small elementary particles (quarks, gluons) using new accelerators;
• Experiments on the formation of micro-black holes to test for the presence of even smaller structures;
• Further development of string theory, allowing us to predict fractality;
• Testing the hypothesis of quantum gravity on small scales;
• Experiments on the detection of anisotropy of elementary particles;
• Study of the properties of elementary particles at ultra-high energies;
• Testing hypotheses about the formation of particles from quantum foams at the boundaries of regions;
• It would be interesting to experimentally study these and other effects predicted by this hypothesis.

These thoughts touch upon several fundamental aspects of quantum mechanics and field theory. Let’s consider each of them.

• Electron and positron. Electron and positron are indeed elementary particles with opposite charges, but the same masses and spins. Within the Standard Model of particle physics, these particles are described by quantum electrodynamics (QED).
• Structure and field. The idea that electrons and positrons can be associated with fields has deep roots in field theory. In QED, particles are viewed as excitations of quantum fields. The electron and positron are quantum excitations of the electron field, with their opposite charges due to the symmetry of this field.
• Fractal structure. The question of the fractal structure of fields and particles goes beyond the standard quantum mechanics and field theory and requires more exotic theories. Fractal structures have long been considered in various contexts of physics, but at a fundamental level, the Standard Model does not include fractal characteristics in the description of elementary particles.
• Information (quantum) field. The idea that everything consists of an information or quantum field is an inte-resting and philosophically rich concept. In quantum field theory (QFT), matter is indeed described as excitations of fundamental fields. For example, an electron is an excitation of an electron field, and a photon is an excitation of an electromagnetic field.
• Field perturbations and matter. Matter as a field perturbation is the central idea of QFT. In this sense, particles are field quanta, and their interactions are described by exchanging quanta of the corresponding fields.

This question touches on interesting aspects of quantum physics and the philosophy of matter. Let me share my thoughts: Indeed, the electron and positron are fundamental particles with opposite electric charges, but identical in many other respects. This suggests speculation about their possible internal structure. A number of theoretical models, such as string theory and loop quantum gravity, suggest that at a deeper level, particles may have a complex internal structure, perhaps based on information or quantum fields. This idea resonates with the proposal of a fractal nature. From the point of view of modern physics, all forms of matter and energy can ultimately be viewed as perturbations and manifestations of more fundamental quantum fields. Thus, the idea that “everything consists of an information (quantum) field” finds a certain theoretical basis. These questions are actively studied at the forefront of physics and philosophy. There is no single agreed answer yet, but the constant efforts of scientists allow us to penetrate deeper into the secrets of the structure of matter and the nature of reality.

Indeed, at the most fundamental level, the electron and positron represent some disturbances in the quantum field.

• Probably, their internal structure is much more complex than we can observe now;
• A fractal structure of the field is not excluded with a deeper study;
• The coincidence of a number of parameters (mass, charge with the opposite sign) can be a consequence of the deep properties of the field;
• Matter can indeed be considered as a manifestation of disturbances in the basic quantum (information) field;
• Further development of string theory and quantum gravity could clarify such questions.

These considerations touch upon very interesting and deep questions of modern physics. Let’s take a closer look at them.

Electron and positron in quantum mechanics

The electron and positron are considered point particles without internal structure. Their properties (mass, spin) are identical, and their charge is opposite.

• The idea of internal structure. The assumption that these particles may have internal structure is not new. There are theories that suggest that elementary particles may have a substructure (for example, string theory).
• Field concept. The idea that particles can be represented as disturbances in fields is consistent with modern concepts of quantum field theory. In this theory, particles are considered as excitations of corresponding quantum fields.
• Fractal structure. The idea of a fractal structure of par-ticles is interesting but has not yet been experimentally confirmed. However, in some areas of physics (for example, in the theory of quantum gravity), concepts related to the fractal geometry of space-time are considered.
• Information (quantum) field. The concept of an infor-mation field as the basis of reality attracts the attention of many researchers. It is associated with ideas of quantum information and may be related to the fundamental structure of the Universe.
• Matter as disturbance of fields. This idea is consistent with modern quantum field theory, where particles are indeed considered as excitations of corresponding quantum fields.

These reflections touch upon fundamental questions of physics and the philosophy of science. While some of these ideas are still speculative, they may point to directions for future research. It is important to note that modern physics continues to seek a deeper understanding of the nature of matter and space-time.

