(zenodo.org)

correction dated 13 May

The full version of the work is freely available at https://zenodo.org/records/15383291

The philosophical work that gave rise to the idea of the wave structure of matter is available at https://zenodo.org/records/15291534

Channel on dzen.ru where there are additional articles, not included in the main work, but indirectly related to the process of formation of this work, available at https://dzen.ru/id/66820d78f6faca1d3feac4b8

Introduction

In this chapter, a mathematical apparatus is constructed, according to which elementary particles can be described through standing waves with different number of nodes. This allows us to relate their properties to scaling in space and interaction through wave resonance between fractal levels

To understand and justify what an elementary particle in space is, the paper devotes Appendices 1, 2 and 3. Appendix 6 states the principle of scalability (fractality) in space at finite propagation speed of interactions.

Under elementary particles in this paper we consider only four known, stable particles — neutrino, electron, neutron and proton, as well as their anti-analogues. It is assumed that only four stable stable standing waves in space can exist.

Despite the fact that the free neutron is not stable, within the framework of this model it is considered as a standing wave, under certain conditions preserving its integrity. Its half-life can be explained by its peculiarity of energy rotation within its structure, which leads to entanglement in the birth of neutron and antineutron. Since it is subject to frequent interactions due to its sufficient mass, a change in the state of one of the pair leads to a change in the state of the other.

It is assumed that the rule should be preserved — the smaller the wavelength, the higher the mass value. The larger the number of nodes, the smaller the wavelength, since we have a limited speed of propagation of interactions, the correspondingly higher the mass. This is an analogue of energy quantisation in standing waves, as in a string: energy increases with the number of nodes, while the length shortens.

It is known from experience that of the listed particles the electron and the proton have charge. In this work they should correspond to standing waves with an even number of nodes. Proceeding from this and from the known from experiments masses of particles, it is possible to assume sequences of particles in the order of increasing number of nodes of the standing wave — neutrino, electron, neutron and proton. Here only the neutron is out of the general picture. It is considered that its mass is greater than that of the proton, which based on this paper should not be. But taking into account that it has in its structure the effect of internal rotation of energy, which, when measuring the mass, in fact, is similar to the manifestation of charge between charged particles, it can lead to a misinterpretation of the obtained result. I.e., the neutron mass obtained in the experiment may be overestimated, and the interaction of rotational energy may be added to it, which leads to an overestimated mass value.

Relationship between dimensions and weight

Initial data:

There are two interconnected waves — transverse and longitudinal. The cross section of their interaction is considered.

• The speed of interaction is c (the speed of light).
• The wave propagates not along one axis (x) (longitudinal wave), but along two axes — x (size, longitudinal wave) and y (energy/mass, transverse wave). The transverse component of the wave is considered as a carrier of amplitude information related to the energy density, which by the formula E= mc² allows us to pass to mass
• Since the interaction speed is a constant, the constant c² becomes important in this case, not just the propagation speed c, i.e. the sum of the squares of the speeds in both directions is important in this case.

Deriving dimensional and mass limits

If the wave is travelling simultaneously along the x-axis and y-axis, and the maximum possible velocity is c, then by Pythagoras’ theorem:

If we talk about a standing wave, it requires that all points of the wave can remain in interaction, i.e. the maximum distance between points must not exceed the path that the signal can travel in a given time. If we were to consider only the propagation of the wave along the x-axis, then at Δt≤1/c we would observe an interaction between the initial and final points of propagation. Since we consider propagation along two orthogonal directions, the interaction time limit is replaced by the condition Δt≤1/| c²||ₙᵤₘ. which corresponds to the interaction limit when both the geometric (x) and energy (y) components are fully engaged. Here and further I will omit the dimensions in order not to create confusion.

In this case, we can speak about the maximum size of standing waves in the region of space, which will take place at once for both types of waves — transverse and longitudinal components. Their size cannot exceed the size:

• The constancy of c² is a balance between size (geometry) and mass nature (amplitude, energy).
• The greater the mass (or amplitude), the smaller the allowable geometric dimension.
• Maximum size for the formation of a standing wave:

If the interaction propagates with velocity c and is decomposed along two axes, size (x) and energy/mass (y), the maximum size of the standing wave will be limited not only by 1/c but also by the mass contribution. The larger the mass contribution, the smaller the allowed size. And the constant becomes not c, but c².

The maximum size for standing wave like particles we have defined. It should not exceed 1/c. Now we need to determine the lower limit.

If we repeat this logic, taking 1/c as the unit size, we find that a size smaller than 1/c of the unit size will produce its own interconnected system of longitudinal and transverse waves. In such a case it makes sense to speak about the size 1/c² as the minimum possible for standing waves in the region of space.

I.e. the minimum possible limit for the existence of a standing wave in the form of a particle is possible at:

Hence standing waves (particles) must be in the size range:

Now we need to decide on the limits for mass.

It is no secret that photons, being electromagnetic waves, have some effective mass. So mass is a characteristic associated with the electromagnetic wave. It’s related to the y-axis. The electromagnetic wave itself propagates on a sphere, so the energy is related to the length of the circle and is related to the size in space through 2π. In this case we get (dimensionless):

The manifestation of energy in space is mass. We know that:

Then, we can obtain the range of possible values for the mass in the region of space as the amplitude of the longitudinal wave, getting rid of the signs of proportionality, since the mass belongs already to the region of space:

Here we see a value (constant) of 2π/c⁴, which is remarkably close to the value of Planck’s constant:

If:

Тогда:

The difference from the experimentally obtained value of Planck’s constant is connected with that at calculation of the total mass of a standing wave, similarly to GR, it is necessary to take into account that gravitation «feels» the integral picture of density. If in the gaps between compactifications there is a void, then instead of just summation of masses it is necessary to take into account redistribution of work (energy) in space. This will be discussed below. This imposes its imprint on the value of the experimentally obtained Planck constant.

Thus, for the structure of an elementary particle to be in interaction, it is necessary to take into account both the mass scale and the size scale, which are related by multiples of the degree of the speed of light. Thus, the maximum size should not exceed:

and the minimum mass:

The dimensions are omitted here, only numerical values are taken. It is important to take into account that a longitudinal wave in space is responsible for the size, and an electromagnetic wave propagating on the surface of a sphere, in a cross section — a circle, is responsible for the mass. This imposes an additional condition on the ratio between the radius of the particle and the length of the circle. Therefore, it is necessary to take into account the coefficient 2π.

Further considerations are given in Appendix 6. Briefly we can say that the organicity of the speed of light creates restrictions leading to resonance for oscillatory processes in space. Therefore, matter, being a result of standing waves is subject to the general resonance, which leads to the necessity of equality of geometrical work enclosed under each half-wave of standing waves.

If we assume that the oscillation profile is a sinusoid, the area under one half-wave can be defined as:

Where:

• M₀ is the amplitude of the wave, in our model corresponding to the mass;
• λ is the wavelength (characteristic size of one oscillation);
• S is the area, i.e. the work done to create the half-wave.

Thus, a one-half-wave job:

This means that size and mass are in a relationship. Since energy arrives in portions defined by standing wave nodes, the portion of size change must correspond to the portion of mass change.

Node quantisation and particle existence limits

Thus we see that the variation in both size and energy is in the range:

I.e. we have a step dependence on the speed of light, as well as it is necessary to take into account the coefficient 2π for the area of energy, as it is related to the circle in the area of space.

Standing wave nodes for a circle must occur when the velocities along the x-axis and y-axis are equal. This condition can be written as:

Thus, on a circle of length 2π we obtain exactly 4 directions in which the velocity components are equal — these are angles 45°, 135°, 225°, 315°. These points correspond to symmetric positions on the circle where the counter waves can cancel each other with the same phase projections along both axes.

Physically, this means that a standing node is formed only when the wave in its circular dynamics propagates energy uniformly along both spatial dimensions. If one of the velocity components becomes dominant (or equal to zero), either a purely translational wave (without nodes) or an unstable system incapable of stable self-consistent resonance arises.

This gives a natural quantum constraint:

In a two-dimensional structure on a circle, there are only 4 stable positions at which a standing wave can form.

Hence, no more than four stable standing wave types are allowed, corresponding to 1, 2, 3 and 4 nodes.

The appearance of the fifth node breaks the equality of the velocity components, and the energy starts to go in the longitudinal direction — forming not a wave but a mass contribution.

Quantisation and the transition to mass:

Thequantisation is by a value of 2π/| c|ₙᵤₘ covering the whole range of energy change. But since the dependence is expressed in terms of degree, the minimum quantum of change is defined as:

Here the reverse transition takes place — from size to mass. The fifth degree is taken because the 4 points correspond to standing waves in space, and the fifth step is connected with the birth of mass — that is, going beyond the wave state.

For the amplitude of the longitudinal wave| M₀|ₙᵤₘ, the wavelength | λ₀|ₙᵤₘ and the standing wave size d₀, we obtain the formulas:

Remark:
In this chapter, mass refers to the amplitude of a longitudinal wave. The total mass of a particle as a wave system consisting of several zones will be considered in the following chapters.

Extreme states: motion and spherical wave

In addition to the four types of standing waves corresponding to the number of nodes (n=1,2,3,4), there are two limiting cases: n=0 and n=5. These states represent extreme forms of energy change.

At n=0 no standing wave is formed, the whole interaction velocity belongs to the region of space. In this case we get electromagnetic waves. And under certain conditions — photons.

At n=5 the whole interaction velocity belongs to the energy field. In this case we obtain the phenomenon of energy compression, gravitational waves propagating in all directions. This state corresponds to the energy transition to the region with the size less than 1/c². Such a region exists inside each elementary particle and can be interpreted as an analogue of a black hole — a transition to a new, deeper level fractalisation of space. It is this region that can be considered as point matter and its behaviour can be described by wave functions, which was mentioned earlier. Its behaviour in space can be described by wave functions in space, because around it there is a region — a standing wave, due to which all interactions will have a wave character.

