Relativistic Quantum Mechanics Research Articles

Non-inertial interpretation of the Dirac oscillator

Non-inertial physics is seldom considered in quantum mechanics and this contrasts with the ubiquity of non-inertial reference frames. Here, we show an application to the Dirac oscillator which provides a fundamental model of relativistic quantum mechanics. The model emerges from a term linearly dependent on spatial coordinates added to the momentum of the free-particle Dirac Hamiltonian. The definition generates peculiar features (mutating vacuum energy, non-Hermitian momentum, accidental degeneracies of the spectrum, etc). We interpret these anomalies in terms of inertial effects. The demonstration is based on the decoupling of the Dirac equation from the stereographic projection that maps the 3D geometry of the dynamical problem to the complex plane. The decoupling shows that the fundamental mechanical model underpinning the Dirac oscillator reduces to the representation of the oscillator in the rotating reference frame attached to the orbital angular momentum. The resulting Coriolis-like contribution to the Hamiltonian accounts for the peculiarities of the model (mutating vacuum energy, form of the non-minimal correction to the momentum, classical intrinsic spin and gain of its quantum value, accidental degeneracies of the energy spectrum, supersymmetric potential). The suggested interpretation has an interdisciplinary character where stereographic geometry, classical physics of the Coriolis effect and quantum physics of Dirac particles contribute to the definition of one of the few exactly soluble models of relativistic quantum mechanics.

Dispersion Relations Alone Cannot Guarantee Causality.

We show that linear superpositions of plane waves involving a single-valued, covariantly stable dispersion relation ω(k) always propagate outside the light cone unless ω(k)=a+bk. This implies that there is no notion of causality for individual dispersion relations since no mathematical condition on the function ω(k) (such as the front velocity or the asymptotic group velocity conditions) can serve as a sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emerge from a careful cancellation that occurs when one superimposes all the excitation branches of a physical model. This happens automatically in local theories of matter that are covariantly stable. Hence, we find that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need for antiparticles in relativistic quantum mechanics.

Relational Quantum Mechanics, quantum relativism, and the iteration of relativity

The idea that the dynamical properties of quantum systems are invariably relative to other systems has recently regained currency. Using Relational Quantum Mechanics (RQM) for a case study, this paper calls attention to a question that has been underappreciated in the debate about quantum relativism: the question of whether relativity iterates. Are there absolute facts about the properties one system possesses relative to a specified reference, or is this again a relative matter, and so on? It is argued that RQM (in its best-known form) is committed to what I call the Unrestricted Iteration Principle (UIP), and thus to an infinite regress of relativisations. This principle plays a crucial role in ensuring the communicability and coherence of interaction outcomes across observers. It is, however, shown to be incompatible with the widespread, conservative reading of RQM in terms of relations, instead necessitating the adoption of the more unorthodox notion of perspectival facts. I conclude with some reflections on the current state of play in perspectivist versions of RQM and quantum relativism more generally, underscoring both the need for further conceptual development and the importance of the iteration principle for an accurate cost-benefit analysis of such interpretations.

Derivation of Dirac equation from the stochastic optimal control principles of quantum mechanics.

In this paper, we present a stochastic approach to relativistic quantum mechanics. We formulate the three fundamental principles of this theory and derive the Dirac equations based on them. This approach enables us to bring more insight into the nature of Dirac's spinors. Furthermore, we provide a physical interpretation of the stochastic optimal control theory of quantum mechanics.

Dirac equation for photons in a fibre: Origin of polarisation

Spin is a fundamental degree of freedom, which was discovered by Dirac for an electron in his relativistic quantum mechanics, known as the Dirac equation. The origin of spin for a photon is unclear because Maxwell's equations in a vacuum are Lorentz invariant without introducing the concept of spin. Here, the propagation of coherent rays of photons in a graded-index optical fibre is considered to discuss the origin of polarisation for photons using exact solutions of the Laguerre-Gauss and Hermite-Gauss modes. The energy spectrum is massive, and the effective mass is a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional (1D) non-relativistic Schrödinger equation, which is equivalent to the 2D space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the 2D Dirac equation with spin. The spin expectation values of photons correspond to the polarisation state on the Poincaré sphere. As an application of the theory, a polarisation interferometer is proposed, whose energy spectrum shows a Dirac cone in the Stokes parameter space.

