We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.
We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
The Sum of Resonances (Σ) and the Product of Primes (Π) presents a resonance-field interpretation of Euler’s identity and the Riemann zeta function within the theoretical framework of R-theory. In this work, Σ and Π are redefined not as abstract mathematical operations, but as the two core phases of the universal resonance field: • Σ(τ) — the non-spatiotemporal superposition of all possible resonant modes in the zero-field (𝕆) • Π(t) — the anchored, phase-locked configurations that form stable structures such as particles, atoms, and geometric order By interpreting Euler’s identity ζ(s) = Σ n=1→∞ n^{-s} = Π over primes (1 - p^{-s})^{-1} as a coherence bridge between Σ and Π, the paper shows that number theory, quantum interference, and the formation of matter arise from one underlying resonance mechanism. The Riemann critical line is reinterpreted as the physical balance between expansion (Σ) and condensation (Π), while the double-slit experiment is shown to be a direct spacetime projection of the ζ(s) duality. A key contribution of the paper is the introduction of τ, the internal cycle parameter of the zero-field, which governs pre-spatiotemporal resonance evolution. Physical time t emerges only when coherence-closure occurs, mapping Σ(τ) into Π(t). This provides a unified way to describe wave superposition, resonance nodes, and measurement without invoking particles as fundamental entities. In summary, the paper proposes that reality is not constructed from matter, but from the transformation: Σ(τ) → Π(t) and that primes, resonance, interference, and spacetime structure are all manifestations of a single coherence principle. This work forms part of the broader development of R-theory, a resonance-based unification framework aimed at reconciling quantum behaviour, field structure, and the emergence of physical reality.
Simultaneous observation of the wave and particle natures of a quantum object under one experimental arrangement represents the latest development regarding wave-particle duality. In this work, we propose a simple scheme using linear-optics devices to observe either the particle nature or wave nature or the wave-particle superposition (WPS) of photons by controlling a relevant classical parameter. Our setup consists of two stages, the preparation and measurement stages, with the latter stage containing an additional classical variable whose value will be given only after the first stage in order to avoid any possible communication between the two stages and thus rule out the classical hidden-variable model. We not only develop the theory but also perform the corresponding experiments. The experimental observations perfectly agree with the theoretical predictions. Our findings are the following. The frequency (wavelength) of both single- and two-photon systems is independent of whether the photon(s) is wavelike or in a WPS state, with the common fact being that the frequency (wavelength) of two photons as a whole entity is twice (half) that of a single photon. As for the interference-pattern visibility, it is always lower in the case of WPS than in the case of a wavelike state for a single photon as well as for two photons. Remarkably, with a given value of the control parameter, the interference visibility in the case of WPS is always higher in the single-photon scenario than in the two-photon one. Although here we are dealing with photons, the obtained results apply equally well to matter particles.
Classical physics identified matter with particles and radiation with waves. Although these descriptions accounted for a large number of very important facts, it became apparent at the beginning of the twentieth century that other phenomena, such as blackbody radiation and photoelectric effect, are satisfactorily explained only if radiation is described in terms of particles. Louis de Broglie developed a theory according to which, conversely, matter has wave properties. As an outcome of this theory, he was able to write down a formula relating the corresponding wavelength with the momentum of the particle. Diffraction patterns characterize waves, and the basic fact that corroborated the validity of the notion of matter waves was that the Davisson and Germer experiment proved electrons can be diffracted. In classical optics the simplest experiment displaying this phenomenon is the well-known Young two-slit experiment, in which a beam of light is passed through a diaphragm with two slits.
Abstract We report the experimental realization of the prime number quantum potential VN(x), defined as the potential entering the single-particle Schrödinger Hamiltonian with eigenvalues given by the first N prime numbers. Using computer-generated holography, we create light intensity profiles suitable to optically trap ultracold atoms in these potentials for different N values. As a further application, we also implement a potential whose spectrum is given by the lucky numbers, a sequence of integers generated by a different sieve than the familiar Eratosthenes’s sieve used for the primes. Our results pave the way toward the realization of quantum potentials with arbitrary sequences of integers as energy levels and show, in perspective, the possibility to set up quantum systems for arithmetic manipulations or mathematical tests involving prime numbers.