The case of n= −1 is also interesting. This boundary condition defines the maximum possible wavelength of a photon. Waves with lengths longer than this value cease to be directed quanta of energy and pass into the category of classical electromagnetic waves. Thus:

• The maximum wavelength of a photon:

• The minimum wavelength of a photon:

Photons can occur only in the presence of energy source motion — for example, when an electron transitions between levels. A localised change of energy state leads to the formation of a directed electromagnetic wave — a photon.

Photons do not have rest mass in the usual sense. However, due to a sufficiently high frequency, the electromagnetic wave manages to create a longitudinal wave, which, although not standing, still exists, which can be interpreted as equivalent to mass

At further decreasing of photon wavelengths they already represent closed structures, the size of which is commensurate with the size of their wavelengths. They behave more like particles. Their birth is possible due to the motion of protons and also due to the decay of elementary particles such as neutrons. This is the so-called gamma rays.

Photons with wavelength up to λ≈10⁻⁹ is not a closed figure, but a structure with a shell whose shape is close to a parabola. The spatial wave, propagating, forms a trace in the form of a transverse electromagnetic wave. Since matter is in resonance with the scale of space, the photon shell is actually stretched, interacting with the entire region bounded by the speed of light. Thus, the photon’s centre of mass does not coincide with its geometric centre, due to the curvature of its boundary. This curvature, asymmetrically distributing energy, can be interpreted as a manifestation of the photon mass.

From the above we can explain why photons have spin. Their boundary structure sets symmetry in the direction of motion. I.e. spin in this case does not characterise the rotation, it characterises the non-perfection of the surface. But along the axis of motion the photon can be twisted.

Features of elementary particles

If n=1, we get a neutrino. It will have the largest size, commensurate with the size of an atom. Minimal mass. All of which makes it minimally interacting. It will have an internal spinning of energy. It is created by adding a portion of energy from the dimension of space to the dimension of an electromagnetic wave. Since the electromagnetic wave propagates on a sphere, it causes the effect of energy rotation internally.

At n=2 we get the electron. Here the particle receives a second portion of energy from the space side, which removes the rotation and gives an understanding of charge. Charge characterises work on the space side. That’s why it remains a constant for elementary particles. Charge characterises the work on the part of the surrounding space to create one half-wave of a standing wave. As a result, we either have a rotation of energy inside or a charge.

At n=3 we get a neutron. The newly received portion of work again breaks the structure of the particle in favour of the electromagnetic component, which leads to the effect of energy rotation inside the particle.

At n=4 we get the last stable particle, the proton. The situation is similar to that of the electron. The resulting particle will be characterised by the presence of charge, but now of the opposite sign.

Here also it should be noted that particles are born in pairs so that the law of conservation of energy is not violated. Therefore it is necessary to note the birth of neutral particles. Neutron and antineutron have a rotation, which conditions their internal state and leads to the connection between the pair-born particles through the rotational momentum. This spin is important for the particles to maintain their stability and neutrality. They are born entangled with each other, which leads to interesting consequences.

Since neutron and antineutron are opposites that are formed simultaneously (in a pair), their states appear quantum entangled. A change of the state of one of these objects (for example, an external influence) will lead to a change of the state of the other, which is due to their energy interdependence.

The decay of the neutron can be explained through the mechanisms of its interaction with an external influence. If an external influence (e.g., a collision with other particles or a field) is imposed on the neutron, it will cause a change in its state. Since the neutron and antineutron form a pair with quantum entanglement, any impact on one of the particles must cause a change for the other.

The interrelation between neutron and antineutron by the principle of quantum entanglement can be the main mechanism that explains their decay under certain conditions. Standing waves with an odd number of knots create quantum entanglement between particles, which affects their mutual state and decay.

Neutrinos are also born quantum entangled. But they interact very weakly with the surrounding space, so they remain quite stable.

What elementary particles look like.

In the ideal case, elementary particles are a spherical object, within which alternate regions of altered energy density. They can be represented as a rubber ball, inside of which there will be alternating areas of compression/stretching. Inside the neutral particles there will also be an area of twisting along some axis, or we can say an area of whirlpool — a spiral of energy density.

But in practice, since the birth of particles is accompanied by changes in the distribution of energy density in the surrounding space, since the particle has a certain size, its boundary will be affected by different forces from the side of space, because of which there will be a curvature of the shell of the particle, which creates the phenomenon of spin, as well as to the appearance of interaction forces.

As already mentioned elementary particles will be of two kinds — charged and neutral. Neutral ones are characterised by internal rotation of energy. Charged ones at their boundary will have a region of space with either increased energy density or decreased energy density. Space expends some work to create each half-wave, and this work will be the same for all half-waves because space is unified and the particles are bound by resonance. The charge, as it was already mentioned in Appendix 3, it makes sense to interpret it as a value characterising the direction and magnitude of the work done by space in the formation of a standing wave. Thus, the charge characterises the work expended on the part of space, to create a standing wave, for one half-wave of the particle. It is for this reason that the charge will always have a constant value, independent of the speed of motion. Mass, on the other hand, being a characteristic related to the amplitude of the longitudinal standing wave, will always change with a change in size to keep the work constant. That is why when the velocity of an elementary particle changes, its size and as a consequence the mass of the particle will change. In the case of neutral particles, the internal rotation will also be a characteristic of the work on the space side. This internal rotation of energy in experiments on determination of mass of a particle will give additional share of mass at interaction with matter, as it will unlike charge show itself always uniformly. Charges have been determined due to their different types of interaction and so this is taken into account when determining mass. In the case of internal rotation of energy, it manifests itself always in the same way, therefore it is attributed to the total mass, because like gravitation it manifests itself always as a force of attraction.

Standing wave interference

Elementary particles are formed in such a way that their internal structure utilises the maximum possible speed of space interaction at a given resonance. This means that «penetration» of one particle into another is impossible without destroying their structure — as observed, for example, in nuclear reactions.

In this model, particles do not interact by passing through each other, but by exchanging energy due to the work of space, which tends to minimise the total energy stress of the system.

• Dissimilar charges are attracted because their interaction reduces the total work of space.
• Single-aimed — repelled to reduce the work required.
• Uncharged rotating particles exhibit an attraction and confinement effect due to energy momentum, and additional work is required to separate them.

Interference is possible only when the influence of the work of space is either minimal or has already been overcome. This is exactly what is observed in experiments with a single slit: the wave is not completely standing and passes through the slit interacting with minimal resistance of the medium.

Thus, interference is not simply the result of superposition of waves, but a consequence of specific conditions under which the wave structure and the workings of space come into resonance.

Work as the basis of charge: geometrical interpretation

In the previous chapters it was shown that elementary particles are standing waves in space formed under the influence of external energy. The space does work, creating an oscillatory structure — a standing wave with a certain number of nodes. This work in each elementary portion of the wave is expressed in the form of a half-wave. The question arises: what is the exact physical meaning of the work done in this process and how does it relate to the charge of the particle?

Geometric nature of the work

Let us represent an elementary particle as a standing wave consisting of n+1 half-waves. Each half-wave is a sinusoidal oscillation arising from the energy interaction of space with a local region.

If we assume that the oscillation profile is a sinusoid, the area under one half-wave can be defined as:

Where:

• M₀ is the amplitude of the wave, in our model corresponding to the mass ;
• λ is the wavelength (characteristic size of one oscillation);
• S is the area, i.e. the work done to create the half-wave.

Thus, a one-half-wave job:

Now we substitute into the above equations the formulas for calculation of amplitude and length of the longitudinal wave from the chapter «7.1.3 Quantisation of knots and particle existence boundaries», we obtain:

The last expression shows that the work on the space side will be an invariant quantity, independent of the number of nodes of the standing wave. This is actually expected, since the formulas for calculating the amplitude and length of the longitudinal wave were created on the basis of work invariance.

Charge coupling

Since the charge in the model is understood as a characteristic determining the interaction of a particle with the field, and the interaction is the result of energy work, it is logical to consider that the charge is proportional to this work. Only one boundary half-wave of the standing wave will be responsible for the charge, exactly it will create close interaction between particles. But there is also a very important point to consider. Everything that is smaller than the size 1/|c|ₙᵤₘ creates an internal interaction, but what exceeds this size — refers to the manifestation of interactions in space. Charge, by definition, must manifest its effects in external space. Therefore, the action of charge will only begin to manifest itself at distances greater than 1/|c|ₙᵤₘ. The charge must then characterise the work density in the volume of a sphere of diameter 1/|c|ₙᵤₘ. In such a case we can write:

The most interesting thing is that if we multiply the obtained value by the fine structure constant α≈1/137.03599, we get a value very close to the value of the elementary charge:

The electron charge is the geometric work density at the boundary half-wave multiplied by the manifestation factor of the interaction in space.

The electric charge is the structural work of space on the boundary half-wave normalised by the probability of exchange interaction with the external space, expressed through the fine structure constant α.

Since space is unified, it acts according to a universal principle — expending the same work for each half-wave, and hence for each elementary unit of charge. This explains why the charge does not depend on the number of knots, speed or other parameters: it reflects the very fact of space’s action to create a stable wave unit.

Nature of mass and calculation of its values through longitudinal wave amplitudes.

1. What is mass? The mass of a particle is a manifestation of a local increase of the energy amplitude of space. In conditions of resonance it can occur in the process of only portionally possible work. We can speak about manifestation of local increase of energy density. In the framework of the proposed model, each particle is formed due to a longitudinal wave in space, where density compression creates a region with increased energy. This region defines mass as a gravitational characteristic.

