(zenodo.org)

Introduction

In this paper we construct a mathematical apparatus according to which elementary particles can be described through standing waves with different number of nodes. This allows to connect their properties with scaling in multidimensional space and interaction through wave resonance between fractal levels.

The wave nature of elementary particles

Here under elementary particles we understand only long-lived particles: neutrino (standing wave with one node), electron (two nodes), neutron (three nodes), proton (four nodes), as well as their antiparticles.

Standing waves are formed on the basis of wave resonance between different fractal levels. In the previous article (https://zenodo.org/records/15094660) it was shown that scaling occurs in multiples:

where| ℏ|ₙᵤₘ is the theoretically calculated analogue of Planck’s reduced constant, and c is the speed of light. This suggests that the scaling of wave resonance occurs as a function of size by a multiple of the fourth power of the speed of light. We also know that wavelength is related to mass through the same Planck constant. So size and mass are related to each other through the same numerical value.

Relationship between size and weight

In order for the structure of an elementary particle to be in interaction, it is necessary to consider 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 the longitudinal wave in space is responsible for the size, and the electromagnetic wave propagating on the surface of a sphere, in the cross-section — a circle, is responsible for the mass. It is worth noting that in the structure of elementary particles there are two types of waves: longitudinal (spatial) and transverse (electromagnetic). This imposes an additional condition on the ratio of the radius of the particle to the length of the circle. Therefore, the 2π coefficient must be taken into account.

This means that size and mass are in a relationship. Since energy arrives in portions defined by standing wave nodes, a portion of the size change must correspond to a portion of the mass change. The scaling structure shows that one size scale contains four mass scales.

Node quantisation and particle existence limits

From the scaling we can predict that the maximum possible number of elementary particle nodes does not exceed 4. At zero number of nodes the motion occurs only in space, and at five nodes all energy goes into mass, which leads to the disappearance of size.

The quantisation is by the value of 2π/c, but since the dependence is expressed in terms of degree, the minimum quantum of change is defined as:

Here there is an inverse process of transition of size into mass, on it the root of the fifth degree from the characteristic of space — size is taken, that at five knots all energy of space has passed into energy of mass.

In such a case, a relationship can be constructed for the mass:

Calculation and experimental data

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

The obtained values agree quite well with the experimental data, although there are small deviations. The most surprising thing is that if we compare the obtained wavelength and calculate it based on the mass obtained in the calculation and de Broglie’s formula for the speed of light (only take the refined value of Planck’s constant 7.757×10⁻³⁴), the wavelengths will coincide, although the calculation is carried out using different formulas.

Extreme states: motion and spherical wave

Besides the known four standing states of elementary particles (n=1,2,3,4), there are two limiting cases: n=0 and n=5. These states represent extreme forms of existence of energy, when it either completely passes into directed motion, or propagates in all directions, but does not form a localised mass.

At n=0 the standing wave is not formed and all energy passes to the space dimension. This means that the particle, as an object with mass, does not arise, and energy acquires directed motion. This is how photons exist — a pure electromagnetic wave travelling at the speed of light.

At n=5, on the contrary, the energy is not concentrated in motion, but is completely distributed over the sphere, creating a pure electromagnetic wave propagating in all directions. In this case, there is no dedicated direction of motion in space as there is for a photon. An example of such a state are electromagnetic waves whose nature corresponds to spherical propagation of energy.

These two limit states demonstrate the fundamental connection between energy, motion and structure. Perhaps, it is the transition between them that plays a key role in the processes of birth and transformation of matter. For example, the collision of photons can produce particles, turning n=0 into n=1. And the reverse process — the transition of mass into a pure electromagnetic wave (n=5) — may explain the mechanisms of energy release observed in extreme astrophysical conditions.

Photons, being pure energy, have no mass of their own at rest. However, their finite velocity of propagation in space creates a transverse electromagnetic wave. Although this wave cannot diverge in all directions, it forms a certain potential, which manifests itself as an equivalent mass. Thus, the photon itself has no mass, but its motion in space leads to an equivalent gravitational effect. This process emphasises the fundamental connection between energy, motion and the concept of mass.

A photon 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 is similar to a parabola, which sets symmetry in the direction of motion. I.e. spin in this case does not characterise rotation, it characterises not ideality of the surface.

Features of elementary particles

Knowing what mass depends on, let’s look at what energy is:

We also know that:

That is, from the two formulas we have obtained an absolutely identical expression.

From the law of conservation of energy, energy represents the radius of a circle. That is, we get that the radius of the circle of energy measurement is equal to:

Here we must realise that the interaction takes place in two dimensions — the longitudinal wave in the dimension of space (gravity) and the transverse wave propagating on the sphere, electromagnetic must balance each other.

Since the interaction is associated with a spherical structure, the equality of forces will be observed only at four points — at a 90° rotation. These points characterise the appearance of nodes, which imposes a restriction on the possible states of the system. Thus, the stability of the structure is ensured precisely by these discrete positions where gravitation and electromagnetic interaction are balanced. Therefore, the born elementary particles will also have from 1 to 4 nodes.

Elementary particles are the result of the intersection of these two planes of measurement. The elementary particles here are the neutrino, electron, neutron and proton.

The creation of a standing wave is due to the work done by the external space. Space can only do this work in equal portions. This process must be in mutual resonance.

At n=0 we have photon, it exclusively belongs to space.

If n=1, we get a neutrino. It will have the largest size, commensurate with the size of an atom. Minimal mass. All this makes it minimally interacting. Its feature will be internal rotation of energy. It’s created by adding a portion of energy from the space dimension to the electromagnetic wave dimension. As the electromagnetic wave propagates through the sphere, it produces the effect of internal rotation of energy.

At n=2 we get the electron. Here the particle receives a second portion of energy from the side of space, which removes the rotation and gives an understanding of charge. Charge characterizes the work on the space side. That’s why it remains constant for elementary particles. Charge characterizes the work on the part of the surrounding space to create an elementary particle. As a result, we either have a spin 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 favor of the electromagnetic component, which leads to the effect of energy rotation inside the particle.

The obtained particle will be characterized by the presence of charge, but now with the opposite sign.

Here it is also worth noting that the 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 possess rotation, which conditions their internal state and leads to connections between 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 are quantum entangled. A change in the state of one of these objects (e.g., external influence) will lead to a change in 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., collision with other particles or a field) is imposed on the neutron, it will lead to a change in its state. Since the neutron and antineutron form a pair with quantum entanglement, any effect on one of the particles must cause a change in 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 nodes create quantum dependence 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.

Conclusion

Thus, theoretical calculations based on scaling and standing waves allow us to refine experimental data and predict the properties of elementary particles. This work offers a new understanding of the relation between mass and dimensionality, and also opens new possibilities for studying quantum effects in the context of the fractal structure of the Universe.