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This study explores the age-old quest to construct a geometric model of a quantum particle. While static classical particle models have largely been dismissed, the focus has now shifted to intricate dynamic models that hold the promise of reconciling general relativity with quantum mechanics. We propose that matter particles can be described as radiation confined within dynamically curved spacetime regions, without the need for quantization of space and time, and using standard field equations and natural Planck units. Specifically, we investigate a cyclic or oscillating radiation-dominated micro-cosmos undergoing repeated bouncing. Our methodology employs integration, with carefully defined initial conditions. The results include several observable properties characteristic of quantum particles. We calculate the total mass, revealing a compelling inverse proportionality between mass and radius identical with the de Broglie relationship. Applying this model to protons, we discover a profound and surprisingly simple relationship between the proton’s radius and mass expressed in Planck units. This enables a definition of the proton radius that aligns remarkably well with the 2018 CODATA value. Furthermore, our analysis demonstrates that the radial density profile of the proton (or nucleon), averaged over a cycle time, increases toward the center. The problem of embedding the micro-cosmos within a background spacetime is also described. These results underscore the relevance of general relativity in the domain of nuclear physics. Moreover, the model offers a fresh perspective that can stimulate new ideas in the ongoing quest to unify general relativity with quantum physics.

We study the corrections induced by the theory of noncommutativity, in both space–space and space–time versions, on the spectrum of hydrogen-like atoms. For this, we use the relativistic theory of two-particle systems to take into account the effects of the reduced mass, and we use perturbation methods to study the effects of noncommutativity. We apply our study to the muon hydrogen with the aim to solve the puzzle of proton radius [R. Pohl et al., Nature466, 213 (2010) and A. Antognini et al., Science339, 417 (2013)]. The shifts in the spectrum are found more noticeable in muon H(μH) than in electron H(eH) because the corrections depend on the mass to the third power. This explains the discrepancy between μH and eH results. In space–space noncommutativity, the parameter required to resolve the puzzle θ_ss≈(0.35 GeV)^-2, exceeds the limit obtained for this parameter from various studies on eH Lamb shift. For space–time noncommutativity, the value θ_st≈(14.3 GeV)^-2 has been obtained and it is in agreement with the limit determined by Lamb shift spectroscopy in eH. We have also found that this value fills the gap between theory and experiment in the case of μD and improves the agreement between theoretical and experimental values in the case of hydrogen–deuterium isotope shift.

The effect of the particle-number symmetry restoration on the root mean square (rms) proton and neutron radii of neutron-deficient nuclei is studied in the isovector pairing case. As a first step, an expression of the nuclear radii which includes the neutron–proton pairing effects and which strictly conserves the particle-number has been established using the SBCS (Sharp BCS) method. It is shown that this expression generalizes the one obtained in the pairing between like-particles case. As a second step, the proton and neutron rms radii are numerically evaluated for even–even nuclei such as 16⩽Z⩽56 and 0⩽(N-Z)⩽4 using the single-particle energies of a Woods–Saxon mean-field. The results are compared with experimental data when available and with the results obtained when one considers only the pairing between like-particles.

The new experimental data on breakup of oxygen-16 nucleus on four doubly charged ($Z=+2$) fragments in ${}^{16}$O$+p$ interactions at 3.25 GeV/$c$ per nucleon are presented. The number of such interaction events (with the formation of four doubly charged fragments) with simultaneous production of ${\pi }^{+}+{\pi }^{-}+p_{\mathrm{\text{fr}}}$ with no recoil proton has proved to be equal to the number of events with simultaneous production of ${\pi }^{+}+{\pi }^{-}$ as well as $p_{\mathrm{\text{fr}}}+{\pi }^{-}$ in events with a recoil proton. For the first time, the experimental cross-section of the diffractive dissociation of the oxygen-16 nucleus on four $\alpha$-particles in ${}^{16}$O$+p$ interactions at 3.25 GeV/$c$ per nucleon has been determined to be $\sigma$ (diffr.)$=1.42\pm 0.25$ mb. The temperature of the diffractive breakup of oxygen nucleus on four $\alpha$-particles has been calculated to be $7.6\pm 0.7$ MeV within the framework of the statistical model of fast fragmentation of Feshbach–Huang–Goldhaber. Using the obtained slope of the integral distribution on the square of a transverse momentum of a recoil proton in the laboratory system for the events of diffractive breakup of an oxygen nucleus on four $\alpha$-particles, we have calculated the radius of a target proton to be $0.84\pm 0.05$ fm, which agrees very well with the corresponding radius, $0.831\pm 0.007\pm 0.012$ fm, determined a few years ago in the Jefferson Lab Proton Radius (PRad) experiment.

The prediction of quantum electrodynamics for elastic electron–proton scattering needs further modifications to agree with experimental data. Two scattering mechanisms that may occur are the forward and backward scattering angles for four-momentum transfer $Q^2=0.2$ up to 0.8 (GeV/$c$)². This is due to two different scattering centers. But the forward scattering is observed only at relatively high values of $Q^2\ge 1$(GeV/$c$)². The modifications related to the electric and magnetic form factors characterize the proton’s internal structure. These form factors’ dependence on $Q^2$ provides good agreement with other analysis methods. The proton’s charge radius, which corresponds to the maximum value of the scattering radius, is $0.81\pm 0.04$fm. The scattered electron is sensitive to proton internal constituents, especially at high values of $Q^2$ in which both scattering radius and photon wavelength decrease.

High-precision laser spectroscopy of atomic energy levels enables the measurement of nuclear properties. Sensitivity to these properties is particularly enhanced in muonic atoms which are bound systems of a muon and a nucleus. Exemplary is the measurement of the proton charge radius from muonic hydrogen performed by the CREMA collaboration which resulted in an order of magnitude more precise charge radius as extracted from other methods but at a variance of 7 standard deviations. Here, we summarize the role of muonic atoms for the extraction of nuclear charge radii, we present the status of the so called “proton charge radius puzzle”, and we sketch how muonic atoms can be used to infer also the magnetic nuclear radii, demonstrating again an interesting interplay between atomic and particle/nuclear physics.