Feshbach Villars Research Articles

Solution of Relativistic Feshbach–Villars Spin-1/2 Equations

Solution of Relativistic Feshbach–Villars Spin-1/2 Equations

Weak equivalence principle violation for mixed scalar particles

We investigate the non-relativistic limit of the Klein–Gordon equation for mixed scalar particles and show that, in this regime, one unavoidably arrives at redefining the particle’s inertial mass. This happens because, in contrast to the case when mixing is absent, the antiparticle sector contribution cannot be neglected for particles with definite flavor. To clearly demonstrate this feature, we adopt the Feshbach–Villars formalism for Klein–Gordon particles. Furthermore, within the same framework, we also demonstrate that, in the presence of a weak gravitational field, the mass parameter that couples to gravity (gravitational mass) does not match the effective inertial mass. This, in turn, implies a violation of the weak equivalence principle. Finally, we prove that the Bargmann’s superselection rule, which prohibits oscillating particles on the basis of the Galilean transformation, is incompatible with the non-relativistic limit of the Lorentz transformation and hence does not collide with the results obtained.

Applications of the Klein–Gordon equation in the Feshbach–Villars representation in the non-inertial cosmic string space–time

We study the relativistic quantum motion of a spinless particle using the Feshbach–Villars (FV) formalism in the spinning cosmic string spacetime. The equations of motion are derived using the first-order FV formulation of the Klein–Gordon (KG) equation. We apply the equation of motion (a) to study the motion of the particle confined to a rigid-wall potential, (b) motion in the presence of a Coulomb-type potential, and (c) particle interacting with the Feshbach–Villars oscillator (FVO). The energy levels and wave functions are obtained for the three cases. Our study focused on the impact of rotation and curvature on the energy levels of the particle.

Statistical properties of the two dimensional Feshbach–Villars oscillator (FVO) in the rotating cosmic string space–time

This paper focuses on the quantum mechanical dynamics of a massive and spinless relativistic particle in space–time generated by a rotating cosmic string. The equations of motion are found by the use of the Feshbach–Villars formulation of the Klein–Gordon equation. The wave functions and related energies have been deduced. Subsequently, the partition function was approached with the help of the representation of the Zeta function. Through this function, thermal properties are reported. The behavior of these properties in relation to the physical parameters of the model such as rotation, curvature and quantum numbers was investigated. Moreover, the impact of the topological defect on the quantum system under investigation is discussed.

General pseudo self-adjoint boundary conditions for a 1D KFG particle in a box

We consider a 1D Klein–Fock–Gordon particle in a finite interval, or box. We construct for the first time the most general set of pseudo self-adjoint boundary conditions for the Hamiltonian operator that is present in the first order in time 1D Klein–Fock–Gordon wave equation, or the 1D Feshbach–Villars wave equation. We show that this set depends on four real parameters and can be written in terms of the one-component wavefunction for the second order in time 1D Klein–Fock–Gordon wave equation and its spatial derivative, both evaluated at the endpoints of the box. Certainly, we write the general set of pseudo self-adjoint boundary conditions also in terms of the two-component wavefunction for the 1D Feshbach–Villars wave equation and its spatial derivative, evaluated at the ends of the box; however, the set actually depends on these two column vectors each multiplied by the singular matrix that is present in the kinetic energy term of the Hamiltonian. As a consequence, we found that the two-component wavefunction for the 1D Feshbach–Villars equation and its spatial derivative do not necessarily satisfy the same boundary condition that these quantities satisfy when multiplied by the singular matrix. In any case, given a particular boundary condition for the one-component wavefunction of the standard 1D Klein–Fock–Gordon equation and using the pair of relations that arise from the very definition of the two-component wavefunction for the 1D Feshbach–Villars equation, the respective boundary condition for the latter wavefunction and its derivative can be obtained. Our results can be extended to the problem of a 1D Klein–Fock–Gordon particle moving on a real line with a point interaction (or a hole) at one point.

Nonlinear Klein\u2013Gordon equation and the Bose\u2013Einstein condensation

The interest in the Klein–Gordon equation with different potentials has increased in recent years due to its possible applications in Cosmology, Hadron Physics and High-Energy Physics. In this work, we investigate the solutions of the Klein–Gordon equation for bosons under the influence of an external potential by using the Feshbach–Villars method. We present detailed results for two cases: the Coulombic potential and the harmonic potential. For the latter case, we studied the effects of self-interacting particles by adopting a mean-field approach. We show that our results converge smoothly to the solution of the Schrödinger equation for the same systems as the relativistic effects diminish.

