An approach to MBPT calculations using positive- and negative-energy electrons

Negative-energy solutions of the Dirac equation have been difficult to incorporate into calculations involving positive-energy many-electron systems. Chief among these difficulties include what is known as "degeneracy collapse.'" The primary way of avoiding such difficulties has been to exclude negative-energy states from the many-electron Dirac Hamiltonian using positive-energy-only projection operators. This paper outlines a different approach that both incorporates negative-energy solutions and suppresses their contribution in a rigorous way. By requiring that negative-valued probabilities associated with a conserved current vanish for a linear combination of positive- and negative-energy electron state wavefunctions, an electron wavefunction with both positive- and negative-energy contributions is uniquely defined. Significantly, by balancing a negative-valued probability of a positive-energy state against the positive-valued probability of a negative-energy state, the contribution from the negative-energy state can be inherently suppressed—without the use of projection operators. The use of these electron states is illustrated in barrier-scattering problems. Hydrogenic solutions are also constructed and central-field solutions are considered. Unique aspects of these states are discussed.