Positive-energy states

Also arising from relativistic quantum mechanics is the fact that there should be both negative and positive energy states. One of these corresponds to electron energies and the other corresponds to the electron antiparticle, the positron. 

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... 

Both FW and ESC methods have been applied to decouple the positive energy states from the negative and mixed states of the two-fermion DBC Hamiltonian [47-49]. Since we are only interested in positive energy solutions, we may start from the original two-fermion DBC Hamiltonian without positive and negative energy projection operators. The two-fermion DBC Hamiltonian for stationary states can be written as... 

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. 

A more appropriate spin-orbit coupling Hamiltonian can be derived if electron-positron pair creation processes are excluded right from the beginning (no-pair approximation). After projection on the positive energy states, a variationally stable Hamiltonian is obtained if one avoids expansion in reciprocal powers of c. Instead the Hamiltonian is transformed by properly chosen... 

Since equation (3.83) contains even operators only, we may set p equal to 1, and substituting for g and G we obtain the Hamiltonian which is appropriate to positive energy states only,... 

The creation/annihilation operators aj /a, denote the one-particle operators which diagonalize the Hamiltonian Hen. The summation indices i, j, k, l denote the usual set of one-electron quantum numbers and run over positive-energy states only. The quantities Vjju are two-electron Coulomb matrix elements and the quantities biju denote two-electron Breit matrix elements, respectively. We specify their static limit (neglecting any frequency dependence) ... 

There is a catch the construction of energy projectors is simple for free electrons, but in atomic models the only rigorous construction requires that all the wanted positive energy states be already available ... 

A natural way to generalize the non-relativistic many-body Schrodinger equation is to combine the one-electron Dirac operators and Coulomb and Breit two-electron operators. However such an equation would have serious defects. One of them is the continuum dissolution first discussed by Brown and RavenhaU [36]. This means that the Schrodinger-type equation has no stable solutions due to the presence of the negative energy Dirac continuum. A constrained variational approach to the positive energy states becomes therefore necessary. 

The decomposition of the eigenvalue W into a rest mass contribution mc and a remainder E is only meaningful if IT > 0 and IE] energy close to the onset of the positive-energy continuum, in particular for a bound positive-energy state. If we were interested in a state near the negative-energy continuum, we would define E as VT - - mc vide infra). 

One should, nevertheless, be aware, that it has been essential for the proof of the holomorphicity of G z) = G z+mc ) to consider this resolvent for values of 2 in the neighborhood of the energies of the bound positive-energy states, i.e. for 2 Levy-Leblond equation one gets a different limit of G z) for positronic states, i.e. for 2 = 2 — mc, and no limit at all for ultrarelativistic states. 

While the next step for the FW transformation was a renormalization by means of the transformation Wb in order to achieve an overall unitary transformation (which can actually be done before or after the projection to positive energy states), we now apply a transformation that simultaneously block-diagonalizes the Hamiltonian for electrons, such that it isolates a block corresponding to the model space, reestablishes unitarity of this... 

In choosing a finite-dimensional model space or a direct sum of such spaces the FW singularities are avoided. They only arise if the model space is the entire space of positive-energy states. 

The rest mass energy for the positive energy states can be removed by... 

The rules of perturbation theory associated with the relativistic no-pair Hamiltonian are identical to the well-known rules of nonrelativistic many-body perturbation theory, except for the restriction to positive-energy states. The nonrelativistic rules are explained in great detail, for example, in Lindgren and Morrison [30]. Let us start with a closed-shell system such as helium or beryllium in its ground state, and choose the background potential to be the Hartree-Fock potential. Expanding the energy in powers of V) as... 

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