Relativistic Quantum Mechanics Research Articles

Classical characters of spinor fields in torsion gravity

Abstract We consider the problem of having relativistic quantum mechanics re-formulated with hydrodynamic variables, and specifically the problem of deriving the Mathisson-Papapetrou-Dixon equations (describing the motion of a massive spinning body moving in a gravitational field) from the Dirac equation. The problem will be answered on a general manifold with torsion and gravity. We will demonstrate that when plane waves are considered the MPD equations describe the general relativistic wave-particle duality with torsion, but we will also see that in such a form the MPD equations become trivial.

Tunnelling amplitudes through localised external potentials from Feynman diagram summation

AbstractCurrently there is no general theory of quantum tunnelling of a particle through a potential barrier which is compatible with QFT. We present a complete calculation of tunnelling amplitudes for a scalar field for some simple potentials using quantum field-theoretic methods. Using the perturbative S-matrix formalism, starting with the Klein–Gordon Lagrangian, we show that an infinite summation of Feynman diagrams can recover tunnelling amplitudes consistent with relativistic quantum mechanics. While this work does not include many-particle effects arising from a fully quantised QFT, it is necessary to investigate QFT corrections to tunnelling amplitudes.

Quantum corrections to tunnelling amplitudes of neutral scalar fields

Though theoretical treatments of quantum tunnelling within single-particle quantum mechanics are well-established, at present, there is no quantum field-theoretic description (QFT) of tunnelling. Due to the single-particle nature of quantum mechanics, many-particle effects arising from quantum field theory are not accounted for. Such many-particle effects, including pair-production, have proved to be essential in resolving the Klein-paradox. This work seeks to determine how quantum corrections affect the tunnelling probability through an external field. We investigate a massive neutral scalar field, which interacts with an external field in accordance with relativistic quantum mechanics. To consider QFT corrections, we include another massive quantised neutral scalar field coupling to the original via a cubic interaction. This study formulates an all-order recursive expression for the loop-corrected scalar propagator, which contains only the class of vertex-corrected Feynman diagrams. This equation applies for general external potentials. Though there is no closed-form analytic solution, we also demonstrate how to approximate the QFT corrections if a perturbative coupling to the quantised field is assumed.

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

The Particle of Haag's Local Quantum Physics: A Critical Assessment.

Rudolf Haag's Local Quantum Physics (LQP) is an alternative framework to conventional relativistic quantum field theory for combining special relativity and quantum theory based on first principles, making it of great interest for the purposes of conceptual analysis despite currently being relatively limited as a tool for making experimental predictions. In LQP, the elementary particles are defined as species of causal link between interaction events, together with which they comprise its most fundamental entities. This notion of particle has yet to be independently assessed as such. Here, it is captured via a set of propositions specifying particle characteristics and then compared to previous particle notions. Haag's particle differs decisively with respect to mechanical intuitions about particles by lacking, among other things, even an approximate independent space-time location. This notion is thus found to differ greatly even from those of relativistic quantum mechanics and quantum field theory, which have been applied to the known elementary particles.

Analytical Solution of the Klein-Gordon coupled Equations via the Differential Transform Method

The basic equations in physical applications are the Klein-Gordon equation, as it gives a description of numerical particles in the fields of quantum and quantum mechanics, through the use of the differential transformation method and combining it with the Idomian method, in order to provide an integrated method for solving this equation, and it has been widely used to solve differential equations. Partial, to get rid of nonlinear terms and complex terms by integrating these two methods, which led to reducing the error and thus increasing the efficiency and accuracy of the solution using these numerical methods and the theoretical analysis of the method.. Researchers can learn more about quantum field dynamics using our method, which makes it easier to study a wide range of physical phenomena including scalar particles. In addition to improving the computational toolset in relativistic quantum mechanics, this work offers a useful foundation for investigating basic interactions in the field of particle physics and other fields. Conclusions: Adomain decomposition method (ADM) and the Differential Transform Method (DTM) together provide a powerful approach to solving the Klein-Gordon system equation. The effectiveness of this integrated method in efficiently and precisely solving complex differential equations controlling scalar particle dynamics has been proven by our research. The nonlinearities and boundaries of the Klein-Gordon system equation have proven to be formidable obstacles, but we have overcome them by combining the two approaches, and the findings are solid and trustworthy. Scalar particle behaviour in relativity quantum mechanics can be better understood with the help of this comprehensive methodology, which also improves computational efficiency. The DTM technique is a powerful and flexible tool for academics in many fields, including applied mathematics, particle physics, and quantum field theory, as our results show. The future looks bright for tackling ever more complex systems and improving our understanding of basic physical processes as we continue to investigate and enhance this integrated approach.

