Dirac Weyl representation

Apart of historical reasons, there are several features of the Dirac-Pauli representation which make its choice rather natural. In particular, it is the only representation in which, in a spherically-symmetric case, large and small components of the wavefunction are eigenfunctions of the orbital angular momentum operator. However, this advantage of the Dirac-Pauli representation is irrelevant if we study non-spherical systems. It appears that the representation of Weyl has several very interesting properties which make attractive its use in variational calculations. Also several other representations seem to be worth of attention. Usefulness of these ideas is illustrated by an example. 

This particular choice of the (4 x 4)-matrices is neither unique nor is the dimension 4 the only dimension, which allows us to fulfill the required properties. Higher even dimensions are also possible. How one may find other representations of these four parameters will be discussed in section S.2.4.3. The particular choice of Eqs. (5.37) is called the standard representation of the Dirac matrices a. and j6. We give as a further example the Weyl representation... 

Basically, DKH theory starts from the standard representation of the Dirac Hamiltonian introduced in section 5.2. The fact that the Dirac matrices a and j6 are not uniquely defined suggests that a different representation of the Dirac operator might be more advantageous. In order to shed light on this question, we consider the Weyl representation of Eq. (5.38), which produces a Dirac Hamiltonian of a very different structure, namely the following... 

The relations 04,0 1 + 0.104- = 2Ski, Clifford algebra, for which we choose the Dirac representation. In a phase space language, employing Weyl quantisation, this Hamiltonian can be written as (see, e.g., (Dimassi and Sjostrand, 1999))... 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

The canonical quantization of the field has introduced by Dirac [1] (see also Refs. 2-4,10,11,14,15,26,27) is provided by the substitution of the photon operators, forming a representation of the Weyl-Heisenberg algebra, into the... 

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