2. Basic Mechanism. In the past chapters, it has been shown that:

• For all elementary particles, the geometric work under one half-wave is constant and equal to 2c⁻⁵.
• The charge depends on what structure is formed at the particle boundary.
• The mass depends on the central energy density in the particle.
• For neutral particles the internal rotation of energy is also taken into account.
• For antiparticles, antigravity will be observed.

If the longitudinal wave forms a single zone with increased density at the centre (as in the case of an electron), the mass of the particle is directly proportional to the amplitude of the wave responsible for the compression:

where M₀ is the amplitude of the longitudinal wave, which is calculated by formulas from the chapter «7.1.3 Node quantisation and particle existence limits».

Explanation of the origin of the mean particle mass

In the model of a particle with several zones of increased energy density (for example, for the proton) we observe the following structure:

1. In the centre is the zone with maximum density, amplitude M₀.
2. Then — gaps with reduced density (conventionally «voids»).
3. At the edges are again zones with increased M₀ density.

If we consider the cross-section of the particle along one diameter (from the centre), we can see that the structure alternates: a zone of increased density — a zone of decreased density — a zone of increased density.

However, for mass estimation, it is important to consider that:

• Gravity «feels» the integral density pattern.
• If there is a void in the spaces between the seals, then instead of just summing the amplitude, we need to consider the redistribution of work (energy) in space

Transition to the mean distribution:

One can imagine that if the «void» between the seals is removed, the work previously concentrated on one half-wave (one density) must now be distributed over the whole wave. To keep the geometric work constant, the amplitude must halve.

Which is to say:

• Centre: density M₀.
• Outer part: density M₀/2.

As a result, we must add up to get the total mass concentrated in the centre of the resulting structure:

This is where the factor 3/2​ comes from to get the total mass of the elementary particle with 4 standing wave nodes, the proton.

Taking into account this approach to the calculation of mass, one can speak about the «modified» wavelength of the particle — that is, about the appearance of some average value of the wavelength due to the integral character of mass. In such a case, the de Broglie formula linking mass and wavelength will use a modified (averaged) constant linking these quantities. This is why the numerical value of Planck’s constant turns out to be slightly smaller than the calculated value of 1/| c⁴|ₙᵤₘ. Appendix 7 discusses «Assessment of possible distortion of fundamental constants due to integral effects». There it is shown that the error in calculations of elementary particle masses does not exceed 4 per cent, which corresponds to the level of deviation of the «corrected» speed of light, if we accept the experimental value of Planck’s constant.

Why is the contribution of lower density zones not taken into account?

Gravity responds to positive energy density. That is: gravitational attraction is created where energy is concentrated, not where it is «rarefied».

Although the phenomenon of antigravity for antiparticles has not yet been detected in experience, but from this paper it follows as an obligatory element, so similarly to the previous paragraph we can conclude the following:

Antigravity responds to negative energy density. That is: antigravity repulsion is created where energy is discharged, not where it is «concentrated»

In the equations of the general theory of relativity (GTR), mass is part of the energy-momentum tensor:

Tμν

And the gravitational field is defined where Tμν​ is positive, that is, where there is positive energy. Sparse areas where the energy density is less — they don’t create «negative mass», they just reduce the average density.

In other words:

• Density increase → active contribution to mass.
• Density reduction → passive influence: stretches density zones, affects «average mass» but does not create negative mass.

Reduced density does not «subtract» mass because:

• Gravity arises from the presence of energy densification, not rarefaction.
• The void only increases the «smearing» of the mass, making it more distributed, but does not destroy the densification itself in density zones.

In other words, the lower density zones do not add negative mass, they only reduce the average mass density in space. It is the redistribution that leads to the need to correct the amplitude in the calculations.

Antimatter and antigravity

For antimatter the structure is opposite: In the centre of the particle a region with reduced energy density is formed, which leads not to attraction but to repulsion, to antigravity.

• In matter: centre attracts → mass is positive.
• In antimatter: the centre repels → mass manifests as «negative» gravity.

This explains the natural «scattering» of antimatter and its rarity in the observable Universe.

Mass calculation for an odd number of nodes

Within the framework of the proposed model, mass is not a fixed amount of matter, but an energetic manifestation of structure and internal processes. The smaller the size of the standing wave, the higher the energy density, and hence the larger the mass. However, if the internal structure of a particle contains rotational elements, this leads to the concentration of energy closer to the centre, creating the effect of an increase in the «effective mass».

This rotation represents an internal circulation of the energy density in the wave structure, which can show up as an additional contribution to the mass in measurements, as it creates an additional gravitational force.

In order to obtain values of masses of neutral particles coinciding with the experimental ones, it is necessary to make a correction. At odd n there appears an internal rotation due to one of the half-waves, which goes into twisted motion, concentrating the energy closer to the centre of the structure. In the calculations, this half-wave is already taken into account as the maximum of the wave and contributes to the mass through the amplitude, but in the experiment the rotation appears as an additional energy density, which leads to an overestimation of the mass.

Thus, the mass obtained from experimental measurements includes not only the geometric component, but also the effect of rotation, which adds another «virtual» part of the mass.

It is important to note that the neutron mass has never been measured directly as a mechanical weighting. It is determined indirectly through energy balances in β-decay reactions, where the neutron decays into a proton, an electron, and an antineutrino. The mass calculation is based on the energy and mass difference of these products. In such processes, the internal energy and spin of the neutron are actually interpreted as part of its mass. This means that the experiment captures not only the geometric mass, but also the additional rotational energy — which exactly corresponds to the «virtual» mass in the framework of the proposed model.

To harmonise the theoretical and experimental masses, one can additionally increase the theoretical one by one minimal quantum of change, this will help to get closer to the values obtained in experience:

Thus, the rotation effect, already taken into account in the calculations as the maximum of the wave, additionally appears in the experimental measurement as an added mass associated with the internal energy density. This correction makes it possible to harmonise the calculated data with the experimental data without violating the logic of the standing wave.

For a neutron at n=3, we get:

M₀=3,142×10⁻²⁹×34,3343= 1,078783706×10⁻²⁷

which is already comparable to the experimentally measured value of the neutron mass.

Now similarly to consideration of the proton, we also need to take into account the redistribution of work on the space side. Here the picture is a little different. If we consider relative to the center, we will have, on the one hand, one positive half-wave with amplitude M₀, it does not need to be corrected. On the other hand, the positive wave and the center are separated by one half-wave with a negative half-wave. Here we must increase the wavelength by a factor of two and correspondingly decrease its amplitude by a factor of two as well. As a result, like the proton, we get 3M₀/2. But the neutron is smaller in diameter than the proton, which leads to an increased mass value in the experiment. That is, the neutron mass will be determined not less than the proton mass:

m₀=1,5 ×1,078783706×10⁻²⁷=1,618175559×10⁻²⁷

Features of neutrino masses

The case of neutrinos deserves special attention. According to the proposed wave model, the mass cannot be less than a certain lower limit due to the fundamental quantity 2π/c⁴. The neutrino possesses at least one half-wave, which can be interpreted as a contribution to the mass. Even ignoring the energy spin within the neutrino, this gives a minimum mass estimate of 2,664×10⁻³²kg. However, experimental estimates of the neutrino mass are much lower, up to 8,6×10⁻³⁷ kg. This discrepancy in experience is due to the fact that the neutrino has a significantly longer wavelength than other particles (about 10⁻¹¹ m versus 10⁻¹⁵ m for the proton). Since mass is associated with a specific point, matter, and β-decay always involves more massive particles like the proton or neutron, they are the ones that set the scale of measurement. Their wavelength determines the scale to which the neutrino is «bound» in the measurement. As a result, the mass associated with the wave structure of the neutrino is «smeared» on a much larger scale, and the perceived value decreases in proportion to the increase in scale:

which is within the experimental constraints. Thus, the proposed model does not contradict the observed data and explains why the neutrino mass appears to be so small: it reflects the energy amplitude distributed in the external space. This confirms the idea that the mass is not a fundamental quantity, but a consequence of the geometry of the wave configuration and its local concentration.

Measuring galaxy masses

In the case of galaxies, mass measurements are not direct either. Mass is determined from the rotational velocities of stars in the periphery and from the gravitational effects on surrounding objects. These methods automatically include the effects created by the dynamic structure of the galaxy itself, including internal flows and vortices. Thus, the mass estimate includes not only the «real» mass of matter, but also the gravitational equivalent of the rotational energy — similar to the neutron.

In this regard, when scaling from neutron to galaxy size (Appendix 8, the calculation is based on the formulas of the next chapter «7.10 Fractality of the Universe structure.»), it is necessary to use the corrected mass, taking into account the effect of internal rotation, to obtain a correspondence with the observed characteristics.

Fractality of the structure of the Universe.

In deriving the formulas, constraints for mass and for dimensions were introduced:

For the sizes smaller than 1/c² the interaction speed equal to the speed of light is not able to create wave processes with given parameters, in this connection there is a jump to another level, smaller. The limit of transition will be the value 2π/c⁴ (Planck’s constant), quantisation by mass and by size occurs. Quantisation of all interconnected quantities should occur multiples of this value. In this case, using the known expression:

You can rewrite it:

It should be emphasised that, within this model, the speed of light remains a fundamental constant at each particular level of fractality. However, due to differences in energy density between levels, the relative perception of the speed of interactions changes. This does not mean that c changes in the physical sense — only the scale of perception of interactions in different energy layers of the Universe structure changes.

The laws of physics and fundamental constants remain unchanged and identical at every level of organisation of matter. The speed of light determines the limiting speed of electromagnetic interactions, depending on the energy density at a given scale.