The physical interpretation of point interactions in one-dimensional relativistic quantum mechanics

We investigate point interactions (PIs) in one-dimensional relativistic quantum mechanics using a distributional approach based on Schwartz’s theory of distributions. From the properties of the most general covariant distribution describing relativistic PIs (RPIs) we obtain the physical parameters associated with the point potentials that behave as a scalar, a pseudo-scalar and a vector under Lorentz transformations. Then, we establish a one-to-one relationship between these physical parameters and the well-known set of four parameters giving the boundary conditions at the singular point(s), which define a self-adjoint Hamiltonian. By considering the non-relativistic limit, we obtain the most general PI in the Schrödinger equation in terms of these four physical point potentials. Finally, we study the symmetries of the RPIs under space inversion, time reversal and charge conjugation, and investigate how requirements of invariance under these symmetry transformations can be used to restrict the set of physical parameters.

Foundations of time-symmetric physics 1. Time-symmetric relativity and relativistic quantum mechanics

Time-symmetric theory (TST) is formulated as a theoretical foundation of time-symmetric physics (TSP), classical, relativistic and quantum. TST is based on the widely used in particle physics fact that the description of positive energy antiparticles moving forward in space and time is equivalent to the description of negative energy particles moving backward in space and time (Zisman 1940, Stückelberg 1941, Feynman 1949). In TST, this fundamental fact is generalized by formulating the equivalence principle of particle physics and applying it to all physical phenomena in the rest frames of negative energy particles. As a result, the group of transformations TST also includes such inertial frames that move backward in space and time with coordinates related to the ordinary ones by 4-inversion. This led to the formulation of time-symmetric relativity theory and corresponding relativistic quantum mechanics. In the latter, the probabilities remain positive, but the bilinear forms of the wave functions are associated with probability currents, the signs of which depend on the direction of movement along the coordinate axes, including the time axis. In TST, therefore, the Klein-Gordon equation is a consistent equation for probability amplitudes, and the fermion theory contains the necessary corrections, which make it physically more consistent, unchanging its main observational consequences. Applications of TST to quantum fields will be considered in the second article.

New spin and p-spin symmetries of the deformed Dirac equation for the IMRQHPs including the improved YLT interaction in three-dimensional extended relativistic quantum mechanics symmetries

In this paper, the modified Dirac equation (MDE) has been solved approximately for improved Manning–Rosen plus quasi-Hellman potentials (IMRQHPs), including the improved Yukawa tensor interaction under new spin symmetry and pseudospin symmetry for arbitrary spin–orbit number [Formula: see text], an appropriate approximation employed on the centrifugal terms in the symmetries of three-dimensional extended relativistic quantum mechanics (3D-ERQM). This potential is a superposition of Manning–Rosen plus quasi-Hellman potentials and some other exponential terms. Through the application of both new approximations to deal with the centrifugal term, Bopp’s shift method and the independent-time perturbation theory method, the recent modified approximate energy corrections under IMRQHPs are obtained for nuclei [Formula: see text]O and [Formula: see text]F and (O2, I2, N2, H2, LiH, CO and NO) diatomic molecules in 3D-ERQM and three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries. The recent values that we get appear sensitive to [Formula: see text], which are known as the discrete atomic quantum numbers, the mixed potential depths ([Formula: see text], [Formula: see text]) and [Formula: see text], the range of the potential [Formula: see text] and non-commutativity parameters [Formula: see text]. In both 3D-ERQM and 3D-ENRQM symmetries, we show that the recent corrections obtained on the new bound states under IMRQHPs are infinitesimal to the principal part of energy in the ordinary cases of 3D-RQM and 3D-NRQM symmetries. In the new symmetries of 3D-ERQM symmetries, it is not possible to get the exact analytical solutions [Formula: see text] and [Formula: see text], only the approximate solutions are possible. In addition, we discussed in detail the non-relativistic study of the studied mixed potentials. Four special cases, i.e. we investigated some special cases in the context of modified Schrödinger theories. In the framework of known relativistic quantum mechanics, we have clearly demonstrated that the modified Schrödinger equation can be represented as the Dirac equation under the influence of IMRQHPs.

Variational Approach to Relativistic Fluid and Quantum Mechanics

In previous papers we have shown how Schrödinger’s equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields.This program was applied also to Pauli’s equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. Here we repeat the analysis for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.

