2026-07-15
Introduction to Wave Electricity
A popular science review article
Modern physics boasts several highly successful theories. Classical mechanics describes the motion of bodies, electrodynamics describes electric and magnetic fields, the theory of relativity describes the relationship between space, time, energy, and momentum, and quantum mechanics describes the behavior of matter at the microscopic level. However, these theories employ different mathematical languages, and some fundamental characteristics of particles, including mass, charge, spin, and magnetic moment, are often introduced as initial properties.
The section "Wave Electricity" represents an attempt to address these issues from a unified geometric perspective. The entire series of papers is based on a new u-basis constructed from two mutually complementary idempotents. After a complex expansion, this basis forms a four-dimensional real structure in which the constant, translational, and internal periodic components of motion are naturally separated.
It is this new u-basis that forms the common foundation of all subsequent approaches. Internal rotation and norm conservation arise from its algebra. From the geometry of rotation—the state vector, mass, and time-dependent Schrödinger equation. Integration of the motion vector leads to a unified representation of the wave and particle. Applying the same construction to the electron relates the internal frequency, characteristic radius, energy, and magnetic moment. Finally, the closure condition for the internal motion allows us to transition to discrete states and the generalized Rydberg law.
This article serves as an introduction and a navigation map for the entire section. It presents the minimum formulas necessary for understanding the general idea. Detailed proofs, calculations, and physical hypotheses are discussed in separate works, references to which are given throughout the presentation.
Theory Construction Outline
 The New I-Basis as a Common Foundation for Wave Electricity Theory |
Chapter 1. Why Does Research Begin with a New Basis?
Physical theory is determined not only by experimental data, but also by the mathematical language in which these data are described. Euclidean geometry turned out to be the natural language of classical mechanics, complex numbers – of oscillatory processes and quantum mechanics, and four-dimensional space-time – of the theory of relativity.
A new mathematical language does not necessarily change the known laws. It can show that several seemingly distinct phenomena are manifestations of a single, more general structure. This is precisely the goal posed in the series of papers on wave electricity.
The research does not begin with a ready-made model of the electron, photon, or atom. First, a new algebra is constructed. Then, its geometric properties are studied. And only then are possible physical interpretations introduced.
The initial construction of the algebra and its basic operations is presented in the articles
Chapter 2. The New U-Basis
The algebra is based on two mutually complementary idempotents:
\[ \tag{1} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] An idempotent preserves itself when squared. Furthermore, the product of the two idempotents under consideration is zero. This allows us to represent them as two independent and mutually complementary directions of a single algebra.
The difference of idempotents forms a hyperbolic unit:
\[ \tag{2} \j=\ep-\em, \qquad \j^2=1. \] Therefore, the same structure can be written either in the basis \(\{1,\j\}\) or in the idempotent basis \(\{\ep,\em\}\). The second option is particularly convenient because it separates the state into two independent components.
The next key step is that the coefficients of the idempotents become complex. The result is a four-dimensional real basis:
\[ \tag{3} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}. \] Each idempotent direction forms its own complex plane. Therefore, the four-dimensional structure appears not as an additional assumption about physical space, but as a mathematical consequence of the complex expansion of two idempotent components.
A detailed description of the basis, multiplication rules, and transitions between different functionsThe notational forms are given in the works "A New Cartesian Basis of Two Idempotents", "Supplement to the New Cartesian Basis", and "A New Cartesian Basis. Part 2".
Chapter 3. Fractional Power of a Hyperbolic Unit
An important stage in the development of algebra was the derivation of an expression for the fractional power of a hyperbolic unit. In the idempotent representation, it takes a particularly simple form:
\[ \tag{4} \j^{\alpha} = \ep + \em e^{i\pi\alpha}. \] This formula shows that one idempotent component remains unchanged, while the second acquires a complex phase. This is where the fundamental geometric mechanism of the entire theory arises: the combination of constant direction and internal rotation.
The parameter \(\alpha\) can be viewed as a continuous quantity determining the phase state of the second component. As the parameter changes, the end of this component moves along a circle in the plane \(\{\em,i\em\}\).