Thus, the scaling of spatial-energy characteristics leads to natural portioning of energy at all levels of the matter structure. This explains why energy is transferred discretely, and Planck’s constant, in its turn, turns out to be not a fundamental constant, but a consequence of the structure of wave interaction.

With this approach, the Universe acquires a potentially infinite fractal structure. In mathematical expression, the variable n has no strict restrictions and can take both positive and negative values. At the moment, the fundamental principles imposing restrictions on n are unknown.

The relative frequency at all levels remains the same, but the speed of light changes

If the frequency remains common to all levels, it means that resonance is possible between them.

Resonance is key because:

1. Interaction between levels — if two levels share a common frequency, they can exchange energy even if their interaction rates and wavelengths are different.
2. Information transfer between scales — this explains why the structure of matter at different scales retains common patterns. For example, galaxies, atoms and elementary particles may obey the same wave laws.
3. Scale effects — since the rate of interactions changes, but the frequency remains unchanged, we can assume that on large scales (e.g. at the level of galaxies) space behaves like a quantum system, but with other values of fundamental constants.

The wavelength is related to these parameters by the equation:

If we denote the parameters at our level as c₀,νand R₀​ , and at another level as c,νand R, then, since the relative frequency does not change, we are left with:

Wavelength ratio:

If we assume that the scaling dimension R is proportional to the wavelength λ, then

Then the relative scale will be:

Whence it follows that the size ratio between levels is defined as:

Thus, quantisation is expressed in stepwise change of speed of light that automatically sets scale transitions and energy density.

Theorem (on fractal energy scaling)

Prerequisites:

1. Matter possesses a fractal structure, which scales in multiples of the number n, where n∈ Z is the level of matter nesting (fractality).
2. Basic physical constants — such as the speed of light and mass — scale according to the wave structure of space, which depends on the energy density at level n.
3. The energy density and the rate of interactions at each level are related by a resonance relation, and energy transfer occurs discretely — in portions.

Formulation:

The energy of the structure at the level of fractality n is expressed through the scale wave:

Appendix 8 discusses the calculation of neutron parameters at the larger fractal level, its mass and size parameters are very close to the known parameters of the Milky Way.

Influence of the fractal structure of space on the mass of particles

The formulas for calculating the size and amplitude of standing waves obtained in chapter «Quantisation of knots and particle existence limits» are valid only for idealised conditions when the average amplitude level in space is homogeneous. However, space has a fractal structure: wave amplitude fluctuations appear at different scales. In such conditions, the resulting mass of an elementary particle can vary depending on the energy region of space in which it is located.

This effect is confirmed by the experimentally observed «trembling» of the mass of particles placed in strong fields: the mass can both increase and decrease, indicating its sensitivity to external energy conditions.

We are inside the arm of a spiral galaxy — a region with increased concentration of matter. In the framework of this model, this means the presence of an enhanced energy background, which, in turn, should lead to a change in the mass of elementary particles compared to the calculations performed for ideal conditions. This is confirmed by the analysis: the experimental masses do not coincide with the theoretical ones.

In addition, the relative size of the particle plays an important role. The closer the particle diameter is to the 1/c² limit, the larger is the contribution of background fluctuations to its mass. Thus, for the electron (whose diameter is much larger than 1/c²), the error between the calculated and experimental masses is only 0.44%. For the neutron, which has the smallest diameter among the particles considered, the discrepancy reaches 3.59%, whereas for the proton (whose diameter is larger than that of the neutron) it is 3.46%.

It was also shown in chapter «Quantisation of knots and particle existence limits» that stable standing waves occur in only four variants in which the condition is satisfied:

Not only the geometrical constraint, but also the sign of the discrepancy between the calculated and observed masses follows from this relation. Thus, for particles with nodes 1 and 2 (neutrino and electron) the calculated mass is larger than the observed one. For particles with nodes 3 and 4 (neutron and proton), the opposite is true: the calculated mass is smaller than the measured mass. This can be interpreted as a result of the phase position of the standing wave relative to the background — for the upper half of the circle (positive y-axis) the background energy contribution can be subtractive, while for the lower half (negative y-axis) it can be additive.

Frequency as a basis for perception and interaction

Initially, we proceeded from the postulate that the speed of light is constant, and this is what supposedly sets the structure. But if we look deeper, resonance is not related to speed, but to frequency. Perhaps the speed of energy propagation depends on the frequency of the wave, not the other way round.

Analogy with water: one medium — many frequencies

Let’s imagine a bathtub of water. It is possible to excite different waves in it:

• Mechanical vibrations are slow but large.
• Sound waves are faster and shallower.
• Electromagnetic oscillations are even faster and more compact.

All of them exist in the same medium, but at different levels of frequency and speed.
They do not interact with each other if they are not in resonance, and the multiplicity of the speed of propagation of interactions can act as an indicator of resonance.

We exist at certain frequencies — and everything we observe is in resonance with us. And what is not in resonance simply does not exist for us.

Hence the logical conclusion:

The world can be much more complex and diverse, but due to the fact that different waves can affect each other only under certain conditions, for which the frequency of the wave is responsible, we cannot observe everything happening around us. These parallel worlds can theoretically be infinitely numerous. The speed of interaction (speed of light) does not have to be boundary or static in principle. It remains constant only for a particular world of frequencies.

Conclusion

In the presented work, a mathematical model is constructed in which elementary particles are considered as standing waves formed as a result of resonance between different fractal levels of space. The key position of the model is the statement that the mass and size of a particle are interrelated parameters depending on the energy distribution in space. The energy, according to the obtained relations, comes in portions, which provides discreteness of possible states.

The work spent by space to create one half-wave is either to create a rotation of energy density inside the particle or to create a charge — a uniform gradient of energy density from its boundary to the centre. Charge is similar to the effect of gravitational force, but inside the particle. Since space is one and everything is connected by resonance, the work to create one half-wave in all particles is the same. That is why the charge is a constant value, independent of the speed of motion.

The model shows that the maximum size and minimum mass of a possible standing wave are limited by fundamental values related to the speed of light. In this case, the constancy of the value of c² (and not just c) acts as an invariant of the interaction between the size and the mass component, ensuring the stability and closedness of the wave structure.

A formula linking mass, interaction rate and energy, depending on the level of fractality:

allows not only to explain the fractality of the observed Universe, but also to describe it as a nested structure, where the same laws operate at all levels — while preserving the frequency of interaction and changing the speed of wave propagation.

This approach provides a natural explanation for the origin of mass, spin, charge and quantum entanglement, treating them as consequences of the resonant wave nature of matter. In addition, it explains the large-scale similarity of the structure of matter — from elementary particles to galaxies — as a manifestation of the same principle of resonance operating at different levels.

Thus, the proposed model unites the physics of the microcosm and macrocosm within a single fractal-wave structure, where each scale is a reflection of the other, and the observed diversity of forms is the result of nesting and resonance in a single energy field.

Appendices

Appendix 1. Longitudinal waves of energy in the mechanism of electromagnetic wave propagation

Introduction

Usually electromagnetic waves are considered as transverse waves that do not require a medium for propagation. However, the very fact of their finite propagation speed makes us wonder: what happens to the energy in space during the wave’s motion? If energy cannot change instantaneously, it means that at some point in space there must be a gradient of its change. And if so, it is logical to assume that an electromagnetic wave generates not only transverse but also longitudinal fluctuations of energy.

In classical physics, the electromagnetic wave is described as strictly transverse. However, in the framework of the present model, which takes into account the finite propagation speed of interactions and the energy structure in space, it becomes necessary to introduce a longitudinal component associated with the energy density gradient. This does not cancel the classical picture, but extends it by describing additional subtle effects.

P1.1 Energy gradient and its implications

When an electromagnetic wave leaves the source, it changes the energy density in the surrounding space. As it propagates, the energy at the source point gradually returns to its original value, but at a distance R it remains changed. This means that there is an energy gradient between these points, which can propagate as a longitudinal wave. It is not considered in classical physics, but this does not mean that it is absent. Rather, its influence may be too subtle to be easily noticed.

We can put it another way: space, being energetically homogeneous at the beginning, becomes inhomogeneous after the wave passes through. And if the energy is redistributed, it means that there is a process that can be described as a longitudinal oscillation.

P1.2 Relationship to the de Broglie hypothesis

De Broglie introduced the idea that particles have wave properties, but did not give a specific mechanism of their formation. If a standing wave is indeed the basis of an elementary particle, then it is logical to ask: what exactly creates this wave? If an electromagnetic wave generates longitudinal energy fluctuations, then it is these that can stabilise the standing wave, making it a particle. Thus, a particle can be not just an abstract probability wave, but a real structure in space caused by wave processes.

P1.3 Conclusion

We are used to thinking that an electromagnetic wave is an exclusively transverse process. But if we consider it from the point of view of changes in energy density, it becomes obvious that transverse oscillations cannot exist in isolation. They inevitably cause longitudinal redistribution of energy, which opens new possibilities for understanding physical processes. Perhaps it is longitudinal waves of energy that will help explain not only the propagation of electromagnetic waves, but also the structure of particles and the nature of some unexplained phenomena. This does not require a revision of the whole physics, but gives an opportunity to look at the known processes from a different angle.

Appendix 2. Rethinking the Michelson-Morley Experience

One of the most famous experiments in the history of physics, the Michelson-Morley experiment, is traditionally interpreted as proof of the absence of the aether. However, what if this experiment does not reject the existence of the medium at all, but merely points to its unusual properties?

Albert Einstein created a very beautiful and coherent theory. It really describes everything very well. It is absolutely correct, so it cannot be refuted. It has only one drawback — it is difficult to understand and to describe. But it’s absolutely true. It also contains a postulate about the invariability of the speed of light, which is not explained in any way. And it uses the Lorentz transformation, which is also introduced as an axiom. These are flaws in the theory, which introduce a limitation on the understanding of the world.

