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The Quantum Theory of the Electron (Part II)
Proceedings of the Royal Society A
... The Lévy-Leblond equation is the non-relativistic limit of the Dirac equation [1]. Dirac introduced his equation to describe a relativistic spin-1/2 particle and was the first quantummechanical equation to account for both spin and special relativity [2]. To obtain his equation, Dirac took a 'square root' of the Klein-Gordon equation (which describes a relativistic spin-0 ...
... Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. particle) in order to replace the second-order time derivative with a first-order derivative [2]. This square root required the introduction of gamma matrices (which generate Clifford algebras) and naturally led to the introduction of spin into the equation. ...
... (with δ jk the Kronecker delta). Using (2), the square of the Lévy-Leblond operator is ...
The Lévy-Leblond equation with free potential admits a symmetry algebra that is a Z2×Z2-graded colour Lie superalgebra (see Aizawa–Kuznetsova–Tanaka–Toppan, 2016). We extend this result in two directions by considering a time-independent version of the Lévy-Leblond equation. First, we construct a Z23-graded colour Lie superalgebra containing operators that leave the eigenspaces invariant and demonstrate the utility of this algebra in constructing general solutions for the free equation. Second, we find that the ladder operators for the harmonic oscillator generate a Z2×Z2-graded colour Lie superalgebra and we use the operators from this algebra to compute the spectrum. These results illustrate two points: the Lévy-Leblond equation admits colour Lie superalgebras with gradings higher than Z2×Z2 and colour Lie superalgebras appear for potentials besides the free potential.
... The Dirac equation [1,2] has acquired special attention in the last few years, since it governs the unusual behavior of electrons at the Fermi energy in graphene [3], where the valence and conduction bands are different, except at one point. This type of band dispersion where two linear bands appear to cross each other in a forbidden band is known as the Dirac cone. ...
... The Dirac equation [1,2] for an electron in an internal electromagnetic field, characterized by a scalar and a vector potentials, is given by ...
... The matrices proposed by Dirac for his equation [1] are ...
The energies of an electron in a one-dimensional crystal are studied with both the Schrödinger and Dirac equations using the plane wave expansion method. The crystalline potential sensed by the electron in a cell was calculated by accounting for the Coulombic (electrostatic) interaction between the electron and the surrounding cores (immobile positive ions at the center of the crystal cells). The energies and wave functions of the electron were calculated as a function of four parameters: the period ap of the lattice, the dimension ndim of the matrix in the momentum space, the partition number lpa in which the unit cell is divided to calculate the potential and the number of cores nco that affect the electron. It was found that 8000 cores (surrounding the electron) were needed to reach our convergence criterion. An analytical equation that accurately describes the behavior of the energies in function of the cores that affect the electron was also found. As case studies, the energies for pseudo-lithium and pseudo-graphene were obtained as a first approximation for one-dimensional lattices. Subsequently, the energies of an isolated dimer nanoparticle were also calculated using the supercell method.
... The absence of manifest Lorentz covariance is unsatisfying, as historically, physics has advanced through symmetry-based formalisms, such as the Dirac [22] and Becchi-Rouet-Stora-Tyutin [23][24][25] formalisms. Therefore, it is desirable to develop a manifestly Lorentz-covariant wave-packet formalism. ...
... is the Lorentz-covariant volume element. We stress that σ is not summed nor integrated in the identity (22) and that the identity holds for any fixed σ. Let us consider a "time-slice frame"X of the centralposition space in whichΣŇ ;T becomes an equal-time ...
... generalizing the completeness relation of the scalar wave packet (22). Once the wave-packet state is defined, its Lorentz transformation law is obtained aŝ ...
We propose a novel formulation for a manifestly Lorentz-covariant spinor wave-packet basis. The traditional definition of the spinor wave packet is problematic due to its unavoidable mixing with other wave packets under Lorentz transformations. Our approach resolves this inherent mixing issue. The wave packet we develop constitutes a complete set, enabling the expansion of a free spinor field while maintaining Lorentz covariance. Additionally, we present a Lorentz-invariant expression for zero-point energy.
Published by the American Physical Society 2024
... The Dirac equation [1] is one of the fundamental equations of modern theoretical physics. It is in service near 100 years (1928-2024). ...
... Dirac's derivation. At first, one should note the elegant derivation given by Paul Dirac in his book [17] (of course, this consideration is based on origins of [1]). Until today it is very interesting for the readers to feel Dirac's way of thinking and to follow his logical steps. ...
... J.D. Bjorken and S.D. Drell in their well-known book [59] added some words to the original derivation [1], which make it possible to extend the derivation to Ndimensional case and, further in [60], to the N -dimensional space-time. ...
More then 35 approaches to the Dirac equation derivation are presented. The various physical principles and mathematical methods are used. A review of well-known and not enough known contributions to the problem is given, the unexpected and unconventional derivations are presented as well. Three original approaches to the problem suggested by the author are considered as well. They are (i) the generalization of H. Sallhofer derivation, (ii) the obtaining of the massless Dirac equation from the Maxwell equations in maximally symmetrical form, (iii) the derivation of the Dirac equation with nonzero mass from the relativistic canonical quantum mechanics of the fermion-antifermion spin s=1/2 doublet. Today we are able to demonstrate new features of our derivations given in original papers. In some sense the important role of the Dirac equation in contemporary theoretical physics is demonstrated. A criterion for the usefulness of one or another derivation of the Dirac equation has been established
... and the scaling vacuum Einstein equations in the presence of a cosmological constantΛ, (67) andR =ḡ µνR µν , with Einstein notation. Therefore, we conclude that the scaling Einstein field equations in the absence of any matter and in the presence of a cosmological constantΛ require ζ α to satisfy ...
