Reflections, spinors, and projections on a Minkowski space underlie Dirac's equation

Maxwell's equations, Lorentz's equations of motion, and Dirac's equations are investigated on a cosmological background. Maxwell's equations are conform invariant and the equations of motion can easily be made conform invariant. Then, without introducing any new assumptions, Dirac's equations are seen to be conform invariant. In open universes the solutions of Maxwell's and Dirac's equations are the same as in flat Minkowski space. The behavior of these equations is different in closed universes by reasons of topology. In closed universes both Maxwell's and Dirac's equations provide new eigenvalue problems. Dirac's equations exhibit a marked difference between an elliptic and a spherical closed universe.

In this paper, we want to better understand the causality reflection that arises under a subset of Abelian local gauge transformations in geometrodynamics. We proved in previous papers that in Einstein–Maxwell spacetimes, there exist two local orthogonal planes of gauge symmetry at every spacetime point for non-null electromagnetic fields. Every vector in these planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. The vectors that span these local orthogonal planes are dependent on electromagnetic gauge. The local group of Abelian electromagnetic gauge transformations has been proved isomorphic to the local groups of tetrad transformations in these planes. We called LB1 the local group of tetrad transformations made up of SO(1, 1) plus two different kinds of discrete transformations. One of the discrete transformations is the full inversion two by two which is a Lorentz transformation. The other discrete transformation is given by a matrix with zeroes on the diagonal and ones off-diagonal two by two, a reflection. The group LB1 is realized on this plane, we call this plane one, and is spanned by the time-like and one space-like vectors. The other local orthogonal plane is plane two and the local group of tetrad transformations, we call this LB2, which is just SO(2). The local group of Abelian electromagnetic gauge transformations is isomorphic to both LB1 and LB2, independently. It has already been proved that a subset of local electromagnetic gauge transformations that leave the electromagnetic tensor invariant induces a change in sign in the norm of the tetrad vectors that span the local plane one. The reason is that one of the discrete transformations on the local plane one that belongs to the group LB1 is not a Lorentz transformation, it is a flip or reflection. It is precisely on this kind of discrete transformation that we have an interest since it has the effect of changing the signature and the causality. This effect has never been noticed before.

In this paper we explore the problem of antiparticles in DSR1 and $\kappa$-Minkowski space-time following three different approaches inspired by the Lorentz invariant case: a) the dispersion relation, b) the Dirac equation in space-time and c) the Dirac equation in momentum space. We find that it is possible to define a map $S_{dsr}$ which gives the antiparticle sector from the negative frequency solutions of the wave equation. In $\kappa$-Poincar\'e, the corresponding map $S_{kp}$ is the antipodal mapping, which is different from $S_{dsr}$. The difference is related to the composition law, which is crucial to define the multiparticle sector of the theory. This discussion permits to show that the energy of the antiparticle in DSR is the positive root of the dispersion relation, which is consistent with phenomenological approaches.

The common opinion that the Lorentz transformation distorts the Minkowski spacetime arises from a mathematically incorrect neglect of its non-diagonal terms. When the non-diagonal Lorentz transformation is properly treated, the time dilation and the space distortion disappear. Consequently, the Lorentz metric produced by the Lorentz transformation is identical to the Minkowski metric. The spacetime distortion in moving inertial frames is properly described by the so-called Doppler metric produced by the Doppler transformation. This transformation assumes the existence of a preferred frame and depicts the Doppler shift of light for a moving light source and/or observer. The Doppler metric is Lorentz invariant and leaves the Maxwell’s equations unchanged from their form in the Minkowski spacetime. Moreover, the Doppler metric is experimentally confirmed because it yields the null result in the Michelson–Morley experiment and other interferometric experiments. The suitability of the Doppler metric for describing light properties in Special and General Relativity is supported by recent cosmological observations of the cosmic microwave background.

A non-canonical correspondence of the complete sets of solutions to the Dirac and Klein-Gordon free equations in Minkowski space-time is established. This allows for a novel viewpoint on the relationship of relativistic equations for different spins and on the origin of spinor transformations. In particular, starting from a solution to the Dirac equation, one obtains a chain of other solutions to both Dirac and Klein-Gordon equations. A comparison with the massless case is performed, and examples of non-trivial singular solutions are presented. A generalization to Riemannian space-time and inclusion of interactions are also discussed.

It has been almost one hundred years since Einstein formulated his special theory of relativity in 1905. He showed that the basic space-time symmetry is dictated by the Lorentz group. It is shown that this group of Lorentz transformations is not only applicable to special relativity, but also constitutes the scientific language for optical sciences. It is noted that coherent and squeezed states of light are representations of the Lorentz group. The Lorentz group is also the basic underlying language for classical ray optics, including polarization optics, interferometers, the Poincaree sphere, one-lens optics, multi-lens optics, laser cavities, as well multilayer optics.