Anti-foundationalist Coherentism as an Ontology for Relational Quantum Mechanics

There have been a number of recent attempts to identify the best metaphysical framework for capturing Rovelli’s Relational Quantum Mechanics (RQM). All such accounts commit to some form of fundamentalia, whether they be traditional objects, physical relations, events or ‘flashes’, or the cosmos as a fundamental whole. However, Rovelli’s own recommendation is that ‘a natural philosophical home for RQM is an anti-foundationalist perspective' (Rovelli in Philos Trans R Soc 376:10, 2018). This gives us some prima facie reason to explore options beyond these foundationalist frameworks, and take seriously a picture that lacks fundamentalia. I construct an argument from elimination in favour of an anti-foundationalist interpretation of RQM. The argument notes that priority monism and priority pluralism are exhaustive foundationalist options, and then shows that there are reasons to reject their union with RQM. I finish by recommending metaphysical coherentism as a promising anti-foundationalist alternative, which captures the key characteristics of RQM through accepting symmetrical dependence, whilst avoiding challenges by jettisoning any commitment to fundamental entities.

Quantum Big-Bounce as a phenomenology of RQM in the Mini-superspace

We investigate the emergence of a quantum Big-Bounce in the context of an isotropic Universe, filled by a self-interacting scalar field, which plays the role of a physical clock. The bouncing cosmology is the result of a scattering process, driven by the scalar field potential, which presence breaks down the frequency separation of the Wheeler-DeWitt equation, treated in strict analogy to a relativistic quantum system. Differently from previous analyses, we consider a really perturbative self-interaction potential, affecting the dynamics in a finite range of the time labeled by the scalar clock (and in particular we remove the divergent character previously allowed). The main result of the present analysis is that, when the Relativistic Quantum Mechanics formalism is properly implemented in the Mini-superspace analogy, the probability amplitude for the bounce is, both in the standard and polymerized case, characterized by a maximum in correspondence of the quasi-classical condition of a Universe minimum volume.

Quantum rate electrodynamics and resonant junction electronics of heterocyclic molecules

The quantum rate theory provides a framework to understand electron-transfer reactions by correlating the electron-transfer rate constant (ν) with the quantum capacitance (Cq) and the molecular conductance (G). This theory, which is rooted in relativistic quantum electrodynamics, predicts a fundamental frequency ν=E/h for electron-transfer reactions, where E is the energy associated with the density of states Cq/e2. This work demonstrates the applicability of the quantum rate theory to the intermolecular charge transfer of push-pull heterocyclic compounds assembled over conducting electrodes. Remarkably, the observed differences between molecular junction electronics formed by push-pull molecules and the electrodynamics of electrochemical reactions on redox-active modified electrodes can be attributed solely to the adiabatic setting of the quantum conductance in push-pull molecular junctions. The electrolyte field-effect screening environment plays a crucial role in modulating the resonant quantum conductance dynamics of the molecule-bridge-electrode structure. In this context, the intermolecular electrodynamics within the frontier molecular orbital of push-pull heterocyclic molecules adhere to relativistic quantum mechanics, consistent with the predictions of the quantum rate theory.

[formula omitted] symmetry in hydrogen-atom-like model with spin

As the simplest atom in nature, the hydrogen atom has been explored thoroughly from the perspective of non-relativistic quantum mechanics to relativistic quantum mechanics. Among the research on hydrogen atom, its energy level is the most basic, which can be obtained more conveniently predicated on the SO(4) symmetry than the wave-equation resolution. Moreover, “spin” is another indispensable topic in quantum mechanics, appearing as an intrinsic degree of freedom. In this work, we generalize the quantum Runge-Lenz vector to a spin-dependent one, and then extract a novel Hamiltonian of hydrogen-atom-like model with spin based on the requirement of SO(4) symmetry. Furthermore, the energy spectrum of hydrogen-atom-like model with spin potentials is also determined by the remarkable approach of SO(4) symmetry. Our findings extend the ground of hydrogen atom, and may contribute to other complicated models based on hydrogen atom.

Relational Quantum Mechanics and Contextuality

This paper discusses the question of stable facts in relational quantum mechanics (RQM). I examine how the approach to quantum logic in the consistent histories formalism can be used to clarify what infomation about a system can be shared between different observers. I suggest that the mathematical framework for Consistent Histories can and should be incorporated into RQM, whilst being clear on the interpretational differences between the two approaches. Finally I briefly discuss two related issues: the similarities and differences between special relativity and RQM and the recent Cross-Perspectival Links modification to RQM.

Comment on Aurélien Drezet’s Defense of Relational Quantum Mechanics

Aurélien Drezet has attempted in Found. Phys. 54(1), 5 (2023) to defend Relational Quantum Mechanics (RQM) against our recent critique, entitled Relational Quantum Mechanics is incompatible with quantum mechanics, published in Quantum 7, 1015 (2023). Drezet not only misrepresents our work, but he also misconstructs the very theory (RQM) that he claims to defend.