... These conditions are the same as the ones for scaling Einstein field equations in the absence of any matter and scaling vacuum Einstein equations in the presence of a cosmological constant. Following the same procedure, the Dirac equation [67,68] [iγ µ ∂ ∂x µ − mc ℏ ]ψ = 0 (77) ...
The study of physics at the Planck scale has garnered significant attention due to its implications for understanding the fundamental nature of the universe. At the Planck scale, quantum fluctuations challenge the classical notion of spacetime as a smooth continuum, revealing a complex microstructure that defies traditional models. This study introduces a novel scaling-based framework to investigate the properties of spacetime microstructures. By deriving a scaling-characterized metric tensor and reformulating fundamental equations—including the geodesic, Einstein field, Klein-Gordon, and Dirac equations—into scaling forms, the research reveals new properties of local spacetime dynamics. Remarkably, the golden ratio emerges naturally in linear scale measurements, offering a potential explanation for the role of the Planck length in resolving ultraviolet (UV) divergence. Furthermore, the study demonstrates how scale invariance in spacetime can restore classical geometric stability through the renormalization group equations. These findings significantly revise classical geometric intuitions, providing a fresh lens for understanding quantum fluctuations and offering promising insights for advancing quantum gravity theories.
... 37,38 Finalmente, en 1928 Dirac logró la síntesis entre la mecánica cuántica y la relatividad. 39,40 Él demostró que el acoplamiento spin órbita era un efecto puramente relativístico que todos los electrones, incluidos los s, tenían un momento angular, que consistía en la suma del momento angular del orbital y del spin. ...
... Ambos efectos, la contracción del orbital 4f y la contracción relativística de las capas s/p, son de la misma magnitud general, y la última se hace más importante para los elementos mucho más pesados que los lantánidos. 40,41 Sin embargo, en el caso de los orbitales 5d el efecto relativístico está en dirección opuesta: la capa 4f llena contrae al orbital 5d, pero los efectos relativísticos lo expanden. De todos modos, se puede considerar la superposición de los efectos relativísticos y los orbitales particularmente fuertes encontrados en el platino, el oro y el mercurio, como los responsables de las peculiaridades observadas en estos elementos. ...
... The electron's spin degree of freedom appears naturally in the framework of relativistic quantum mechanics governed by the Dirac equation [30]. A classical description of the electron spin may be found phenomenologically or via a correspondence principle, which is applicable when the typical length scale of the electromagnetic fields is larger than the position uncertainty of the particle. ...
Various classical models of electrons including their spin degrees of freedom are commonly applied to describe the electron dynamics in strong electromagnetic fields. We demonstrate that different models can lead to different or even contradicting predictions regarding how the spin degree of freedom modifies the electron's orbital motion when the electron moves in strong electromagnetic fields. This discrepancy is rooted in the model-specific energy dependency of the spin-induced Stern-Gerlach force acting on the electron. The Frenkel model and the classical Foldy-Wouthuysen model are compared exemplarily in the nonrelativistic and the relativistic limits in order to identify parameter regimes where these classical models make different predictions. This allows for experimental tests of these models. In ultrastrong laser setups in parameter regimes where effects of the Stern-Gerlach force become relevant, radiation-reaction effects are also expected to set in. We incorporate the radiation reaction classically via the Landau-Lifshitz equation and demonstrate that although radiation-reaction effects can have a significant effect on the electron trajectory, the Frenkel model and the classical Foldy-Wouthuysen model remain distinguishable also if radiation-reaction effects are taken into account. Our calculations are also suitable to verify the Landau-Lifshitz equation for the radiation reaction of electrons and other spin-1/2 particles.
... The study of double β decay is a unique method of inquiry into the nature of neutrinos, as the observation of a neutrinoless branching ratio is the only known avenue to establish that neutrinos are Majorana fermions as opposed to Dirac fermions [1][2][3][4][5][6]. ...
We introduce high-resolution solid-state imaging detectors for the search of neutrinoless double decay. Based on the present literature, imaging devices from amorphous Se evaporated on a complementary metal-oxide-semiconductor (CMOS) active pixel array could have the energy and spatial resolution to produce two-dimensional images of ionizing tracks of utmost quality, effectively akin to an electronic bubble chamber in the double decay energy regime. Still to be experimentally demonstrated, a detector consisting of a large array of these devices could have very low backgrounds, possibly reaching /(kg y) in the neutrinoless decay region of interest (ROI), as it may be required for the full exploration of the neutrinoless double decay parameter space in the most unfavorable condition of a strongly quenched nucleon axial coupling constant.
... This research helps identify distinct imprints arising from background properties, such as spacetime topology and curvature. The exploration of relativistic dynamics in f f systems began shortly after the introduction of the Dirac equation [1]. Breit was the first to analyze f f systems, initially exploring them using the Darwin potential as an inter-particle interaction [2]. ...