Feshbach–Villars equation in a κ-Minkowski spacetime

In this paper, by using the Dirac derivatives the Klein–Gordon (K-G) equation is determined in a [Formula: see text]-Minkowski spacetime. The dispersion relation and the first-order approximation case are deduced. The Feshbach–Villars (FV) equation is derived by applying the new linearization process to the time. We then study the effect of magnetic interaction on energies spectrum in a [Formula: see text]-Minkowski spacetime as an application, as a result we found that the energies spectrum are not symmetrical. We also study the case of hydrogen atom in non-relativistic limit by using perturbation theory. The upper bound of the [Formula: see text]-deformation parameter is evaluate, on the basis of the experimental data for [Formula: see text] transition frequency.

Bosonic Schwinger model for spinning Feshbach–Villars particle in step potential

Propagator for spinning particle described in the Feshbach–Villars formalism is set up using the bosonic coherent state path integral. The boson model for the charge-spin symmetry is presented following the Schwinger recipe. The calculations are explicitly evaluated in the case of the step potential. The perturbation technique is used and the series are exactly summed.

Relativistic Spin-0 Feshbach–Villars Equations for Polynomial Potentials

We propose a solution method for studying relativistic spin-$0$ particles. We adopt the Feshbach-Villars formalism of the Klein-Gordon equation and express the formalism in an integral equation form. The integral equation is represented in the Coulomb-Sturmian basis. The corresponding Green's operator with Coulomb and linear confinement potential can be calculated as a matrix continued fraction. We consider Coulomb plus short range vector potential for bound and resonant states and linear confining scalar potentials for bound states. The continued fraction is naturally divergent at resonant state energies, but we made it convergent by an appropriate analytic continuation.

Fermionic Schwinger model for spinning Feshbach–Villars particle in magnetic field

Propagator for spinning particle described in the Feshbach–Villars formalism is set up using fermionic coherent state path integral. The fermionization of the charge-spin symmetry is presented following the Schwinger recipe. The explicit calculations are done in the free case and magnetic interaction using the Foldy–Wouthuysen tranformation. The spectrum and wave functions are deduced.

Relativistic particle in a box: Klein–Gordon versus Dirac equations

The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantised energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein–Gordon equation in the Feshbach–Villars formalism. We compare the solutions and the energy spectra obtained with the corresponding ones from the Dirac equation for a spin one-half relativistic particle. We note the similarities and differences, in particular the spin effects in the relativistic energy spectrum. As expected, the non-relativistic limit is the same for both kinds of particles, since, for a particle in a box, the spin contribution to the energy is a relativistic effect.

Linearization of wave equations

We explore the ways to linearize the wave equations. Special emphasis is paid to the Klein–Gordon equation for a spin-0 relativistic particle and the Helmholtz equation governing scalar optics. Owing to the mathematical similarity, both of these equations are linearized using the Feshbach–Villars procedure. Maxwell's equations are linear but coupled and constrained. So, a matrix representation is presented. New formalisms of beam optics are presented using the linearized form of the Helmholtz equations and the Dirac-like matrix form of Maxwell's equations respectively. It is shown that the matrix formulation of the Maxwell optics naturally leads to a unified treatment of beam optics (including aberrations to all orders) and light polarization, from a single parent Hamiltonian. The non-traditional treatments using quantum methodologies lead to wavelength-dependent modifications of the traditional prescriptions. In the limit of low wavelength, the non-traditional prescriptions of both Helmholtz optics and Maxwell optics presented here reproduce the ‘Lie algebraic formalism of light beam optics’. The accompanying machinery of the Foldy–Wouthuysen transformation technique is described. From the new prescriptions of light beam optics, it is seen that the Hamilton's optical-mechanical analogy persists in the wavelength-dependent regime.

Feynman Propagator for a Spinless Relativistic Particle in Feshbach–Villars Representation

In this paper we used the Feynman path integral technique to calculate propagators for the Feshbach–Villars equation and the generalized Feshbach–Villars equation. We calculate these propagators for three cases: for a free particle, for a particle in a constant magnetic field and in an inhomogeneous magnetic field. Some special relations between Pauli matrices, which appear in the Feshbach–Villars equation, allow us to express the Feynman propagator for the free particle in a compact form, and in the one-dimensional case we can calculate this propagator in explicit form in terms of Bessel functions. For the second case, i.e. for a particle in a constant magnetic field, using a relation between the Feshbach–Villars Hamiltonian and a certain associated Schrodinger Hamiltonian, which appears in the Feshbach–Villars representation, we are able to express the path integral propagator for Feshbach–Villars equation in explicit form by means of Landau levels for the associated Schrodinger Hamiltonian. In the third case, i.e. for inhomogeneous magnetic field, in the same way as in the second case, we can express the path integral propagator for Feshbach–Villars equation in the explicit form in terms of eigenfunctions of problem.

