Relativistic States Research Articles

Covariant Representation of Spin and Entanglement—A Review and Reformulation

A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector nμ and then inducing a representation of SL(2,C), based on pμ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space {(xμ,pμ),(ζμ,πμ)}, inducing a representation on the momentum πμ, thus providing a novel dynamical interpretation of the timelike unit vector nμ=πμ/M. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.

Innovative approache for the nonlinear atangana conformable Klein-Gordon equation unveiling traveling wave patterns

The aim of current work is to establish novel traveling wave solutions of the nonlinear Atangana conformable Klein - Gordon equation using a new extended direct algebraic technique. The Klein - Gordon equation is the relativistic state of the Schrödinger equation with a second - order time derivative and zero spin. Complex wave variable transformation is used to convert Atangana conformable nonlinear differential equation into an ordinary differential equation. Using the proposed technique based on Maple software structure, various types of solutions, such as, generalized trigonometric, generalized hyperbolic, and exponential functions, are established. When special parameteric values are considered for this method, solitary wave solutions can be obtained through other methods, such as the (G′G)-expansion method, the modified Kudryashov method, the sub-equation method, and so forth. A physical explanation is provided for the solutions under consideration to enhance comprehension of the physical phenomena resulting from the obtained solutions, provided that the physical parameters are set appropriately using 3D, 2D, and contour simulations. The results demonstrated that the new extended direct algebraic method provides a more potent mathematical tool for solving numerous more nonlinear partial differential equations with the aid of symbolic computation.

Nonlinear electrodynamics and its possible connection to relativistic superconductivity: an example

Abstract This work presents an illustrative example suggesting a potential connection between the Heisenberg–Euler (HE) model of nonlinear electrodynamics (NLED) and relativistic superconductivity. Within a cylindrical coordinate system, we derive nonlinear Maxwell’s equations from the HE Lagrangian to the fourth order in the electromagnetic (EM) field. By considering static electric and magnetic fields with Gaussian radial profiles, we compute the corresponding HE current density. In parallel, we explore relativistic type-II superconductivity in analogy with the Ginzburg–Landau (GL) theory. We calculate the vortex-supercurrent density of the condensate and propose a link between this current and the HE current. This connection enables the derivation of the relativistic order parameter state. The problem considered in this article may contribute to the understanding of superconductivity in strong EM environments within the broader framework of NLED.

The exact relativistic scalar quasibound states of the dyonic Kerr–Sen black hole: quantized energy, and Hawking radiation

We consider Klein–Gordon equation in the Dyonic Kerr–Sen black hole background, which is the charged rotating axially symmetric solution of the Einstein–Maxwell–Dilaton–Axion theory of gravity. The black hole incorporates electric, magnetic, dilatonic and axionic charges and is constructed in 3+1 dimensional spacetime. We begin our investigations with the construction of the scalar field’s governing equation, i.e., the covariant Klein–Gordon equation. With the help of the ansatz of separation of variables, we successfully separate the polar part, and find the exact solution in terms of Spheroidal Harmonics, while the radial exact solution is obtained in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that gives rise to discrete complex-valued energy levels for massive scalar fields. The real part is the scalar field relativistic quantized energy, while the imaginary part represents the quasibound states’s decay. We present all of the sixteen possible exact energy solutions for both massive and massless scalars. We also present the investigation the Hawking radiation of the Dyonic Kerr–Sen black hole’s apparent horizon, via the Sigurd–Sannan method by making use of the obtained exact scalar wave functions. The radiation distribution function, and the Hawking temperature are also obtained.

One-Particle Operator Representation over Two-Particle Basis Sets for Relativistic QED Computations.

This work is concerned with two-spin-1/2-fermion relativistic quantum mechanics, and it is about the construction of one-particle projectors using an inherently two-particle, "explicitly correlated" basis representation necessary for good numerical convergence of the interaction energy. It is demonstrated that a faithful representation of the one-particle operators, which appear in intermediate but essential computational steps, can be constructed over a many-particle basis set by accounting for the full Hilbert space beyond the physically relevant antisymmetric subspace. Applications of this development can be foreseen for the computation of quantum-electrodynamics corrections for a correlated relativistic reference state and high-precision relativistic computations of medium-to-high-Z helium-like systems, for which other two-particle projection techniques are unreliable.