Classical and quantum gravity from the axioms of relativistic quantum mechanics

It is common practice to describe elementary particles by irreducible unitary representations of the Poincaré group. In the same way, multi-particle systems can be described by irreducible unitary representations of the Poincaré group. Representations of the Poincaré group are characterised by fixed eigenvalues of two Casimir operators corresponding to a fixed mass and a fixed angular momentum. In multi-particle systems (of massive spinless particles), fixing these eigenvalues leads to correlations between the particles. In the quasi-classical approximation of large quantum numbers, these correlations take on the structure of a gravitational interaction described by the field equations of conformal gravity. A theoretical value of the corresponding gravitational constant is calculated. It agrees with the empirical value used in the field equations of general relativity.

Non-Markovian cost function for quantum error mitigation with Dirac Gamma matrices representation

This paper investigates the non-Markovian cost function in quantum error mitigation (QEM) and employs Dirac Gamma matrices to illustrate two-qubit operators, significant in relativistic quantum mechanics. Amid the focus on error reduction in noisy intermediate-scale quantum (NISQ) devices, understanding non-Markovian noise, commonly found in solid-state quantum computers, is crucial. We propose a non-Markovian model for quantum state evolution and a corresponding QEM cost function, using simple harmonic oscillators as a proxy for environmental noise. Owing to their shared algebraic structure with two-qubit gate operators, Gamma matrices allow for enhanced analysis and manipulation of these operators. We evaluate the fluctuations of the output quantum state across various input states for identity and SWAP gate operations, and by comparing our findings with ion-trap and superconducting quantum computing systems' experimental data, we derive essential QEM cost function parameters. Our findings indicate a direct relationship between the quantum system's coupling strength with its environment and the QEM cost function. The research highlights non-Markovian models' importance in understanding quantum state evolution and assessing experimental outcomes from NISQ devices.

On the Wave-Particle Duality of the Photons and the Matter-Photon Particle Mixture Model

Relativistic quantum mechanics and many experimental results and observations show that electron-positron annihilation can produce photons; photon-matter and photon-photon interaction can also create electrons and positrons. The photons have no rest mass, e.g. relative to Lab frame, and always travel at the speed of light in the vacuum. In this paper, a rotational moving electric dipole model with negative and positive charges was proposed for photons. In a vacuum, the electric dipoles are moving in a twisted helical motion around their propagation axis (their center of mass). Photon particles are traveling at the speed of light along the propagation axis. This pair of negative and positive charged particles exerts both electrostatic and magnetic force on each other. Both forces are attractive and act as a centripetal force to keep the electric dipole in a continuous helical motion around the axis of rotation. With this model, the wave-particle duality of photons can be described simultaneously. The space is filled by matter (having rest mass relative to Lab frame) and photons (without rest mass); it is a multicomponent mixture fluid. In "vacuum", though there is no rest matter particle, but there are still photon gas particles, the total energy-momentum tensor should include rest matter particles and photon gas particles.

New Eigensolution of the Klein--Gordon and Shrödinger equations for improved modified Yukawa-Kratzer potential and its applications using Boop's Shift method and standard perturbation theory in the 3D-ERQM and 3D-ENRQM symmetries

The deformed Klein-Gordonequation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the improved modified Yukawa-Kratzer potential (IMYKP) model under the influence of the deformation space-space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the IMYKP model it was shown to be sensitive to the atomic quantum numbers (j, l, s, m), mixed potential depths (V0, De, re), the screening parameter’s inverse α and noncommutativity parameters (Θ,τ ,χ). In addition, we analyzed the nonrelativistic energy values by applying the well-known transmission rules known in the literature. In addition, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the improved modified Kratzer potential, the improved generalized Kratzer potential, the improved Kratzer potential, the improved modified Kratzer plus screened Coulomb potential, the improved Hellmann potential, the improved Yukawa potential, and improved inversely square Yukawa potential. We noticed that these particular results are identical to our previous work and other known works in the literature. The study is further extended to calculate the mass spectra of mesons of charmonium (cc) and bottomonium (bb) within the framework of the IMYKP model in three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries.  

A Fluid Perspective of Relativistic Quantum Mechanics.

