Relativistic Quantum Mechanics Research Articles

Magnetic susceptibility of Dirac fermions, Bi-Sb alloys, interacting Bloch fermions, dilute nonmagnetic alloys, and Kondo alloys

Wide ranging interest in Dirac Hamiltonian is due to the emergence of novel materials, namely, graphene, topological insulators and superconductors, the newly-discovered Weyl semimetals, and still actively-sought after Majorana fermions in real materials. We give a brief review of the relativistic Dirac quantum mechanics and its impact in the developments of modern physics. The quantum band dynamics of Dirac Hamiltonian is crucial in resolving the giant diamagnetism of bismuth and Bi-Sb alloys. Quantitative agreement of the theory with the experiments on Bi-Sb alloys has been achieved, and physically meaningful contributions to the diamagnetism has been identified. We also treat relativistic Dirac fermion as an interband dynamics in uniform magnetic fields. For the interacting Bloch electrons, the role of translation symmetry for calculating the magnetic susceptibility avoids any approximation to second order in the field. The expressions for magnetic susceptibility of dilute nonmagnetic alloys give a firm theoretical foundation of the empirical formulas used in fitting experimental results. The unified treatment of all the above calculations is based on the lattice Weyl-Wigner formulation of discrete phase-space quantum mechanics. For completeness, the magnetic susceptibility of Kondo alloys is also given since Dirac fermions in conduction band and magnetic impurities exhibit Kondo effect.

Which Proof We Have against Continuous Trajectories for Particles?

It is in general accepted that the concept of continuous trajectories for particles is at odds with the relativistic quantum mechanics. Namely, when examining the evolution of entangled quantum objects according to frames of coordinates in relative move-ment, one gets contradictory trajectories. Such a situation is typically derived from the famous “Hardy’s paradox”. However, it is argued here that if the rationale ignores the principle of quantum contextuality, as happens typically when using Hardy’s thought-experiment, the conclusion—rejection of the assumption of trajectories—is questionable. The issue is exemplified by an additional example: the 101 property of spin 1 bosons implies conflicting trajectories when the singlet state of two such bosons is examined according to frames in relative movement. It is concluded here that in the absence of a rationale which doesn’t violate the quantum contextuality, there are no sufficient arguments for refuting the possibility of a substructure of the quantum mechanics, consisting in particles following continuous trajectories.

From Space-Time Quantization to Dark Matter

We generalize relativistic quantum mechanics and the Standard Model of elementary particle physics by considering a finite limit for the smallest measurable length. The resulting theory of Space-Time Quantization is logically consistent and accounts for all possible particle states by means of four new quantum numbers. They specify possible variations of wave functions at the smallest possible scale in space and time, while states of motion are defined by their large-scale variations. This theory also provides insight into the nature and properties of dark matter particles. It can facilitate their detection and identification because of a very strict conservation law.

Matter-induced magnetic moment and neutrino helicity rotation in external fields

The induced magnetic moment that arises due to the propagation of neutrinos in a dispersive medium can affect the dynamics of the neutrino spin in an external electromagnetic field. In particular, it can cause a helicity flip of a massive neutrino in a magnetic field. In some astrophysical media, this helicity transition mechanism could be more effective than a similar process caused by the anomalous magnetic moment of the neutrino. If the neutrino energy is sufficiently high, the two helicity transition mechanisms mentioned above can compensate each other. Then a helicity flip in an external field will not occur. Calculations are carried out using both the methods of relativistic quantum mechanics and the quasiclassical Bargmann-Michel-Telegdi equation.

UNIVERSE IN A BLACK HOLE IN EINSTEIN–CARTAN GRAVITY

ABSTRACT The conservation law for the angular momentum in curved spacetime, consistent with relativistic quantum mechanics, requires that the antisymmetric part of the affine connection (torsion tensor) is a variable in the principle of least action. The coupling between the spin of elementary particles and torsion in the Einstein–Cartan theory of gravity generates gravitational repulsion at extremely high densities in fermionic matter, approximated as a spin fluid, and thus avoids the formation of singularities in black holes. The collapsing matter in a black hole should therefore bounce at a finite density and then expand into a new region of space on the other side of the event horizon, which may be regarded as a nonsingular, closed universe. We show that quantum particle production caused by an extremely high curvature near a bounce can create enormous amounts of matter, produce entropy, and generate a finite period of exponential expansion (inflation) of this universe. This scenario can thus explain inflation without a scalar field and reheating. We show that, depending on the particle production rate, such a universe may undergo several nonsingular bounces until it has enough matter to reach a size at which the cosmological constant starts cosmic acceleration. The last bounce can be regarded as the big bang of this universe.

