In this paper, we investigate the thermodynamic properties of a set of neutral Dirac particles in the presence of an electromagnetic field in contact with a heat bath for the relativistic and non-relativistic cases. In order to perform the calculations, the high-temperature limit is considered and the Euler-MacLaurin formula is taking into account. Next, we explicitly determine the behavior of the main thermodynamic functions of the canonical ensemble: the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. As a result, we verified that the mean energy and the heat capacity for the relativistic case are two times the values of the non-relativistic case, thus, satisfying the so-called Dulong-Petit law. In addition, we also verified that the Helmholtz free energy and the entropy in both cases increase as a function of the electric field. Finally, we note that there exists no influence on the thermodynamic functions due to the magnetic field.
The present paper is the continuity of the previous papers "Non-linear field theory" I and II. Here on the basis of the electromagnetic representation of Dirac's electron theory we consider the geometrical distribution of the electromagnetic fields of the electron-positron. This gives the posibility to obtain the explanation and solution of many fundamental problems of the QED.
The fundamental solution of the Dirac equation for an electron in an electromagnetic field with harmonic dependence on space-time coordinates is obtained. The field is composed of three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency. Each standing wave consists of two eigenwaves with different complex amplitudes and opposite directions of propagation. The fundamental solution is obtained in the form of the projection operator defining the subspace of solutions to the Dirac equation.
The exact bound state solutions of the Dirac-Pauli equation are studied for an appropriate position-dependent mass function by using the Nikiforov-Uvarov method. For a central electric field having a shifted inverse linear term, all two kinds of solutions for bound states are obtained in closed forms.
High approximations of semiclassical trajectory-coherent states (TCS) and of semiclassical Green function (in the class of semiclassically concentrated states) for the Dirac operator with anomalous Pauli interaction are obtained. For Schrodinger and Dirac operators trajectory-coherent representations are constructed up to any precision with respect to h, h-->0.
It is shown a complex function $Φ$ defined to be the product of a real Gaussian function and a complex Dirac delta function satisfies the Cauchy-Riemann equations. It is also shown these harmonic $Φ$-functions can be included in the solution of the classical electromagnetic field equations to generate the quantum field as a many-particle solution such that the $Φ$-functions represent the particle states. Creation and destruction operators are defined as usual to add or subtract photons from the particle states. The orbital angular momentum of the $Φ$-states is interpreted as spin since it emer…
A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac Quantum Walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a well-known $3D$ walk on the cubic lattice. Recently introduced DQWs on the triangular and honeycomb lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the…
It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of $1/c$ in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreemen…
The fundamental solution of the Dirac equation for an electron in an electromagnetic field with harmonic dependence on space-time coordinates is obtained. The field is composed of three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency. Each standing wave consists of two eigenwaves with different complex amplitudes and opposite directions of propagation. The fundamental solution is obtained in the form of the projection operator defining the subspace of solutions to the Dirac equation. It is illustrated by the analysis of the ground state and the spin p…
Exact solutions are presented of the Dirac equation of a charged particle moving in a classical monochromatic electromagnetic plane wave in a medium of index of refraction n < 1. The found solutions are expressed in terms of new complex trigonometric polynomials, which form a doubly infinite set labeled by two integer quantum numbers. These quantum numbers represent quantized spectra of the energy-momentum components of the charged particle along the polarization vector and along the propagation direction of the applied electromagnetic plane wave field (which is considered as a laser field of …
Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics exists. Given the growing number of applications of high energy electron beams interacting with a variety of quantum systems in laser fields, novel methods for finding exact solutions to the Dirac equation are called for. We present a method for building up solutions to the Dirac equation employing a recently introduced approach for the description of spinorial f…