In previous papers, it has been shown how Schrödinger's equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour is derived from Fisher information terms added to the classical Lagrangian, showing that a quantum mechanical system is driven by information and not only electromagnetic fields. This program was applied to Pauli's equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful, there were terms that did not admit interpretation, a number of which can be easily traced to the relativistic Dirac theory. Here, this analysis is repeated for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.

Revealing inherent quantum interference and entanglement of a Dirac particle

Although originally predicted in relativistic quantum mechanics, Zitterbewegung can also appear in some classical systems, which leads to the important question of whether Zitterbewegung of Dirac particles is underlain by a more fundamental and universal interference behavior without classical analogs. We here reveal such an interference pattern in phase space, which underlies but goes beyond Zitterbewegung, and whose nonclassicality is manifested by the negativity of the phase space quasiprobability distribution, and the associated pseudospin-momentum entanglement. We confirm this discovery by numerical simulation and an on-chip experiment, where a superconducting qubit and a quantized microwave field respectively emulate the internal and external degrees of freedom of a Dirac particle. The measured quasiprobability negativities agree well with the numerical simulation. Besides being of fundamental importance, the demonstrated nonclassical effects are useful in quantum technology.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The deformed Dirac equation has been investigated, in the context of 3D-relativistic noncommutative space (3D-RNCS) symmetries, using the improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model under the conditions of spin symmetry and pseudospin symmetry. The ISDM(CP-CLTI) model is the combining the spatially dependent mass Coulomb potential with the Coulomb-like tensor interaction (CP-CLTI) and the two central terms that are generated to the topological defects of spacespace. Within the confines of the parametric Bopp shift method and conventional perturbation theory, the new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms ( H-atoms), such as He+,Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived. The novel values Encsp(n,C,m0,m1,H,Θ,τ,χ,j,l,s,m) and Encps(n,C,m0,m1,H,Θ,τ,χ,j,l,l̃,s̃,m̃) that we discovered examined to be dependent on the noncommutativity parameters (NP) (Θ,σ,χ), mixed potential depths C/q,m0,m1,H), and quantum atomic discrete quantum numbers (j,k,l,s,m,l̃,s̃,m̃). We have obtained several interesting special examples within the framework of relativistic extended quantum mechanics, which we believe will be of interest to the expert researcher. We were able to retrieve the typical results of relativistic and non-relativistic examples in the literature when we applied the three simultaneous constraints (Θ,σ,χ)→(0,0,0). Compared to previous models that are known from the literature, our new model had novel physical characteristics.

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

New relativistic and non-relativistic investigation of deformed Klein–Gordon and Schrödinger equations for new improved screened Kratzer potential in the framework of non-commutative space

The deformed Klein–Gordon equation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the newly improved screened Kratzer potential (NISKP) model under the influence of the deformation space–space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the NISKP model were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text], control parameter [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the non-relativistic energy values by applying the well-known transmission rules known in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the new screened cosine Kratzer potential, the new screened Kratzer potential (SKP), the new Kratzer potential, the new coshine Yukawa potential, the new coshine Yukawa potential, the new shifted improved SKP and the new shifted SKP. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system such as charmonium [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text] that have the quark and antiquark flavor within the framework of the NISKP model in 3D-ENRQM symmetries.

Generalized equations and their solutions in the (1/2,0)+(0,1/2) representations of the Lorentz group

We present explicit examples of generalizations in relativistic quantum mechanics. First of all, we discuss the generalized spin-1/2 equations for neutrinos. They have been obtained by means of the Gersten-Sakurai method for derivations of arbitrary-spin relativistic equations. Possible physical consequences are discussed. Next, it is easy to check that both Dirac algebraic equations Det(ˆp − m) = 0 and Det(ˆp + m) = 0 for u− and v− 4-spinors have solutions with p0 = ±Ep = ± p p2 + m2. The same is true for higher-spin equations. Meanwhile, every book considers the equality p0 = Ep for both u− and v− spinors of the (1/2, 0) ⊕ (0, 1/2) representation, thus applying the DiracFeynman-Stueckelberg procedure for elimination of the negative-energy solutions. The recent Ziino works (and, independently, the articles of several others) show that the Fock space can be doubled. We re-consider this possibility on the quantum field level for both S = 1/2 and higher spin particles. The third example is: we postulate the non-commutativity of 4-momenta, and we derive the mass splitting in the Dirac equation. The applications are discussed.