Relativistic Quantum Mechanics Research Articles

On the bosonic symmetries of the Dirac equation with nonzero mass

Bosonic symmetries of the Dirac equation with nonzero mass, which existence is under consideration after our publications in the years 2011–2015, are proved here on the basis of two different methods. The first one appeals to the 64-dimensional gamma matrix representation of the Clifford algebra Cℓℝ(0,6) over the field of real numbers and the 28-dimensional gamma matrix representation of the algebra SO(8) (over the field of real numbers as well). The second way of proof is based on the start from the relativistic canonical quantum mechanics of spin (1,0) particle multiplet and its relationship with the Dirac equation, which is given by the extended Foldy-Wouthuysen transformation suggested by us in 2014–2017. Both the Lorentz and Poincaré bosonic symmetries are considered. The 31-dimensional algebra of invariance is found. The bosonic solutions and conservation laws are found as well. The considered phenomenon is called the Fermion-Boson duality of the Dirac equation according to P. Garbaczewski’s titles suggested in 1986.

Classical and Quantum Mechanical Calculations of the Stacking Interaction of NdIII Complexes with Regular and Mismatched DNA Sequences.

The design of organometallic complexes used as selective intercalators to bind and react at DNA mismatch sites has concentrated efforts in the last few years. In this context, lanthanides have received attention to be employed as active optical centers due to their spectroscopic properties. Despite the fact that there are several experimental data about synthesis and DNA binding of these compounds, theoretical analyses describing their interaction with DNA are scarce. To understand the binding to regular and mismatched DNA sequences as well as to determine the effect of the intercalation on the spectroscopic properties of the complexes, a complete theoretical study going from classical to relativistic quantum mechanics calculations has been performed on some lanthanide complexes with phenanthroline derivatives synthesized and characterized herein, viz. [Nd(NO3)3(H2O)(dppz-R)] with R = H, NO2-, CN- and their [Nd(NO3)3(H2O)(dpq)] analogue, which was computationally modeled. The results were in correct agreement with the available experimental data showing that dppz complexes have higher binding affinities to DNA than dpq one and supporting the idea that these complexes are not selective to mismatch sites in the sampled time scale. Finally, the spectroscopic analysis evidence an intercalative binding mode and made possible the elucidation of the emission mechanism of these systems. This approach is proposed as a benchmark study to extend this methodology on similar systems and constitutes the first theoretical insight in the interaction between DNA and lanthanide complexes.

Magnetic dipole moment in relativistic quantum mechanics

We investigate relativistic quantum mechanics (RQM) for particles with arbitrary magnetic moment. We compare two well known RQM models: a) Dirac equation supplemented with an incremental Pauli term (DP); b) Klein-Gordon equations with full Pauli EM dipole moment term (KGP). We compare exact solutions to the external field cases in the limit of weak and strong (critical) fields for: i) homogeneous magnetic field, and ii) the Coulomb $1/r$-potential. For i) we consider the Landau energies and the Landau states as a function of the gyromagnetic factor ($g$-factor). For ii) we investigate contribution to the Lamb shift and the fine structure splitting. For both we address the limit of strong binding and show that these two formulations grossly disagree. We discuss possible experiments capable of distinguishing between KGP and DP models in laboratory. We describe impact of our considerations in astrophysical context (magnetars). We introduce novel RQM models of magnetic moments which can be further explored.

Quantum walk search on the complete bipartite graph

The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of $N$ vertices for one of $k$ marked vertices with different initial states. We prove intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one timestep to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches 1/2, but if they are in different partite sets, it instead reaches $1$. This differs from the complete graph, which is the quantum walk formulation of Grover's algorithm, where the success probability with two marked vertices is 8/9. This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schr\"odinger's equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at sub-nucleon distance scales. Hadronic states in different inertial reference frames are needed to compute current matrix elements that probe hadronic structure and dynamics. Relativistic invariance is an important consideration as the resolution of the probe is increased. Many different treatments of relativistic dynamics are used in practice, including Poincar\'e covariant methods, Lorentz covariant methods, Euclidean covariant methods and methods based on quantum fields. Wave functions are typically matrix elements of interacting relativistic states in a basis of non-interacting relativistic states. The purpose of this work is to develop the relation between these different representations of relativistic states that are used in different applications from a unified point of view, starting with positive mass irreducible representations of the Poincar\'e group.

Ground state binding energies and RMS radii of Λ in hypernuclei in the presence of spin-dependent potential

In recent years, the physics of hypernuclei has been in progress dramatically both in experiments and theories. In this regard, there have been considerable interests in the study of [Formula: see text]-particle 1s state in hypernuclei using relativistic quantum mechanics. Many theoretical results have been obtained numerically in the relativistic frameworks. In this work, using the relativistic Dirac equation, we study the properties of [Formula: see text]-particle in the hypernuclei analytically and, specifically, we determine its binding energies of ground state in a single [Formula: see text]-hypernuclei. The [Formula: see text]-N potential is considered as a spin-dependent term and the radial dependence of potential is approximated by a Gaussian function. For a more complete analysis, we will also compute the hypernuclei mass, its wave function and the root mean square (RMS) radii of [Formula: see text] orbits in [Formula: see text]-hypernuclear system.

