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The potential energy of attraction between hydrogen atoms spaced by distance R on the order of aB/α, where aB is the Bohr radius and α = e2/ $$\hbar $$ c = 1/137 is the fine structure constant, has been determined. The energy depends of R according to the law of R–6 at distances aB ≪ R ≪ aB/α and according to the law of R–7 at distances R ≫ aB/α.

The multicomponent wave function of the spin and vector fields is presented as a one-component function depending on the position and orientation of a zero-size moving rotating observer. It is shown that the Dirac and Maxwell equations and the fine structure constant are the result of the connection of two spaces: the Minkowski space and the space of orientations (the observer), and this relationship is not mathematical, but physical in nature.

How can it be that uranium, thorium etc. are key in star dynamism with nuclear fission, when these atoms are barely visible in spectrometry data ? The answer of wavelength deformations of the uranium, thorium… gamma rays in the plasma, making data undecipherable directly, is drawn from previous works of the author. The paper also explains why, however, 235U appears in “excess” in meteors by developing simple comments on the fission cross-sections and building as well on earlier works of the author and on super-symmetry theory. In both cases, the concept of squeezed states and the associated anti-squeezed states, in application of Heisenberg’s uncertainty relation, happen to be extremely helpful to the demonstration. This eventually helps explaining the dynamism of the collapse before explosion in standard supernovas. More elements on super-symmetry are discussed in conclusion together with a return on the Chapter 1 of From an Einstein Syndrome to the People, where the use of the fine structure constant is justified, self-repressed behaviour (with as key the fear of anal penetration, in men) is anti-squeezed.

A quantum solution of the classical electrodynamics equations has been found. It is shown that all information on the multiparticle process of creation of scalar pairs of particles by a nonstationary self-acting electric field is contained in solutions of the d’Alembert single-particle equation. The existence of a quantum solution of the d’Alembert equation is determined by the Ehrenfest theorem. In this case, the corresponding solution is independent of the Planck constant. The process of conversion of thermal and acceleration energies into radiation has been investigated. It is demonstrated that a self-acting electric field exhibits elastic behavior. The connection of classical electrodynamics with geometrodynamics is established. The geometrodynamic justification of the appearance of the Hubble magnitude in classical electrodynamics is provided. The probabilities of creation of one-dimensional space and charge during tunneling in time of the effective Planck particle are determined. It has been shown that the fine structure constant α can be interpreted as the probability $$ \alpha ={e}^{-\frac{\pi^2\gamma }{2}} $$ of charge creation without charge and real mass. This implies that the so-called fine structure of mathematical constants can contain information on the interactions of matter, which can be used to solve the problem of information loss in black holes.

We present updated constraints on the variation of the fine structure constant, $\alpha_{\rm EM}$, and effective electron rest mass, $m_{\rm e}$, during the cosmological recombination era. These two fundamental constants directly affect the ionization history at redshift $z\simeq 1100$ and thus modify the temperature and polarisation anisotropies of the cosmic microwave background (CMB) measured precisely with {\it Planck }. The constraints on $\alpha_{\rm EM}$ tighten slightly due to improved {\it Planck} 2018 polarisation data but otherwise remain similar to previous CMB analysis. However, a comparison with the 2015 constraints reveals a mildly discordant behaviour for $m_{\rm e}$, which from CMB data alone is found below its local value. Adding baryon acoustic oscillation data brings $m_{\rm e}$ back to the fiducial value, $m_{\rm e}=(1.0078\pm0.0067) m_{\rm e,0}$, and also drives the Hubble parameter to $H_0=69.1\pm 1.2$ [in units of ${\rm km \, s^{-1} \, Mpc^{-1} }$]. Further adding supernova data yields $m_{\rm e}=(1.0190\pm0.0055) m_{\rm e,0}$ with $H_0=71.24\pm0.96$. We perform several comparative analyses using the latest cosmological recombination calculations to further understand the various effects. Our results indicate that a single-parameter extension allowing for a slightly increased value of $m_{\rm e}$ ($\simeq 3.5\sigma$ above $m_{\rm e,0}$) could play a role in the Hubble tension., Comment: 8 pages, 8 figures, 4 tables; modified to copy the accepted MNRAS version

The present paper derives the radiative corrections for the electron and positron cores in the Planck vacuum theory, leading to the electron and position particles in a 7-dimension spacetime. The physical meaning of the fine structure constant is twofold: it is the ratio of the electron spin coefficient to the electron-core spin coefficient; and it is the probability that an electron or positron will emit or absorb a photon. The nature of this photon defines the spin quanta in the PV theory. Results suggest that the Feynman and Schwinger QED calculations refer to the PV state. Finally, the closed-loop electron-positron pairs in the Feynman diagrams are explained.