This paper offers educational insight into the Dirac equation, examining its historical context and contrasting it with the earlier Schrödinger and Klein-Gordon equations. The comparison highlights their Lorentz transformation symmetry and potential probabilistic interpretations. We explicitly solve the free particle dynamics in Dirac's model, revealing the emergence of solutions with negative energy. In this discussion, we examine the Dirac Sea Hypothesis and explore the inherent helicity of the solutions. Additionally, we demonstrate how the Dirac equation accounts for spin and derive the Pauli equation in the non-relativistic limit. Through the Foldy-Wouthuysen transformation, we reveal how the equation incorporates spin-orbit interaction and other relativistic effects, ultimately leading to the hydrogen fine structure. A section on relativistic covariant notation is included to emphasize the invariance of the Dirac equation, along with more refined formulations of both the KG and Dirac equations. Designed for undergraduate students with an interest in the Dirac equation, this resource provides a historical perspective without being purely theoretical. Our approach underscores the significance of a pedagogical method that combines historical and comparative elements to profoundly understand the role of the Dirac equation in modern physics.

The Dirac equation is commonly demonstrated under stringent hypotheses and after considerable math work made in relativistic quantum mechanics and quantum field theory. Here, a purely geometric approach free from hypotheses is suggested. The suggestion draws inspiration from the technique of stereographic projection that was developed before the quantum era to solve gyroscopic problems of classical mechanics. The projected variable is the generalized (or canonical) momentum vector. Its undetermined geometric orientations define a sphere in the momentum space and the projection onto the equatorial plane generates the Pauli matrices as soon as the conventional stereographic matrix is introduced. The associated eigenvalue problem results in the Dirac equation and the eigenvector (bispinor) has components that are related to geometric elements of the momentum space. The procedure has the advantage of revealing the correct form of the Dirac matrices without the mathematical effort that characterizes the presentation of the equation in traditional approaches. The other remarkable advantage is that, unlike the common reduction to the case of free space, the spatial inhomogeneity due to interaction potentials is included in the demonstration from the very beginning. The whole suggestion has an interdisciplinary character (relativity, complex analysis, rotation of rigid bodies, Pauli matrices) and can be useful in teaching the equation to students who lack in sufficient knowledge of quantum mechanics. Students equipped with more advanced education can benefit from the purely geometric perspective of this work if used to complement their studies about the equation.

The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.

We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q-holomorphic functions is considered. The precise form of Cauchy-Szego kernel and the orthogonal projection operator is obtained. The fundamental solution for the operator Δλ is found.

In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom.

We consider the Dirac equation in Minkowski space-time in Cartesian coordinates with external wave fields of different tensor structures, namely in the presence of scalar, vector, pseudoscalar, pseudovector, coupled vector and tensor, and coupled vector and pseudovector waves, i.e. we consider the generalized Volkov problem. The solution of the Dirac equation is reduced to that of a system of four equations, two of which are first-order ordinary differential equations and two are algebraic ones. Series of new solutions of the Dirac equation are obtained.

We discuss a geometric approach to confining a Dirac neutral particle with a permanent magnetic dipole moment interacting with external fields to a hard-wall confining potential in the Minkowski spacetime through noninertial effects. We discuss the behaviour of external fields induced by noninertial effects, and a case where relativistic bound states can be achieved in analogous way to having a Dirac particle confined to a quantum dot. We show that this confinement of a Dirac neutral particle analogous to a quantum dot arises from noninertial effects that give rise to the geometry of the manifold playing the role of a hard-wall confining potential. We also discuss the possible use of this mathematical model in studies of noninertial effects on condensed matter systems described by the Dirac equation.

Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function [Formula: see text] with [Formula: see text] is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half-space, meaning that the Minkowski spacetime is cut off before [Formula: see text]. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at [Formula: see text] can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in [Formula: see text] dimensions.

The authors examine the gravitational coupling of Klein-Gordon and Dirac fields to matter vorticity and spacetime torsion, in the context of Einstein-Cartan theory. The background spacetime is endowed with a Godel-type metric, characterized by two real parameters ( Omega , l2); the source of spacetime curvature is a Weyssenhoff-Raabe fluid with spin vector parallel to the vorticity field. They show that torsion and matter vorticity have identical effects on the physics of particle fields. Complete sets of solutions are obtained, satisfying boundary conditions connected to the test-field character of the solutions. The energy spectrum obtained is discrete in general, except for the case of hyperbolic Godel-type geometries (l2>0) where a continuum region in the energy spectrum may appear: if 0< Omega 2<l2 a continuum region is present in the upper part of the spectrum; if Omega 2>or=l2, Dirac solutions may present, under certain conditions, a continuum region in the lower part of the spectrum. The correspondence between classical geodesic motion and Klein-Gordon solutions is established, and used as a guide to select the correct boundary conditions for the test fields. Matter vorticity and/or spacetime torsion split the energy spectrum of Dirac particles. These effects are additive and result from the existence of the same constant of motion for both cases. This constant of motion generates a trivial symmetry of the system in Minkowski spacetime, the associated degeneracy of which is raised by matter vorticity and/or torsion fields, producing the above-mentioned split.