Quadrupole moments of the ρ meson and the S -wave deuteron and some general constraints on quadrupole moments of spin-1 systems

We costruct the relativistic operator of the quadrupole moment of two-particle composite spin one systems with zero orbital moment of the relative motion and derive explicit analytical expression for the quadrupole moment using the approach to relativistic composite systems based on our version of the instant-form relativistic quantum mechanics (RQM). We calculate the quadrupole moments of the rho meson and of the S-wave deuteron without any free parameters, using our unified pi&rho model (Phys. Rev.D 93, 036007 (2016); 97, 033007 (2018)) and our previous results on deuteron. Our calculation gives Q_rho=-0.158+-0.04 GeV^-2 and Q_d=-1.4*10^-4 GeV^-2. Having in our disposition the rather general form of the quadrupole-moment operator we for the first time formulate the problem of the upper and lower bounds for possible values of the quadrupole moment of a two-particle system with indicated quantum numbers for a large range of constituent masses, and partially solve it.

Compton Scattering of \u03b3-Ray Vortex with Laguerre Gaussian Wave Function

In this work, we report calculation for Compton scattering of a γ-ray vortex with a wave function of Laguerre Gaussian on an electron in the framework of the relativistic quantum mechanics. We consider the coincidence measurement of the scattered photon and the scattered electron from each Compton scattering. The momentum of the scattered photon distributes outside of the reaction plane determined by the incident photon and the scattered electron, and the energy of the scattered photon also distributes, when the scattered angle of the electron is simultaneously measured. These distributions depend on the angular momentum and the node number of the Laguerre Gaussian function of the incident photon. Thus, the coincident measurement for Compton scattering is useful to identify the nature of the vortex photon wave function.

Two body relativistic wave equations

The relativistic quantum mechanics of two interacting particles is considered. We first present a covariant formulation of kinematics and of reduced phase space, giving a short outline of the classical results. We then quantize the systems for the scalar–scalar, fermion–scalar and fermion–fermion cases. We study the spectrum and the spherical waves solutions of the free case. The interaction with central scalar and vector potentials is introduced and the explicit equations are deduced. The one particle and the non relativistic limits are recovered and the general lines for the solution of the boundary value problems are given. We make a numerical analysis of the first two cases with Coulomb interaction. For the two fermions we largely revisit the model we had previously derived in order to uniformize the description for all the three cases. In order to give a complete review we report in the Appendix some of the most interesting results obtained for atomic and mesonic systems with Coulomb and Cornell potential interactions respectively.

De Broglie clock, electron channeling, and time in quantum mechanics

De Broglie’s association of a wave to particles is a fundamental concept in the quantum mechanical description of nature. The wave oscillation is referred to alternatively as the “de Broglie clock”, the “Compton clock”, or the “de Broglie periodic phenomenon”. In the present paper it is shown that Dirac’s relativistic quantum mechanics, complemented with the dynamical time operator recently introduced, provides a consistent theoretical description of: (i) the generation of the de Broglie wave through Lorentz boosts; and (ii) the characteristics of the resonance observed in electron channeling through thin crystals as responding to both the periodicity derived from the adjustment of the de Broglie period to the crystal interatomic distance (resonance energy) and the periodicity of the predicted trembling motion (Zitterbewegung). One can conclude that the channeling experiments provide the first direct evidence of the electron Zitterbewegung, and that the de Broglie period is an intrinsic property of matter arising from a self-adjoint dynamical time operator.

Relativistic Quantum Mechanics of the Majorana Particle

This article is a pedagogical introduction to relativistic quantum mechanics of the free Majorana particle. This relatively simple theory differs from the well-known quantum mechanics of the Dirac particle in several important aspects. We present its three equivalent formulations. Next, so called axial momentum observable is introduced, and general solution of the Dirac equation is discussed in terms of eigenfunctions of that operator. Pertinent irreducible representations of the Poincar\'e group are discussed. Finally, we show that in the case of massless Majorana particle the quantum mechanics can be reformulated as a spinorial gauge theory.

Quantum Interference without Quantum Mechanics

A recently proposed model of the Dirac electron, which describes observed properties of the particle correctly, is in the present paper shown to be also able to explain quantum interference by classical probabilities. According to this model, the electron is not point-like, but rather an "entity" formed by a fast periodic motion of a quantum whose energy is equal to the rest energy of the electron. Only after a time span equal to the period of that periodic motion the "entity" becomes the electron, with its properties, mass, spin, charge, etc.. When in motion with respect to the observer, the "dynamic substructure" of the electron described in this way, leads to a certain time structure of its detection probability, if the space-time point of detection is taken as the space-time point of the quantum. In the typical "two slit" experimental situation, this leads to a periodic detection probability with a frequency of twice the De Broglie frequency. This result is identical to the result obtained by the quantum mechanical description of the moving electron by the free particle wave function. The different interpretations of the established interference pattern, inherent in the two alternative theoretical descriptions is outlined, and the relation between the two descriptions discussed. It is concluded that quantum interference is well explained with classical probabilities, without quantum mechanics and without paradoxes. In view of the demonstrated merits of the model on the one hand, and the new aspects regarding the established theories it implies on the other, a more thorough investigation of its role in relation to relativistic quantum mechanics, and to quantum field theory is suggested. Some of the interesting aspects are summarized.