Localized solutions of the Dirac equation for an electron moving in free space and electromagnetic field lattices with periodic dependence on space-time coordinates (electromagnetic space-time crystals) are treated using the expansions in basis wave functions. The techniques for calculating these functions with any prescribed accuracy are presented. It is shown that in the crystals created by two counterpropagating plane electromagnetic waves with the same or the opposite circular polarizations, the Dirac equation describing the basis functions reduces to matrix ordinary differential equations…
A system evolving under the driven Jaynes--Cummings model will undergo a phase transition at a critical driving field amplitude. This transition is foreshadowed by a collapse of the quasienergy level spectra of the system and remains present as the model is extended to include a counter-rotating interaction. We study this critical response and obtain the eigenvalues and eigenstates of the extended model by presenting a correspondence between the Jaynes--Cummings model and a charged Dirac particle subject to an external electromagnetic field. Under this correspondence, the field and two-level s…
The family of solutions to the Dirac equation for an electron moving in an electromagnetic lattice with the chiral structure created by counterpropagating circularly polarized plane electromagnetic waves is obtained. At any nonzero quasimomentum, the dispersion equation has two solutions which specify bispinor wave functions describing electron states with different energies and mean values of momentum and spin operators. The inversion of the quasimomentum results in two other linearly independent solutions. These four basic wave functions are uniquely defined by eight complex scalar functions…
The quantum mechanical states of the neutral particle endowed with a magnetic moment in the combination of electromagnetic vortex field together with the constant magnetic field are dealt with. It is shown that this system of fields is capable to capture the particle in the perpendicular direction and guide it along the propagating wave. The quantum evolution is subject to tunneling processes, which can destroy the delicate trapping mechanism. The probability of these processes show that it should be in principle possible to catch and guide the particle for the time of several microseconds. Th…
In the previous papers (quant-ph/0204037), (quant-ph/0204134) on the basis of Dirac's equation we have considered the electromagnetic interpretation of the quantum theory of electron. Here we continue the electron structure study. Since the Dirac equation is also the equation of the other leptons, in the present paper from the electromagnetic point of view we analyse the structure of neutrino. We will show that not only the electron but also all the other leptons have the electromagnetic structure.
The Dirac equation is invariant under rotations with a constant frequency and invariable cylindrical radius. 3D transformation for rotating frames is found with help of this invariance. Exact localized solutions of the Dirac equation in the field of a traveling circularly polarized electromagnetic wave and constant magnetic field exist and possess unusual properties. Temporal changes of spin pertaining to the solutions are studied.
Based on the fluctuation-electromagnetic theory, we have calculated the retarded force of attraction, frictional moment and heating rate of a neutral particle rotating near a polarizable surface. The particle and surface are characterized by arbitrary material properties and temperatures. The surface is assumed to be in thermal equilibrium with vacuum environment.
This paper presents new analytic solutions to the Dirac equation employing a recently introduced method that is based on the formulation of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. A first family of solutions describe the shape-preserving translation of a wavepacket along any desired trajectory in the x-y plane. In particular, we show that the dispersionless motion of a Gaussian wavepacket along both elliptical and circular paths can be achieved with rather simple electromagnetic field configurations. A second family of solutions involves a plan…
Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator
Analogous to Landau quantization related to a neutral particle possessing an electric quadrupole moment, we generalize such Landau quantization to include position-dependent mass (PDM) neutral particles. Using cylindrical coordinates, the exact solvability of PDM neutral particles with an electric quadrupole moment moving in electromagnetic fields is reported. The interaction between the electric quadrupole moment of a PDM neutral particle and a magnetic field in the absence of an electric field is analyzed for two different radial cylindrical PDM settings. Next, two particular cases of radial…
We consider Pauli--Dirac fermion submitted to an inhomogeneous magnetic field. It is showed that the propagator of the neutral Dirac particle with an anomalous magnetic moment in an external linear magnetic field is the causal Green function $S^{c}(x_{b},x_{a})$ of the Pauli--Dirac equation. The corresponding Green function is calculated via path integral method in global projection, giving rise to the exact eigenspinors expressions. The neutral particle creation probability corresponding to our system is analyzed, which is obtained as function of the introduced field $B'$ and the additional s…