Separation of Dirac's Hamiltonian by Van Vleck transformation

ABSTRACTThe now classic Foldy–Wouthuysen transformation (FWT) was introduced as successive unitary transformations. This fundamental idea has become the standard in later developments such as the Douglas–Kroll transformation (DKT) – but it is not the only possibility. FWT can be seen as a simple special case of the general Van Vleck transformation (VVT) which besides the successive version has another, known as the canonical because of a series of nice mathematical properties discovered gradually over time. The aim of the present paper is to compare the two approaches – which give identical results in the lower orders, but not in the higher. After having recapitalised both, we apply them to Dirac's Hamiltonian for the electron in a constant electromagnetic field, written with so few assumptions about the operators that the mathematical techniques stand out separated from the terminology of relativistic quantum mechanics. FWT for a free particle is dealt with by a recent geometric approach to VVT. The original FWT is continued through the next non-zero orders. DKT is considered with special weight on equivalent formulations of the generalised and the optimised forms introduced by Wolf, Reiher and Hess.

Zero-mass limit of a Dirac spinor with general spin orientation

The helicity eigenstates that describe fermions with a special spin orientation (parallel or antiparallel to the direction of momentum) provide a considerable simplification in calculations. Hence, it is generally preferred to use the helicity basis during calculations in relativistic quantum mechanics or the quantum field theory. Possibly for the above reason, Dirac spinors describing a general spin orientation have been ignored in many textbooks. Although the helicity eigenstates give an almost complete understanding of the behavior of the free Dirac solutions, the zero-mass limit is one of its exceptions. The zero-mass behavior of the free spinor with general spin orientation and its relation with chirality eigenstates has been skipped in textbooks and hence it deserves a clear, detailed investigation. In this paper we obtain the free Dirac spinors describing a general spin orientation and examine their zero-mass limit. We also briefly discuss some of the implications of this small-mass behavior of the spinors on particle physics.

Bound States for Pseudoharmonic Potential of the Dirac Equation with Spin and Pseudo-Spin Symmetry via Laplace Transform Approach

In the field of relativistic quantum mechanics, the Dirac equation plays an important role for spin-1/2 particles. This equation has been used extensively to study the relativistic heavy ion collisions, heavy ion spectroscopy and more recently in laser–matter interaction [1] and condensed matter physics [2]. The most interesting feature for the Dirac equation is the concept of spin and pseudospin symmetries. Although these symmetries of the Dirac Hamiltonian were discovered long ago, there has been a renewed interest in obtaining the solutions of the Dirac equations for some typical potentials under spin symmetry and pseudo-spin symmetry cases. The idea about spin symmetry and pseudo-spin symmetry with the nuclear shell model has been introduced in [3]. This idea has been widely used in explaining a number of phenomena in nuclear physics and related areas. Spin and pseudo-spin symmetric concepts have been used in the studies of certain aspects of deformed and exotic nuclei. Spin symmetry is relevant to meson with one heavy quark, which is being used to explain the absence of quark spin–orbit splitting (spin doublets) observed in heavy-light quark mesons [4] and pseudo-spin symmetry concept has been successfully used to explain different phenomena in nuclear structure including deformation, superdeformation, identical bands, exotic nuclei, and degeneracies of some shell model orbitals in nuclei (pseudospin doublets) [5, 6]. In recent times, many works have been done to solve the Dirac equation to obtain the bound states energy spectra and the corresponding eigenfunctions. Ginocchio [7–12] deduced that a Dirac Hamiltonian with scalar S(r) and vector V (r) harmonic oscillator potentials when V (r) = S(r) possesses a spin symmetry as well as a U(3) symmetry, whereas a Dirac

Lorentz-covariant quantum 4-potential and orbital angular momentum for the transverse confinement of matter waves