It appears possible to deduce black holes, general relativity and the standard model of elementary particles from one-dimensional strands that fluctuate at the Planck scale. This appears possible as long as only switches of skew strand crossings are observable, but not the strands themselves. Woven fluctuating strands behave like horizons and imply black hole entropy, the field equations of general relativity and cosmological observations. Tangled fluctuating strands in flat space imply Dirac’s equation. The possible families of unknotted rational tangles produce the spectrum of elementary particles. Fluctuating rational tangles also yield the gauge groups U(1), broken SU(2), and SU(3), produce all Feynman diagrams of the standard model, and exclude any unknown elementary particle, gauge group, and Feynman diagram. The conjecture agrees with all known experimental data. Predictions for experiments arise, and the fundamental constants of the standard model can be calculated. Objections are discussed. Predictions and calculations allow testing the conjecture. As an example, an ab initio estimate of the fine structure constant is outlined.

The technique of quantum electrodynamics (QED) calculations of energy levels in the helium atom is reviewed. The calculations start with the solution of the Schrödinger equation and account for relativistic and QED effects by perturbation expansion in the fine structure constant α. The nonrelativistic wave function is represented as a linear combination of basis functions depending on all three interparticle radial distances, r1, r2 and r = |r→1−r→2|. The choice of the exponential basis functions of the form exp(−αr1−βr2−γr) allows us to construct an accurate and compact representation of the nonrelativistic wave function and to efficiently compute matrix elements of numerous singular operators representing relativistic and QED effects. Calculations of the leading QED effects of order α5m (where m is the electron mass) are complemented with the systematic treatment of higher-order α6m and α7m QED effects.

We investigate the history of dark energy to explain the present magnitude. We assume the dark energy is the residual cosmological constant. The most important channel in the reheating process is the gluon pair productions by QCD trace anomaly. We argue dark energy decays rapidly by gluon pair emissions during the reheating and after the big bang. The reheating temperature is determined by the decay width of dark energy Gamma and the Planck mass M_p as sqrt{M_P Gamma} ~ 10^6GeV. It is the consequence of Friedmann's equation and an equilibrium condition Gamma~ H. As the Universe cools below the hadronic scale, dark energy density is almost frozen. Nevertheless the dark energy further decreases by emitting two photons. We have estimated the current decay rate of dark energy from the QED trace anomaly. The consistent solution of Friedmann equation is in an excellent agreement with the observations. The suppression factor of dark energy scale is the product of fine structure constant alpha and curvature perturbation P as 10^{-30}=(\alpha^2P/4\pi)^2. We argue the conformal symmetry breaking in the both UV and IR are necessary unless dark energy is subtracted. We also investigated lepto-genesis by adding massive right handed neutrinos. The realistic lepto-genesis takes place during reheating process., Comment: 14 pages, 1 figure, typos corrected, physical consequences are emphasized in the concluding section. Lepto-genesis and delta N formalism are mentioned. The suppression factors of dark energy is identified. Reference is added. The version to be published

There must exist a reformulation of quantum field theory which does not refer to classical time. We propose a pre-quantum, pre-spacetime theory, which is a matrix-valued Lagrangian dynamics for gravity, Yang-Mills fields, and fermions. The definition of spin in this theory leads us to an eight dimensional octonionic space-time. The algebra of the octonions reveals the standard model; model parameters are determined by roots of the cubic characteristic equation of the exceptional Jordan algebra. We derive the asymptotic low energy value 1/137 of the fine structure constant, and predict the existence of universally interacting spin one Lorentz bosons, which replace the hypothesised graviton. Gravity is not to be quantized, but is an emergent four-dimensional classical phenomenon, precipitated by the spontaneous localisation of highly entangled fermions., Comment: 9 pages, honorable mention in Gravity Research Foundation Essay Contest 2021, v2: references updated. arXiv admin note: substantial text overlap with arXiv:2110.02062

We substantiate the need for account of the proper electromagnetic field of the electron in the canonical problem of hydrogen in relativistic quantum mechanics. From mathematical viewpoint, the goal is equivalent to determination of the spectrum of everywhere regular solutions to the self-consistent system of Dirac and Maxwell equations (with external Coulomb potential). We demonstrate that only particular classes of solutions, "nonlinear" analogues of s- and p-states, can be obtained through decomposition of a solution in a series, with respect to the fine structure constant parameter $\alpha$. In the zero approximation at $\alpha \rightarrow 0$ the reduction to the self-consistent non-relativistic system of Schr\"odinger-Poisson equations takes place. For the latter, using both numerical and variational methods, we obtain the solutions corresponding to the ground and set of excited states. Spectrum of the binding energies with remarkable precision reproduces the "Bohrian" dependence $W_n = W/ n^2$. For this, the ionization energy $W$ proves to be universal yet about two times smaller than its observed value. Possibility of the renormalization procedure and the problem of account for relativistic corrections to the binding energies of order $\alpha^2$ are considered, Comment: 23 pages, 8 figures, 2 tables