Ro-vibrational relativistic states for the q-deformed hyperbolic barrier potential

Ro-vibrational relativistic states for the q-deformed hyperbolic barrier potential

Dimensional regularization and two-loop vacuum polarization operator: Master integrals, analytic results, and energy shifts

We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in D=4−2ε dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order ε, to be combined with the 1/ε divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for 1S, 2S, and 2P states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the Zα-expansion. We compare the results obtained to the existing literature. Published by the American Physical Society 2024

Relativistic massive and massless scalar fields bound to the Bumblebee gravity's black hole

In this letter, we examine massive and massless scalar quasibound states bound in the static spherically symmetric black hole solution of the Einstein-Bumblebee space-time. We start with constructing the governing relativistic fields' dynamical equation, i.e. the Klein-Gordon equation, component by component. With the help of the ansatz of separation of variables, we find the exact solution of the angular part in terms of Spherical Harmonics while the radial exact solution is discovered in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that reveals a complex-valued energy levels expression for the case of massive scalar, where the real part is the scalar particle's energy and the imaginary part represents the quasibound state's decay. And for a massless scalar, a pure imaginary energy levels is obtained. The quasibound states, thus, describe decaying relativistic states bound in the black hole's gravitational potential well. At the end, we investigate the Hawking radiation of the static Bumblebee black hole's horizon by deriving and investigating the radiation distribution function.

High-entropy engineering of the crystal and electronic structures in a Dirac material

Dirac and Weyl semimetals are a central topic of contemporary condensed matter physics, and the discovery of new compounds with Dirac/Weyl electronic states is crucial to the advancement of topological materials and quantum technologies. Here we show a widely applicable strategy that uses high configuration entropy to engineer relativistic electronic states. We take the AMnSb2 (A = Ba, Sr, Ca, Eu, and Yb) Dirac material family as an example and demonstrate that mixing of Ba, Sr, Ca, Eu and Yb at the A site generates the compound (Ba0.38Sr0.14Ca0.16Eu0.16Yb0.16)MnSb2 (denoted as A5MnSb2), giving access to a polar structure with a space group that is not present in any of the parent compounds. A5MnSb2 is an entropy-stabilized phase that preserves its linear band dispersion despite considerable lattice disorder. Although both A5MnSb2 and AMnSb2 have quasi-two-dimensional crystal structures, the two-dimensional Dirac states in the pristine AMnSb2 evolve into a highly anisotropic quasi-three-dimensional Dirac state triggered by local structure distortions in the high-entropy phase, which is revealed by Shubnikov–de Haas oscillations measurements.

Statistical characterization of the collective synchrotron radiation power emitted by non-ideal magnetized plasma fluids in relativistic jets

The problem of determining the collective synchrotron radiation power emitted by non-ideal magnetized plasma fluids at kinetic equilibrium in relativistic jets is addressed. A covariant statistical kinetic approach is implemented based on a novel solution for the corresponding non-isotropic kinetic distribution function (KDF). This is expressed by a Gaussian-like solution that is consistent with relativistic magnetic moment conservation holding in collisionless magnetized plasmas and predicts tensorial equation of state and pressure anisotropy which are specific for these systems. Notably, the same equilibrium admits also a convergent integrable Chapman–Enskog series expansion around a leading-order Juttner distribution, which affords the analytical calculation of continuum fluid fields. In this reference, it is shown that the statistical average of total synchrotron power evaluated over the non-isotropic KDF differs significantly from the corresponding ensemble estimate that would be trivially obtained if the underlying velocity distribution were purely isotropic. It is pointed out that the knowledge of such a statistical discrepancy on the radiation-power curve could provide an independent framework for the characterization of the physical properties of the relativistic plasma state or of the background magnetic field that permeates these astrophysical scenarios.