In this study, we aim to investigate the interaction dynamics of a fermion-antifermion (f f) pair within a curved three-dimensional space-time with a non-zero cosmological constant (Λ > 0). Our primary goal is to understand the relative motion of this pair by deriving a radial equation set and finding analytical solutions using the covariant many-body Dirac equation. We begin by formulating the relevant equation, which leads to a 4 × 4 matrix equation governing the motion of the pair. We present a non-perturbative second-order wave equation for the quantum system and demonstrate that it can be solved under the assumption that Λ is small. This assumption allows us to derive analytical solutions using well-known special functions. Accordingly, we explore the impact of the spacetime background on the dynamics of the coupled pairs.
... Essa ideia ajudou a explicar anomalias nos espectros atômicos, como o desdobramento fino das linhas espectrais. Além disso, a equação de Dirac previu a existência de antipartículas, uma previsão confirmada pela descoberta do pósitron por Carl Anderson em 1932 [6,8]. ...
... Relativistic effects play a crucial role for the quantitative description of heavy-element containing molecules [1,2]. The usual starting point for a relativistic description of the electronic motion in a molecule is the free-particle Dirac equation [3,4] combined with a classical Coulomb-type potential for the description of leading order particle-particle interactions [5]. This formalism describes the motion of electrons in accord with special relativity, but not its interactions, which are due to the Coulomb potential instantaneous. ...
Vacuum polarisation (VP) and electron self energy (SE) are implemented and evaluated as quantum electrodynamic (QED) corrections in a (quasi-relativistic) two-component zeroth order regular approximation (ZORA) framework. For VP, the Uehling potential is considered, and for SE, the effective potentials proposed by Flambaum and Ginges as well as the one proposed by Pyykk\"o and Zhao. QED contributions to ionisation energies of various atoms and group 2 monofluorides, group 1 and 11 valence orbital energies, and transition energies of Li-, Na-, and Cu-like ions of nuclear charge Z = 10, 20, ..., 90 as well as and transition energies of BaF and RaF are presented. Furthermore, perturbative and self-consistent treatments of QED corrections are compared for Kohn--Sham orbital energies of gold. It is demonstrated, that QED corrections can be obtained in a two-component ZORA framework efficiently and in excellent agreement with corresponding four-component results.
... The proper description of molecules containing 4f elements requires a careful consideration of relativistic effects, i.e., one has to go beyond the nonrelativistic Schrodinger equation. 30,31 The most rigorous way to include relativity is to use Dirac's formalism, where the wave function is a four-component object (known as bispinors). 32,33 However, this approach is computationally too demanding and cannot be applied to larger molecules even on today's high-performance computers. ...
This study employed relativistic methods to investigate the connection between the conformation and bonding properties of 45 lanthanide trihalides LnX 3 (Ln: La−Lu; X:F, Cl, Br). Our findings reveal several insights. The proper symmetry exhibited by open-shell LnX 3 requires the inclusion of spin−orbit coupling, achieved with 2-component relativistic Hamiltonians. Fluorines (LnF 3) primarily exhibit pyramidal structures, while chlorides and bromides tend to yield planar conformations. For a given halide, the strength of Ln−X bonds increases across the lanthanide series, another outcome of the lanthanide contraction. Both strength and covalency of Ln−X bonds decrease upon the halide, i.e., LnF 3 > LnCl 3 > LnBr 3. We introduced a novel parameter, the local force constant associated with the dihedral β(X−Ln−X−X), k a (β), which quantifies the resistance of these molecules to conformational changes. We observed a correlation between k a (β) and the covalency of the Ln−X bond, with higher k a (β) values indicating a stronger covalent character. Finally, the degree of pyramidalization in the LnX 3 structures is connected to (i) the extent of charge donation within the molecule and (ii) the greater covalency of the Ln−X bond. These findings provide valuable insights into the interplay between the electronic structure and molecular geometry in LnX 3. ■ INTRODUCTION Lanthanide trihalides LnX 3 , where Ln represents one of the 15 lanthanide elements, 1,2 ranging from lanthanum (La, atomic number 57) to lutetium (Lu, atomic number 71) in the formal oxidation state +3, and X represents a halogen (fluorine, chlorine, bromine, or iodine), have attracted much attention over the past decades due to their unique chemical, electronic, and optical properties. The broad application spectrum of Lanthanide trihalides and their complexes include their use as luminescent materials 3,4 e.g. in phosphors for lighting and display technologies, or scintillators for radiation detection; 5 optic materials; 6 magnetic materials, e.g., in magnetic refrigeration technology; 7 or precursors for the synthesis of lanthanide-containing semiconductors, 2 to name a few. Lanthanides trifluorides have raised interest in the nuclear industry 6 as these compounds can be formed in certain nuclear reactors. In computational chemistry, LnX 3 often serve as models for larger Ln coordination compounds of higher complexity, and as test targets benchmarking for relativistic methodologies and basis sets 8−10 featuring 4f elements. Despite their popularity which has initiated numerous experiments and theoretical studies, 6,11−14 there are still open questions regarding their structure, bonding, and molecular properties, as discussed in a recent communication. 15 There is still an ongoing debate on which factors determine if an LnX 3 compound is planar (D 3h symmetry) or pyramidal (C 3v symmetry). The small conversion barriers between the two conformations, e.g., 0.41 kcal/mol for LaF 3 , make both experimental and computational investigations of this question rather difficult. 13,14 Most authors associate LnF 3 with pyramidal and heavier halides with a planar structure for the majority of Ln(III). 14,16,17 However, the question of what causes a lanthanide trihalide to be planar or pyramidal has not been fully answered yet, as it results from a complex interplay between steric hindrance and electronic effects typical of heavy metals. 18 Possible explanations for favoring pyramidalization are polarization and d/f orbital participation in bonding, whereas planar arrangements seem to be favored by larger halides with lower polarizability, exhibiting pronounced ionic Ln−X bonds. 14,17 Molnar and Hargittai 12 proposed a model based on a number of competing factors, namely the asphericity of the 4f shell, the Ln polarizability, and the halide electronegativity. According to their empirical model, the f electron shape predominantly influences the conformation of
... Where R is the Radial Direction (perpendicular to our 3D Universe), is the nonrelativistic force and (5) is the time dilation interpretation. ...