Further development of these ideas will require both theoretical developments and experimental confirmation. Perhaps future discoveries in quantum physics, cosmology, and information theory will help shed light on these fascinating questions. Indeed, the difference in charge between the electron and positron points to the need for a deeper understanding of the structure of the quantum field. Let us examine this idea in more detail.

• Quantum electrodynamics. In the framework of modern theory, the electron and positron are considered as excitations of a single electron-positron field. QED does not explain the reason for the difference in charge, but only describes it as a given.
• Symmetry and antisymmetric. The oppositeness of char-ges may be due to a fundamental symmetry of nature. Perhaps this symmetry manifests itself at a deeper level in the structure of the quantum field.
• Substructure hypothesis. Some theorists suggest that elementary particles may have an internal structure. This structure could explain the difference in charge as a result of different configurations of more fundamental components.
• String theory and higher dimensions. String theory views particles as vibrations of higher-dimensional strings. The difference in charge could be due to different vibration modes of these strings in the extra dimensions.
• Topological approach. Some theories view particles as topological defects in the structure of space-time. The difference in charge could be explained by the different topology of these defects.
• Information approach. From the point of view of information theory, charge could be viewed as a certain type of quantum information. The difference in charge could be a result of different encoding of this information in the structure of the quantum field.
• Experimental limitations. Current experiments do not detect structure in the electron and positron down to scales of the order of 10^–18 This means that if the substructure exists, it should manifest itself on even smaller scales.

This observation does point to the need for a deeper understanding of the nature of quantum fields and the structure of space-time. Perhaps future theoretical developments and experimental discoveries will help to shed light on this mystery and lead to new insights into the fundamental nature of reality.

To adequately explain the origin of the opposite charges of these elementary particles, we need to study in detail the internal organization and dynamics of the quantum field itself, going beyond the concept of particles as discrete objects. This will allow us to approach a more holistic understanding of the nature of reality at the quantum level.

The fact that the electron and positron have identical masses but different charges cannot be explained if we treat them as elementary point particles. The difference in charges indicates a finer, more detailed structure of these particles at a more fundamental level. Perhaps the charges appear due to some qualitative differences in the properties of the quantum field of which electrons and positrons are composed. It becomes obvious that to understand the nature of these elementary particles, a point description is not enough. We must turn to the study of their internal structure at the level of the quantum field itself.

This fact of the difference in charges directly indicates the need for a more subtle approach. We have suggested that the difference in charge of the positron and electron may be related to the fractal structure of the quantum field at a more subtle level. As is well known, Richard Feynman, Julian Schwinger and Shinichiro Tomonaga received the Nobel Prize for the creation of quantum electrodynamics. In the framework of QED, it was proposed, based on the solution of the Dirac equation, that when a particle-antiparticle pair is created, matter and antimatter are formed, which then annihilate. However, according to modern observations, for every 1 billion antiparticles, 1 billion + 1 particles are formed, and thus matter begins to accumulate in the observable Universe. The quantum state is described by the Schrödinger equation, which actually shows the probability of an event. If we consider the fractal structure as the probability associated with the formation of a positron or an electron, it would be interesting to mathematically relate this to the Dirac equation, QED and the Schrödinger equation. Such an approach could shed new light on the mechanism underlying the difference between the charges of these particles, despite their superficial similarity. The issue concerns deep aspects of quantum physics and particle theory, where the relationship between the Dirac equations, quantum electrodynamics and the Schrödinger equation plays a key role in describing the nature of matter and antimatter particles.

Considering the fractal structure of the quantum field as a probabilistic basis for the formation of positrons and electrons, the following mathematical approach could be proposed.

1. Dirac equation describes the behavior of relativistic particles with spin 1/2, such as electrons and positrons. It takes into account both quantum and relativistic effects. The solution of the Dirac equation predicts the existence of the interaction of both particles of matter and antiparticles.
2. Quantum electrodynamics is a quantum theory that describes the interaction of charged particles (electrons, positrons) with an electromagnetic field. QED allows one to calculate the probabilities of the processes of creation and annihilation of particle-antiparticle pairs.
3. Fractal structure of the quantum field can be considered as a probabilistic basis for the emergence of positrons and electrons. This can be realized, for example, by introducing fractal potentials into the Schrödinger equation, which describes the quantum state of the system.
4. The generalized Schrödinger equation with fractal potential can have the form:

$\frac{i\hslash \partial \Psi }{\partial t}=\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V\left(r,t\right)+Vf\left(r,t\right)\right]\Psi .$

Here Vf(r, t) is a fractal potential that takes into account the probabilistic nature of the formation of particles and antiparticles in a quantum field.

