Negative-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. 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

The prediction, and subsequent discovery, of the existence of the positron, e+, constitutes one of the great successes of the theory of relativistic quantum mechanics and of twentieth century physics. When Dirac (1930) developed his theory of the electron, he realized that the negative energy solutions of the relativistically invariant wave equation, in which the total energy E of a particle with rest mass m is related to its linear momentum V by... 

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... 

For a free particle (A = cp = 0), Eq. (3.6.13) reduces to Eq. (3.6.8) we now must, alas, accept the possibility of a set of negative-energy solutions. Do particles with negative energies exist Yes, they are the so-called antiparticles. In other words, the existence of the positron was predicted by the Dirac equation. 

Therefore the negative-energy solutions for the Dirac equation are not a mathematical fiction In principle, each fundamental particle does have its corresponding antiparticle (which has the opposite electrical charge, but the same spin and the same nonnegative mass). Equation (3.6.15) also shows the formation of a transient Coulomb-bound electron-positron pair ("positronium"), whose decay into two photons is more rapid if the total spin is S = 0 than if it is S = 1, and is dependent on the medium. 

Note that this relationship is in agreement with our previous equation (3.6) connecting the energy and momentum. Now equation (3.22) reveals the existence of positive and negative energy solutions, i.e. 

We will now consider the non-relativistic limit of the above quantities. We will do so by considering the non-relativistic limit of (119). Evidently this limit (c- ) can not be taken directly as the Dirac operator contains operators linear and even quadratic in c. However, the desired limit can be obtained by a change of metric, that is a non-unitary transformation. The choice of transformation will be depend on whether we seek the non-relativistic limit of positive or negative energy solutions. As we are interested in bound electrons we choose the transformation... 

In the case of the many-body terms the neglect of vacuum corrections is no longer uniquely defined. Two possible approaches can be distinguished, both set up within the KS Furry picture in order to be consistent with (70). In the no-pair approximation the contribution of the negative energy solutions to all intermediate sums over states are ignored. For instance, the DPT analog of the... 

The computational procedure usually also yields, for given effective potential, the negative-energy solutions for the Dirac equation Eq. (14), for which j <2c. These are not used to construct the Kohn-Sham reference function xs (s Eq. (1)) and likewise, all vacuum... 

For most chemical applications, one is not interested in negative energy solutions of a four-component Dirac-type Hamiltonian. In addition, the computational expense of treating four-component complex-valued wave functions often limited such calculations to benchmark studies of atoms and small molecules. Therefore, much effort was put into developing and implementing approximate quantum chemistry methods which explicitly treat only the electron degrees of freedom, namely two- and one-component relativistic formulations [2]. This analysis also holds for a relativistic DFT approach and the solutions of the corresponding DKS equation. 

The imphed wave nature of elementary matter furthermore clarifies their mode of interaction through standing waves generated by the interference between advanced and retarded wave components. The negative-energy solutions of relativistic wave equations first indicated the existence of antimatter, as later confirmed experimentally. To avoid the annihilation of matter and antimatter on a cosmic scale an involuted structure of the vacuum, consistent with projective space-time, is inferred. 

A result is that for a vector to be representative of the wave function it must have four components or, alternatively, that must contain a variable taking on four values. Dirac has explained why the electron has spin, which was known as requiring the wave function W to have two components, and that this number must be doubled because the quasi-linear Eq. (2.11), which is equivalent to the quadratic Eq. (2.10) under the conditions (2.13), has additional, negative-energy solutions, which he assigned to an antielectron having opposite charge [5]. 

The existence of negative-energy solutions is therefore an intrinsic feature of every relativistic theory, although they cannot be interpreted in classical mechanics and thus have to be discarded here. In quantum mechanics, we will have to find suitable means and interpretations to cope with this peculiarity (cf. chapters 5 and 7). 

---