The baryon acoustic oscillations (BAO) have proven to be an invaluable tool in constraining the expansion history of the Universe at late times and are characterized by the comoving sound horizon at the baryon drag epoch $r_\mathrm{s}(z_\mathrm{d})$. The latter quantity can be calculated either numerically using recombination codes or via fitting functions, such as the one by Eisenstein and Hu (EH), made via grids of parameters of the recombination history. Here we quantify the accuracy of these expressions and show that they can strongly bias the derived constraints on the cosmological parameters using BAO data. Then, using a machine learning approach, called the genetic algorithms, we proceed to derive new analytic expressions for $r_\mathrm{s}(z_\mathrm{d})$ which are accurate at the $\sim0.003\%$ level in a range of $10\sigma$ around the Planck 2018 best-fit or $\sim0.018\%$ in a much broader range, compared to $\sim 2-4\%$ for the EH expression, thus obtaining an improvement of two to three orders of magnitude. Moreover, we also provide fits that include the effects of massive neutrinos and an extension to the concordance cosmological model assuming variations of the fine structure constant. Finally, we note that our expressions can be used to ease the computational cost required to compute $r_\mathrm{s}(z_\mathrm{d})$ with a Boltzmann code when deriving cosmological constraints using BAO data from current and upcoming surveys., Comment: 8 pages, 2 figures, 2 tables. Changes match published version

Recent observations suggest a time variation of the fine structure constant $$\alpha =e^2/\hbar c$$ on a time scale of the age of the Universe. We conjecture that the dependence is to be attributed to the Planck constant $$\hbar \rightarrow \hbar _0f(t)$$ , define the cosmological time $$\tau = \int _0^t f(t') \mathrm{d}t'$$ and derive a form of the Schrodinger equation which takes into account the variation of $$\hbar $$ . As a first result, the solution of the equation for an atom interacting with a long resonant laser field predicts a sizeable variation of the population of the atomic levels. As a second result, we find that the time dependence of the Planck parameter introduces a new form of chaotic behaviour which is not due to the initial conditions.

The fine-structure constant (FSC) measures the coupling strength between photons and charged particles and is more strongly associated with quantum electrodynamics than with atomic and molecular physics. Here we present an elementary derivation that accurately predicts the electronic polarizability of atoms A from their geometric van-der-Waals (vdW) radius RvdW and the FSC α through the compact formula A = (4πe0/a04) × α4/3RvdW7, where e0 is the permittivity of free space and a0 is the Bohr radius. The validity of this formula is empirically confirmed by estimating the value of α from nonrelativistic quantum calculations of atomic polarizabilities and atomic vdW radii obtained from both theory and experiment. Our heuristic derivation based on empirical data extends the influence of FSC from quantum electrodynamics and specific materials properties such as the visual transparency of graphene to atomic electronic properties throughout the periodic table of elements.

Condensed matter systems provide alternative `vacua' exhibiting emergent low-energy properties drastically different from those of the standard model. A case in point is the emergent quantum electrodynamics (QED) in the fractionalized topological magnet known as quantum spin ice, whose magnetic monopoles set it apart from the familiar QED of the world we live in. Here, we show that the two greatly differ in their fine-structure constant $\alpha$, which parametrizes how strongly matter couples to light: $\alpha_{\mathrm{QSI}}$ is more than an order of magnitude greater than $\alpha_{\mathrm{QED}} \approx 1/137$. Furthermore, $\alpha_{\mathrm{QSI}}$, the emergent speed of light, and all other parameters of the emergent QED, are tunable by engineering the microscopic Hamiltonian. We find that $\alpha_{\mathrm{QSI}}$ can be tuned all the way from zero up to what is believed to be the \textit{strongest possible} coupling beyond which QED confines. In view of the small size of its constrained Hilbert space, this marks out quantum spin ice as an ideal platform for studying exotic quantum field theories and a target for quantum simulation. The large $\alpha_{\mathrm{QSI}}$ implies that experiments probing candidate condensed-matter realizations of quantum spin ice should expect to observe phenomena arising due to strong interactions.

The Hubble constant Ho represents the speed of expansion of the universe and various cosmological observations and modeling methods were utilized by astronomers for a century to pin down its exact value. Determining Ho from cosmological observations is a long and tedious process requiring highly accurate datasets. To circumvent this need, a simple theoretical approach is introduced in this study which uses the concept of gravitational weakening and seismic-induced recession. As tremors occur among celestial objects, their gravitational fields would also change. This resulted in a fundamental relation of Ho and the computed rate of recession that gives a theoretical value for Ho=69.921 Km/s/Mpc. Using the newly discovered seismic-induced gravitational weakening and time dilation, it is possible that various astrophysical methods using different measurement methods would converge to this theoretical Ho value when cosmological distances and time delay measurements are corrected with the simple formulas we derived. The new model assumes that, as quakes occur in celestial objects, luminosity-induced acceleration and high-energy collision of protons and electrons may produce a massive number of neutrinos, quarks and other subatomic particles. Furthermore, the fine structure constant was found to be inversely proportional to Ho-squared and that the fine-structure constant obtained in this study gives a new physical interpretation of α. New relations for the speed of light, orbital velocity, gravitational force and the Hubble constant were further derived from the new recession constant using approximate relations for the Newtonian and electric force constant. This resulted in a modified gravitational law that is both repulsive and attractive and a theoretical explanation of the phenomenon of light-induced gravitation analogous to the electromagnetic force where photon is the force-carrier. Finally, the fundamental forces of gravitation, electromagnetism and strong nuclear force are now unified.