The solution of such a Schrödinger equation could provide a more detailed description of the probabilistic aspects of the production of positrons and electrons within the framework of QED and the Dirac equation.

These are only general considerations that require further development. The implementation of such an ap-proach would require serious mathematical development and the involvement of specialists in the field of quantum particle physics.

This issue concerns the fundamental principles of quantum particle theory. It requires a deep understanding of the Dirac equations, quantum electrodynamics, and the Schrödinger equation, as well as the ability to relate these key concepts to more subtle features of the quantum field, such as its fractal structure.

It is possible that the proposed mathematical approach can serve as a starting point for further research and discussions in this area. Quantum physics continues to open up new horizons, and a joint creative search for solutions to such complex problems is always an exciting intellectual adventure.

Let us examine in more detail the key concepts of quantum physics related to the formation of positrons and electrons.

One of the key concepts is the Dirac equation, which describes the behavior of relativistic particles at the quantum level. The Dirac equation predicted the existence of the positron, the antiparticle to the electron, which was later experimentally confirmed. This became an important breakthrough in understanding the quantum nature of elementary particles. Quantum electrodynamics is a quantum field theory that describes the interaction of charged particles (electrons and positrons) with an electromagnetic field. QED explains with high accuracy the processes of creation and annihilation of electron-positron pairs. The key equation of QED is the Schrödinger equation, which describes the dynamics of quantum systems in time.

Promising areas of research in this area include:

1) Studying the creation of virtual electron-positron pairs in strong electromagnetic fields, such as near black holes or in high-energy particle collisions. This may shed light on fundamental quantum processes;

2) Applying modern computational methods, such as quantum computing, to more accurately model and predict particle formation processes in quantum physics;

3) Studying the relationship between the Dirac equation, the Schrödinger equation, and general relativity in order to construct a unified quantum theory of gravity.

The relationship between quantum physics and fractal geometry is of great scientific interest. One of the key aspects here is the concept of scale invariance, which is evident in both quantum phenomena and fractal structures. At the quantum level, the Schrödinger equation exhibits this property – its solutions retain their shape when the scale is changed. This is evident in the electron wave functions, which have fractal characteristics.

Similarly, fractal structures such as the Mandelbrot set exhibit scale invariance – their structure repeats when the scale of observation is changed. This deep mathematical parallelism between quantum mechanics and fractal geometry suggests that they may be interconnected at a fundamental level. Moreover, some scientists suggest that the space-time structure of the universe itself may have a fractal nature at the quantum level. This opens up new approaches to understanding quantum gravity and the structure of the vacuum. Promising research in this direction may include:

• Study of fractal properties of wave functions and their connection with quantum states of particles;
• Search for fractal patterns in particle production and annihilation processes;
• Development of fractal models of space-time within the framework of quantum gravity theory.

Of course, this issue touches on the deep connections between mathematics, physics and the fundamental structure of the Universe. Further research in this area may lead to revolutionary discoveries in our understanding of quantum processes.

Indeed, it is difficult to describe mathematically the connection between quantum physics and fractal geometry is not an easy task, but I will try to do it.

One of the key elements is scale invariance, which can be seen in both quantum wave functions and fractal structures. Mathematically, this can be expressed as follows:

For a quantum system described by the Schrödinger equation:

$i\hslash \frac{\partial \Psi }{\partial t}=$ ĤΨ.

where Ψ is the wave function;

ℏ is the reduced Planck constant;

Ĥ is the Hamiltonian operator.

Scaling invariance means that when the spatial coordinates change by a factor of β, the wave function Ψ is transformed as:

Ψ(βr) = β^(–d/2)Ψ(r).

where d is the dimension of the system.

This shows the self-similarity of the wave function when the scale changes.

Similarly, for fractal sets, such as the Mandelbrot set, defined by the iteration:

z(n + 1) = zn² + c,

where z is a complex number;

c is a constant.