The Hypergeometrical Universe Theory (HU) introduces a novel model of the universe and matter, fundamentally altering our understanding of cosmology, quantum mechanics, and classical forces[Smarandache, F. (2007)][Smarandache, F., & Christianto, V. (2007)][Pereira, M., 2017][Pereira, M., 2018]. Central to HU is the concept of the Fundamental Dilator (FD), a quantum mechanical wave generator responsible for shaping space through metric waves. HU posits that all particles are polymers of FDs, existing as shapeshifting space deformation solitons spinning in 4D and traveling at the speed of light along a hyperspherical locus in a 4D spatial manifold. This model redefines matter and replaces conventional particle-wave dualism with the Quantum Trinity of the FD, the dilaton field, and the Quantum Lagrangian Principle (QLP). HU reinterprets spacetime as a mere proxy for events occurring in a 4D spatial manifold, introducing an absolute 4D reference frame and reinterpreting Lorentz transformations in reciprocal space. So, HU replaces all the discussion on metric, spacetime with rotation matrices in a 4D spatial manifold affecting not space but 4D k-vectors. In HU, forces are carried by the dilation field (4D metric waves). Waves' 4D k-vectors transform according to Lorentz transformations. So, HU solves the dynamics problem in the inertial frame and then reverts the solution to the Absolute Reference frame. By doing so, HU derives the Laws of Nature from first principles and resolves fundamental issues in cosmology, dismissing the need for dark matter, dark energy, and inflation [Guth, A. ,1981]. The theory explains the horizon problem with an initial hyperspherical uniform mass distribution and galaxy dynamics with idiosyncratic mass distributions while showing that time dilation is an artifact of diminishing forces as absolute velocities approach the speed of light. One of HU's significant contributions is the derivation of the laws of gravitation and electromagnetism, demonstrating that both follow the same Lorentz force format. The radial dependence of gravitational forces is shown to be a consequence of reference frame selection, such as one centered on the Sun, where radial symmetry dictates the force's behavior. This revolutionary insight has profound implications for electrodynamics, promising to transform the design of magnetic bottles, stellarators, tokamaks, and space propulsion technologies. Additionally, HU interprets gravitation as a Van der Waals force, where the carrier dilaton field oscillates at a frequency of 1E24 Hertz. This high-frequency process leads to the dynamic screening of gravitational effects, unifying gravitation and electromagnetism. The theory also facilitates non-perturbative Quantum Chromodynamics (HU-QCD) by mapping its particle model to the Pati-Salam [Pati, J. C., & Salam, A.,1974] SU(4) GUT model and eliminating the 1 need for an integral functional in the Lagrangian Principle through the use of the Quantum Lagrangian Principle (QLP).
... The discovery of the Dirac equation stands as a foundational achievement of modern physics [1]. As the first quantum mechanical theory fully compatible with special relativity, the Dirac equation resolved several inconsistencies that beset the, otherwise widely successful, Schrödinger equation [2]. ...
The Dirac equation has resided among the greatest successes of modern physics since its emergence as the first quantum mechanical theory fully compatible with special relativity. This compatibility ensures that the expectation value of the velocity is less than the vacuum speed of light. Here, we show that the Dirac equation admits free-particle solutions where the peak amplitude of the wave function can travel at any velocity, including those exceeding the vacuum speed of light, despite having a subluminal velocity expectation value. The solutions are constructed by superposing basis functions with correlations in momentum space. These arbitrary velocity wave functions feature a near-constant profile and may impact quantum mechanical processes that are sensitive to the local value of the probability density as opposed to expectation values.
Published by the American Physical Society 2024
... In this appendix, we will linearize the stochastic HJB equation for a particle in an electromagnetic field, as derived in section 4. As pointed out in our previous work [16], to linearize the stochastic HJB equation, we use a similar approach to that which Dirac [50] If we substitute equations (A2), (A3), and(A4) into the stochastic HJB equation (A1), and remove the minimization function since the used four-velocity is the optimal one, we obtain: ...
Recent studies have extended the use of the stochastic Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies often assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers when finding optimal controls. This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the stochastic movement of quantum particles within the framework of stochastic optimal control theory. We obtain the complex diffusion coefficient in the stochastic equation of motion using the Cauchy-Riemann theorem, considering that the particle’s stochastic movement is described by two perfectly correlated real and imaginary stochastic processes. During the development of the covariant form of the HJB equation, we demonstrate that if the temporal stochastic increments of the two processes are perfectly correlated, then the spatial stochastic increments must be perfectly anti-correlated, and vice versa. The diffusion coefficient we derive has a form that enables the linearization of the HJB equation. The method for linearizing the HJB equation, along with the subsequent derivation of the Dirac equation, was developed in our previous work [V. Yordanov, Scientific Reports 14, 6507 (2024)]. These insights deepen our understanding of quantum dynamics and enhance the application of stochastic optimal control theory to quantum mechanics.
