Stability of QED Vacuum and 3+1 Dimensional Scattering Problem in the Presence of a Coulomb Scalar Potential and Vector Field

Abstract

We show that, in the presence of a scalar field the range of the value of external field parameters a and b, at which corresponding Hamiltonian operator is hermitian, essentially wider than in its absence. It allows us to study precisely the question on stability of QED vacuum in the presence of a strong electric field of a point charge Z|e| and external scalar Coulomb field with respect to electron-positron pair production. Also, we consider the scattering of Dirac particle by the specified fields in 3 + 1 dimensions. The phase shift and wave functions are obtained exactly. We calculate the scattering amplitude in a quasiclassical approximation as a partial wave series. By means of figures obtained for the cross section σ (θ) in general and special cases, such as a = b and a = b, we find that σ (θ) is not exactly symmetric about θ = π.

Relativistic Coulomb Problem for Z Larger Than 137

“…In the following section, we define the problem and present our approach to the solution. This work is an extension to, and/or departure from earlier work on the subject wherein a scalar Coulomb coupling is introduced in addition to the vector Coulomb coupling [7]. We show that there is a physical difference between the "pseudo Coulomb potential" introduced in this work and the scalar Coulomb potential.…”

“…Therefore, the physical content of the potential coupling in the Dirac Hamiltonian is a combination of the potential ( ) is a scalar potential; both having equal magnitudes, αμ . We should also note that this constitutes a departure from earlier work on this problem, where a pure scalar Coulomb potential is added to the vector Coulomb potential (see, for example [7], and references therein). Moreover, the contribution of a pure scalar potential survives the non-relativistic limit, whereas this pseudo potential does not [8].…”

“…To obtain the bound states energy spectrum for this equivalent relativistic Coulomb problem, we first establish the parameter map between the relativistic equation (7) and the nonrelativistic equation (9). This map reads as follows: 0…”

“…(7) for positive energy (the top sign) is obtained by taking m E ε ≅ + , whereE m << . This results in the same Schrödinger equation (9) with an electric charge Z ν μ = + .…”

“…16) into(7) and using the differential equation of the Laguerre polynomial[10], we obtain 2 spinor component (19) into the kinetic balance relation (6) with the top sign and using the differential and recursion properties of the Laguerre polynomials[10], we obtain the lower component as follows components of the spinor wavefunction obtained above in (19) and (21) are for positive energies. To obtain the corresponding negative energy solutions, we apply on them the map(8).…”