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

The Feshbach–Villars equations, like the Klein–Gordon equation, are relativistic quantum mechanical equations for spin-0 particles.We write the Feshbach–Villars equations into an integral equation form and solve them by applying the Coulomb–Sturmian potential separable expansion method. We consider boundstate problems in a Coulomb plus short range potential. The corresponding Feshbach–Villars CoulombGreen’s operator is represented by a matrix continued fraction.

NONCOMMUTATIVE PATH INTEGRAL FOR SPINLESS RELATIVISTIC EQUATION IN THE TWO-COMPONENT THEORY

In this paper, we have constructed the Green function of the Feshbach–Villars (FV) spinless particle in a noncommutative (NC) phase-space coordinates, where the Pauli matrices describing the charge symmetry are replaced by the Grassmannian odd variables. Subsequently, for the perform calculations, we diagonalize the Hamiltonian governing the dynamics of the system via the Foldy–Wouthuysen (FW) canonical transformation. The exact calculations have been done in the cases of free particle and magnetic field interaction. In both cases, the energy eigenvalues and their corresponding eigenfunctions are deduced.

Discrete Symmetry in Relativistic Quantum Mechanics

EPR experiment on $K^0-\bar{K}^0$ system in 1998\cite{1} strongly hints that one should use operators $\hat{E}_c=-i\hbar\frac{\partial}{\partial t}$ and $\hat{\bf p}_c=i\hbar\nabla$ for the wavefunction (WF) of antiparticle. Further analysis on Klein-Gordon (KG) equation reveals that there is a discrete symmetry hiding in relativistic quantum mechanics (RQM) that ${\cal P}{\cal T}={\cal C}$. Here ${\cal P}{\cal T}$ means the (newly defined) combined space-time inversion (with ${\bf x}\to -{\bf x}, t\to-t$), while ${\cal C}$ the transformation of WF $\psi$ between particle and its antiparticle whose definition is just residing in the above symmetry. After combining with Feshbach-Villars (FV) dissociation of KG equation ($\psi=\phi+\chi$)\cite{2}, this discrete symmetry can be rigorously reformulated by the invariance of coupling equation of $\phi$ and $\chi$ under either the combined space-time inversion ${\cal P}{\cal T}$ or the mass inversion ($m\to -m$), which makes the KG equation a self-consistent theory. Dirac equation is also discussed accordingly. Various applications of this discrete symmetry are discussed, including the prediction of antigravity between matter and antimatter as well as the reason why we believe neutrinos are likely the tachyons.

The Green function for the Duffin–Kemmer–Petiau equation

We present a calculation of the Green function for the Duffin–Kemmer–Petiau equation in the case of scalar and vectorial particles interacting with a square barrier potential, and relate it to that of the Klein–Gordon equation. A formal Hamiltonian of the Duffin–Kemmer–Petiau theory is first developed using the Feshbach–Villars analogy and the Sakata and Taketani decomposition. The coefficients of reflection and transmission are deduced.

NEGATIVE ENERGY SOLUTIONS AND SYMMETRIES

We revisit the negative energy solutions of the Dirac (and Klein–Gordon) equation, which become relevant at very high energies in the context of the Feshbach–Villars formulation, and study several symmetries which follow therefrom. Significant consequences are briefly examined.

Klein–Gordon equation in hydrodynamical form

We follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.

Pair creation of spin-1/2 particles in Feshbach–Villars formalism

The Feshbach–Villars equation (FV1/2) is studied by comparing it with the Dirac equation and by examining its behaviour in the creation process of pairs of particles from vacuum while being perturbed by an external field using two approaches: (i) the effective action method of Schwinger and (ii) the Bogoliubov transformation. In the first approach, the transition vacuum–vacuum amplitude is determined by treating the FV1/2 field as a fermionic field. In the second approach, the anticommutation relations are used contrary to the case of spin 0, in order to obtain meaningful probabilities. The two approaches show that the expressions of pair creation probability as well as those relating to the vacuum remaining vacuum are exactly similar to those obtained by either FV1/2 or Dirac equations.