... Dirac equation, proposed by Paul Dirac in 1928 as the first reconciliation of special relativity and quantum mechanics, has a profound impact on the development of many aspects of modern physics [1]. While the Dirac equation was mainly studied in the context of particle physics and quantum field theories in the early years, people later found that Dirac physics can also be accessed in crystals with band degeneracies called Dirac points [2]. ...
Relativistic quasiparticle excitations arising from band degeneracies in crystals not only offer exciting chances to test hypotheses in particle physics but also play crucial roles in the transport and topological properties of materials and metamaterials. Quasiparticles are commonly described by low-energy Hamiltonians that are Hermitian, while non-Hermiticity is usually considered detrimental to quasiparticle physics. In this work, we show that such an assumption of Hermiticity can be lifted to bring quasiparticles into non-Hermitian systems. We propose a concrete lattice model containing two non-Hermitian Dirac cones, with one hosting amplifying Dirac quasiparticles and the other hosting decaying ones. The lifetime contrast between the Dirac cones at the two valleys imposes an ultra-strong valley selection rule not seen in any Hermitian systems: only one valley can survive in the long time limit regardless of the excitation, lattice shape and other details. This property leads to an effective parity anomaly with a single Dirac cone and offers a simple way to generate vortex states in the massive case. The non-Hermitian feature of the bulk Dirac cones can also be generalized to the boundary, giving rise to valley kink states with valley-locked lifetimes. This makes the kink states effectively unidirectional and more resistant against inter-valley scattering. All these phenomena are experimentally demonstrated in a non-Hermitian electric circuit lattice.
... The measurement of the 2S 1/2 → 2P 1/2 energy transition, known as the Lamb shift, was accomplished for the first time in atomic hydrogen in 1947 [41,42]. This energy splitting is not predicted by the Dirac theory [43], which considers the principles of quantum mechanics and special relativity. The observation of the non-degenerate 2S 1/2 and 2P 1/2 states led to the development of quantum electrodynamics (QED) [44]. ...
The potential of circulating antideuterons () in the AD/ELENA facility at CERN is currently under investigation. Approximately 100 per bunch could be delivered as a beam based on measured cross-sections. These could be further decelerated to using the GBAR scheme, enabling the synthesis of antideuterium () via charge exchange with positronium, a technique demonstrated with antiprotons for antihydrogen production. While this would demonstrate the production of low-energy , higher fluxes are required to facilitate spectroscopic studies. We propose enhancing the anti-atom production by using laser-excited positronium in the 2P state within a cavity, which is expected to increase the production cross-section by almost an order of magnitude for with energy. The ELENA team is currently studying the possibility of increasing the rate using an optimized new target geometry. We present the projected precision for measuring the antideuterium Lamb shift and extracting the antideuteron charge radius, as a function of the beam flux.
... The next major breakthrough was achieved when it was found that matter in the condensed state consisted of tiny nuclei surrounded by a structured electron medium, i.e., the electron field [113]. Finally, the joint theory of electron-positron and electromagnetic fields (quantum electrodynamics) was developed [374][375][376][377][378][379][380][381][382][383][384][385] primarily by Paul Adrien Maurice Dirac [386]. Significant contributions of other scientists and further progress in quantum electrodynamics can be traced in Refs. ...
A short historical review of the development of tunneling concept in low-temperature condensed matter physics, physical electronics, nuclear physics, chemistry, and biology is given. It is shown how the preceding classical physics is related to the quantum mechanical tunneling phenomenon. The emphasis is placed on the common features of various tunneling manifestations in nature. The triumph of the Faraday–Maxwell–Einstein idea of the physical field has been demonstrated.
... The quantum state of a qubit can be written as a linear combination of basis states |0⟩ = 6 1 0 7 and |1⟩ = 6 0 1 7 as | ⟩ = 6 7 = 6 1 0 7 + 6 0 1 7 = |0⟩ + |1⟩ in Dirac notation [40], in which and are complex numbers. After measurement, |0⟩ and |1⟩ always convert into classical bits 0 and 1, respectively. ...
Blockchains have gained substantial attention from academia and industry for their ability to facilitate decentralized trust and communications. However, the rapid progress of quantum computing poses a significant threat to the security of existing blockchain technologies. Notably, the emergence of Shor's and Grover's algorithms raises concerns regarding the compromise of the cryptographic systems underlying blockchains. Consequently, it is essential to develop methods that reinforce blockchain technology against quantum attacks. In response to this challenge, two distinct approaches have been proposed. The first approach involves post-quantum blockchains, which aim to utilize classical cryptographic algorithms resilient to quantum attacks. The second approach explores quantum blockchains, which leverage the power of quantum computers and networks to rebuild the foundations of blockchains. This paper aims to provide a comprehensive overview and comparison of post-quantum and quantum blockchains while exploring open questions and remaining challenges in these domains. It offers an in-depth introduction, examines differences in blockchain structure, security, privacy, and other key factors, and concludes by discussing current research trends.