In two recent papers exact Hermite-Gaussian solutions to relativistic wave equations have been obtained for both electromagnetic and particle beams that include Gouy phase. The solutions for particle beams correspond to those of the Schr\"{o}dinger equation in the non-relativistic limit. Here, distinct canonical and kinetic 4-momentum operators will be defined for quantum particles in matter wave beams. The kinetic momentum is equal to the canonical momentum minus the fluctuating terms resulting from the transverse localization of the beam. Three results are obtained. First, the total energy of a particle for each beam mode is calculated. Second, the localization terms couple into the canonical 4-momentum of the beam particles as a Lorentz covariant quantum 4-potential originating at the waist. The quantum 4-potential plays an analogous role in relativistic Hamiltonian quantum mechanics to the Bohm potential in the non-relativistic quantum Hamilton-Jacobi equation. Third, the orbital angular momentum (OAM) operator must be defined in terms of canonical momentum operators. It is further shown that kinetic 4-momentum does not contribute to OAM indicating that OAM can therefore be regarded as a pure manifestation of quantum 4-potential.

Optimization of a relativistic quantum mechanical engine.

We present an optimal analysis for a quantum mechanical engine working between two energy baths within the framework of relativistic quantum mechanics, adopting a first-order correction. This quantum mechanical engine, with the direct energy leakage between the energy baths, consists of two adiabatic and two isoenergetic processes and uses a three-level system of two noninteracting fermions as its working substance. Assuming that the potential wall moves at a finite speed, we derive the expression of power output and, in particular, reproduce the expression for the efficiency at maximum power.

Spacetime replication of continuous variable quantum information

The theory of relativity requires that no information travel faster than light, whereas the unitarity of quantum mechanics ensures that quantum information cannot be cloned. These conditions provide the basic constraints that appear in information replication tasks, which formalize aspects of the behavior of information in relativistic quantum mechanics. In this article, we provide continuous variable (CV) strategies for spacetime quantum information replication that are directly amenable to optical or mechanical implementation. We use a new class of homologically constructed CV quantum error correcting codes to provide efficient solutions for the general case of information replication. As compared to schemes encoding qubits, our CV solution requires half as many shares per encoded system. We also provide an optimized five-mode strategy for replicating quantum information in a particular configuration of four spacetime regions designed not to be reducible to previously performed experiments. For this optimized strategy, we provide detailed encoding and decoding procedures using standard optical apparatus and calculate the recovery fidelity when finite squeezing is used. As such we provide a scheme for experimentally realizing quantum information replication using quantum optics.

Euclidean Closed Linear Transformations of Complex Spacetime and generally of Complex Spaces of dimension four endowed with the Same or Different Metric

Relativity Theory and the corresponding Relativistic Quantum Mechanics are the fundamental theories of physics. Special Relativity (SR) relates the frames of Relativistic Inertial observers (RIOs), through Linear Spacetime Transformation (LSTT) of linear spacetime. Classic Special Relativity uses real spacetime endowed with Lorentz metric and the frames of two RIOs with parallel spatial axes are always related through Lorentz Boost (LB). This cancels the transitive attribute in parallelism, when three RIOs are related, because LB is not closed transformation, causing Thomas Rotation. In this presentation, we consider closed LSTT of Complex Spacetime, so there is no necessity for spatial axes rotation and all the frames are chosen having parallel spatial axes. The solution is expressed by a 4x4 matrix (Λ) containing components of the complex velocity of one Observer wrt another and two functions depended by the metric of Spacetime. Demanding isometric transformation, it emerges a class of metrics that are in accordance with the closed LSTT and the transformation matrix contains one parameter ω depended by the metric of Spacetime. In case that we relate RIOs with steady metric, it emerges one steady number (ωI) depended by the metric of Spacetime of the specific SR. If ωI is an imaginary number, the elements of the Λ are complex numbers, so the corresponding spacetime is necessarily complex and there exists real Universal Speed (UI). The specific value ωI=±i gives Vossos transformation (VT) endowed with Lorentz metric (for gii=1) of complex spacetime and invariant spacetime interval (or equivalently invariant speed of light in vacuum), which produce the theory of Euclidean Complex Relativistic Mechanics (ECRMs). If ωI is a real number (ωI#0) the elements of the Λ are real numbers, so the corresponding spacetime is real, but there exist imaginary UI. The specific value ωI=0 gives Galileo Transformation (GT) with the invariant time, in which any other closed LSTT is reduced, if one RIO has small velocity wrt another RIO. Thus, we have infinite number of closed LSTTs, each one with the corresponding SR theory. In case that we relate accelerated observers with variable metric of spacetime, we have the case of General Relativity (GR). For being that clear, we produce a generalized Schwarzschild metric, which is in accordance with any SR based on this closed complex LSTT and Einstein equations. The application of this kind of transformations to the SR and GR is obvious. But, the results may be applied to any linear space of dimension four endowed with steady or variable metric, whose elements (four- vectors) have spatial part (vector) with Euclidean metric.

