Negative-energy states filling

A naive application of Fermi statistics would require that for the ground state 4>,5) of this system all levels below the Fermi energy Cp are occupied. This would imply that in addition to the finite number of discrete levels below Sp all negative energy states are filled,... 

There are several points to be noted about this operator. First, the second term creates an electron-positron pair, and the third term annihilates an electron-positron pair. This means that the Hamiltonian connects states with different particle numbers, that is, particle number is not conserved, though charge is. The existence of these terms embodies the idea of an infinitely-many-body problem that arose from the filling of the negative-energy states in Dirac s interpretation. Second, the order of the operators in the fourth term means that the vacuum expectation value of this operator is not zero, but... 

Thus the vacuum has an energy that is equal to the sum of the energies of the negative solutions of the Dirac equation, as is expected from Dirac s interpretation. Note that the matrix elements are the same as in the Dirac equation, so the sum is negative and infinite. This Hamiltonian operator therefore represents the first stage of the Dirac reinterpretation with the negative-energy states all filled. 

Only the last term contributes, but this yields an infinite charge, consistent with all negative-energy states being filled. However, if we use the normal-ordered... 

This is not quite the end of the matter. What we have assumed in the QED approach is that the reference vacuum is that of the current guess. If we were to take an absolute reference, such as the free-particle vacuum, the normal ordering should take place with respect to this fixed vacuum, and then the QED approach would give the same results as the filled Dirac approach, in which rotations between the negative-energy states and the unoccupied electron states affect the energy. By this means a vacuum polarization term has been introduced into the procedure, but without the renormalization term. In atomic structure calculations in which QED effects are introduced, the many-particle states employed are usually the Dirac-Fock states (Mittleman 1981), that is, those that result from the empty Dirac picture. We will therefore take as our reference the QED approach with the floating vacuum —a vacuum defined with respect to the current set of spinors. 

According to Eq. (1.14), negative curvature of the valence band would mean a negative electron mass, which is physically not acceptable. It has therefore been concluded that occupied orbitals in the valence band correspond to holes. A hole acts in an applied electric or magnetic field as though it were a particle with a positive charge. This concept has been experimentally proved by Hall measurements (see Section 1.6). However, it only makes sense if nearly all energy states are filled by electrons. It should be further mentioned that the effective mass of holes may be different from that of electrons. A selection of values is listed in Appendix D. 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

As common in relativistic electronic structure theory, one invokes the so-called no-sea approximation where one neglects all vacuum contributions of the filled negative energy continuum [41]. The only remaining effect of the sea is the restriction for electrons to occupy only states of positive energy. Then the density is constructed from DKS one-electron orbitals of a single-determinant A-electron wave function ... 

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