The Michelson-Morley experiment (1887) was the decisive factor at that time. The wave nature of elementary particles had not yet been discovered at that time. At that time there was a wave theory of light, but the corpuscular-wave dualism had not yet been formulated. The very fact that light can behave as a particle began to be discussed later — at the beginning of the XX century.

Einstein, developing the special theory of relativity (1905), took as a basis the negative result of the Michelson-Morley experiment; his approach was to reject the ether concept rather than to consider the wave nature of matter. At that time it was accepted that light was an electromagnetic wave (according to Maxwell), but in a sense it was considered a separate nature from matter.

The reason why Einstein did not view everything as waves is that at that time there was neither experience nor mathematical apparatus to support this view. Corpuscular-wave dualism only began to take shape after 1924, when Louis de Broglie hypothesised the wave properties of matter. And quantum mechanics, which explained it, did not develop until the 1920s and 1930s.

Now, knowing the wave properties of matter, I propose to revisit the result of the Michelson-Morley experiment.

The Michelson-Morley experiment (1887) was conceived to discover the ether, a hypothetical medium in which light was then assumed to propagate, just as sound propagates in air.

Purpose of the experiment: To test whether there is a so-called «ether wind» — the movement of the ether relative to the Earth, which should affect the speed of light in different directions.

Basic idea: If the Earth moves through the ether:

• Light travelling along the Earth’s motion must travel the path in one time.
• Light travelling across has to travel the path in a different time.
• This should cause interference when comparing the two beams.

Michelson and Morley built an interferometer to compare the two beams.

The Michelson-Morley experiments did show small deviations, but:

• Small shifts of the interference pattern were recorded, about 10 times smaller than expected in the presence of an ether wind with the Earth’s velocity (about 30 km/s).
• These deviations were not of a magnitude or direction that could be explained by the motion of the Earth through the ether.

How the result was interpreted:

• Michelson and Morley considered these small deviations to be within the errors of measurement, and not evidence of an ether wind.
• The work was later repeated with increased precision (including by Michelson in the 1920s, as well as by Miller), and although Miller claimed to have found the effect, his results were not confirmed in other experiments.

I.e. experience has disproved the existence of ether. In the light of what we know from physics, we speak about particles as some point that behaves in space, obeying wave functions. But what if we extend the concept of a particle? Let’s give the particle a certain size in which the standing wave will be located. The size is fine, it does not contradict the known data in physics, but with the understanding of the standing wave there are uncertainties associated with the absence of a medium. In the previous chapter it was shown that the electromagnetic wave, due to the limitation of its propagation speed, must create a longitudinal wave, otherwise the limit of the propagation speed of electromagnetic interactions is violated. In this connection particles can be represented as longitudinal standing waves of energy in space.

If particles are waves of energy, then all matter is nothing but waves of energy in space. As we know, waves do not transport matter, but only energy. In this case, the Michelson-Morley experiment is not capable of creating and registering an «ether wind», since all we know is that it is waves of energy, in which case we can only study the energy itself. Of course, the presence of waves implies the presence of the medium, but since all we know is waves, we cannot influence the medium itself.

If we accept that everything in the world, including matter, observer, devices and light are forms of wave processes of energy in the structure of space, then the very question about the movement of the medium loses its meaning. If the medium is everything, there is no external background against which its motion could be fixed.

Even if this structure «moves», we, being a part of it, cannot measure it — because our metre, our clock, and light itself obey the same internal laws. Everything «floats» together, and there is no relativity within this structure.

This means that the negative result of the Michelson-Morley experiment does not at all exclude the existence of a structure in which waves propagate — it only shows that we cannot observe the absolute motion of this structure if we ourselves are waves of energy.

Conclusion

The Michelson-Morley experiment does not prove the absence of medium, but only shows that the Earth is not capable of creating an etheric wind because it is itself composed of waves. If elementary particles are waves and matter is their aggregate, then all matter is a wave structure.

That means:

• The speed of light remains the same because the speed of wave propagation is independent of the source.
• For construction of mathematical apparatus in physics, the ether is not necessary, as physics at the present moment is able to study only energy and its change.
• The forces of nature are wave processes of energy change because they are directly related to matter, which is a wave structure of energy.

The Michelson-Morley experiment confirms the wave nature of all matter, in which light and elementary particles simply follow the laws of wave propagation. That also speaks about impossibility to fix the motion of the medium from inside, if the matter itself is a wave form in this medium.

Appendix 3. Standing waves of energy and structure of elementary particles

Introduction

Modern physics describes elementary particles as point objects or perturbations of quantum fields, but another interpretation is possible. This chapter considers the structure of elementary particles as a longitudinal standing wave of energy, and their properties can be explained through de Broglie waves.

We will also consider how particle birth can be explained within this model and why the law of conservation of energy leads to the symmetry of matter and antimatter.

P3.1 Standing waves of energy density and particles

To date, science does not describe the origin of electric charge. What is its nature? Why is its value constant for elementary particles? Charge does not change its sign or its value regardless of the environment. Why does a change in velocity change the mass of a body, but nothing happens to the charge? What can be the cause of charge?

It is known that:

1/c = αћ/e²

where

e is the charge of the electron,

ћ is the reduced Planck constant,

c is the speed of light.

One constant is expressed through another constant.

Can be reviewed at the link (http://nuclphys.sinp.msu.ru/misc/constants.htm).

The physical quantity that is the inverse of speed is called pace. The pace shows how long it takes you to cover the desired distance.

This parameter is clearly a characteristic of an elementary particle, not an interaction, but a characteristic of a particle. It is clearly related to the rate from the speed of light. Question — what and how is overcome in the particle? What can we talk about?

It is necessary to understand what and where it is moving and why the tempo can take values «+1», «-1» and 0 (I have divided the tempo byαћ/e², because it is all a constant, but I have kept the sign of the charge, it will be very useful for us), let it be only numbers that tell us only about the direction. I.e.: «+1» means something is travelling somewhere at the speed of light in one direction, «-1» is movement in the opposite direction, and «0» is as if there is no movement in either direction.

As was shown in Chapter 7.1, the electromagnetic wave, being a transverse wave, also generates longitudinal waves of energy during its propagation. These two waves are interconnected. One propagates in the dimension of space, the other in the sphere. Energy fluctuations will occur in independent coordinates. It can also be said that the planes in which these two waves oscillate are orthogonal to each other, which means that to describe the overall process, both of these processes must be taken into account. The two processes will be related, but will be described in their own coordinates.

Since the particle is isolated in the dimension of space, it is logical to assume that it is a standing wave in the dimension of space. The particle is a standing wave, longitudinal, born by the propagation of an electromagnetic wave.

Let’s consider a few possible cases. First option:

Fig. 1 The first elementary particle, the neutral, neutrino

The wave has one node. The wave has run once up, and once down. The result is zero. This is the first and neutral particle. Let’s assume it’s a neutrino. If there is a standing wave node at the centre of the circle, the particle will be neutral. With an odd number of nodes the particle will be neutral, with an even number of nodes the particle will be «charged».

Fig. 2 Electron and positron

I will assume that this pair of particles is an electron and a positron. For the electron, the wave has travelled down twice and up once. Two of the beams are negative and one is positive. If we add them up, we get the tempo.

I think this is what characterises the tempo in the case of elementary particles. For the electron, it’s -1. For the positron, it’s +1.

Thus, the charge is related to the standing wave’s tempo. For standing waves, the tempo will be a number integer from the array {-1, 0, 1}. That’s why it is a constant. This is only true for standing waves, which can exist indefinitely. The important thing to understand is that it is not the electromagnetic wave itself, but the energy distribution wave in the region of space that the electromagnetic wave created during its propagation. In the following chapters it will be explained that in conditions of resonance, each half-wave of elementary particles will be geometrically responsible for the same work on the part of space to maintain it, which just so can later explain the constancy of charge, as well as its properties.

Let us interpret electric charge as a quantity that characterises the direction and magnitude of the work done by space in forming a standing wave. Thus, the charge is an internal characteristic of the wave structure due to the ratio of longitudinal and transverse energy fluctuations.

The characteristics that will eventually change are the radius of the circumference of the particle, the number of nodes or bundles, and the amplitude of the wave.

Now I will not explain about the principle of comparing the number of nodes and the name of particles, in the sequel I think it will become clear.

It is worth paying attention to the fact that for matter, «charged» particles in the centre always have a positive value of change in energy density on the space side. The amplitude of energy on the side of space increases, relative to some initial level, we can say there is a process of «compression». For antimatter, it is always negative. The energy of electromagnetic wave is spent on decreasing the amplitude of energy on the side of space, there is a process of «stretching» of space. You can see the relationship between tempo and charge. I have omitted the figures. The main thing is to understand the physics of the process. The increase of energy amplitude can be considered as a process of increasing energy density on the side of space, and the process of decreasing amplitude can be considered as a process of decreasing energy density on the side of space. Further in the text it is the phrases increase of energy density and decrease of energy density that will be often used.

Consider the following possible particle. Suppose that it is a proton and an antiproton:

Fig. 3 Proton and antiproton

It is known that the proton consists of three quarks in the case of the simple model. But also in some experiments and theories the proton can be considered as a system of five quarks. The five-quark (5-quark) model is used for a more accurate description of the internal structure and properties of the proton, but does not cancel the 3-quark basis. This is clearly seen when considering the structure of the proton as a wave of changing energy density. Quarks in this case, are nothing but half-waves of curvature of space, half-waves of change of energy density in a region of space. Therefore quarks by themselves cannot exist, the meaning of a standing wave is lost. These half-waves are different, three of one sign and two of the other. In this case, it turns out that the proton is made up of five quarks. The concept of quark is very convenient for describing the interactions within the atom, will help to describe the sublevels for electrons, will describe the absorption and release of energy in the form of quanta, at the transition of electrons on sublevels, as well as allow you to accurately calculate and describe the processes of nuclear reactions.