We present a self-consistent mean-field model based on a two-component Pauli-like equation that incorporates quantum and relativistic effects (up to second-order in 1/c) for both external and internal electromagnetic fields. By taking the semi-relativistic limit of the Dirac-Maxwell equations in the presence of an external electromagnetic field we obtain an analytical expression of a coherent light-induced mean-field Hamiltonian. The latter exhibits several mechanisms that involve the internal mean fields created by all the electrons and the external electromagnetic field (laser). The role pla…
We analyse the electromagnetic coupling in the Kemmer-Duffin-Petiau (KDP) equation. Since the KDP--equation which describes spin-0 and spin-1 bosons is of Dirac-type, we examine some analogies and differences from the Dirac equation. The main difference to the Dirac equation is that the KDP equation contains redundant components. We will show that as a result certain interaction terms in the Hamilton form of the KDP equation do not have a physical meaning and will not affect the calculation of physical observables. We point out that a second order KDP equation derived by Kemmer as an analogy t…
We discuss a geometric approach to confining a Dirac neutral particle with a permanent magnetic dipole moment interacting with external fields to a hard-wall confining potential in the Minkowski spacetime through noninertial effects. We discuss the behaviour of external fields induced by noninertial effects, and a case where relativistic bound states can be achieved in analogous way to having a Dirac particle confined to a quantum dot. We show that this confinement of a Dirac neutral particle analogous to a quantum dot arises from noninertial effects that give rise to the geometry of the manif…
We present a new step in the foundation of quantum field theory with the tools of scale relativity. Previously, quantum motion equations (Schrödinger, Klein-Gordon, Dirac, Pauli) have been derived as geodesic equations written with a quantum-covariant derivative operator. Then, the nature of gauge transformations, of gauge fields and of conserved charges have been given a geometric meaning in terms of a scale-covariant derivative tool. Finally, the electromagnetic Klein-Gordon equation has been recovered with a covariant derivative constructed by combining the quantum-covariant velocity operat…
The relativistic semiclassical evolution of the position of an electron in the presence of an external electromagnetic field is studied in terms of a Newton equation that incorporates spin effects directly. This equation emerges from the Dirac equation and allows the identification of scenarios where spin effects are necessary to understand the main characteristics of the electron trajectories. It involves the eigenvalues of the non-Hermitian operator $Σ_{μν}F^{μν}$ with $Σ_{μν}$ and $F^{μν}$ as the spin and electromagnetic tensors. The formalism allows a deeper understanding on the physics be…
We study the relations between massless Dirac fermions in an electromagnetic field and atoms in quantum optics. After getting the solutions of the energy spectrum, we show that it is possible to reproduce the 2D Dirac Hamiltonian, with all its quantum relativistic effects, in a controllable system as a single trapped ion through the Jaynes--Cummings and anti-Jaynes--Cummings models. Also we show that under certain conditions the evolution of the Dirac Hamiltonian provides us with Rashba spin-orbit and linear Dresselhaus couplings. Considering the multimode multiphoton Jaynes-Cummings model int…
In a recent work, we have shown that all solutions to the Weyl equation and a special class of solutions to the Dirac equation are degenerate in the sense that they remain unaltered under the influence of a wide variety of different electromagnetic fields. In this study, our previous work is significantly extended, providing a wide class of degenerate solutions to the Dirac equation for massless particles. The electromagnetic fields corresponding to these solutions are calculated and examples regarding both spatially constant electromagnetic fields and electromagnetic waves are also provided. …
We show that the de Haas van Alphen effect can be induced in a two dimensional atomic gas by the He-McKellar-Wilkens interaction mediated via an electric dipole moment. Under an appropriate field-dipole configuration, we show that the neutral atoms subject to a synthetic magnetic field arrange themselves in Landau levels. An experimental arrangement for observation of the atomic analog of dHvA oscillations is proposed. In a strong effective magnetic field regime we present the quantum oscillations in the energy and effective magnetization of the two dimensional atomic gas. From the dHvA period…
A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show how to achieve dispersionless rotation and translation of wave packets. Additionally, this formalism can handle control interactions beyond electromagnetic. This work reveals unexpected flexibility of the Dirac equation for control applications, which may open new prospects for quantum technologies.
