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Relational Time Flow from Particles’ Internal Cyclic Times: Why Elementary Cycles Theory Works

Donatello Dolce B

Published Jul 27, 2026• 0 citations

Abstract

Elementary particles exhibit intrinsic phase recurrences, implicit in their undulatory description, so that each can serve, in principle, as an ultra-precise relativistic reference clock, with ticking rate universally fixed by the particle’s mass through the Compton relation. From this perspective, Rovelli’s “timeless” viewpoint is best read not as denying time, but as denying the fundamentality of any preferred external time coordinate. The physically relevant temporal variables are identified here with the particles’ intrinsic cyclic times, covariantly modulated by energy exchange and relativistic transformations, whereas the ordinary external time axis is reconstructed operationally through cycle counting and records. The present work is therefore not merely a reinterpretation of relational time, but addresses a specific consistency problem: how compact microscopic temporal variables, fixed by de Broglie–Compton periodicity, can yield the ordinary non-compact relativistic time flow observed in nature. Macroscopic time flow and its effective irreversibility then arise through coarse graining and the accumulation of records, allowing a relational-clock reformulation of the usual tension between Hamiltonian time reversal and the second law of thermodynamics, including the Loschmidt-paradox viewpoint. The proposal involves no adjustable fine-tuning, no hidden Planck-scale assumptions, and no arbitrary choice of clock variables. The internal consistency of describing time flow in terms of internal cyclic times is established here, while its connection with covariant modulations, interactions, and exact quantization has been developed in previous work on Elementary Cycles Theory.

DOI: 10.20944/preprints202602.1131.v3

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Elementary particles exhibit intrinsic phase recurrences, implicit in their undulatory description, so that each can serve, in principle, as an ultra-precise relativistic reference clock, with ticking rate universally fixed by the particle’s mass through the Compton relation. From this perspective, Rovelli’s “timeless” viewpoint is best read not as denying time, but as denying the fundamentality of any preferred external time coordinate. The physically relevant temporal variables are identified here with the particles’ intrinsic cyclic times, covariantly modulated by energy exchange and relativistic transformations, whereas the ordinary external time axis is reconstructed operationally through cycle counting and records. The present work is therefore not merely a reinterpretation of relational time, but addresses a specific consistency problem: how compact microscopic temporal variables, fixed by de Broglie–Compton periodicity, can yield the ordinary non-compact relativistic time flow observed in nature. Macroscopic time flow and its effective irreversibility then arise through coarse graining and the accumulation of records, allowing a relational-clock reformulation of the usual tension between Hamiltonian time reversal and the second law of thermodynamics, including the Loschmidt-paradox viewpoint. The proposal involves no adjustable fine-tuning, no hidden Planck-scale assumptions, and no arbitrary choice of clock variables. The internal consistency of describing time flow in terms of internal cyclic times is established here, while its connection with covariant modulations, interactions, and exact quantization has been developed in previous work on Elementary Cycles Theory.

Elementary particles exhibit intrinsic phase recurrences, \rev{implicit in the undulatory description}, so each can serve as a ``virtually perfect'' reference relativistic clock. From this perspective, Rovelli's ``timeless'' viewpoint is best read not as denying time, but as denying the fundamentality of any preferred external time coordinate: time persists as internal cyclic variables carried by particles, covariantly modulated by energy exchange and relativistic transformations. \rev{Then,} macroscopic flow arises from records and thermodynamic coarse-graining. This Letter shows that cyclic internal times of elementary systems are fully compatible with the ordinary non-compact relativistic time flow observed in nature. In particular, it identifies the fundamental ``internal'' variables underlying physical relativistic time with the particles' intrinsic cyclic times and their relational covariant modulations. Supported by theoretical and phenomenological results established in previous works, intrinsic temporal periodicity constitutes the fundamental principle of Elementary Cycles Theory and acts as exact quantization condition.