Emission of twisted photons by a Dirac electron in a strong magnetic field

We study the spontaneous emission of a photon during the transitions between relativistic Landau states of an electron in a constant magnetic field that can reach the Schwinger value of Hc=4.4×109 T. In contrast to the conventional method, in which detection of both the final electron and the photon are implied in a certain basis, here we derive the photon state as it evolves from the process itself. It is shown that the emitted photon state represents a twisted Bessel beam propagating along the field axis with a total angular momentum (TAM) projection onto this axis ℓ−ℓ′, where ℓ and ℓ′ are the TAM of the initial electron and of the final one, respectively. Thus, the majority of the emitted photons turn out to be twisted, with ℓ−ℓ′≳1, even when the magnetic field reaches the critical value of H∼Hc. The transitions without a change of the electron angular momentum, ℓ′=ℓ, are possible, yet much less probable. We also compare our findings with those for a spinless charged particle and demonstrate their good agreement for the transitions without change of the electron spin projection even in the critical fields, while the spin-flip transitions are generally suppressed. In addition, we argue that whereas the ambiguous choice of an electron spin operator affects the differential probability of emission, this problem can partially be circumvented for the photon evolved state because it is the electron TAM rather than the spin alone that defines the TAM of the emitted twisted photon. Published by the American Physical Society 2024

Abnormal states with unequal constituent masses

The Bethe–Salpeter equation for system of two oppositely charged particles not only reproduces the Coulomb spectrum, but, for enough large coupling constant C>π4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal C}>\\frac{\\pi }{4}$$\\end{document}, predicts additional levels not predicted by the Schrödinger equation. These relativistic states (called abnormal), in contrast to the normal ones, are dominated, for more than 90–99%, by Fock states involving the exchange particles—the photons, whereas contribution of two massive charged particles themselves is rather small (1–10%). Since the carrier of a large (positive) charge is a heavy ion, and the negative charge is provided by electron, the masses of two constituents are very different. It is shown that in a system with so different masses the abnormal states still exist. Moreover, the effect of unequal masses is attractive. The balance between photons and charged constituents is weakly sensitive to the mass ratio, so the photons still predominate.

The fermionic entanglement entropy and area law for the relativistic Dirac vacuum state

The fermionic entanglement entropy and area law for the relativistic Dirac vacuum state

Calculation of Relativistic Single-Particle States

Calculation of Relativistic Single-Particle States

Deep anharmonicity on the relativistic spin-0 particles within the spherical regime

We present an oscillator model of relativistic spin-0 charges moving in quantum states with minimal electromagnetic field coupling. Rather than using a perturbative approach, we implemented anharmonicity directly under the integer-dependent levels. In this way, the rest mass energy is kept at 280[Formula: see text]MeV. Within the extended Pekeris approximation, we have also improved the deep approximation to the third and fourth orders near equilibrium at 7.5[Formula: see text]fm with a width range of [Formula: see text]. By taking into account the Morse potential energy, the improved approximation provides a model for the relativistic quantum states of the spatially independent rest mass without an external magnetic field. We considered an extra-energy addition that results in shifted Morse potentials in the depth range of 80–100[Formula: see text]MeV, yielding positive and negative values for particles and antiparticles, respectively. As a result of the shift, it has been concluded that the potential depth of the charged particle affects the relativistic energy levels, where we have found about 200[Formula: see text]MeV for particles and nearly [Formula: see text]10[Formula: see text]MeV for antiparticles. In addition to the negative energy states, the wave functions ([Formula: see text], [Formula: see text]) and ([Formula: see text], [Formula: see text]), which correspond to the energy levels, have been followed by the typical probability form, which shows charge distribution.

Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules.

Materials such as graphene and topological insulators host massless Dirac fermions that enable the study of relativistic quantum phenomena. Single quantum dots and coupled quantum dots formed with massless Dirac fermions can be viewed as artificial relativistic atoms and molecules, respectively. Such structures offer a unique testbed to study atomic and molecular physics in the ultrarelativistic regime (particle speed close to the speed of light). Here we use a scanning tunnelling microscope to create and probe single and coupled electrostatically defined graphene quantum dots to unravel the magnetic-field responses of artificial relativistic nanostructures. We observe a giant orbital Zeeman splitting and orbital magnetic moment up to ~70 meV T-1 and ~600μB (μB, Bohr magneton) in single graphene quantum dots. For coupled graphene quantum dots, Aharonov-Bohm oscillations and a strong Van Vleck paramagnetic shift of ~20 meV T-2 are observed. Our findings provide fundamental insights into relativistic quantum dot states, which can be potentially leveraged for use in quantum information science.