In the Standard Model of elementary particles, the rest mass, magnetic moment and electric charge of the electron are considered to be natural constants. In this article it is proposed that these properties can be calculated in a simple way from the corresponding Planck units using the well-known period doubling phenomenon in nonlinear dynamical systems. Periods corresponding to the values of the electron and positron properties belong to a subset of stable periods. The periodic structures of the rest energy and magnetic moment consist of three internal degrees of freedom, whereas the Coulomb energy of the electric charge consists of four. The number of period doublings for the elementary charge determines the value of the fine structure constant alpha.

The fine structure constant $\alpha$ and the ratio $h/m_{\mathrm{u}}$ between the Planck constant and the unified atomic mass are keystone constants for the determination of other fundamental physical constants, especially the ones involved in the framework of the future International System of units. This paper presents how these two constants, which can be deduced from one another, are measured. We will present in detail the measurement of $h/m_\mathrm{Rb}$ performed by atomic interferometry at the Laboratoire Kastler Brossel in Paris. This type of measurement also allows a test of the standard model to be carried out with unparalleled accuracy., Comment: arXiv admin note: text overlap with arXiv:1309.3393

In this short note we present a possible connection between the proton radius and the proton mass using the fine structure constant. The Hagedorn temperature is related to the energy levels assumed to be required to free the quarks from the proton, where hadronic matter is unstable. We also speculate that there could be a connection between the Hagedorn temperature and the Planck temperature through the fine structure constant. Regarding whether or not there is something to this (or if it is purely a coincidence), we will leave to others and future research to explore. However, we think these possible relationships are worth further investigation.

We have measured the fine-structure constant via the ratio h/mCs of the Planck constant and the mass of the cesium atom. In this proceedings, we will comment on prospects for further improvements.

The metric system appeared as the system of units designed for macroscopic (laboratory scale) measurements. The progress in accurate determination of the values of quantum constants (such as the Planck constant) in SI units shows that the capabilities in high-precision measurement of microscopic and macroscopic quantities in terms of the same units have increased substantially recently. At the same time, relative microscopic measurements (for example, the comparison of atomic transition frequencies or atomic masses) are often much more accurate than relative measurements of macroscopic quantities. This is the basis for the strategy to define units in microscopic phenomena and then use them on the laboratory scale, which plays a crucial role in practical methodological applications determined by everyday life and technologies. The international CODATA task group on fundamental constants regularly performs an overall analysis of the precision world data (the so-called Adjustment of the Fundamental Constants) and publishes their recommended values. The most recent evaluation was based on the data published by the end of 2014; here, we review the corresponding data and results. The accuracy in determination of the Boltzmann constant has increased, the consistency of the data on determination of the Planck constant has improved; it is these two dimensional constants that will be used in near future as the basis for the new definition of the kelvin and kilogram, respectively. The contradictions in determination of the Rydberg constant and the proton charge radius remain. The accuracy of determination of the fine structure constant and relative atomic weight of the electron has improved. Overall, we give a detailed review of the state of the art in precision determination of the values of fundamental constants. The mathematical procedure of the Adjustment, the new data and results are considered in detail. The limitations due to macroscopic properties of material standards (such as the International prototype of the kilogram) and the isotopic composition of substances involved in precision studies in general (as standard measures for the triple point of water) and, in particular, in the determination of the fundamental constants are discussed. The perspectives of the introduction of the new quantum units, which will be free from the mentioned problems, are considered. Many physicists feel no sympathy for the International system of units (SI), believing that it does not properly reflect the character of physical laws. In fact, there are three parallel systems, namely the systems of quantities, system of their units and the related standards. The definition of the units, in particular, the SI units, above all, reflects our ability to perform precision measurements of physical values under certain conditions, in particular, to create appropriate standards. This requirement is not related to the beauty of fundamental laws of nature. More accurate determination of the fundamental constants is one of the areas where we accumulate such experience.

Modern precision measurements on Hubble expansion H ( z ) introduce two integral constraints on background models derived from early and late-time cosmology, based on Planck observations of the BAO and, respectively, H 0 , here normalized to ΛCDM. For H ( z ) coupled to Λ = ( 1 − q ) H 2 of zero wave-number modes of cosmological spacetime with deceleration parameter q ( z ) , we infer a coupling constant g 1 for H 0 in tension with ΛCDM. Results from the Local Distance Ladder give g ≃ 1 − ξ α with ξ = 0.49 ± 0.10 . This result appears natural upon identifying H 0 -tension with breaking of T-duality in the Friedmann scale factor a ( t ) to the same order.

Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.

