Dirac Equation For Particles Research Articles

Dirac Equation on the Kerr–Newman Spacetime and Heun Functions

By employing a pseudoorthonormal coordinate-free approach, the Dirac equation for particles in the Kerr–Newman spacetime is separated into its radial and angular parts. In the massless case to which a special attention is given, the general Heun-type equations turn into their confluent form. We show how one recovers some results previously obtained in literature, by other means.

Degenerate solutions to the Dirac equation for massive particles and their applications in quantum tunneling

In a recent work we have proven the existence of degenerate solutions to the Dirac equation, corresponding to an infinite number of different electromagnetic fields, providing also some examples regarding massless particles. In the present article our results are extended significantly, providing degenerate solutions to the Dirac equation for particles with arbitrary mass, which, under certain conditions, could be interpreted as pairs of particles (or antiparticles) moving in a potential barrier with energy equal to the height of the barrier and spin opposite to each other. We calculate the electromagnetic fields corresponding to these solutions, providing also some examples regarding both spatially constant electromagnetic fields and electromagnetic waves. Further, we discuss some potential applications of our work, mainly regarding the control of the particles outside the potential barrier, without affecting their state inside the barrier. Finally, we study the effect of small perturbations to the degenerate solutions, showing that our results are still valid, in an approximate sense, provided that the amplitude of the electromagnetic fields corresponding to the exact degenerate solutions is sufficiently small.

Dirac equation in (1 + 1) dimensional curved space-time: Bound states and bound states in continuum

We study Dirac equation in (1 + 1) dimensional curved space-time. In particular, it is shown that bound states can be found in the presence of interactions in certain curved spaces. It has also been shown that solutions of the free particle Dirac equation when mapped to some suitable curved spaces become bound states in continuum. A number of examples have been considered and complete solutions have been obtained analytically in each case.

Massless fermions on static general prolate metrics and their Heun solutions

Employing a pseudo-orthonormal coordinate-free approach, we write down the Dirac equation in spacetimes with static general prolate metrics. As examples, we consider the electrically charged C-metric, the vacuum C-metric, the Reissner–Nordström and Schwarzschild spacetimes and the BBMB black hole and show that the solutions to the Dirac equations for particles in these spacetimes can be derived in terms of Heun’s general functions and their confluent and double confluent forms. By imposing boundary conditions to the radial solutions, the so-called resonant frequencies are obtained.

Heun Functions Describing Bosons and Fermions on Melvin’s Spacetime

Employing a pseudo-orthonormal coordinate-free approach, the solutions to the Klein–Gordon and Dirac equations for particles in Melvin spacetime are derived in terms of Heun’s biconfluent functions.

Emergent tilt order in Dirac polymer liquids.

We study a liquid of zigzagging two-dimensional directed polymers with bending rigidity, i.e., polymers whose conformations follow checkerboard paths. In the continuum limit the statistics of such polymers obey the Dirac equation for particles of imaginary mass. We exploit this observation to investigate a liquid of these polymers via a quantum many-fermion analogy. A self-consistent approximation predicts a phase of tilted order, in which the polymers may develop a preference to zig rather than zag. We compute the phase diagram and key response functions for the polymer liquid, and comment on the role played by fluctuations.

New solutions to the Dirac equation for particles in a magnetic field and a medium

We briefly review the application of the method of exact solutions of quantum equations for the description of charged particle motion in external fields. We develop a quantization technique by using the lowering and raising operators to describe the states of fermions moving in a magnetic field and a dense medium. For the first time we have posed and solved the problem of a charged fermion moving in a medium and an external magnetic field by taking into account the anomalous magnetic moment of the particle: we have obtained exact solutions for the wave functions and energy spectrum of the corresponding modified Dirac equation. We discuss a quasi-classic interpretation of the solutions obtained and, in particular, find a correction for the synchrotron radiation intensity dependent on medium density.

Special function solutions of the free particle Dirac equation

The Dirac equation is one of the fundamental equations in physics. Here we present and discuss two novel solutions of the free particle Dirac equation. These solutions have an exact analytical form in terms of Airy or Mathieu functions and exhibit unexpected properties including an enhanced Doppler effect, accelerating wavefronts and solutions with a degree of localization.