A detailed derivation of the fractional power, its group properties, and its relationship to the logarithm of the hyperbolic unit is given in the article "Geometry of the Fractional Power of the Hyperbolic Unit".
Chapter 4. Internal Rotation as a Consequence of a New Basis
If the phase parameter is made time-dependent, a dimensionless internal state vector arises:
\[ \tag{5} J(t) = \ep + \em e^{-i\omega t}. \] The first component of this vector remains constant. The second rotates in its own complex plane. Thus, a single object simultaneously contains both an unchanging direction and an internal periodic process.
This separation is the most important property of the new u-basis. It allows us to describe translational and internal motion not as two independent objects, but as different components of a single state.
Multiplying \(J(t)\) by velocity yields a motion vector. Integrating it yields a displacement containing both linear and periodic parts. In the usual geometric interpretation, this corresponds to the translational motion of the system's center and the rotation of the internal component.
The development of this construction and its mapping into ordinary space are discussed in the articles "The New Cartesian Basis. Part 2" and "The New Cartesian Basis. Part 3".
Chapter 5. Norm Conservation and Geometry of Motion
Circular motion can be viewed as a consequence of vector length conservation. If the norm does not change, the vector's derivative is tangent to the trajectory and is orthogonal to the vector itself.
In the new u-basis, the magnitude of the rotational component \(e^{-i\omega t}\) is always equal to unity. Therefore, a change in time only changes its phase, not its length.
This leads to the following sequence:
norm conservation → orthogonality of the state and its change → internal rotation.
Thus, rotation does not necessarily have to be introduced as an independent postulate. In the algebra under consideration, it arises naturally when the norm of the complex idempotent component is preserved.
The mathematical justification for this result is given in the article "Rotation as a Consequence of Norm Conservation in an Idempotent Basis".
Chapter 6. The Mass and Time Schrödinger Equation
Differentiating the internal component of the vector \(J(t)\) yields its angular frequency. Multiplying by Planck's constant yields the quantity \(\hbar\omega\), which has the dimension of energy.
For the rotational component, we obtain an expression that matches the time part of the Schrödinger equation for a steady state:
\[ \tag{6} i\hbar \frac{\partial \Psi}{\partial t} = E\Psi. \] In this representation, the time operator of quantum mechanics receives a geometric interpretation. The derivative measures the rate of change of the internal phase, the factor \(i\) returns the direction to the original rotating component, and \(\hbar\) converts frequency to energy.
If the energy of the internal eigenmode is identified with the rest energy, a relationship arises:
\[ \tag{7} \hbar\omega_0=mc^2. \] Then mass can be considered as an energetic measure of the natural frequency of an internal state. The higher the natural frequency, the greaterThe rest energy associated with it.
It is necessary to distinguish between the mathematical result and the physical interpretation. The new basis does indeed generate an internal frequency and the value \(\hbar\omega\). However, identifying this energy with \(mc^2\) is a separate physical assumption of the model.
A detailed derivation and the limits of applicability of the result are discussed in the article "The Schrödinger Equation and the Origin of Mass in a New Idempotent Basis".
Chapter 7. The Lorentz Factor and the Energy Invariant
Another branch of the theory arises from changing the ratio between the two idempotent components. If their coefficients become mutually inverse, a hyperbolic structure arises, analogous to the structure of relativistic transformations.
The internal imbalance of the components can be related to the dimensionless velocity \(\beta=v/c\). The normalization coefficient then takes the form of the usual Lorentz factor:
\[ \tag{8} \gamma = \frac{1}{\sqrt{1-\beta^2}}. \] After multiplying the hyperbolic state by the rest energy, its components are interpreted as energy and momentum. The norm of such a state reproduces the well-known energy invariant of special relativity.
In this interpretation, the Lorentz factor appears not as an external coordinate transformation coefficient, but as a characteristic of the relationship between the mutually complementary components of a single idempotent state.
This approach is described in detail in the article "Origin of the Lorentz Factor and the Energy Invariant from the Hyperbolic Unit".