The fine-structure constant α [1] is a constant in physics that plays a fundamental role in the electromagnetic interaction. It is a dimensionless constant, defined as: (1) being q the elementary charge, e0 the vacuum permittivity, h the Planck constant and c the speed of light in vacuum. The value shown in (1) is according CODATA 2014 [2]. In this paper, it will be explained that the fine-structure constant is one of the roots of the following equation: (2) being e the mathematical constant e (the base of the natural logarithm). One of the solutions of this equation is: (3) This means that it is equal to the CODATA value in nine decimal digits (or the seven most significant ones if you prefer). And therefore, the difference between both values is: (4) This coincidence is higher in orders of magnitude than the commonly accepted necessary to validate a theory towards experimentation. As the cosine function is periodical, the Equation (2) has infinite roots and could seem the coincidence is just by chance. But as it will be shown in the paper, the separation among the different solutions is sufficiently high to disregard this possibility. It will also be shown that another elegant way to show Equation (2) is the following (being i the imaginary unit): (5) having of course the same root (3). The possible meaning of this other representation (5) will be explained.

The Einstein equivalence principle in the electromagnetic sector can be violated in modifications of gravity theory generated by a multiplicative coupling of a scalar field to the electromagnetic Lagrangian. In such theories, deviations of the standard result for the cosmic distance duality relation, and a variation of the fine structure constant are expected and are unequivocally intertwined. In this paper, we search for these possible cosmological signatures by using galaxy cluster gas mass fraction measurements and cosmic chronometers. No significant departure from general relativity is found regardless of our assumptions about cosmic curvature or a possible depletion factor evolution in cluster measurements., 11 pages, 4 figures, I table, Accepted Version for Publication in JCAP

Sommerfeld in 1916 introduced the dimensionless fine structure constant, $$\alpha $$ , in to the context of atomic physics, in the course of working out the relativistic theory of the H atom, under the old quantum theory of Bohr. He was able to account for the fine structural detail of the atomic line spectrum of H by introducing this dimensionless constant which emerged naturally from his relativistic theory of the H atom. Since this time, the fine structure constant has emerged in several other contexts within experimental and theoretical physics. It has attained a status of being a mysterious number in physics that defies understanding as to its experimentally verified magnitude and identity. Being physically dimensionless, such a number invites a suggestion (or approximation) of its value in terms of mathematical constants in some formulation. Feynman most famously has conjectured that it might be possible to account for $$\alpha $$ in some type of series or product expression in “e”, the base of natural logarithms, and “ $$\pi $$ ” the familiar circular constant. Here we propose an infinite series in the product $$\mathrm{e} \cdot \pi $$ that converges, within a few terms, to better than 9999 parts in 10,000 of the true value of $$\alpha $$ .

Recent cosmological measurements tend to confirm that the fine structure constant {\alpha} is not immutable and has undergone a tiny variation since the Big Bang. Choosing adequate units, this could also reflect a variation of Planck's constant h. The aim of this Letter is to explore some consequences of such a possible change of h for the pure and mixed states of quantum mechanics. Surprisingly enough it is found that not only is the purity of a state extremely sensitive to such changes, but that quantum states can evolve into classical states, and vice versa. A complete classification of such transitions is however not possible for the moment being because of yet unsolved mathematical difficulties related to the study of positivity properties of trace class operators., Comment: To appear in Phys. Lett. A

We reviewed an effective consistent ab initio approach to relativistic calculation of the spectra for multi-electron heavy and superheavy ions with an account of relativistic, correlation, nuclear, radiative effects is presented. The method is based on the relativistic gauge-invariant (approximation to QED) perturbation theory (PT) and generalized effective field nuclear model with using the optimized one-quasiparticle representation firstly in theory of the hyperfine structure for relativistic atom. The wave function zeroth basis is found from the Dirac equation with potential, which includes the core ab initio potential, the electric and polarization potentials of a nucleus. The correlation corrections of the high orders are taken into account within the Green functions method (with the use of the Feynman diagram’s technique). There have taken into account all correlation corrections of the second order and dominated classes of the higher orders diagrams (electrons screening, particle-hole interaction, mass operator iterations). The magnetic inter-electron interaction is accounted in the lowest on α parameter (α is the fine structure constant), approximation, the self-energy part of the Lamb shift is taken effectively into consideration within the Ivanov-Ivanova non-perturbative procedure, the Lamb shift polarization part - in the generalized Uehling-Serber approximation with accounting for the Kallen-Sabry α2 (αZ) and Wichmann-Kroll α(αZ)n corrections.

Arnold Sommerfeld extended Niels Bohr’s quantum model of the atom in order to explain the splitting of spectral lines in magnetic (Zeeman effect) and electric fields (Stark effect). Elaborating the atomic theory of spectra became a concerted effort that involved Walther Kossel (X-ray spectra), Paul Epstein and Karl Schwarzschild (Stark effect), Wilhelm Lenz (fine structure constant), Adalbert Rubinowicz (intensity and selection rules), and a close collaboration with Friedrich Paschen (optical spectroscopic precision measurements) and Manne Siegbahn (X-ray spectroscopy). Within a few years of the publication of Bohr’s model in 1913, and despite World War I, atomic and quantum theory was transformed into a budding research field. In 1919, Sommerfeld reviewed these achievements in Atombau und Spektrallinien, a book that was soon regarded as the “bible” of atomic physics.