Dirac equation for particles with arbitrary half-integral spin

The sets of -component irreducible and Clifford algebraic Hermitian and unitary matrices through the two-component Pauli matrices are suggested, where s = 1/2, 3/2, 5/2, … . Using these matrix sets, the eigenvalues of which are , the -component generalized Dirac equation for a description of arbitrary half-integral spin particles is constructed. In accordance with the correspondence principle, the generalized Dirac equation suggested arises from the condition of relativistic invariance. This equation is reduced to the sets of two-component matrix equations the number of which is equal to . The new relativistic invariant equation of motion leads to an equation of continuity with a positive-definite probability density and also to the Klein–Gordon equation. This relativistic equation is causal in the presence of an external electromagnetic field interaction. It is shown that, in the case of nonrelativistic limit, the relativistic equation presented is reduced to the Pauli equation describing the motion of half-integral spin particle in the electromagnetic field.

Introductory review on quantum field theory with space–time resolution

We review the steps needed to reduce quantum field theory to quantum mechanics as its limiting case. While processes involving a changing number of particles require a quantum field theoretical description, some works used quantum mechanical data based on the single particle Dirac equation to predict the outcome. These non-quantum field theoretical approaches have led to conceptual controversies that are often related to the negative energy states, which are characteristic of the Dirac equation. As a result, the issues about the physical reality of the relativistic localisation phenomenon, the Zitterbewegung and the Klein paradox are widely discussed in many textbooks. In order to contribute to this discussion, we review recent computer simulations of oversimplified model systems based on quantum field theory. The direct visualisation of intrinsically quantum field theoretical process with full spatial and temporal resolutions can provide us with some first insight into the spontaneous electron–positron pair creation from the vacuum triggered by a strong external (supercritical) force field and with the resolution of the Klein paradox.

Relativistic effects in quantum walks: Klein's paradox and zitterbewegung

Quantum walks are not only algorithmic tools for quantum computation but also non-trivial models describing various physical processes. The Letter compares one-dimensional version of the free particle Dirac equation with the discrete time quantum walk (DTQW). It is shown that two relativistic effects associated with the Dirac equation, namely zitterbewegung (quivering motion) and Klein's paradox, are manifested in DTQW. A special case of DTQW for Lorentz invariance not satisfied in the corresponding continuous limit is considered. The effects are examined.

Fractal space-time and the statistical mechanics of random walks

We illustrate some of the ideas involved in fractal space-time using familiar deterministic fractals. Starting with the objective of reproducing the Heisenburg uncertainty principle for point particles, we use the Peano-Moore curve to help visualize the qualitative behaviour of particles moving on fractal trajectories in space and time. With this qualitative picture in mind we then explore exactly solvable models to verify that our ideas are mathematically consistent. We find that the Schrödinger equation describes ensembles of classical particles moving on fractal random walk trajectories. This shows that the Schrödinger equation has a straightforward microscopic model which is not, however, appropriate for quantum mechanics. The free particle Dirac equation is also derivable in terms of ensembles of classical particles and this unites the two equations conceptually in a very direct way. In both cases what we discover is a many-particle simulation of quantum mechanics and this confirms in a graphic way that the mysteries surrounding quantum mechanics lie not in the equations, but in interpretation and the theory of measurement. Finally, we discuss an exactly solvable model which incorporates fractal time. The calculation produces the Dirac equation in 1 + 1 dimensions and because of intrinsic space-time loops, constitutes a model with the potential to exhibit the wave-particle duality found in nature.

A Finite and Discrete Model for Single Fermion Mass Renormalization: Derivation of the Free Particle Dirac Equation

We assume that a single particle of mass m cannot be localized to better than +h/2mc. Using our understanding of finite and discrete measurement accuracy, the single particle transition in 1+1 space-time from (O, O) to (z, t) can then be characterized by a scale factor N and two integers r, 1defined by z = N(r – l)h/mc and ct = N(r + l)h/mc 2. The average velocity over this finite interval is v = fic. The square of the average momentum is p2 = #mc2. We show that the solution of the free particle Dirac equation with these boundary conditions can be derived by assuming that the (unobserved) trajectories connecting the two endpoints are all constructed from Nr steps to the right and NZ steps to the left, with velocity +C or —c respectively; each single step has length h/me. We attribute this Zitterbewegung to the emission and absorption of transverse photons to and from the background radiation, each of which necessarily flips the spin. We assert that the symmetry . ...-. condition on the background radiation that this radiation be undetectable in free particle motion, plus the assumption that the starting and ending spin state must be the same, constitutes the essential requirement for successful single particle mass renormalization in our simple model. We then show that these requirements suffice to determine finite series which uniquely correspond to the (truncated) series . . solution of the corresponding free particle Dirac equation with the same boundary conditions. We sketch how to extend the model to 3+1 dimensions. The connection of oti model to the derivation of Maxwell’s equations from finite and discrete spacetime measurement accuracy is briefly discussed.

