Positive-energy solutions

It is a characteristic feature of all these relativistic equations that in addition to positive energy solutions, they admit of negative energy solutions. The clarification of the problems connected with the interpretation of these negative energy solutions led to the realization that in the presence of interaction, a one particle interpretation of these equations is difficult and that in a consistent quantum mechanical formulation of the dynamics of relativistic systems it is convenient to deal from the start with an indefinite number of particles. In technical language this is the statement that one is to deal with quantized fields. 

A Lorentz invariant scalar product can be defined in the linear vector space formed by the positive energy solutions which makes this vector space into a Hilbert space. For two positive energy Klein-... 

We have noted that in the F W-representation the operator / plays the role of the sign of the energy. Hence, the proper Schrddinger equation describing a relativistic spin particle is obtained by restricting in Eq. (9-391) to contain only positive energy solutions. It is given by... 

One verifies that the vector = 1/V2(8 — ), constructed from positive energy solutions of (9-470), (9-471), (9-472), and (9-473), corresponds in the case of a photon of definite energy, to the photon having its spin parallel to its direction of motion, i.e., positive helicity (s-k = + k ). 

Before doing so we note a certain peculiarity of negative energy spinors. Let (p,a) be a positive energy solution of the Dirac equation corresponding to helicity s so that... 

Similarly, if hjtp) is a normalizable positive energy solution of ... 

The limit c —can now be taken provided that (1) V is everywhere non-singular, which is true for finite nuclei [42] but not point nuclei, and that (2) E < c which is true for the (shifted) positive-energy solutions only. With this procedure all relativistic effects are eliminated and one obtains the four-component non-relativistic Levy-Leblond equation [34,43]... 

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... 

The convention adopted here is that, for the positive-energy solutions of the Dirac equation, the large components correspond to p = 1,2 and the small components to p = 3,4. Some authors, notably Slater, use the opposite convention. 

When working on a positive energy solution, the operators A+ give unity, but on a negative energy solution, they give zero. The eigenstates to a matrix representation of Eq. (40) can conveniently be used to express the A operators. The electron-electron interaction operator, V12/ is dominated by the Coulomb interaction, which when complex rotation is used is written... 

This equation is the same as the classical Hamiltonian for a slowly moving electron, except for the middle term. This term represents an additional potential energy and may be interpreted as arising from the electron having a magnetic moment —(eh/2m)a. It was shown in equations (3.25) and (3.26) that the <,C and Vm components of the wave function are large in comparison with the f and 2 components for the positive energy solutions. Now the relations... 

A brief glance at the various terms involved in the RKS-scheme reveals a selfconsistency problem of considerable complexity The evaluation of both jy and requires summation over all negative and positive energy solutions and appropriate renormalisation in each step of the iterative procedure. Moreover, all potentials exhibit a four component structure. As the solution of this selfconsistency problem is at best tedious (if at all possible), one is bound to consider a suitable hierarchy of (hopefully) useful approximations. 

All the derivations made up to now have been rigorous. We are interested in comparing the relativistic and the non-relativistic Hamiltonian terms, which can be done approximately in the following way. The Dirac Hamiltonian is assumed in the form of the dominating term for the positive energy solutions E0 and the small perturbation HNR due to a slow-moving electron... 

The Dirac equation represents a proper relativistic form of the characteristic equation for energy. It is fulfilled for state vectors in a form of a four-component spinor. Since the upper two-component spinor dominates in the positive energy solutions for an electron, a decomposition of the Dirac equation is appropriate. 

In practice, this looks as follows. The many-electron wave function (let us focus our aUmtiim on a two-electron system only) is constructed from those trispinors, which correspond to positive energy solutions of the Dirac equation. For example, among two-electron functions built of such bispinors, no function corresponds to E, and, most importantly, to E". This means that carrying out computations with such abasis set, we do not use the full DC Hamiltonian, but insleacL its projection on the space of states with positive energies. 

For ultra-relativistic particles with p me the rest energy contribution to the energy E is negligible and we therefore find E c p. The energy-momentum relation (3.118) also holds for massless particles with m = 0 and reads E = c p (for positive-energy solutions). 

The intrinsic failure of the Foldy-Wouthuysen protocol is therefore without doubt related to the ill-defined 1/c expansion of the kinetic term Ep, which does not bear any reference to the external potential V. However, in the literature the ill-defined behavior of the Foldy-Wouthuysen transformation has sometimes erroneously been attributed to the singular behavior of the Coulomb potential near the nucleus, and even the existence of the correct nonrelativistic limit of the Foldy-Wouthuysen Hamiltonian is sometimes the subject of dispute. Because of Eqs. (11.82) and (11.83) and the analysis given above, the nonrelativistic limit c —> oo, i.e., X —> 0 is obviously well defined, and for positive-energy solutions given by the Schrodinger Hamiltonian /nr = / 2me + V. 

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