... Derivation of the relativistic energy momentum of an electron can be given by the scalar magnitudes given that the rest energy is the total energy of a free particle [5,6]: ...
The Dirac equation is a cornerstone of quantum mechanics that fully describes the behaviour of spin ½ particles. Recently, the energy momentum relationship has been reconsidered such that |E| 2 = |(m0c^2)|^2 + |(pc)|^2 has been modified to: |E|^2 = |(m0c^2)|^2-|(pc)|^2 where E is the kinetic energy, moc^2 is the rest mass energy and pc is the wave energy for the spin ½ particle. This has been termed the 'Hamiltonian approach' and with a new starting point, the original Dirac equation has been derived, and the modified covariant form found where h/2π = c = 1. The behaviour of spin particles is found to be the same as for the original Dirac equation . The Dirac equation will also be expanded by setting the rest energy as a complex number.
... For spin 1=2 fields, which satisfy the Dirac equation [123], the polarization vectors with helicity λ ¼ AE1 are given by ...
We develop a systematic approach to analyze polarization correlations of two baryons B 1 B ¯ 2 produced in the electron-positron annihilation process. With spin density matrices for arbitrary spin particles established in the standard, the Cartesian, and the helicity forms, we provide analyses of polarization correlations for two baryons with various spin combinations. This framework can be applied to determine the spin and the parity of excited baryons, and therefore offers opportunities for the investigation of baryon spectrum and transition form factors in present and future electron-positron annihilation experiments.
Published by the American Physical Society 2024
The standard cosmological model, based on Cold Dark Matter and Dark Energy ( Λ CDM), faces several challenges. Among these is the need to adjust the scenario to account for the presence of vast voids in the large-scale structure of the universe, as well as the early formation of the first stars and galaxies. Additionally, the observed matter–antimatter asymmetry in the universe remains an unresolved issue. To address this latter question, Andrei Sakharov proposed a twin universe model in 1967. Building upon this idea and introducing interactions between these two universe sheets through a bimetric model, we propose an alternative interpretation of the large-scale structure of the universe, including its voids and the acceleration of cosmic expansion.
The domain of Charge Parity Time (CPT) symmetries has intrigued physicists for the better part of a century. Whilst the CPT theorem—a core principle asserting the combined invariance of charge conjugation, parity transformation and time reversal in quantum field theories—is foundational in modern theoretical physics, its implications are rooted deeply in the experiments that have confirmed or refuted our understanding. This paper delves into the world of CPT from an experimental physicist’s viewpoint.
Основываясь на результатах Ф. Вильфа о необходимости учета в уравнении Дирака для электрона принятых в квантовой механике правил соответствия, было показано, что уравнение, получаемое при придании физического смысла α-операторам Дирака, следует рассматривать как феноменологическое уравнение для частицы ненулевого размера – ЕМ-полярона, ранее введенного автором. Это позволило разрешить присущий уравнению Дирака парадокс, состоящий в равенстве скорости перемещаемых частиц скорости света в вакууме c , что a priori нереализуемо, а также понять физическую сущность спина как собственного механического момента ЕМ-полярона. Было показано также, что уравнение Дирака–Вильфа в случае одного пространственного измерения может рассматриваться как обобщение уравнения Шредингера на случай релятивистских энергий. PACS: 03.65.-w; 03.65. Pm; 03.75.-b
A biological reduction method for silver nanoparticles was employed using Cassia alata extract from plant leaves, which functions as a reducing agent and the metallo‐surfactant [Co(dqn)2(C12H27N)2]³⁺ (dqn = dipyrido[3,2‐f:2′,3′‐h]‐quinoxaline; C12H27N)2 = dodecylamine) acting as both stabilizing and capping agent. The ratio of Ag nanoparticles (AgNPs) formation was adjusted to be equal to the amount of AgNO3, along with variations in the amount of plant leaf extract, the metallo‐surfactant, pH, surrounding temperature, and the length of interaction periods. High‐resolution transmission electron microscopy (HRTEM), field emission scanning electron microscopy (FE‐SEM), energy‐dispersive spectroscopy (EDS), and energy‐dispersive X‐ray analysis (EDAX) were used to confirm the formation of AgNPs. The infrared results indicate the presence of hydroxyl, amine, and carboxylate groups in the extract plays a crucial role in the reduction process. Additionally, the metallo‐surfactant acts as a capping agent for the silver nanoparticles, preventing agglomeration. By adjusting the acidity of the solution and the quantity of the additive metallo‐surface active agent utilized, the size of AgNPs can be precisely regulated. The relativistic effects observed in this metallo‐surfactant‐assisted silver nanoparticle demonstrate excellent reduction capabilities for nitro compounds, effective dye degradation, and mercury sensing applications.
A short historical review of the development of tunneling concept in low-temperature condensed matter physics, physical electronics, nuclear physics, chemistry, and biology is given. It is shown how the preceding classical physics is related to the quantum mechanical tunneling phenomenon. The emphasis is placed on the common features of various tunneling manifestations in nature. The triumph of the Faraday–Maxwell–Einstein idea of the physical field has been demonstrated.
The purpose of this work is to present the atomic model proposed by Sommerfeld. We outline the classical calculation of the elliptical orbit and then apply the Wilson–Sommerfeld quantization rules to obtain expressions for the quantization of energy and orbit semi-axes. We then apply the relativistic theory to solve the problem of degenerate orbits. Thus, we see that the Sommerfeld atom is a very rich topic, involving the integration of different subjects ranging from Lagrangian mechanics to relativity, as well as plane geometry and differential equations.