Quantum methodologies in Helmholtz optics

It is well-known that the Helmholtz equation describing scalar optics has a striking mathematical similarity with the Klein–Gordon equation of relativistic quantum mechanics. Exploiting this similarity, quantum methodologies are applied to the Helmholtz equation leading to a new formalism of scalar optics. Paraxial and aberrating Hamiltonians are derived for a general spatially varying refractive index. The beam-optical Hamiltonians thus derived give rise to wavelength-dependent contributions. Example of the graded-index medium is covered in detail. Explicit formulae relating the incident and emergent light rays for the general case of a thick lens are also presented.

Consistency of multi-time Dirac equations with general interaction potentials

In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac’s multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.

Neutrino induced magnetic moment and spin precession

When propagating through a dispersing medium, a massive neutrino acquires an induced magnetic moment that may give rise to a helicity flip in an external magnetic field with a larger probability than that caused by the anomalous magnetic moment. This phenomenon is investigated in the framework of relativistic quantum mechanics and of the generalized Bargmann–Michel–Telegdi equation.

Dirac and Weyl Materials: Fundamental Aspects and Some Spintronics Applications

Dirac and Weyl materials refer to a class of solid materials which host low-energy quasiparticle excitations that can be described by the Dirac and Weyl equations in relativistic quantum mechanics. Starting with the advent of graphene as the first prominent example, these materials have been attracting tremendous interest owing to their novel fundamental properties as well as the great potential for applications. Here we introduce the basic concepts and notions related to Dirac and Weyl materials and briefly review some recent works in this field, particularly on the conceptual development and the possible spintronics/pseudospintronics applications.

An extended Dirac equation in noncommutative spacetime

Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.

The role of shape invariance potentials in the relativistic quantum mechanics

The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We want to show that some of the non-relativistic solvable potentials with shape-invariant symmetry can be related to the radial Dirac equation. Using this method, the idea of supersymmetry and shape invariance can be expanded to the relativistic quantum mechanics. The spinor wave functions for some of the obtained four-component electromagnetic potential are given in terms of special functions such as Jacobi, generalized Laguerre and Hermite polynomials. The relativistic bound-states spectrum for each case is also calculated in terms of the bound-states spectrum of the solvable potentials.

Peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus

We consider the peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus and discuss the long history of an incorrect interpretation of this problem in the case of a pointlike nucleus and its current correct solution. We consider the spectral problem in the case of a regularized Coulomb potential. For some special regularizations, we derive an exact equation for the point spectrum in the energy interval (-m,m) and find some of its solutions numerically. We also derive an exact equation for charges yielding bound states with the energy E = -m; some call them supercritical charges. We show the existence of an infinite number of such charges. Their existence does not mean that the oneparticle relativistic quantum mechanics based on the Dirac Hamiltonian with the Coulomb field of such charges is mathematically inconsistent, although it is physically unacceptable because the spectrum of the Hamiltonian is unbounded from below. The question of constructing a consistent nonperturbative second-quantized theory remains open, and the consequences of the existence of supercritical charges from the standpoint of the possibility of constructing such a theory also remain unclear.

Logical inference approach to relativistic quantum mechanics: Derivation of the Klein–Gordon equation

The logical inference approach to quantum theory, proposed earlier De Raedt et al. (2014), is considered in a relativistic setting. It is shown that the Klein–Gordon equation for a massive, charged, and spinless particle derives from the combination of the requirements that the space–time data collected by probing the particle is obtained from the most robust experiment and that on average, the classical relativistic equation of motion of a particle holds.