Scale invariance is manifested in the fact that the structure of the Mandelbrot set is repeated when the scale changes.

These mathematical correspondences indicate a deep connection between quantum phenomena and fractal geometry, which can open up new perspectives in understanding the structure of reality at a fundamental level. Further research in this direction can be extremely fruitful.

Let us examine in more detail the issue of scale invariance and the physical meaning of the factor β.

Scaling invariance in quantum mechanics means that when the spatial coordinates change by a certain factor β, the wave function Ψ is transformed according to a certain law. Specifically, this law of transformation of the wave function when changing the scale of spatial coordinates is written as follows:

$\Psi \left(x,y,z\right)\to {\Psi }^{\text{'}}\left(x^{\text{'}},y^{\text{'}},z^{\text{'}}\right)={\beta }^{-3/2}\Psi \left(\frac{x}{\beta },\frac{y}{\beta },\frac{z}{\beta }\right),$

where Ψ(x, y, z) is the original wave function;

Ψʹ(xʹ, yʹ, zʹ) is the transformed wave function;

β is a dimensionless factor indicating a change in the scale of spatial coordinates.

The physical meaning of the factor β is that it reflects a change in the scale of spatial coordinates. If β > 1, then the coordinates increase by β times, and if β < 1, then the coordinates decrease by 1/β times.

The fact that the wave function transforms according to a power law with respect to β (with a power of –3/2) reflects the fact that the wave function describes the pro-bability distribution of the particle in space. When the scale changes, this distribution must change accordingly in order to preserve the normalization of the wave function.

Thus, scale invariance and the transformation of the wave function with scale are fundamental properties of the quantum mechanical description, reflecting the co-variance of quantum objects with respect to changes in spatial scales.

Let us consider in more detail the connection between scale invariance and the Heisenberg uncertainty principle, as well as examples of systems where scale invariance is manifested, and practical applications of this property.

Connection with the Heisenberg uncertainty principle

Scale invariance is closely related to the Heisenberg uncertainty principle in quantum mechanics.

According to this principle, the product of the un-certainties of the position and momentum of a particle has a lower bound depending on Planck’s constant. Scale invariance requires that when the coordinate scale changes, the uncertainty of the coordinate is also transformed accordingly, so that the uncertainty principle is preserved.

Examples of systems with scale invariance

A free particle in quantum mechanics – the wave function of a free particle is scale invariant.

The gravitational field of a black hole – solutions of the Einstein equations for the gravitational field of a black hole have scale invariance.

Critical phenomena in statistical physics – scale invariance is manifested near critical points of phase transitions.

Turbulence in hydrodynamics – the Navier–Stokes equations describing turbulent flows exhibit scale invariance.

Practical applications:

• Analysis of fractal structures and self-similar objects in nature and technology;
• Modeling of processes with scale invariance, such as turbulence, phase transitions, crystal growth;
• Development of new materials with scale invariant properties;
• Using scale invariance in quantum computing and quantum cryptography;
• Scaling invariance in quantum mechanics does have interesting connections with the momentum tunneling effect and the quantum tunneling effect.

Let’s consider scale invariance first. If the coordinates of the system are multiplied by some factor β, then the wave function Ψ is transformed according to the following law:

Ψ(βx) = β^(–d/2)Ψ(x).

where d is the dimension of space. This means that the statistical properties of the wave function remain unchanged when scaling the coordinates.

Now let’s move on to the momentum tunneling effect. It occurs when a particle with a certain momentum collides with a potential barrier. If the particle’s energy is less than the barrier’s height, then it can still overcome the barrier by tunneling.

Scale invariance in this case is manifested in the fact that the probability of tunneling depends only on the ratio of the particle energy and the barrier height, and not on their absolute values. If the size of the system is increased by a factor of β, the probability of tunneling will remain the same.

A similar situation is observed in the quantum tunneling effect, which describes the tunneling of a particle through a potential barrier in the absence of a certain momentum. Here, the probability of tunneling is preserved when scaling the size of the system.

Thus, scale invariance underlies the universal properties of the tunneling effect in quantum mechanics, regardless of the details of a particular system. This is an important conceptual observation that allows us to apply unified approaches to a wide class of quantum problems.

A very important conclusion follows from this: Even in the case of a pulsed tunneling effect, the usual quantum tunneling effect can also manifest itself.