International audience; The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler's triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.

The project of \emph{"quantum spacetime phenomenology"} focuses on searching pragmatically for the Planck scale quantum features of spacetime. Among these features is the existence of a characteristic length scale addressed commonly by effective approaches to quantum gravity (QG). This characteristic length scale could be realized, for instance and simply, by generalizing the standard Heisenberg uncertainty principle (HUP) to a \emph{"generalized uncertainty principle"} (GUP). While usually it is expected that phenomena belonging to the realm of QG are essentially probable solely at the so-called Planck energy, here we show how a GUP proposal containing the most general modification of coordinate representation of the momentum operator could be probed by a \emph{"cold atomic ensemble recoil experiment"} (CARE) as a low energy quantum system. This proposed atomic interferometer setup has advantages over the conventional architectures owing to the enclosure in a high finesse optical cavity which is supported by a new class of low power consumption integrated devices known as \emph{"micro-electro-opto-mechanical systems"} (MEOMS). The proposed system comprises of a micro mechanical oscillator instead of spherical confocal mirrors as one of the components of high finesse optical cavity. In the framework of a bottom-up QG phenomenological viewpoint and by taking into account the measurement accuracy realized for the fine structure constant (FSC) from the Rubidium ($^{87}$Rb) CARE, we set some constraints as upper bounds on the characteristic parameters of the underlying GUP. In the case of superposition of the possible GUP modification terms, we managed to set a tight constraint as $0.999978, Comment: 12 pages, 2 figures, Revised Version, Accepted for publication in Progress of Theoretical and Experimental Physics

We study the quantum mechanical many-body problem of $$N \ge 1$$ non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $$K \ge 0$$ static nuclei. We model the dynamics of the electrons and their self-generated electromagnetic field with the so-called many-body Maxwell–Pauli equations. Here we construct time global, finite-energy, weak solutions to the many-body Maxwell–Pauli equations under the assumption that the fine structure constant $$\alpha $$ and the nuclear charges are not too large. The particular assumptions on the size of $$\alpha $$ and the nuclear charges ensure that we have energetic stability of the many-body Pauli Hamiltonian, i.e., the ground state energy is finite and uniformly bounded below with lower bound independent of the magnetic field and the positions of the nuclei. This work serves as an initial step towards understanding the connection between the energetic stability of matter and the well-posedness of the corresponding dynamical equations.

In this paper we determine in second order in the fine structure constant the energy levels of Weber's Hamiltonian admitting a quantized torus. Our formula coincides with the formula obtained by Wesley using the Schr\"odinger equation for Weber's Hamiltonian., Comment: 15 pages, 1 figure

In this paper, we will derive the following formula for the value of the gravitational constant G: (1). This equation has only 0.81% error compared to the common accepted value [1]. The parameters in the equation are the following: the fine structure constant, q the elementary charge, the mass of the electron, the permittivity of the free space, e the exponential function and the relation between a circumference and its diameter. Values attached: [2], [1], [1], [1], [3], [4], (Planck constant) [1], (speed of light) [1]. As it can be checked, all of them are electromagnetic or mathematical constants and properties of the electron. No constant related to gravity has been used to arrive to this value. This formula has only 0.81% error numerically and has the correct units. And it is the output of this paper. To get to this formula, we will consider space as being composed by the particles that occupy it. These particles include mass particles and force carriers (as photons). The non-uniform distribution of these particles creates distortions in space. These distortions create the warping of space, making gravity appear as an emergent phenomenon. Having in mind these considerations for a special case (an isolated electron emitting photons) we get to this formula that validates the assumptions.

In this paper, we study the equation of the form of which can also be written as . Apart from the trivial solution x = y, a non-trivial solution can be expressed in terms of Lambert W function as . For y > e, the solutions of x are in-between 1 and e. For integer y values between 4 and 12, the solutions of x written in base y are in-between 1.333 and 1.389. The non-trivial solutions of the equations and written in base y are exactly one and two orders higher respectively than the solutions of the equation . If y = 10, the rounded nontrivial solutions for the three equations are 1.3713, 13.713 and 137.13, i.e. 100.13713 = 1.3713. Further, ln(1.3713)/1.3713 = 0.2302 and W(-0.2302) = -2.302. The value 137.13 is very close to the fine structure constant value of 137.04 within 0.1%.

