Negative Energy States

The appearance of negative energy states was initially considered to be a fatal flaw in the Dirac theory, because it renders all positive energy states... 

The DC or DCB Hamiltonians may lead to the admixture of negative-energy eigenstates of the Dirac Hamiltonian in an erroneous way [3,4]. The no-virtual-pair approximation [5,6] is invoked to correct this problem the negative-energy states are eliminated by the projection operator A+, leading to the projected Hamiltonians... 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

The definition of the no-sea approximation for is not completely unambiguous. As discussed in Appendix B we define it through neglect of all vacuum fermion loops in the derivation of an approximate [/]. Alternatively, one could project out all negative energy states, thus generating a direct equivalent of the standard no-pair approximation. As one would expect the differences between these two schemes to be small, we do not differentiate between these approximations here. 

Here D denotes the subspace of negative-energy states in both n) and n) spaces. 

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