Chapter 8. A Unified Geometric Model of Waves and Particles
One of the most important directions in the development of the theory was the unified description of particles and waves. In the conventional representation, a particle is associated with localized motion, and a wave with a distributed periodic process. In the new u-basis, both regimes can be represented as different states of a single motion vector.
In the partial regime, a pronounced translational direction is preserved. In the wave regime, the internal rotational components become dominant, and their integral displacement over the entire period can vanish.
The general condition for the closure of internal motion is written as:
\[ \tag{9} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0. \] This condition means that after a full period, the internal motion returns to its original state and creates no residual displacement. This closure allows us to consider the periodic regime as a wave.
The velocity vector can have a constant norm equal to the speed of light, while its observed projection changes. A particle and a wave differ not in the total velocity of internal motion, but in the way this velocity is distributed between the translational and periodic components.
Integrating the velocity vector produces a displacement vector. Its components in the \(\{\em,i\em\}\) plane can be associated with two mutually perpendicular field components. In the wave regime, these components propagate together and form a single periodic process.
Thus, the particle-wave transition is viewed not as the disappearance of one object and the emergence of another, but as a change in the geometric regime of the same state in a new i-basis.
Chapter 9. Geometric Model of the Internal Structure of the Electron
After constructing the general state vector, the model is applied to the electron. Its motion is separated into a translational component and the internal periodic motion of the charge.
The natural frequency of the internal motion determines the characteristic geometric radius. For the frequency chosen in the model, this scale is close to the classical radius of the electron.
This coincidence is used as the basis for the hypothesis that the classical radius may characterize not the external size of the electron, but the scale of the internal periodic motion of its charge component.
The model also allows for a separation of the center of translational motion and the center of rotation of the charge. This concept has a certain similarity with the idea of zitterbewegung in relativistic quantum mechanics, although the mathematical origin of the internal motion is different here.
In Dirac's theory, the trembling motion arises from the structure of the relativistic wave equation. In the new u-basis, the internal rotation is directly contained in the form of the vector \(J(t)\).
If the rotational component corresponds to charge motion,It forms an internal circular current and creates a magnetic moment. Under additional assumptions about internal relativistic dynamics, the model reproduces the fundamental scale of the electron's magnetic moment—the Bohr magneton.
The first part of the model, including the matrix representation and connection with the Pauli matrices, is presented in the article "Geometric Model of the Internal Structure of the Electron. Part 1".
The eigenfrequency, double orbit, classical radius, energy, fine structure constant, and magnetic moment are discussed in the continuation: "Geometric Model of the Internal Structure of the Electron. Part 2".
The relationship between internal motion and an additional magnetic field is also discussed in the article "The Second Magnetic Field of the Electron. A Mathematical Model".
Chapter 10. The Fine Structure Constant
The fine structure constant is one of the most important dimensionless constants in physics. In standard theory, it characterizes the strength of electromagnetic interaction.
The geometric model of the electron offers an additional interpretation: the fine structure constant is considered as the reciprocal of the hypothetical Lorentz factor of internal motion.
In this case, it determines several relationships at once:
— the ratio of the rest energy to the total energy of the internal periodic process;
— the ratio of the classical electron radius to the reduced Compton length;
— the relationship between the geometric motion of the internal charge and the fundamental scale of the magnetic moment;
— the small difference between the internal velocity and the speed of light.
So far, the numerical value of the fine structure constant has not been independently derived from algebra. It is taken from known physics, after which its possible geometric content is examined. Therefore, this result should be considered an interpretation, not a definitive derivation of a fundamental constant.
A detailed discussion of this hypothesis is contained in the article "Geometric Model of the Internal Structure of the Electron. Part 2".
Chapter 11. Discreteness of Internal Motion
The next fundamental question is the origin of discrete states. In standard quantum mechanics, quantum numbers are introduced as characteristics of solutions of the wave equation with given boundary conditions.
In the model under consideration, discreteness is associated with the geometric condition for the closure of internal periodic motion. After a complete cycle, the state must return to its initial phase and initial position in a new basis.
