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.
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 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.
Besides the purely digital or analog interpretation of reality there is a third possibility which incorporates important aspects of both. This is the cyclic formulation of elementary systems, in which elementary particles are represented as classical strings vibrating in compact space-time dimensions with periodic boundary conditions. We will address these cyclic solutions as "de Broglie internal clocks". They constitute the deterministic gears of a consistent semi-classical description of quantum relativistic physics, providing in addition an appealing formulation of the notion of time.
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 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.
By investigating the Feynman Path Integral we prove that elementary quantum particle dynamics are directly associated to single compact (cyclic) world-line parameters, playing the role of the particles' internal clock, implicit in ordinary undulatory mechanics and indirectly observed for instance in Time Crystals. This allows us to formulate a novel purely four-dimensional stringy description of elementary particles as possible physics beyond quantum mechanics. The novelty of this approach is that quantum mechanics originates from a non-trivial compact nature of the minkowskian space-time. Our result is a further evidence in support of Elementary Cycles Theory (ECT), which in previous papers has been proven to be consistent with known physics from theoretical particle physics to condensed matter. Here we provide additional conceptual arguments in support to this novel unified scenario of quantum and relativistic physics, potentially deterministic, and fully falsifiable having no fine-tunable parameters. The first evidences of such new physics characterized by ultra-fast cyclic time dynamics will be observed by probing quantum phenomena with experimental time accuracy of the order of 10−21sec. Considerations about the emergence of the arrow of time from the realm of pure, zero temperature, quantum physics governed by intrinsic time periodicity are also provided. Concerning Einstein's dilemma "God does not play dice" we conclude that, all in all, "God" would have no fun playing quantum dice.