For «charged» particles, some gradient of density change in the region of space is formed at their boundary. The energy density will either increase or decrease. Hence there is an effect of interaction of «charged» particles. Like-minded particles repel, like-minded particles attract. This can be explained by the fact that the total energy density of the surrounding WORLD will try to minimise the distortions created by the density of particles at their boundary.

One should not also forget what the «charged» particle carries in its centre. This is very important. The difference of behaviour of matter and antimatter depends on the last factor. For matter in the centre of the particle there will be an increased energy density in the space area compared to the surrounding WORLD, which will lead to the gravitational effect and macro-objects will be created. In the case of antimatter the process will go the opposite way, all particles will try to «move away» from each other as far as possible. But at the same time antimatter will try to group at some distance from the matter particles. This will happen along the sphere. Later it will be clear that antimatter will accumulate along the boundary of the event horizon of the black hole of a larger scale.

A fuller description of these processes, as well as the birth principles of existing interactions, can be found in the philosophical work https://dzen.ru/a/ZoINoVGeh12vIl-E or https://zenodo.org/records/15291541 .

P3.2 De Broglie waves as the basis of particle structure

In the framework of the hypothesis of standing waves of energy density particles can be considered as nodes of such waves. The de Broglie wave associated with a particle does not simply describe its motion, but is its structural element. The de Broglie wavelength is determined by the relation:

Where:

• h is Planck’s constant,
• m is the mass of the particle,
• c is the speed of light.

If a particle is a standing wave, its size must correspond to an integer number of half-waves, which explains energy quantisation.

There was an article written on this topic https://dzen.ru/a/Z7GBl8tL9DbB2x5L or https://zenodo.org/records/14883086 .

P 3.2.1 Results of calculations for the proton:

At the speed of light v=c the relativistic mass goes to infinity and the de Broglie wavelength tends to zero. However, if we consider a limiting approximation, for example, 0.9999999999999c, we can get a finite value. Let’s do the calculations.

Results for the proton at a velocity of 0.9999999999999c:

• De Broglie wavelength: 1.32×10⁻¹⁵ м
• Number of half-waves along the diameter (2r): 2.55

What’s interesting:

The number of half-waves remains finite. Moreover, in the variant proposed in the previous chapter, the proton is a standing wave with four nodes, which means that the number of half-waves should be equal to 2.5, which is very close to the calculated value.

So there’s a good chance that the de Broglie wave is nothing but a standing wave of energy in space.

If the de Broglie wavelength does describe the spatial structure of a standing wave, then:

The wavelength is not the electromagnetic wavelength, but the wave structure of space itself.

P3.3 Fractal structure of particles

If matter is viewed in terms of waves, we can assume the existence of similar structures, but with different scales. The phenomenon of resonance can do this. I.e. we can assume a fractal structure of the Universe. This means that at different levels of fractality particles can look the same. For example, if a neutron is a standing wave with three nodes, then a similar structure may appear on larger scales, such as the size of a galaxy. In the following, a formula will be presented that may be able to describe this process.

P3.4 Particle birth and the law of conservation of energy

P3.4.1 How particles are born

In the framework of the proposed model the birth of particles can be represented as a process of local redistribution of energy density. When in vacuum there arise fluctuations of energy density, they can lead to formation of stable standing waves, which are perceived as particles.

The birth of particles is accompanied by the formation of matter and antimatter particles. This follows from the law of conservation of energy: any local fluctuation must be compensated by an equal and opposite fluctuation.

P3.4.2 Why there is no symmetry breaking of matter and antimatter

It is usually considered that in the Universe there is an excess of matter over antimatter, but in the framework of this model symmetry breaking is not required. If a particle is a standing wave of energy density, its antipode may be a wave with opposite phase. The difference between matter and antimatter may lie in where the maxima and minima of energy density are located.

In a confined space, the redistribution of energy density will be due to the structure of the particles themselves:

• In the centre of matter particles (with an even number of nodes) there is a region with increased energy density, which leads to the effect of creating macro-objects and the emergence of gravity.
• In the centre of antimatter particles (with even number of nodes) there is a region with reduced energy density, which leads to their scattering from each other and formation of antigravity effect.
• Neutral particles are standing waves with an odd number of nodes. There is no density change at their centre, but they have the property of rotation. The antiparticle in this case differs only in the direction of rotation.

Antimatter is not capable of forming macroobjects because of the peculiarities of its structure. While matter particles tend to combine and can lead to the formation of black holes, antimatter is probably not capable of forming atoms more complex than antihydrogen. Instead, it would be distributed in a sphere around the forming black hole, contributing to the balance of energy in space.

Appendix 4: Lorentz transformation: classical derivation and wave interpretation

Introduction

Lorentz transformations play a key role in relativistic physics, describing how the spatial and temporal coordinates of objects change as they move. They were originally introduced as a mathematical tool to explain the invariance of the speed of light, but later became the basis of the special theory of relativity. However, if we consider particles as wave structures, we can arrive at the same transformations through a simple geometrical interpretation.

P4.1 How did Lorentz get his transformation?

The problem Lorenz was solving

Until the end of the 19th century, it was assumed that light propagates in the ether, a hypothetical medium that fills all space.

But the Michelson-Morley experiment (1887) showed that the speed of light does not depend on the motion of the Earth. This was strange: if the Earth moves through the ether, why does not the speed of light change?

Lorentz proposed that objects moving through the ether contract in the direction of motion.

P4.2 Length of the rod in a moving system

Lorentz imagined a rod of length L at rest.

• When it is stationary, its length is simply L.
• When it is travelling at velocity υ, the time at different points on the rod will not be the same due to relativistic time.

To find the new length L′, he used Lorentz transformations (named later in his honour).

P4.3 Output of length reduction

Consider two observers:

• One in a resting system (where the length of the rod is L).
• The second in a moving system (the observing rod moving with velocity υ).

Step 1: Write down the coordinates of the rod ends

• In the rest frame, the ends of the rod are at x₁ and x₂. Then its length is:

L=x₂-x₁

• In a moving system (a frame of reference moving with velocity υ) the coordinates are transformed by Lorentz formulas: x′=γ(x-υt)

Step 2: Calculate the length of the rod in a moving system

Let us write down the new coordinates of the rod ends:

Subtract:

since x₂− x₁=L, we obtain:

Where:

• L is the length of the object in the resting system,
• L’ is the length in the moving system,
• υ is the speed of motion,
• c is the speed of light.

P4.4 Formal mathematical reasoning

Using his coordinate transformations, Lorentz considered two observers:

• One is in a resting system where the rod has length L.
• The second moves with the rod and measures its length as L’.

He derived the length conversion based on the assumption that the speed of light is constant in any frame of reference and more time is required for a signal travelling in a moving system.

P4.5 Wave interpretation of length contraction

P4.5.1 Particle as wave

Instead of considering the particle as a point object, suppose it is a wave structure whose propagation in its own frame of reference is spherical at the speed of light c.

In this case, the particle has two velocities:

• The speed of wave propagation inside the particle (radial), which is c at rest.
• The velocity of the particle itself in space υ.

P4.5.2 Geometric relationship of velocities

If a particle moves, its internal wave structure changes. Since the total velocity cannot exceed c, the radial component of the velocity υ_sub decreases:

Hence:

Since the size of a particle is determined by its internal wave, a reduction in this velocity means a reduction in the effective radius of the particle:

Since the length of an object is proportional to its radius, we get:

which corresponds exactly to the Lorentz formula!

P4.5.3 Conclusions

• Lorentz transformations can be obtained not only through mathematical postulates, but also through the wave nature of particles.
• Length contraction is a natural consequence of limiting the speed of light to internal motion in a particle.
• The wave interpretation makes relativistic effects understandable in terms of the physical structure of particles, not just axioms.

Appendix 5: Energy of a particle as a closed wave structure and the law of conservation of energy

Introduction

Modern physics considers elementary particles as objects possessing both corpuscular and wave properties. In relativistic mechanics the energy of a particle is defined by the expression:

This formula shows that the energy of the particle grows with increasing momentum. However, if the particle is a closed wave structure of an electromagnetic wave and a standing wave in space created by the propagation of the electromagnetic wave, then its energy must be conserved within the system. This leads to an important question: how does a change in velocity affect the internal structure of the wave?

P5.1 De Broglie wavelength and energy redistribution

According to de Broglie’s hypothesis, a moving particle possesses an associated wave with length:

where h is Planck’s constant and p is momentum. An increase in velocity leads to an increase in momentum, and hence to a reduction in wavelength. This means that when a particle accelerates, its wave structure shrinks, changing the energy distribution within the system itself.

P5.2 Particle as a closed object

If a particle is a wave structure localised in space, its energy should not change, but only redistribute. Then:

• For an external observer, the energy of the particle grows due to the growth of momentum.
• Inside the particle system, the energy remains unchanged, changing only its configuration.

If the momentum increases with increasing velocity, the second term must decrease so that the total energy balance remains unchanged. This means that the de Broglie wavelength contraction is not just a consequence of motion, but a mechanism of energy redistribution inside the particle.