Not every value of the internal parameter satisfies this condition. Therefore, only certain closed regimes are admissible. They can be numbered with an integer, which acquires the meaning of a quantum number.
Thus, the quantum number does not necessarily have to be introduced as an independent postulate. It can arise as the number of a geometrically admissible regime of internal rotation.
This result is especially important for the subsequent transition to atomic spectra, since spectral lines require not simply a continuous change in energy, but the existence of individual stable states.
Chapter 12. Geometric Origin of the Generalized Rydberg Law
One of the most recent results of the series was the derivation of a geometric analogue of the Rydberg law. The starting point is the energy difference between the two internal states of a particle.
In the geometric representation, energy is associated with the angle that determines the distribution of the total motion between the internal and observable components. Therefore, the transition between two states is expressed through the difference of the corresponding geometric functions.
After introducing the discreteness condition, the angular states receive integer numbers. Substituting these states into the expression for the energy difference leads to the general spectral formula.
In a suitable limiting case, this formula transforms into the usual Rydberg law:
\[ \tag{10} \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right). \] The main idea of the result is that Rydberg's law arises not as a separately introduced empirical dependence, but as a consequence of three successive steps:
geometry of internal motion → discrete closed states → transition energy difference.
The resulting formula is more general than the classical Rydberg expression, since it preserves the original angular dependence and only in a certain approximation transitions to the difference of the inverse squares of the quantum numbers.
The complete derivation, discreteness condition, and transition to the classical spectral law are given in the article "Geometric Origin of the Generalized Rydberg Law in the U-Basis".
Chapter 13. How Are All Directions Connected?
With the appearance of new works, the development of the theory can no longer be represented as a single linear chain. The new U-basis is a common foundation from which several interconnected branches emerge.
First, an algebra of two idempotents is constructed and its complex expansion is performed. Then, internal rotation and the state vector \(J(t)\) appear. After this, the theory develops in several directions at once.
The first branch is related to the differentiation of the internal state. It leads to frequency, energy, the Schrödinger time equation, and a geometric interpretation of mass.
The second branch is related to the integration of the motion vector. It leads to displacement, closed trajectories, and a unified representation of particle and wave.
The third branch arises from the hyperbolic relation between the idempotent components. It connects the new basis with the Lorentz factor, energy, and momentum.
The fourth branch applies the general construction to the electron and relates the intrinsic frequency, radius, magnetic moment, and fine structure constant.
The fifth branch uses the closure condition of the internal motion. From this, discrete states, spectral transitions, and the generalized Rydberg law arise.
The general architecture of the cycle can be represented as follows:
new u-basis
↓
complex expansion
↓
internal rotation and vector \(J(t)\)
↓
mass and Schrödinger | Geometry of motion | Wave and particle | Electron model | Darr; Discrete states and the generalized Rydberg law.
↓
complex expansion
↓
internal rotation and vector \(J(t)\)
↓
mass and Schrödinger | Geometry of motion | Wave and particle | Electron model | Darr; Discrete states and the generalized Rydberg law.
Chapter 14. Main works of the section
Below is a recommended order for familiarization with the theory.
"New Cartesian basis of two idempotents" — construction of the initial algebra.
"Addition to the New Cartesian Basis" — properties of the basis and clarification of basic operations.
"The New Cartesian Basis. Part 2" — complex expansion and geometry of motion.
"The New Cartesian Basis. Part 3" — development of the motion vector and its representation.
"Fractional hyperbolic unit geometry" — representation of \(\j^\alpha\), intrinsic phase, and rotation.
"Rotation as a consequence of norm conservation" — the geometric origin of rotation.
"Origin of the Lorentz factor and energy invariant" — the hyperbolic branch of the theory.
"The Schrödinger Equation and the Origin of Mass in the New Idempotent Basis" — frequency, energy, mass, and the time-dependent Schrödinger equation.
"Geometric Model of the Internal Structure of the Electron. Part 1" — the internal state and its matrix representation.
"Geometric Model of the Internal Structure of the Electron. Part 2" — frequency, radius, energy, and magnetic moment of the electron.