We consider a long range scalar force that mainly couples to dark matter and unstable Standard Model states, like the muon, with tiny strength. Probing this type of force would present a challenge to observations. We point out that the dependence of the induced background scalar field on dark matter number density can cause the mass of the unstable particles to have spatial and temporal variations. These variations, in turn, leave an imprint on the value of the fine structure constant $\alpha$, through threshold corrections, that could be detected in astronomical and cosmological measurements. Our mechanism can accommodate the mild preference of the Planck data for such a deviation, $(\alpha_{_{\rm CMB}}-\alpha_{\rm present})/\alpha_{\rm present} = (-3.6\pm 3.7)\times 10^{-3}$. In this case, the requisite parameters typically imply that violations of Equivalence Principle may be within reach of future experiments., Comment: 6 pages; 3 figures; V2: published version

The radiative transition wavelengths and oscillator strengths for some Li-like multicharged ions are calculated within the relativistic many-body perturbation theory with the optimized Dirac–Kohn–Sham zeroth approximation and an effective taking the relativistic, exchange-correlation, nuclear, radiative effects into account. All correlation corrections of the second order and dominated classes of the higher orders diagrams have been considered (electrons screening, mass operator iterations etc). The method includes the generalized Glushkov–Ivanov–Ivanova procedure (relativistic energy approach) for generation of the optimal basis set of relativistic electron wave functions with fulfillment of the gauge invariance principle. To reach the latter, we focus on accurate consideration of the QED perturbation theory fourth-order (a second order of the atomic perturbation theory) Feynman diagrams, whose contribution into imaginary part of radiation width Im δE for the multielectron ions accounts for multibody correlation effects. A minimization of the functional Im δE leads to integral–differential Dirac–Kohn–Sham–like density functional equations. The magnetic interelectron interaction is accounted for in the lowest order on α 2 ( α is the fine structure constant) parameter. The Lamb shift polarization part is taken into account in the modified Uehling–Serber approximation. Comparisons of our results on the radiative transition wavelengths and oscillator strengths for some transition in spectra of the Li-like multicharged ions (the nuclear charge Z = 21–30) with other comparable theoretical and experimental results are also given and discussed.

The sqrt of Planck momentum, denoted here as Q, is attributed as a distinct Planck unit and is used to link the mass (gravity) constants with the charge (electric) constants. From these constants, formulas for a magnetic monopole and then an electron frequency in terms of magnetic monopoles and Planck time are derived. The electron frequency formula has units yet is unit-less suggesting that within the electron there is a ratio of units whereby they overlap and cancel (the electron is a mathematical object, units = 1). Consequently of the SI units (kg, m, s, A, K) only 2 are required as from these the rest may be derived. This then permits us to define the least precise constants $G, h, e, m_e, k_B$ in terms of the most precise $c, \mu_0$ (exact values), the fine structure constant $\alpha$ (10-11 digits) and the Rydberg constant $R$ (12-13 digits). Results are consistent with CODATA 2014.

The inverse fine-structure constant 1/α= 137.035 satisfies 1/α = 112 + 42 + 0.035 = 121 + 16 + 0.035 = 137.035, with 11 being the 11 dimensions of M-theory, 4 the number of dimensions of Einstein’s space-time, and 0.035 the 3.5 percent visible Universe. Cosmological information appears to be encrypted linearly as a triplet code in 1/α.

This article explains how a charged body can retain its integrity despite electrostatic repulsion within itself. The study requires derivation of the field equations governing the velocity of any arbitrary element within the system. These relations are obtained from two Lagrangian principles based on obvious and least disruptive modifications of known concepts in classical mechanics. The two Lagrangians lead to a relativistic energy-momentum equation describing the motion and Maxwell's equations quantifying the electromagnetic fields, respectively. The energy-momentum relation shows that rigid body motions creating stationary fields in a rotating frame is a viable solution for the velocity inside the continuum. It also yields a combined potential which remains constant throughout the rotating particle. This constancy coupled with Maxwell's equations and boundary conditions for proper interfacial continuity provides the charge distribution and geometric shape required for continued coherence of the domain. The paper presents specific simulations for slender annuli steadily rotating about the axis of symmetry where the charge density and the axial width are plotted as radial functions. Curiously, when the analysis is extended to systems with different axes of rotation and symmetry, the proportionalities between energy and frequency as well as momentum and wave-number can be established. Such results are very similar to Planck's and de Broglie's laws, even though derived from classical principles. Thus, the presented theory might have deeper implications like, for example, in mathematical computation of fine structure constant.

Two particles interacting by photon exchange, form the bound states predicted by the non-relativistic Schrödinger equation with the Coulomb potential (Balmer series). More than 60 years ago, in the solutions of relativistic Bethe-Salpeter equation, in addition to the Balmer series, were found another series of energy levels. These new series, appearing when the fine structure constant α is large enough (α > π/4), are not predicted by the Schrödinger equation. However, this new (non-Balmer) states can hardly exist in nature, since in order to create a strong e.m. field with α > π/4 a point-like charge Z > 107 is needed. The nuclei having this charge, though exist starting with bohrium, are far from to be point-like. In the present paper, we analyze the more realistic case of a strong interaction created by exchange of a massive particle. It turns out that in the framework of the Bethe-Salpeter equation this interaction still generates a series of new relativistic states, which are similar to those of the massless exchange case, and which are absent in the Schrödinger equation. The properties of these solutions are studied. Their existence in nature seems possible.