In this paper we study in detail the connection between the solutions to the Dirac and Weyl equation and the associated electromagnetic 4-potentials. First, it is proven that all solutions to the Weyl equations are degenerate, in the sense that they correspond to an infinite number of electromagnetic 4-potentials. As far as the solutions to the Dirac equation are concerned, it is shown that they can be classified into two classes. The elements of the first class correspond to one and only one 4-potential, and are called non-degenerate Dirac solutions. On the other hand, the elements of the sec…
The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schrödinger-Pauli wave equation and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the …
In a recent work we have proven the existence of degenerate solutions to the Dirac equation, corresponding to an infinite number of different electromagnetic fields, providing also some examples regarding massless particles. In the present article our results are extended significantly, providing degenerate solutions to the Dirac equation for particles with arbitrary mass, which, under certain conditions, could be interpreted as pairs of particles (or antiparticles) moving in a potential barrier with energy equal to the height of the barrier and spin opposite to each other. We calculate the el…
Dirac's oscillator (DO) is one of the most studied systems in the Relativistic Quantum Mechanics and in the physical-mathematics. In particular, we show that this system has an unique property which it has not ever seen in other known systems: According to its spatial dimensionality, DO represent physical systems with very different electromagnetic nature. So far in the literature, it has been proved using the covariant method the gauge invariance of the Dirac's oscillator potential. It has also shown that in (3+1)dimensions the DO represents a relativistic and electrically neutral fermion wit…
We perform a reduction from three to two spatial dimensions of the physics of a spin-1/2 fermion coupled to the electromagnetic field, by applying Hadamard's method of descent. We consider first the free case, in which motion is determined by the Dirac equation, and then the coupling with a dynamical electromagnetic field, governed by the Dirac-Maxwell equations. We find that invariance along one spatial direction splits the free Dirac equation in two decoupled theories. On the other hand, a dimensional reduction in the presence of an electromagnetic field provides a more complicated theory in…
A new approach is proposed for an electromagnetic field geometrisation. We show that interacting Maxwell and Dirac fields can be considered as a single connected space-time 4-manifold. The Dirac spinors appear wihtin such approach as basic fanctions for the manifold fundamental group representation and electric and magnertic fields appear as components of a curvature tensor of the manifold covering space.
We show that a dilute 2-species gas of Fermi-Dirac alkali-metal atoms in a periodic optical lattice may exhibit fractionization of particle number when the two components are coupled via a coherent electromagnetic field with a topologically nontrivial phase profile. This results in fractional eigenvalues of the spin operator with vanishing fluctuations. The fractional part can be accurately controlled by modifying the effective detuning of the electromagnetic field.
The Dirac Hamiltonian $H^{\left(D\right)}$ for relativistic charged fermions minimally coupled to (possibly time-dependent) electromagnetic fields is transformed with a purpose-built flow equation method, so that the result of that transformation is unitary equivalent to $H^{\left(D\right)}$ and granted to strive towards a limiting value $H^{\left(NW\right)}$ commuting with the Dirac $β$-matrix. Upon expansion of $H^{\left(NW\right)}$ to order $\frac{v^2}{c^2}$ the nonrelativistic Hamiltonian $H^{\left(SP\right)}$ of Schrödinger-Pauli quantum mechanics emerges as the leading order term adding …
The generalized Dirac equation of the second order, describing particles with spin 1/2 and two mass states, is analyzed. The projection operators extracting states with definite energy and spin projections are obtained. The first order generalized Dirac equation in the 20-dimensional matrix form and the relativistically invariant bilinear form are derived. We obtain the canonical energy-momentum tensor and density of the electromagnetic current expressed through the 20-component wave function. Minimal and non-minimal electromagnetic interactions of fermions are considered, and the quantum-mech…
The Dirac equation for massive free electrically neutral spin 1/2 particles in a gravitation field is considered. The secondary quantization procedure is applied to it and the Hilbert space of multiparticle quantum states is constructed.
