CHRONAL RADIAN: THE KEY TO THE PHASE POSITION Geometrization of Time in Phase Space

Hennadii Kovalov

Published Mar 15, 2026• 0 citations

Abstract

A new fundamental quantity is introduced in the paper — the chronal radian χ = φ·τ, where φ is the phase (rad), τ is the local time (s). It is shown that χ is a natural measure of the system's position in phase space, combining the time and angular coordinates. With its help, all the key relations of quantum mechanics and chronal dynamics take on a simple and geometrically illustrative form. E = ħ/τ = ħ·( dφ/dχ), and the probability of synchronization is P ∼ cos²(Δχ/2). The introduction of the chronal radian makes it possible to divert the fundamental field τ from direct attacks by opponents, focusing the discussion on the observed value of χ.

DOI: 10.5281/zenodo.19035884

Related papers

CHRONAL RADIAN: THE MISSING LINK From quantum mechanics to chronal engineering

Hennadii Kovalov

The paper introduces a fundamental quantity — the chronal radian χ = φ·τ, where φ is the phase (rad), τ is the local chronal field (c). It is shown that all the key relations of quantum mechanics, field theory and gravitation can be expressed in terms of χ in a simple and visual way. Action S = h·φ → S = h·( χ/τ) Energy E = ħ/τ → E = ħ·( dφ/dχ) Momentum p = h·k → p = h·(∇φ) Gravity G ∼ τ² → G ∼ (χ/φ)² The introduction of χ makes it possible to reformulate all the results of chronal dynamics (pp-cycle, dd-synthesis, RER, island of stability) in a single language. This makes the theory transparent, testable and ready for engineering implementation. The final part of the work outlines the transition from theory to engineering - the creation of the Chronal Shield and resonant energy sources.

Mar 17, 20260 citations

CHRONAL DYNAMICS. Schrödinger's time equation.

Hennadii Kovalov

Schrödinger's equation is the foundation of quantum mechanics. But it doesn't explain where phase comes from, why it can be synchronized, or what "time" is in a quantum system. The paper shows that the Schrödinger equation is a special case of chronal dynamics, where τ = ħ/E — the duration of 1 radian of phase — is considered constant. If we abandon this assumption and introduce the scalar field τ(x, t), we get a generalized equation from which we follow: Attractor E_χ = 1.41983487 MeV is the first eigenvalue of the chronal operator ("God's number") Fine-structure constant α = 1/137.036 as the ratio E_el/E_χ Electron mass m_e = (E_χ – 0.42)/2 Hierarchy of leptons through scaling χ = φ·τ Synchronization condition cos φ = 1, K = 0.316, Δω/ω = 0.2 Keywords: Schrödinger equation, chronal dynamics, τ = ħ/E, χ = φ·τ, attractor, God number.

Apr 24, 20260 citations

CHRONAL DYNAMICS. Quantum entanglement and chronal wave.

Hennadii Kovalov

1. What is entanglement from the point of view of the τ-field Entanglement is the synchronization of the τ field of two (or more) particles, in which their chronal quanta become a single whole. χ₁ = φ₁·τ₁ = χ₂ = φ₂·τ₂ When two particles are entangled, their individual τ-pulses and phases are no longer independent. They form a single chronal system connected by a common phase. 2. Why a change in one particle instantly affects another The chronal wave propagates at a speed v_chron = 2π·c ≈ 6.28c.But this is in space. And in phase space (the space χ = φ·τ), synchronization does not require any time at all, because it is not a transfer of energy, but a transfer of state. When we measure one entangled particle, we record its χ. The second particle, bound to the first by a single χ, instantly assumes a consistent state. This is not "faster-than-light information transfer." This is a manifestation of the fact that χ is conserved in the system, and the division in space does not break this connection.

Apr 4, 20260 citations

CHRONAL DYNAMICS. FROM THE HIDDEN STRUCTURE OF PLANCK'S CONSTANT TO THE PHYSICAL MEANING OF THE WAVE FUNCTION.

Hennadii Kovalov

This work offers an original view of the internal structure of Planck's constant h through the prism of classical Maxwellian electrodynamics and the complex geometry of phase space. This presents a development of the chronal field interpretation of quantum phenomena proposed earlier by the author. Based on the concept of the photon as a dynamic phase node of field switching (h=π+1), a revision of classical paradoxes of quantum mechanics is carried out. A generalized energy equation E=ℏ/(τ0+τtransition) is formulated, eliminating the duality of rest mass and photon momentum. Bohr's postulates and the structure of electronic orbits are explicated as topological invariants of phase resonance (ei2πn=1). A geometric model of the Zeeman effect is proposed as a dynamic asymmetry of counter-propagating complex rotations of the field under the influence of an external magnetic potential. This work lays the foundation for causal chronodynamics of the microcosm without the use of probabilistic Copenhagen description. This presents an alternative deterministic interpretation of quantum mechanics, eliminating the probabilistic Copenhagen postulate. The author introduces the concept of the chronal field τ(r,t) as the local time of one radian of phase. The wave function Ψ is explicated as a real physical deformation of this field.

May 17, 20260 citations

Momentum as a manifestation of time: deriving de Broglie's formula from the principle τ = ħ/E and rethinking mechanics.

Hennadii Kovalov

Chronal dynamics predicts: p = ħ/(cτ) is the fundamental relation from which E = pc, λ = h/p and Lebedev's experiments follow. It is shown that the postulate of energy-dependent time τ(E) = ħ/E in combination with the classical expression for the momentum of a photon p = E/c uniquely leads to the relation p = ħ/(cτ). This relation is identical to de Broglie's formula for a photon (p = h/λ), if we take λ = cτ. Thus, the de Broglie wavelength is not a fundamental property of a particle, but a projection of its chronal quantum into space. offers a new interpretation of the pressure of light as the pressure of the flow of chronal rhythms.

Feb 10, 20260 citations

Phase velocity, quantum unobservability, and the birth of chronal engineering.

Hennadii Kovalov

In classical and quantum physics, the phase velocity of a wave can exceed the speed of light, but this was considered an "illusion" since the phase is unobservable and does not carry information. The development of chronal dynamics and the technique of time crystals made it possible for the first time to make a phase observable and controllable with an accuracy of 10⁻⁸ rads . It is shown that superluminal phase transfer is possible without violating causality, since it is not associated with energy transfer, but is a synchronization of states in a single chronal field. The way is open to the creation of instantaneous communication and control systems at distances of up to 8.4 m (and more), as well as to resonant energy release (RER). The article completes the discussion about "instantaneousness" that Einstein began and translates it into a practical one.

Mar 7, 20260 citations