P5.3 Derivation of the energy equation

If we assume that the energy of a particle at change of its speed of motion is conserved, then in this case there should be its redistribution between wave components in space and along the sphere. We again obtain the equation of the circle, similar to the equation in the derivation of the Lorentz transformation:

Let’s rewrite the second summand:

Then:

Let’s take out m₀²υ²c² in the first summand:

Now let’s solve the fraction:

Then:

Thus, we end up with the same result as the standard relativistic expression, but we emphasise the splitting of the energy into two contributions:

• One depends on speed and resembles kinetic energy,
• The second reduces the internal energy of the particle as it accelerates.

If we consider the equation for energy in the form:

Then it becomes clear why the photon formally has no rest mass. It would be more correct to say that the summand responsible for the effective mass:

will tend to zero.

P5.4 Consequences

This conclusion confirms that the energy of the particle does not change at acceleration, but only redistributes:

• The first term expresses the momentum-dependent kinetic energy.
• The second term decreases the internal energy of the particle as the velocity increases.

Thus, the motion of a particle can be considered as a process of deformation of its wave structure, which naturally leads to relativistic effects without changing the total energy of the system.

Appendix 6. Wave resonance and scaling. The speed of light as a boundary of physical interactions.

Introduction

Modern physics operates with a number of fundamental constants, among which Planck’s constant h occupies a special place. However, if we consider the process of scaling physical quantities through resonant waves, we can assume that Planck’s constant is not an independent quantity, but is derived from the speed of light and geometrical characteristics of wave processes.

P6.1 Relation of wave processes and scaling

Many physical phenomena are based on resonance. If we consider standing waves at different scales, we can identify their common patterns. One of the key factors is that when scaling the wavelength, the number of nodes is preserved, while the frequency changes inversely proportional to the scale.

The speed of light plays here the role of a fundamental parameter determining the interaction of waves. It is important to note that the interaction velocity remains constant, but it can be decomposed into two components:

• along the x-axis (spatial scale that defines the size);
• along the y-axis (energy-related oscillation frequency).

This leads to a fundamental relationship between the size of the system and its frequency response.

P6.2 Wave resonance and scaling

Resonance occurs when wavelengths or multiples of wavelengths coincide, forming standing waves. In the fractal structure of nature this means that:

where k is the scaling factor between levels.

But an important point: mass and frequency are related, and we know that mass is expressed through the curvature of the wave.

We know that the energy of a single quantum is expressed through the wavelength:

Now let’s find the energy density.

If the energy is distributed over a spherical volume, then:

(Since for a spherical wave, the characteristic scale is the cube of the wavelength).

Then the energy density:

Now let’s look at the density:

If resonance occurs between two levels n and n-1, their energy densities must be matched:

Where R is the transition coefficient between levels. From the resonance condition:

Then:

We substitute λₙ=kλₙ₋₁:

That is, the transition coefficient is related to scaling as:

Since k is related to wavelength, and wavelength depends entirely on the speed of light, it is logical to assume that k is also related to the speed of light.

P6.3 Calculating Planck’s constant from the speed of light

Let R be the characteristic scale of the system associated with wave processes and k be the coefficient associated with the scaling frequency. Then we can write:

Taking k=|c|ₙᵤₘ, we obtain:

Substituting the exact value of the speed of light:

This value is extremely close to Planck’s reduced constant ℏ=h/2π, suggesting that it is not an independent fundamental constant but is determined through the speed of light and the geometry of wave processes.

Если:

Then:

Of course, at the moment this is just an intuitive understanding of where Planck’s constant comes from. So far it’s just numbers, no dimensionality has been taken into account, but here we were deriving a general coefficient, which should be dimensionless by definition. But I think eventually all the constants can be expressed through the speed of light and geometry — in this case the connection comes from the geometry of the circle, the sphere. Since everything around is a wave process, only the speed of light and the π number are likely to be constants. All other constants are likely to be their derivatives.

Since the value of Planck’s constant was measured experimentally, it may contain some error. Taking this into account, we propose to consider the numerical value 1/|c|⁴ as a more accurate basic coefficient from which ℏ can be derived.

Conclusion

Thus, Planck’s constant can be related to wave resonance and scaling through the fundamental interaction speed — the speed of light. This gives a new understanding of quantum effects, connecting them with macroscopic regularities of wave systems. Further investigation of this hypothesis can shed light on the nature of quantum phenomena and the role of scaling in fundamental physics.

We can approach this question from a slightly different angle.

The space in which we live may have no limits, but there is a fundamental limit — the speed of light c. Even if space is infinite, the limiting speed of propagation of interactions imposes a natural limit on the processes occurring in it. This leads to the fact that physical systems cannot exist on arbitrary scales, but must obey certain resonance conditions.

P6.4 Four-dimensionality through a point

In all further calculations |ℏ|ₙᵤₘ means exclusively the numerical value of Planck’s constant abstracted from dimensionality. This allows one to establish possible geometrical relations between fundamental constants through the structure of space, without attempting an exact recalculation of physical units.

We usually say that the world is three-dimensional. However, there is another dimension, or another coordinate, which does not appear as a coordinate in the usual sense. This point is the centre of mass of the system. It plays a key role because:

• All matter interacts through centres of mass.
• Any system localises energy at a point, but the point itself has no size or space.
• Quantum mechanics confirms that energy collected at a single point is not bound to a particular scale of space.

Thus, we can speak about the fourth fundamental dimension, which determines not the coordinates, but the very principle of the organisation of matter.

P6.5 How do you describe a point as a scale centre?

In conventional physics, coordinates are given in 3D space (x,y,z), but if a point is not just a coordinate but a dynamic centre, then:

• The entire energy density must be expressed with respect to the distance to the centre r.
• The scaling of the energies must take into account not just the volume r³, but the interaction through the point itself.

Assumption: the energy around a point is distributed not just in three-dimensional space, but in such a way that an additional term appears in the scaling.

P6.6 Density of states around a point

Usually the density of states is expressed in terms of the volume of available phase space.

In 3D space:

But if a point sets the centre of scaling, then we must consider that:

• The density of states «grows» towards the centre, but cannot become infinite.
• This means that an additional degree of scaling is added due to the interaction through the centre of mass.

Conclusion: if a point plays the role of a fundamental centre, then the possible states scale as:

P6.6 How does this relate to the speed of light?

We know that r is related to c in the fundamental equations because distances are given through the rate of interactions.

If:

then we substitute that into the density of states:

The density of states in the space with a centre point scales as 1/c⁴, which coincides perfectly with what we derived for ℏ! Here again we should note that the dimensionality is not taken into account here, but the proportionality factor is found. That is, we are only talking about the numerical value itself.

Combining the results of these two approaches, we obtain that the scaling factor is proportional to 1/c⁴. Since the constant bar is obtained experimentally, there are reasons to believe that some error may have been made in obtaining it. The value of the speed of light in this sense will be more correctly trusted.

P6.7 Restriction through the speed of light

Since space does not impose rigid boundaries, the only thing that limits physical processes is the limiting speed of interactions.

Wave processes in nature are always subject to the constraints of the medium. For example, sound in a pipe can only exist at certain wavelengths, and electromagnetic waves in a waveguide are also limited by geometry. But if our only limitation is the speed of light, then:

• It sets the natural scale of interactions.
• Any wave system must scale so that the limiting constraint is preserved.

In such a case the density of states of the wave process should depend not on the volume of space, but on the limiting velocity c.

Conclusion

• Space may be infinite, but the limitation of the speed of light creates natural limits to possible interactions.
• A point (centre of mass) is a fundamental dimension that determines the scaling of physical processes.
• The density of wave states in a system bounded by the speed of light leads to a law that is directly related to the speed of light as the fundamental limit of interactions.

Thus, quantum and gravitational effects may simply be a manifestation of a fundamental speed limit on interactions

Appendix 7: Calculation of particle parameters and experimental data.

We perform the calculation for all possible standing wave nodes and compare with the experimental data.

We use formulas from the chapter «7.1.3 Quantisation of knots and bounds of existence of particles»:

P7.1 n=1 (neutrinos):

Taking into account chapter «7.7.6 Particularities of the neutrino mass», a correction for the measured mass in the experiment has to be introduced.

m₀≈6,581×10⁻³⁷

P7.2 n=2 (electron):

For the electron, the total mass is formed by only one centre half-wave, so nothing needs to be corrected.

m₀≈9,14429412189293×10⁻³¹

P7.3 n=3 (neutron):

For the neutron, the mass correction is not so obvious, so we calculate using the formula given in chapter «Calculating the mass for an odd number of nodes».

We increase the amplitude by one quantum and then multiply by a factor of 3/2, related to the geometry of the half-wave arrangement:

m₀≈3,139629376693084×10⁻²⁹×34,3343×1,5 ≈1,616954653622901×10⁻²⁷

P7.4 n=4 (proton):

We have discussed the derivation of the proton mass in chapter «7.7.2 Transition to the mean distribution».

m₀≈1,077969769081934‬×10⁻²⁹×1,5 ≈1,616954653622901×10⁻²⁷

The experimental data was taken from the internet. Based on the data obtained, a table was drawn up:

P7.5 Assessment of possible distortion of fundamental constants due to integral effects

An interesting coincidence is observed when comparing the results obtained using the proposed model of mass and wavelength dependence with the classical de Broglie formula. If we use the obtained wave amplitude from calculated data and substitute it as mass in the de Broglie formula, then despite the difference in theoretical approaches, the numerical values of wavelengths are identical if we substitute not the experimental Planck constant but its expression through fundamental quantities in the form of the de Broglie formula:

This may indicate that the experimentally obtained value of Planck’s constant is an averaged value depending on the measurement methods in which the integral properties of mass, including its gravitational component, are manifested.