We forecast the main cosmological parameter constraints achievable with the CORE space mission which is dedicated to mapping the polarisation of the Cosmic Microwave Background (CMB). CORE was recently submitted in response to ESA's fifth call for medium-sized mission proposals (M5). Here we report the results from our pre-submission study of the impact of various instrumental options, in particular the telescope size and sensitivity level, and review the great, transformative potential of the mission as proposed. Specifically, we assess the impact on a broad range of fundamental parameters of our Universe as a function of the expected CMB characteristics, with other papers in the series focusing on controlling astrophysical and instrumental residual systematics. In this paper, we assume that only a few central CORE frequency channels are usable for our purpose, all others being devoted to the cleaning of astrophysical contaminants. On the theoretical side, we assume LCDM as our general framework and quantify the improvement provided by CORE over the current constraints from the Planck 2015 release. We also study the joint sensitivity of CORE and of future Baryon Acoustic Oscillation and Large Scale Structure experiments like DESI and Euclid. Specific constraints on the physics of inflation are presented in another paper of the series. In addition to the six parameters of the base LCDM, which describe the matter content of a spatially flat universe with adiabatic and scalar primordial fluctuations from inflation, we derive the precision achievable on parameters like those describing curvature, neutrino physics, extra light relics, primordial helium abundance, dark matter annihilation, recombination physics, variation of fundamental constants, dark energy, modified gravity, reionization and cosmic birefringence. (ABRIDGED), Comment: 90 pages, 25 Figures. Revised version with new authors list and references

Motivated by experiments confirming that the optical transparency of graphene is defined through the fine structure constant and that it could be fully explained within the relativistic Dirac fermions in a two-dimensional picture, we investigate in this article how this property is affected by next-to-nearest-neighbor coupling in the low-energy continuum description of graphene. A detailed calculation within the linear response regime allows us to conclude that, somewhat surprisingly, the optical conductivity at zero temperature that determines the transparency remains robust up to this correction., Instituto de Física La Plata

In previous papers, we revisited the Dirac equation and conjectured that the electron can be viewed as a massless charge spinning at light speed, this internal motion being responsible for the rest mass involved in external motions and interactions. Implications of this concept on basic properties such as time, space, electric charge, and magnetic moment were considered. The present paper investigates the deviations of the resulting gyromagnetic factor, fine-structure constant, and gravitational invariant from their integer approximates, and their implication in a better understanding of the electromagnetic, gravitational, and other interactions.

One of the amazing properties of graphene is the ultrarelativistic behavior of its loosely bound electrons, mimicking massless fermions that move with a constant velocity, inversely proportional to a fine-structure constant ${\ensuremath{\alpha}}_{g}$ of the order of unity. The effective interaction between these quasiparticles is, however, better controlled by the coupling parameter ${\ensuremath{\alpha}}_{g}^{*}={\ensuremath{\alpha}}_{g}/\ensuremath{\epsilon}$, which accounts for the dynamic screening due to the complex permittivity $\ensuremath{\epsilon}$ of the many-valence electron system. This concept was introduced in a couple of previous studies [Reed et al., Science 330, 805 (2010) and Gan et al., Phys. Rev. B 93, 195150 (2016)], where inelastic x-ray scattering measurements on crystal graphite were converted into an experimentally derived form of ${\ensuremath{\alpha}}_{g}^{*}$ for graphene, over an energy-momentum region on the $\mathrm{eV}\phantom{\rule{0.16em}{0ex}}\AA{}{}^{\ensuremath{-}1}$ scale. Here, an accurate theoretical framework is provided for ${\ensuremath{\alpha}}_{g}^{*}$, using time-dependent density-functional theory in the random-phase approximation, with a cutoff in the interaction between excited electrons in graphene, which translates to an effective interlayer interaction in graphite. The predictions of the approach are in excellent agreement with the above-mentioned measurements, suggesting a calibration method to substantially improve the experimental derivation of ${\ensuremath{\alpha}}_{g}^{*}$, which tends to a static limiting value of $\ensuremath{\sim}0.14$. Thus, the ab initio calibration procedure outlined demonstrates the accuracy of perturbation expansion treatments for the two-dimensional gas of massless Dirac fermions in graphene, in parallel with quantum electrodynamics.

The essence of electric charge has been a mystery. So far, no theory has been able to derive the attributes of electric charge, which are: bivalency, stability, quantization, equality of the absolute values of the bivalent charges, the electric field it creates and the radii of the bivalent charges. Our model of the electric charge and its field (this paper) enables us (in additional papers), for the first time, to derive simple equations for the radii and masses of the electron/positron muon/anti-muon and quarks/anti-quarks. These equations contain only the constants G, c, ℏ and α (the fine structure constant). The calculated results based on these equations comply accurately with the experimental results. In this paper, which serves as a basis for the other papers, we define electric charge density, based on space density. This definition alone, without any phenomenology, yields the theory of Electrostatics. Electrostatics together with Lorentz Transformation is known to yield the entire Maxwell Electromagnetic theory.