In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.
Recent research on non-Abelian phenomena has cast a new perspective on controlling light. In this work, we provide a simple and general approach to induce non-Abelian gauge field to tremble the light beam trajectory. With in-plane duality symmetry relaxed, our theoretical analysis finds that non-Abelian electric field can be synthesized through a simple real-space rotation of any biaxial material. With orthogonal optical modes excited, their interference leads to an oscillation of the propagating optical beam, which is a direct consequence of the emergence of non-Abelian electric field, influencing light in a manner similar with how electric fields act on charged particles. Our microwave experiments provide unambiguous evidence to the observation of such an optical Zitterbewegung effect where excellent agreement can be found between theorical derivation, numerical simulations and experiments. By extending the idea to optical regime using natural material, we here provide another example to shake the general intuition that light travels in straight lines in homogeneous media.
Weyl fermions are one of the simplest objects that link ideas in geometry and topology to high-energy physics and condensed matter physics. Although the existence of Weyl fermions as elementary particles remains dubious, there is mounting evidence of their existence as quasiparticles in certain condensed matter systems. Such systems are termed Weyl semimetals (WSMs). Needless to say, WSMs have emerged as a fascinating class of materials with unique electronic properties, offering a rich playground for both fundamental research and potential technological applications. This review examines recent advancements in understanding electron transport in WSMs. We begin with a pedagogical introduction to the geometric and topological concepts critical to understanding quantum transport in Weyl fermions. We then explore chiral anomaly, a defining feature of WSMs, and its impact on transport phenomena such as longitudinal magnetoconductance and planar Hall effect. The Maxwell-Boltzmann transport theory extended beyond the standard relaxation-time approximation is then discussed in the context of Weyl fermions, which is used to evaluate various transport properties. Attention is also given to the effects of strain-induced gauge fields and external magnetic fields in both time-reversal broken and inversion asymmetric inhomogeneous WSMs. The review synthesizes theoretical insights, experimental observations, and numerical simulations to provide a comprehensive understanding of the complex transport behaviors in WSMs, aiming to bridge the gap between theoretical predictions and experimental verification.
Recently, topolectrical circuits (TECs) boom in studying the topological states of matter. The resemblance between circuit Laplacians and tight-binding models in condensed matter physics allows for the exploration of exotic topological phases on the circuit platform. In this review, we begin by presenting the basic equations for the circuit elements and units, along with the fundamentals and experimental methods for TECs. Subsequently, we retrospect the main literature in this field, encompassing the circuit realization of (higher-order) topological insulators and semimetals. Due to the abundant electrical elements and flexible connections, many unconventional topological states like the non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional topological states that are challenging to observe in conventional condensed matter physics, have been observed in circuits and summarized in this review. Furthermore, we show the capability of electrical circuits for exploring the physical phenomena in other systems, such as photonic and magnetic ones. Importantly, we highlight TEC systems are convenient for manufacture and miniaturization because of their compatibility with the traditional integrated circuits. Finally, we prospect the future directions in this exciting field, and connect the emerging TECs with the development of topology physics, (meta)material designs, and device applications.
In this work a novel technique to measure the hadronic R value (R = σhad⁰ / σμμ⁰) inclusively is presented. The method employs the initial state radiation (ISR) technique and the study is carried out for the BESIII experiment at the tau-charm factory BEPC-II. Given the large Lorentz-boost of the hadronic system produced in an ISR process, the efficiency to detect at least one of the hadrons (once the ISR photon is reconstructed) is very high, thus relieving the reliance of the measurement on the theoretical model used in the simulations. A first inclusive determination of the R value in the low energy region can, therefore, be achieved, providing a crucial cross check to the current evaluations based on the sum of exclusive measurements.
The asymmetric origin of matter and antimatter in the universe is an important unsolved mystery in science today. In this paper, we briefly review the history of antimatter research and the recent international hotspots of related research. It focuses on the advances in antimatter research made in recent years at the large-scale international RHIC-STAR experiment at the Relativistic Heavy Ion Collider, including the discovery of the first antimatter hypernucleus (anti-hypertriton), antimatter helium 4 and anti-hyperhydrogen 4, the first measurements of antiproton interactions, and the precise measurements of the masses and binding energies of the hypertriton and anti-hypertriton. The antimatter hypertriton nucleus, composed of an antiproton, an antineutron, and an anti-Λ hyperon, is the first anti-hypernucleu to be discovered, extending the three-dimensional nuclide map from the anti-strange quark degree of freedom. Antimatter Helium 4 is the heaviest stable antimatter nucleus yet discovered. Anti-hyperhydrogen 4, just discovered in 2024, is composed of an antiproton, two antineutrons, and an anti-Λ hyperon, and is the heaviest antimatter hypernucleus to date. Equivalence to the proton-proton interaction was established by measurements of the antiproton-antiproton interaction. At the same time, precise measurements of the masses of the hypertriton and anti-hypertriton nuclei confirmed the equivalence of matter and antimatter. And these also fully demonstrate that the CPT symmetry is also valid for antimatter nuclei. Measurements of the binding energy of the hypertriton nucleus indicate that the interaction between Λ and the nucleus of the hypertriton (the deuterium nucleus) is strong, which differs from the earlier common belief that the hypertriton nucleus is a weakly bound system. Furthermore, we discuss different physical mechanisms for the production of (anti)light nuclei, mainly including thermal, coalescence and relativistic kinetic models. Finally, we also present recent results from antihydrogen atom experiments at CERN, antimatter space probes, etc., and discuss the implications of these advances for understanding the structure of matter. In general, current studies of antimatter nuclei and atoms do not yet provide clear evidence for the asymmetric origin of matter and antimatter in the Universe, which pushes to further improve the precision of various types of measurements in the study of antimatter. Of course, other efforts in this direction in nuclear and particle physics are well expected.