A Causal Net approach to Relativistic Quantum Mechanics and Gravitation

In this paper we discuss a causal network model to describe quantum mechanics and gravittation, where space–time is built solely from point events and connecting probabilities. A causal net provides a representation which combines the quantum complementary features of both partticles and waves as each vertex on the causal net represents a possible point event or particle observation. Simultaneity on the causal net is defined by hyperplanes of equivalent proper time. The causal net model when constructed in Minkowksi space–time is shown to lead to a formulation for relativistic quantum mechanics including the Dirac equation. In a curved Riemann space–time the causal paths of the causal net are geodesics and in the local Lorentz frame the net probabilities and the Dirac formalism are preserved. The variation of space–time density of events in the causal paths modifies the metric and provides a space–time curvature leading to the Hilbert action associated with general relativity. Consideration of the Schwarzchild metric shows that in classical limit the perturbation in the density of possible events is proportional to the Newtonian gravitational potential.

Tunable wave localization at the Dirac frequency in a metallic photonic crystal cavity.

In this study, the two-dimensional (2D) triangular lattice metallic photonic crystals (PCs) in visible and infrared bands have been utilized to achieve light confinement at the Dirac frequency. Distinct from the traditional bandgap or total internal reflection cavity modes, the unique photonic localization mechanism leads to an unusual algebraic decay of state and a unique frequency located beyond any bandgaps. This investigation delves into the band structure analysis of 2D metallic PCs, specifically focusing on their distinctive features, such as photonic bandgaps and Dirac cones. The plane wave expansion (PWE) method, enhanced with a linearization technique, is employed for band structure calculations, considering both the frequency-dependent dielectric properties and the intrinsic lossy nature of metallic materials described by the Drude model. The study provides a comprehensive derivation of the PWE equations for metallic PCs and investigates their band characteristics under both TM and TE polarizations. Focusing on TM modes in triangular lattice metallic PCs, it reveals zero density of states (DOS) at K points of the Brillouin corner and the existence of Dirac cones with linearly dispersion and linearly vanishing DOS. The study extends to exploring localized modes at Dirac frequencies, employing a relativistic quantum mechanics approach analogous to graphene's charge carriers. Theoretical predictions are corroborated by numerical simulations, and the potential for tunable Dirac localized modes is highlighted. This research not only deepens the understanding of Dirac properties in graphene-like systems but also lays the groundwork for further exploration of the practical quasi-2D devices, which will provide assistance in the integration of micro- and nano- devices, especially in applications requiring long-range coupling, given the critical importance of optical cavities in contemporary optical technologies.

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

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

Understanding Lorentz Utilizing Galilei: The Emergence of a Friendly Extended Special Relativity Theory that Admits Relativistic Multi-Particle Entanglement

Special relativity theory stems from the Lorentz transformation of signature (1,3). The incorporation into special relativity of the Lorentz transformations of signature (m,n) for all m,n∈ℕ (n = 3 in physical applications) enriches the theory. The resulting enriched special relativity is a friendly extended special relativity that admits multi-particle entanglement, as demanded by relativistic quantum mechanics. The Lorentz transformation of signature (m,n) admits a novel physical interpretation induced by the intuitively clear interpretation of the Galilei transformation of signature (m,n) for all m,n > 1. In this sense we understand Lorentz utilizing Galilei in m temporal and n spatial dimensions, resulting in the emergence of multi-particle entanglement that the enriched special theory of relativity admits. Remarkably, it turns out that, for any m,n∈ℕ, the group of Lorentz transformations of signature (m,n) is the symmetry group that underlies any multi-particle system that consists of m n - dimensional entangled particles.

An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics

An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics

Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models

Solitons and waves with memory effects are two examples of nonlinear wave phenomena that can be studied mathematically using the fractional nonlinear sine-Gordon equation. It is a modification of the traditional sine-Gordon equation that takes memory effects and nonlocal interactions into account by adding fractional derivatives. A more complex explanation of particle dynamics and interactions within relativistic quantum mechanics is made possible by the fractional Klein–Gordon model, a theoretical framework that expands the standard equation to include fractional derivatives. The study uses shifted Legendre–Gauss–Lobatto and shifted Legendre–Gauss–Radau collocation techniques to solve numerically two-dimensional sine-Gordon and Klein–Gordon models. The study handles two-dimensional sine-Gordon and Klein–Gordon models by extending a collocation approach using basis functions. It suggests a collocation technique that uses a suggested basis to automatically satisfy the conditions. The suggested methods’ spectral accuracy and efficiency are validated by numerical results.

Relational Quantum Mechanics: Ozawa’s Intersubjectivity Theorem as Justification of the Postulate on Internally Consistent Descriptions

The Ozawa’s intersubjectivity theorem (OIT) proved within quantum measurement theory supports the new postulate of relational quantum mechanics (RQM), the postulate on internally consistent descriptions. But from OIT viewpoint postulate’s formulation should be completed by the assumption of probability reproducibility. We remark that this postulate was proposed only recently to resolve the problem of intersubjectivity of information in RQM. In contrast to RQM for which OIT is a supporting theoretical statement, QBism is challenged by OIT.

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.