A note on Riemannian geometry and the relativistic quantum mechanics context

This article develops a variational formulation for relativistic quantum mechanics. The main results are obtained through a connection between relativistic and quantum mechanics. Such a connection is established through basic concepts on Riemannian geometry and related extensions for the relativistic context.

Revisiting integer factorization using closed timelike curves

Closed Timelike Curves are relativistically valid objects allowing time travel to the past. Treating them as computational objects opens the door to a wide range of results which cannot be achieved using non relativistic quantum mechanics. Recently, research in classical and quantum computation has focused on effectively harnessing the power of these curves. In particular, Brun (Found. Phys. Lett., 2003) has shown that CTCs can be utilized to efficiently solve problems like factoring and QSAT (Quantified Satisfiability Problem). In this paper, we find a flaw in Brun's algorithm and propose a modified algorithm to circumvent the flaw.

Breakdown of agreement between non-relativistic and relativistic quantum dynamical predictions in the non-relativistic regime

To study quantum dynamics in the non-relativistic regime, the standard practice is to use non-relativistic quantum mechanics, instead of the relativistic theory, because it is thought the approximate non-relativistic result is always close to the relativistic one. Here we present a theoretical argument that this expectation is not true in general. In addition, supporting numerical evidence for the free rotor and hydrogen atom also shows the agreement between the two theories can break down quickly. For the radial Rydberg wave packet in hydrogen atom, the breakdown can occur before spontaneous emission and thus could be tested experimentally. Our surprising result shows relativistic quantum mechanics must be used, instead of the approximate non-relativistic theory, to correctly study quantum dynamics in the non-relativistic regime after the breakdown time. This paradigm shift opens a new avenue of research in a wide range of fields from atomic to molecular, chemical and condensed-matter physics.

On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator

We review the standards of relativistic quantum mechanics such as the Dirac equation under the concept of negative masses. We show that negative energies are acceptable provided the masses are simultaneously negative. Negative energy and mass anti-fermions are obtained from positive energy and mass fermions through a unitary PT transformation.

High-Order Energy and Linear Momentum Conserving Methods for the Klein-Gordon Equation

The Klein-Gordon equation is a model for free particle wave function in relativistic quantum mechanics. Many numerical methods have been proposed to solve the Klein-Gordon equation. However, efficient high-order numerical methods that preserve energy and linear momentum of the equation have not been considered. In this paper, we propose high-order numerical methods to solve the Klein-Gordon equation, present the energy and linear momentum conservation properties of our numerical schemes, and show the optimal error estimates and superconvergence property. We also verify the performance of our numerical schemes by some numerical examples.

The 4D Dirac Equation in Five Dimensions

AbstractThe Dirac equation may be thought as originating from a theory of 5D space‐time. One way to formulate such a theory relies on a special 5D Clifford algebra and a spin‐1/2 constraint equation for describing null propagation in the 5D space‐time. The 5D null formalism breaks down to four dimensions to recover two single‐particle theories: Dirac's relativistic quantum mechanics and a formulation of spin‐1/2 statistical mechanics. Exploring the non‐relativistic limit in five and four dimensions, a new spin–electric interaction is identified, with possible applications to magnetic resonance spectroscopy (within quantum mechanics) and superconductivity (within statistical mechanics).

Limit of Relativistic Quantum Brayton Engine of Massless Boson Trapped 1 Dimensional Potential Well

A quantum Brayton engine using a massless Boson trapped in a one-dimensional potential well as a working substance has been constructed to be studied. We used a modified analogical model by the first law of thermodynamics for a quantum system implementation. The system was built with Klein Gordon equation, relativistic quantum mechanics Hamiltonian, without the mass term of the Hamiltonian and it was trapped in a one-dimensional potential well. Then, it underwent the quantum ideal Brayton cycle process. The quantum Brayton process consisted of adiabatic compression, isobaric expansion, adiabatic expansion, and isobaric compression processes. This relativistic quantum mechanics Brayton engine has found that the efficiency formulation was similar to the classical one. This study obtained the conformity results between the quantum relativistic and classical Brayton engines. By this cycle process, we also invented that the ratio of heat capacity under constant pressure and volume of the system was 2. It could indicate that the efficiency of quantum relativistic Brayton engine is higher than the classical one.

True wavefunctions and antiparticles of Klein–Gordon equation: ℏ-conjugation, elimination of CPT-invariance, and anti-gravitation

While revisiting the Klein–Gordon relativistic quantum equation for spin-0 particles, we predicted that ℏ reverses its sign for negative energies, and formulated a universal symmetry rule, whereby all the parameters that couple particles to external fields reverse their sign along with ℏ at a particle ↔ antiparticle transformation; this in particular implies anti-gravitation between matter and antimatter. Our results suggest that the ℏ-conjugation principle and related invariance may replace CPT-invariance in general relativistic quantum mechanics.