Scalar and spinor quantum mechanics are compared to the vector one. Many interesting analogies and connections emerge. Schrodinger-Pauli and Dirac's equations are found to be derived from vector quantum mechanics. Moreover, the vector wave function is found to be connected with the particle's spin as expressed by Pauli matrices. Quaternionic Dirac equations yield deterministic quantum mechanics that is analogous to Maxwell electrodynamics. While Schrodinger's quantum mechanics deals with a single particle representation, vector quantum mechanics is field-theoretic quantum mechanics. Space and time coordinate rotation when applied in the quantum telegraph equation yield a Dirac-like equation describing four kinds of oscillation. In the small frequency regime, Schrodinger equations for an electron and a positron are derived. The vector quantum mechanics is found to yield the Schrodinger equation with a negative constant potential and high frequency.
We use the Schr"odinger--Newton equation to calculate the regularized self-energy of the particle using a regular self-gravitational and electrostatic potential derived in the string T-duality. The particle mass M is no longer concentrated into a point but it is diluted and described by a quantum-corrected smeared energy density resulting in corrections to the energy of the particle which is interpreted as a regularized self-energy. We extend our results and find corrections to the relativistic particles using the Klein-Gordon, Proca, and Dirac equations. An important finding is that we extract a form of generalized uncertainty principle (GUP) from the corrected energy. The form of GUP is shown to depend on the nature of particles; namely, for bosons (spin 0 and spin 1) we obtain a quadratic form of GUP, while for fermions (spin 1/2) we obtain a linear form of GUP. The correlation we found between spin and GUP may offer insights into investigating quantum gravity.
Radiative corrections which remove the accidental degeneracy in the spectrum of the relativistic hydrogen atom and lead to the modification of the Coulomb law, are calculated within the novel approach, based on the exact solution of the Dirac equation with the Coulomb potential. The energy spectrum of the hydrogen atom is obtained with account for these corrections, and the Lamb shift is calculated for the lowest energy states.
In the Lounesto classification, there are three types of regular spinors. They are classified by the condition that at least one of the scalar or pseudo scalar norms are non-vanishing. The Dirac spinors are regular spinors because their scalar and pseudo scalar norms are non-zero and zero respectively. We construct local and Lorentz-covariant fermionic fields from all three classes of regular spinors. By computing the invariants and bilinear covariants of the regular spinor fields, we show that they are physically equivalent to the Dirac fields in the sense that whatever interactions one writes down using the regular spinor fields, they can always be expressed in terms of the Dirac fields.
The integration of topological insulators in quantum computing introduces novel possibilities for enhancing qubit coherence, gate fidelity, and algorithmic performance. These materials host massless Dirac fermions, which exhibit relativistic-like behavior, leading to unique transport properties, including linear dispersion and spin-momentum locking. One particularly intriguing aspect is the potential for time dilation effects, wherein qubits experience different rates of phase evolution due to their relativistic-like nature. This paper explores how these time dilation effects could impact the coherence and synchronization of qubits, resulting in both challenges and opportunities for quantum computation. By leveraging slower-evolving qubits for memory and stability, and adapting quantum algorithms to account for temporal asymmetry, we propose strategies for mitigating synchronization issues and enhancing error correction protocols. Additionally, we investigate how time dilation could be harnessed for next-generation quantum architectures, offering a new perspective on the intersection of quantum mechanics and relativistic physics in computational systems.
Keywords: time dilation, topological insulators, Dirac fermions, quantum coherence, qubit synchronization, quantum algorithms, relativistic effects, quantum error correction, quantum gate operations, quantum memory.
A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd non-degenerate Clifford algebras is constructed by means of the Kronecker product of a parametrized version of Pauli matrices and by the identification of all possible anticommutation sets. The internal parametrization of the matrix generators allows a straightforward interpretation in terms of rotations, and in the absence of a similarity transformation can be reduced to the canonical representations by an appropriate choice of parameters. The parametric matrix generators of second and fourth-order are linearly decomposed in terms of Pauli, Dirac, and fourth-order Gell–Mann matrices establishing a direct correspondence between the different representations and matrix algebra bases. In addition, and with the expectation for further applications in group theory, a linear decomposition of GL(4) matrices on the basis of the parametric fourth-order matrix generators and in terms of four-vector parameters is explored. By establishing unitary conditions, a parametrization of two subgroups of SU(4) is achieved.
These are the notes for a minicourse taught at the 2022 ICTP summer school Frontiers in Geometry and Topology . The goal is to introduce families of Dirac operators and how they can be used to study interactions between geometry and topology. In particular, we discuss the index theorem for families of twisted Dirac operators, and its applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
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