FOUR-VECTOR VERSUS FOUR-SCALAR REPRESENTATION OF THE DIRAC WAVE FUNCTION

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

MAGNETIC MOMENT μ OF Ω--BARYON USING RICHARDSON POTENTIAL IN A RELATIVISTIC HARTREE–FOCK (RHF) METHOD

A RHF method is used to calculate magnetic moment μ of Ω- using the Richardson potential. Unlike the mass of baryon, μ is very sensitive to the detailed structure of the model since it depends on the wave function. Given any quark–quark (qq) potential we have the RHF wave function for the ground state. We have developed a code for expanding this Dirac wave function in terms of two sets of oscillators. To fit both the mass and magnetic moment simultaneously we find that it is essential to separate out the confining part from the asymptotically free part of the potential and use different parameters for each part. The best fitted results from our model are in good agreement with experiment.

A SIMPLE DIRAC WAVE FUNCTION FOR A COULOMB POTENTIAL WITH LINEAR CONFINEMENT

A simple analytical solution is found to the Dirac equation for the combination of a Coulomb potential with a linear confining potential. An appropriate linear combination of Lorentz scalar and vector linear potentials, with the scalar part dominating, can be chosen to give a simple Dirac wave function. The binding energy depends only on the Coulomb strength and is not affected by the linear potential. The method works for the ground state, or for the lowest state with l=j-1/2, for any j.

Dirac and Maxwell’s Equations Derived from the Density Functions

We present linear transformations between the components of the electric and magnetic fields, E = F1 (H ) and H = F2 (E ), which leave the expression for the density of electromagnetic field energy unchanged. If the coefficients occurring in these transformations are interpreted as the components of the momentum vector of the photon and use is made of the correspondence principle W →∂t and p→∂x, then these transformations turn out to be Maxwell’s equations. In a similar fashion, when starting from the probability density associated with the Dirac wave function, then one can also arrive, via appropriate transformations between the components of the 4- spinor, at the Dirac equation.

Translations as gauge transformations

A local description of space and time in which translations are included in the group of gauge transformations is studied using the formalism of fibre bundles. It is shown that the flat Minkowski space–time may be obtained from a non-flat connection in a de Sitter structured fibre bundle by choosing at least two different cross-sections. The interaction terms in the covariant derivative of a Dirac wave function that correspond to translations may be interpreted as the mass term of the Dirac equation, and then the two cross-sections (gauges) correspond to the description of a fermion and antifermion respectively.