Elementary Cycles Theory is a self-consistent, unified formulation of quantum and relativistic physics. Here we introduce its basic quantum aspects. On one hand, Newton's law of inertia states that every isolated particle has persistent motion, i.e. constant energy and momentum. On the other hand, the wave-particle duality associates a space-time recurrence to the elementary particle energy-momentum. Paraphrasing these two fundamental principles, Elementary Cycles Theory postulates that every isolated elementary constituent of nature (every elementary particle) must be characterized by persistent intrinsic space-time periodicity. Elementary particles are the elementary reference clocks of Nature. The space-time periodicity is determined by the kinematical state (energy and momentum), so that interactions imply modulations, and every system is decomposable in terms of modulated elementary cycles. Undulatory mechanics is imposed as constraint "overdetermining" relativistic mechanics, similarly to Einstein's proposal of unification. Surprisingly this mathematically proves that the unification of quantum and relativistic physics is fully achieved by imposing an intrinsically cyclic (or compact) nature for relativistic space-time coordinates. In particular the Minkowskian time must be cyclic. The resulting classical mechanics are in fact fully consistent with relativity and reproduces all the fundamental aspects of quantum-relativistic mechanics without explicit quantization. This "overdetermination" just enforces both the local nature of relativistic space-time and the wave-particle duality. It also implies a fully geometrodynamical formulation of gauge interactions which, similarly to gravity and general relativity, is inferred as modulations of the elementary space-time clocks. This brings novel elements to address most of the fundamental open problems of modern physics.

We interpret the relativistic and quantum behavior of elementary particles in terms of elementary cycles. This represents a generalization of de Broglie hypothesis of intrinsically "periodic phenomenon". Similarly to a "particle in a box" or to a "vibrating string", the constraint of intrinsic periodicity can be used as semi-classical quantization condition, with remarkable matchings to ordinary relativistic quantum mechanics. In this formalism the retarded and local variations of four-momentum characterizing relativistic interactions can be equivalently expressed in terms of retarded and local variations of "de Broglie internal clock" space-time periodicity.

Wave-particle duality, together with the concept of elementary particles, was introduced by de Broglie in terms of intrinsically periodic phenomena. However, after nearly 90 years, the physical origin of such undulatory mechanics remains unrevealed. We propose a natural realization of the de Broglie periodic phenomenon in terms of harmonic vibrational modes associated to space-time periodicities. In this way we find that, similarly to a vibrating string or a particle in a box, the intrinsic recurrence imposed as a constraint to elementary particles represents a fully consistent quantization condition. The resulting cyclic dynamics formally match ordinary relativistic Quantum Mechanics in both the canonical and Feynman formulations. Interactions are introduced in a geometrodynamical way, similarly to general relativity, by simply considering that variations of kinematical state can be equivalently described in terms of modulations of space-time recurrences, as known from undulatory mechanics. We present this novel quantization prescription from an historical prospective.

Elementary particles, i.e. the basic constituents of nature, are characterized by quantum recurrences in time. The flow of time of every physical system can be therefore decomposed in elementary cycles of time. This allows us to enforce the local nature of relativistic time, yielding interesting unified descriptions of fundamental aspects of modern physics, as shown in recent publications. Every particle can be regarded as a reference clock with time resolution of the order of the Compton time particle, many orders of magnitude more accurate than the atomic clocks. Here we report basic implications about the resulting notion of time.

In Elementary Cycles theory elementary quantum particles are consistently described as the manifestation of ultra-fast relativistic spacetime cyclic dynamics, classical in the essence. The peculiar relativistic geometrodynamics of Elementary Cycles theory yields de facto a unification of ordinary relativistic and quantum physics. In particular its classical-relativistic cyclic dynamics reproduce exactly from classical physics first principles all the fundamental aspects of Quantum Mechanics, such as all its axioms, the Feynman path integral, the Dirac quantisation prescription (second quantisation), quantum dynamics of statistical systems, non-relativistic quantum mechanics, atomic physics, superconductivity, graphene physics and so on. Furthermore the theory allows for the explicit derivation of gauge interactions, without postulating gauge invariance, directly from relativistic geometrodynamical transformations, in close analogy with the description of gravitational interaction in general relativity. In this paper we summarise some of the major achievements, rigorously proven also in several recent peer-reviewed papers, of this innovative formulation of quantum particle physics.