If the mass of an object appears as an integral effect in the gravitational interaction, some part of the wavelength may be «averaged» or unaccounted for in calculations based on gravitational characteristics. This leads to a distortion of the result: the wavelength is shortened, which is equivalent to an increase in the speed of light in the inverse calculations. If we take the form of Planck’s constant proposed in this paper as:

then, using the experimentally determined value of h, it can be seen that the corresponding value of the speed of light would have to be about 312054866 m/s, which is about 4 per cent higher than the generally accepted value of 299792458 m/s.

Given the construction of the mass calculation formula:

it turns out that the change of mass is due to a multiple increase due to the degree of the speed of light. Thus, if the value of the speed of light differs from the true value by 4%, the calculated mass will have at most the same error. This is consistent with the data for proton and neutron, for which the discrepancy is 3-4%. The coincidence of error levels may indicate the systemic nature of the distortion arising in the transition from the wave nature of mass to its gravitational manifestation.

Since most of the experimentally obtained values — mass of elementary particles, electric charge, wavelength — directly or indirectly rely on Planck’s constant, it can be assumed that all these values also include the effect of integral averaging. This makes possible the existence of a closed system of physical measurements in which an error in one of the fundamental quantities is translated to the others. As a result, we obtain a consistent but potentially inaccurate picture of physical parameters, which may be based on a hidden structural error.

As for the wavelengths and sizes of elementary particles, there are still many uncertainties. Until recently, the size of the neutrino was considered to be smaller than 10⁻²² m, and only recently has its quantum extent been obtained: ≥ 6.2 picometres (6.2 × 10⁻¹² m), which is already comparable to the calculated size of 9.715 × 10⁻¹¹. To compare the values calculated here, it is the size of the quantum extension that will be best suited. Exactly it defines the wave structure of the particle, its size.

Here are links to recent studies that discuss the quantum extent of neutrinos:

1. Nature (2025): A lower limit on the spatial extent of the neutrino wave packet, 6.2 picometers, has been established for the first time. This value is much larger than the size of an atomic nucleus and reflects the quantum mechanical nature of neutrinos, where «size» refers to the spatial uncertainty of their wave packet rather than a physical measurement. InFocus Mail+1IXBT+1
2. Phys.org: The study confirmed that the spatial width of the neutrino wave packet is at least 6.2 picometres, which is thousands of times the size of an atomic nucleus. This discovery has important implications for understanding the quantum properties of neutrinos and could influence the development of more efficient neutrino detectors. phys.
3. CERN Courier: The BeEST experiment using unstable beryllium-7 nuclei has set the limit on the spatial localisation of the neutrino wave packet at 6.2 picometres. This value is more than 1000 times the size of an atomic nucleus and provides new constraints on the quantum properties of neutrinos. cerncourier..

Appendix 8: Scaling between the neutron and the Milky Way.

Based on the equation given in chapter «7.8 Fractality of the Universe structure.», the parameters of the neutron analogue will be calculated on a larger scale. The obtained values will be compared with the known parameters of the Milky Way galaxy. As Planck’s constant and neutron parameters the values obtained on experience are taken so that they better agree with the data obtained on experience for galaxies.

The size of the Milky Way:

• Diameter: estimates range from 100,000 to 120,000 light-years (about 30-37 kiloparsecs). znanierussia.ru
• Thickness: about 1,000 light years. techinsider.ru

The mass of the Milky Way:

• Total mass: estimates range from 1 to 2 trillion (10¹²) solar masses, including dark matter. ru.wikipedia.org
• Mass of the stellar component: about 50-60 billion (5-6 × 10¹⁰) solar masses.

P8.1 Size scaling

When going from the neutron to the Milky Way, the level changes towards a lower frequency, i.e. n=-1. Then the scaling of the radius is as follows:

The size of a neutron is about 1 femtometer (fm), which is equivalent to 10-¹⁵ metres elementy.ru.

Substituting the values:

The diameter of the Milky Way in metres:

• Minimum estimate: ≈ 9.46×10²⁰ m
• Maximum score: ≈ 1.14×10²¹ m

The calculated radius of the Milky Way within this model differs slightly from the values accepted in astrophysics (~ 1×10²¹ m). This may be a consequence of several factors:

1. Experimental error in determining the size of both neutron and galaxy.
2. The effect of speed of movement on the size of objects, which is important to consider when comparing scales.
3. The calculation methods in astrophysics are based on models of the expansion of the Universe, which may introduce additional deviations. In the future we will consider the question of how correct it is to take into account the expansion when determining the sizes of objects.

P8.2 Mass scaling

If the mass of a neutron is (ru.wikipedia.org):

That’s the mass of the galactic analogue of the neutron:

Milky Way mass derived from observations:

• Lower estimate: 1.99×10⁴² kg
• Upper estimate: 3.98×10⁴² kg

The mass was slightly less than expected (~ 3×10⁴² kg). This may be due to several factors:

1. Measurement errors arising in the determination of both neutron and galaxy masses.
2. Dependence of mass on speed of motion, which can play an important role when comparing objects at different scales.

P8.3 Analysing the results obtained

The calculated values of the radius and mass of the Milky Way galaxy, obtained on the basis of the fractal coefficient, showed a surprising correspondence with the data of modern astrophysics. The radius calculated using the formula is R=6.626×10¹⁹ m, which is comparable to the observed value of about 1×10²¹ metres. The mass obtained considering the fractal coefficient is M=1.109855×10⁴² kg, while astrophysical estimates give a range of (1.99-3.98)×10⁴² kg. These results confirm that the proposed calculation method takes into account the fundamental principles of matter and space.

However, the question of the accuracy of current measurement methods remains important. In quantum physics, the mass of particles is determined through interaction with fields and depends on the environment. If spatial structures have fractal properties, this can influence the measurement results by introducing systematic errors.

The results obtained indicate that the current methods of mass and size estimation both at the microscale and at the level of galaxies may need to be revised taking into account the fractal structure of the Universe. This opens prospects for refining experimental data and for a deeper understanding of the fundamental processes that shape the world at all scales.

P8.4 Scaling the speed of light

This shows that the speed of light — the limiting speed of electromagnetic interactions — is much smaller at the level of galaxies than at our scale, corresponding to a more rarefied state of energy.

P8.5 The Fractal Structure of the Universe: Galaxies as Elementary Particles

Obtained calculated data on the Milky Way using standard physical formulas with a slight modification and using again known in physics constant — Planck’s constant, can not be a simple coincidence. Of course, it would be possible to assume that it is just a coincidence, but if it was observed with only one parameter. But the fact that both parameters (mass and size) practically coincide already excludes the occurrence of coincidence. There is a strong possibility that the WORLD is fractal. The fact that for the Milky Way the fractalisation formula worked with astonishing accuracy suggests that the Milky Way is an analogue of the neutron. It’s a very good match. This can now be used to study and describe the space around us. The Milky Way galaxy can be taken as a reference.

It remains to be seen how to explain the large number of different kinds of galaxies. Find out whether they are all analogues of elementary particles, or whether some of them arise as a result of their joint interactions.

P8.6 Analogy between spiral galaxies and neutrons

The Milky Way and the Andromeda galaxy have similar masses but different sizes. This may be due to their speed: at lower speeds the galaxy becomes larger, and at higher speeds it becomes more compact. This difference explains the observed differences in size and mass. Moreover, the sizes of galaxies are determined by the visible matter, which will only be observed if the elementary particle is accelerated. If there is no acceleration, it will be difficult to track the size of the particle, as visible matter may not be present, or may not be present in sufficient quantity to determine the true size of the structure. Spiral galaxies are of particular interest because their structure and mass distribution obey certain regularities. When considering galaxies formed as standing waves with an even number of nodes (charged particles), it is difficult to determine the real size of the structure formed. It will consist of alternating regions with increased and decreased energy density. In the regions with increased energy density there may be matter. Interactions with such structures can lead to the formation of various types of galaxies, which are only indirect manifestations of the basic structures.

The cases of spiral galaxies with larger masses than the Milky Way are interesting. For example, ISOHDFS 27 is a spiral galaxy with a mass four times that of the Milky Way, but its size has increased only slightly. This behaviour may indicate that its mass increases by a multiple of the neutron (proton) mass, while its size changes only slightly. This is already reminiscent of nuclear interactions: ISOHDFS 27 is an object similar to a helium nucleus, where the energy density is higher and the mass increases by a multiple of the neutron mass.

P8.6.1 Compact dwarf galaxies and electrons

While spiral galaxies can be compared to neutrons or their interactions with protons, compact dwarf galaxies can be analogues of electrons. The electron can be thought of as a standing wave with a region of higher energy density at its centre. It is in this region that matter can form, which is what is perceived as compact dwarf galaxies. It is worth keeping in mind that the size of an electron refers to its effective size, as determined in scattering experiments. Interestingly, the difference between the size of the electron and neutron nucleus is three orders of magnitude. If the size of the Milky Way is estimated to be 10²⁰ — 10²¹ metres, the size of the electron’s analogue should be of the order of 10¹⁷ — 10¹⁸ metres, which corresponds to the size of compact dwarf galaxies.

P8.6.2 Formation of additional galaxies

There are many galaxies in the Universe, which may not be analogues of elementary particles, but may be the results of interactions between elementary particles of galaxy sizes. As a result of interactions can be formed zones with increased energy density, and given that it will also be accompanied by acceleration processes, it will lead to the emergence of matter in areas with increased energy density, which will give birth to galaxies. Such galaxies may seem to be independent objects, but probably they are just a consequence of redistribution of energy between more fundamental structures.

Thus, analyses of the sizes and masses of galaxies, as well as their interactions, can provide insight into the fundamental structure of the Universe and its analogy to the microcosm.

This approach offers a new perspective on the nature of mass and its relationship to electromagnetic processes. A more detailed discussion of this hypothesis and its philosophical implications can be found in the following works:

— (dzen.ru)

— (Zenodo)