Local thermodynamical equilibrium and relativistic dissipation

We introduce a class of relativistic fluid states satisfying the relativistic local thermodynamical equilibrium postulate [abbreviated as relativistic (LTE) postulate]. States satisfying this postulate are states ``near equilibrium'' (a term defined precisely in the course of the paper), and, they permit one to attach a fictitious ``local thermodynamical equilibrium'' state that fits event by event the actual fluid state. States within this class allow one to single out an admissible class of rest frames relative to which thermodynamical variables like the energy density, thermodynamical pressure, stresses, and particle number density (or densities) measured by observers at rest relative to these frames are becoming frame independent provided second (or higher) order deviations from the fictitious state of local thermodynamical equilibrium are ignored. We have verified this property for a large class of theories of relativistic dissipation that include the Hiscock-Lindblom class of first-order theories, the Eckart and Landau-Lifshitz theories, the Israel-Stewart transient thermodynamics, the Liu-M\"uller-Ruggeri theory, fluids of divergence type, and the latest developed (BDNK) theory. Moreover, for states that are compatible with the relativistic (LTE) postulate, we show that the phenomenological equations describing first-order deviations from any fictitious local thermodynamical equilibrium state satisfy equations that remain form invariant under change of frame within the admissible class of frames. We proved this property for the Hiscock-Lindblom class of first-order theories, the Eckart and Landau-Lifshitz theories, the Israel-Stewart transient thermodynamics, the Liu-M\"uller-Ruggeri theory of relativistic dissipation, and the (BDNK) theory, and we expect the same property to hold for the class of relativistic fluids of divergence type.

Investigation of an energy dependent node creation in graphene quantum states

A quantum state in monolayer graphene can acquire an extra node when its eigenenergy varies relative to the energy range of an applied potential well. In contrast to a non relativistic quantum state, this node creation is not followed by a change in the eigenenergy index. Here, we consider two-component confined states in a cylindrically symmetric graphene system and investigate an energy dependent node creation in a uniform magnetic field. We show that an extra node can be created in one of the two components by increasing the strength of the potential well and the same node can be destroyed by tuning the magnetic field. We demonstrate the importance of the sign of the angular momentum and examine the appropriate conditions which enhance the effect of the node on the quantum states.

Anisotropic Three-Dimensional Quantum Hall Effect and Magnetotransport in Mesoscopic Weyl Semimetals.

Weyl semimetals are emerging to become a new class of quantum-material platform for various novel phenomena. Especially, the Weyl orbit made from surface Fermi arcs and bulk relativistic states is expected to play a key role in magnetotransport, leading even to a three-dimensional quantum Hall effect (QHE). It is experimentally and theoretically important although yet unclear whether it bears essentially the same phenomenon as the conventional two-dimensional QHE. We discover an unconventional fully three-dimensional anisotropy in the quantum transport under a magnetic field. Strong suppression and even disappearance of the QHE occur when the Hall-bar current is rotated away from being transverse to parallel with respect to the Weyl point alignment, which is attributed to a peculiar absence of conventional bulk-boundary correspondence. Besides, transport along the magnetic field can exhibit a remarkable reversal from negative to positive magnetoresistance. These results establish the uniqueness of this QHE system as a novel three-dimensional quantum matter.

Time and Causality: A Thermocontextual Perspective.

The thermocontextual interpretation (TCI) is an alternative to the existing interpretations of physical states and time. The prevailing interpretations are based on assumptions rooted in classical mechanics, the logical implications of which include determinism, time symmetry, and a paradox: determinism implies that effects follow causes and an arrow of causality, and this conflicts with time symmetry. The prevailing interpretations also fail to explain the empirical irreversibility of wavefunction collapse without invoking untestable and untenable metaphysical implications. They fail to reconcile nonlocality and relativistic causality without invoking superdeterminism or unexplained superluminal correlations. The TCI defines a system’s state with respect to its actual surroundings at a positive ambient temperature. It recognizes the existing physical interpretations as special cases which either define a state with respect to an absolute zero reference (classical and relativistic states) or with respect to an equilibrium reference (quantum states). Between these special case extremes is where thermodynamic irreversibility and randomness exist. The TCI distinguishes between a system’s internal time and the reference time of relativity and causality as measured by an external observer’s clock. It defines system time as a complex property of state spanning both reversible mechanical time and irreversible thermodynamic time. Additionally, it provides a physical explanation for nonlocality that is consistent with relativistic causality without hidden variables, superdeterminism, or “spooky action”.