"Geometric Origin of the Generalized Rydberg Law in the U-Basis" — Discreteness of States and Spectral Transitions.
Chapter 15. What is Proven, and What Remains a Hypothesis?
To correctly evaluate a theory, it is necessary to distinguish between mathematical results, conclusions within the accepted model, and physical hypotheses.
15.1. Mathematical Results
The mathematical part includes the properties of two idempotents, the transition between idempotent and hyperbolic representations,Complex expansion to four real directions and an expression for the fractional power of the hyperbolic unit.
The vector \(J(t)\), its derivative, and integral are mathematically defined. The constancy of the magnitude of the rotational component, its periodicity, and the orthogonality of the radial and tangential directions are strictly enforced.
Given these definitions, the hyperbolic invariant, normalization coefficients, and closure conditions for a periodic trajectory also arise mathematically.
15.2. Results within the Model
Within the geometric interpretation, the constant idempotent component is associated with the translational direction, and the complex component with the intrinsic rotation.
Differentiation of the rotational component leads to the time-dependent form of the Schrödinger equation for one stationary mode.
The closure condition of the internal motion leads to the identification of discrete regimes, which can be numbered by integers.
The energy difference between such states leads to a generalized spectral dependence, of which the Rydberg law is a special case.
15.3. Physical Hypotheses
The assertion that the internal rotational component describes the actual motion of the charge or energy of an elementary particle remains a hypothesis.
The physical assumption is the identification of the internal frequency energy with the rest energy and the corresponding interpretation of mass.
The identification of the geometric scale of the internal motion with the classical radius of the electron, the interpretation of the fine structure constant, and the mechanism for the formation of the magnetic moment require additional justification.
It is also necessary to check whether the model can reproduce the full spectrum of quantum phenomena: spinor transformations, fermion statistics, interactions with an external field, anomalous magnetic moment, and the exact results of quantum electrodynamics.
Chapter 16. Directions for Further Research
The first direction is related to constructing a complete space-time wave equation in a new u-basis. This will allow us to move from a single internal mode to a description of the propagation and interaction of fields.
The second direction is to obtain the equations of electrodynamics directly from the geometry of two complex idempotent planes.
The third direction is related to transformations between different observers and different u-basis. Here, it is necessary to study the composition of states and the relativistic law of velocity addition.
The fourth direction is the construction of the Lagrangian and Hamiltonian of the theory. This will allow us to determine symmetries, conservation laws, and possible interactions in the standard form of theoretical physics.
The fifth direction is related to further study of spectra. The generalized Rydberg law should be applied to various series, multielectron systems, and transitions for which the classical formula requires additional corrections.
The sixth direction is the search for experimentally discernible consequences. The independent physical value of the model will be determined by its ability to calculate observable quantities without introducing additional parameters or to predict deviations from known theories.
Conclusion
The "Wave Electricity" section has evolved as a consistent program for the geometric study of the microworld. Its common foundation is a new π-basis, constructed on two mutually complementary idempotents and their complex extension.
The structure of this basis allows us to separate the constant, translational, and intrinsic rotational components of a single state. Intrinsic rotation is associated with norm conservation, its derivative with frequency and energy, and its integral with the geometry of the displacement and the wave regime.
One branch of the theory leads to the Schrödinger time equation and a geometric interpretation of mass. Another relates idempotent components to the Lorentz factor and energy invariant. A third forms a unified representation of particle and wave. A fourth is applied to the internal structure of the electron. A fifth leads to discrete states and a generalized Rydberg law.
Thus, the new u-basis is not one of the separate branches of the theory, but a common mathematical foundation for all its directions.
The main result of the entire cycle is the creation of a unified language in which rotation, wave motion, energy, mass, relativistic coefficients, the internal structure of a particle, and spectral transitions can be considered as different manifestations of a single complex process.extended idempotent state.
Some of the obtained results are rigorous consequences of the chosen algebra. Others depend on the adopted geometric interpretation. The strongest statements about the physical structure of elementary particles remain hypotheses. However, all directions rest on a common mathematical foundation and form a coherent program for further research.