Dirac spectrum

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

If En represents the lowest positive energy eigenvalue of the bound Dirac spectrum, the further evaluation of (23) is straightforward employing standard methods. In the present work this is the case as we consider only the lS 1/2 state. The numerical evaluation scheme is similar to that described already. After including the results for wf irred and AE wf irred, we obtain a total... 

Here r = ri2 and En = En (l— 0), so that the Feynman rules for the integration over u) variables are assumed. All sums run over the complete Dirac spectrum for the electron in the field of the nucleus. The expressions for the counterterms... 

For the evaluation of the sums over the Dirac spectrum the B-spline approximation has been employed. The number of grid points N and the order of splines k have been chosen to be N = 23 and k = 4, respectively. This corresponds to 50 different radial Dirac states which are taken into account for a given Dirac angular-momentum quantum number. 

The mathematical problem associated with the Dirac Hamiltonian, i.e. the starting point of the relativistic theory of atoms, can be phrased in simple terms. The electron-positron field can have states of arbitrarily negative energy. As a general feature of the Dirac spectrum this instability occurs even in the case of extended nuclei and even in the absence of any nucleus (free Dirac spectrum), the energy is not bounded from below. This gives rise to the necessity of renormalization and well-established renormalization schemes have been around for many decades. Despite their successful applications in physics, we may ask instead whether there exist states that allow for positivity of the energy. 

The electron propagator S xiX2) is defined by Eq(122). Inserting these propagators in Eq(165) we have additional double integration over the freqtiency variables and the double summation over the Dirac spectrum m ri2. 

This is mainly due to the development of numerical methods (B-spline approach [62], space discretization [63]) that allow summations to be performed over the complete Dirac spectrum for arbitrary spherically symmetric potentials. 

The electron propagators in the numerical B-spline approach can be treated in the same way as bound propagators with Z=0, so that in total Eq(220) contains the triple summation over the Dirac spectrum. This summation can be performed numerically with modern computer facilities. 

QED provides a framework for describing the role of the negative energy states in the Dirac theory and the divergences which arise in studies of electrodynamic interactions.83-85 In Section 2.3 it will be shown how the use of the algebraic approximation to generate a discrete representation of the Dirac spectrum has opened the way for the transcription of the rules of QED into practical algorithms for the study of many-electron systems. 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

Unless carefully implemented the representation of the Dirac spectrum obtained within the algebraic approximation may exhibit undesirable properties which are not encountered in non-relativistic studies. In particular, an inappropriate choice of basis set may obliterate the separation of the spectrum into positive and negative energy branches. So-called intruder states may arise, which are impossible to classify as being of either positive or negative energy character. The Furry bound state interaction picture of quantum electrodynamics is thereby undermined. 

Figure 22 Schematic representation of the Dirac spectrum and of the Schrodinger spectrum generated in the algebraic approximation...

Exclusion Principle. The energy associated with the filled vacuum is an unobservable constant which should be subtracted from a given physical model. Calculations which go beyond an independent particle model but are carried out using only the positive energy branch of the Dirac spectrum are said to be carried out within the no virtual pair approximation. Such calculations essentially follow the procedures adopted in non-relativistic studies. The relativistic and non-relativistic correlation energy calculations differ only in the model used to defined the reference independent particle model. 

In the limit of c oo the lower component vanishes and 4 terms into nonrelativistic solutions of the Schrddinger equation. Simultaneously the Dirac energy for these solutions becomes equal to the non-relativistic energy. For this reason the part of the Dirac spectrum which has the Schrddinger non-relativistic limit for c oo is referred to as the positive electronic energy spectrum and is associated with the dominant contribution of 4 in the 4 D-spinor (4.17). 

Since the infinite-order two-component theory is based on exact equations, it is obvious that it must reproduce all features of the positive-energy Dirac spectrum. However, the way this theory is used introduces the algebraic approximation. 

Figure 8.1 From the three different parts of the one-electron Dirac spectrum (see Figure 5.1 (b)) six different sets of states — s(++), and — can be con-...

In the preceding two chapters, we dealt with general unitary transformation schemes to produce a one-electron Hamiltonian valid for only the positive-energy part of the Dirac spectrum that governs the electronic bound and continuum states. Evidently, these unitary transformation schemes are elegant but involved. Developments in quantum chemistry always focus on efficient approximations in a sense that the main numerical contribution of some physical effect is reliably captured for any class of molecule or molecular aggregate. The so-called elimination techniques have been very successful in this sense and are therefore discussed in the present chapter. 

A consequence of this conjecture, which survives in more modern formulations of relativistic quantum mechanics, is that even the simple hydrogenic atom is a many-body problem The electrons filling the negative energy branch of the Dirac spectrum are... 

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