A generalized time-dependent relativistic wave equation

A time-dependent, spin-1/2, relativistic wave equation is introduced, so that some difficulties associated with the free particle Dirac equation may be resolved in a natural and systematic way. It has been found possible to establish a connection between the fluctuation of the velocity and a mass function which describes the fluctuation of the rest mass, derived from the Dirac theory. The time-dependent equation obtained by substituting the mass function for the constant mass term allows the description of spin fluctuations, for example, as time-dependent transitions between nonstationary states.

Extension of two-body Dirac equations to general covariant interactions through a hyperbolic transformation

In previous work, Dirac’s constraint mechanics and supersymmetry were used to construct two-body Dirac equations for particles interacting through world scalar and vector potentials. The resulting compatible 16-component wave equations, 𝒮1ψ=γ51(γ1⋅(p1−Ã1) +m1+S̃1)ψ=0, 𝒮2ψ=γ52(γ2⋅(p2−Ã2) +m2+S̃2)ψ=0, yield Schrödinger-like equations in the center-of-momentum system with simpler structure than that possessed by the Bethe–Salpeter equation or any of its standard three-dimensional truncations. For strong interactions, these equations have yielded a relativistic quark model for meson spectroscopy, while for electromagnetic interactions they have yielded a family of exact singlet positronium solutions. This paper uncovers a hyperbolic interaction structure in these equations. This structure is used to generalize these equations to include pseudoscalar, pseudovector, and tensor interactions.

New procedure for introduction of lorentz covariant interactions into two-body dirac equations

In the past, Dirac's constraint mechanics and supersymmetries for two interacting spinning particles has yielded compatible two-body Dirac equations for particles interacting through worldscalar and vector potentials. Here, we extend the procedure to include pseudoscalar, pseudovector, and tensor interactions.

Two-body Dirac equations for particles interacting through world scalar and vector potentials.

In a recent Letter, we used ''two-body Dirac equations'' to make the naive quark model fully relativistic. In this paper, we apply Dirac's constraint mechanics and supersymmetry not to a string but instead to a system of two spinning particles to derive the two coupled Dirac equations that govern the quantum mechanics of two spin-1/2 particles interacting through world scalar and vector potentials. Along the way, we demonstrate that our equations are compatible, that the scalar and vector interaction structures contribute separately to compatibility when both are present, and that compatibility persists even in the presence of relative-momentum-dependent interactions. We show that extrapolation of supersymmetries associated with the ordinary single-particle Dirac equation to the interacting two-body system eliminates spin complications and reduces the compatibility problem to that of the corresponding spinless system. In addition, supersymmetry introduces physically important nonperturbative recoil effects that contribute to the correct perturbative spectrum while rendering nonsingular certain singular terms of standard semirelativistic approximations. This eliminates the need for singularity-softening parameters or finite particle sizes in phenomenological applications. In order to make clear the underlying structure of our equations, we also derive the corresponding coupled Dirac or Klein-Gordon equations that govern a system of two particles,more » one or neither of which has spin.« less

Exact solutions of the Dirac equations in external non-Abel gauge fields

This paper presents solutions to the Dirac equation for particles with isospin T = 1/2 in field defined by non-Abel potentials of two types. The first type consists of constant potentials defining a chromomagnetic field directed along the third isotopic axis; the second type describes plane waves of elliptical polarization. The tensors of fields of both types can be related to a gauge transform of the type where a local revolution is expressed in isospin space.

Solution of the Dirac equation in quaternions

A technique is considered for solving in quaternions the Dirac equation for particles with an anomalous magnetic moment in the field of a plane electromagnetic wave.

Notizen:Eichfeldtheorie zur U3-invarianten Pais-Gleichung/ Gauge Field Theory of an U3-invariant Pais Equation

Abstract We are looking for the gauge field theory which corresponds to the U3-invariance of the recently obtained free particle Dirac equation generalized according to an idea of Pais.