The Brown-Ravenhall disease and related problems

One avoids such unphysical states if one interprets the Dirac operator as acting in a Fock space, with a vacuum, in which no electrons and no positrons are occupied. Annihilation (creation) operators for negative-energy states are then interpreted as positron creation (annihilation) operators. 

The simplest approximation to this Fock space theory consists in projecting the n-particle Hamiltonian to electronic states, i.e. to ignore the creation of virtual electron-positron pair states, whence the name no-pair theory. The next step after a no-pair theory would be a formalism, in which an n-electron state mixes, e.g. with an (n -1- l)-electron-1-positron state etc.. Not the particle number, but the charge is a constant of motion. In this way one takes care of vacuum polarization, which is a real physical effect. It is, however, not recommended to treat it in such a brute-force way, but rather to use the apparatus of QED. 

Fortunately, if we use DPT, the Brown-Ravenhall disease does not show up at all, because we start from an n-electron state in the nrl, and there is no chance for unphysical positronic or ultrarelativistic components to mix in. This is easily seen in the quasidegenerate formalism, where we insist on relating the upper and lower components via the correct X-operator for electrons, which clearly takes care that we only move in the world of 

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