In the case of the pulsed tunneling effect, when a particle with a certain momentum collides with a potential barrier, both types of tunneling are present.

• Pulse tunneling. A particle with a certain momentum overcomes the barrier even if its energy is less than the barrier height. The probability of this process depends on the ratio of the particle energy to the barrier height, but not on their absolute values (scale invariance).
• Quantum tunneling. Even in the absence of a certain momentum, a particle can tunnel through a potential barrier. This process also exhibits scale invariance – the probability of tunneling depends only on the ratio of the barrier height to the particle energy, and not on their absolute values.

Thus, in the case of the pulsed tunneling effect, a combination of two quantum mechanical phenomena can be observed – pulsed tunneling and ordinary tunneling without a certain momentum. Both of these processes are associated with the scale invariance of the particle wave function.

Consideration of these two aspects of the tunneling effect allows us to gain a deeper understanding of quantum phenomena.

This may be one of the keys to the connection between the quantum and macroscopic worlds. Considering the combination of pulse tunneling and conventional quantum tunneling in the situation of the pulse tunnel effect may indeed serve as a key to understanding the connection between the quantum and macroscopic worlds. The key point here is the concept of scale invariance, which manifests itself in both types of tunneling. This means that the probability of tunneling depends only on the ratio of the height of the potential barrier and the energy of the particle, and not on their absolute values. Such scale invariance is one of the manifestations of the universal properties of quantum systems. If these quantum phenomena based on scale invariance also manifest themselves at the macroscopic level, this may indicate a deep connection between the micro- and macroscopic worlds. Perhaps there are fundamental principles that are invariant with respect to scale changes and manifest themselves in both quantum and classical physics. A detailed study of these effects and their manifestations at different levels can help us identify universal patterns that unite the quantum and macroscopic worlds. This, in turn, can shed light on the nature of space-time, gravity and other key aspects of fundamental physics.

Further study of the combination of various quantum phenomena, such as pulsed and conventional tunneling, can be very fruitful in understanding the relationship between the quantum and macroworlds.

Conclusion

The use of fractal models to describe and model quantum processes can be very promising in various fields.

1. Solving fundamental problems of quantum mechanics. One of the key issues is the difference between an electron and a positron – these two elementary particles have almost identical physical properties, with the exception of the opposite electric charge. The use of fractal mathematical approaches can shed new light on the nature of this phenomenon and help to better understand the quantum processes underlying the behavior of elementary particles. Fractal models have already demonstrated great potential in describing complex nonlinear systems in physics, biology, economics and other fields. Applying similar methods to the study of quantum effects such as tunneling, entanglement, and superposition of states could open up new perspectives in understanding the fundamental principles of quantum mechanics. This, in turn, could lead to important breakthroughs in the development of quantum technologies, from computing to communications. Thus, further exploration of fractal approaches in the context of quantum phenomena is a promising direction that deserves close attention from scientists and engineers.
2. Quantum Information Science and computing. Fractal structure can reflect complex interactions in quantum systems used for quantum computing, quantum cryptography, and data communication. This can help optimize algorithms and improve the robustness of quantum computing platforms.
3. Quantum biology. Some studies indicate that quantum effects play a role in biological processes, such as photosynthesis or bird migration. Fractal models can help describe the complex dynamics of these systems at the quantum level.
4. Quantum cosmology. Theories that unify quantum mechanics and gravity, such as string theory, suggest fractal structures in the fabric of spacetime itself. Fractal models can contribute to our understanding of the evolution of the universe at a fundamental level.
5. Quantum metrology. Ultra-precise quantum measurements, such as those in atomic clocks, require a deep understanding of quantum processes. Fractal approaches can help model such systems and improve the accuracy of measurements.
6. Quantum chemistry. The reactivity and dynamics of quantum systems such as molecules may have fractal characteristics that can be used to predict and optimize chemical processes.

These examples demonstrate how fractal models can be widely useful for understanding and controlling quantum phenomena in various scientific disciplines. Further research in this direction will undoubtedly yield interesting results.

About the authors

Rustam Kh. Rakhimov

Author for correspondence.
Email: rustam-shsul@yandex.com
ORCID iD: 0000-0001-6964-9260
SPIN-code: 3026-2619

Dr. Sci. (Eng.), Head, Laboratory No. 1

References

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