Dirac theory representation

The Dirac-Pauli representation is most commonly used in all applications of the Dirac theory to studies on electronic structure of atoms and molecules. Apart of historical reasons, there are several features of this representation which make its choice quite natural. Probably the most important is a well defined symmetry of and in the case of spherically-symmetric potentials V. The Dirac Hamiltonian... 

This requirement does not fix the Dirac matrices uniquely, and thus the whole Dirac theory and all systematic approximations to it could equally well be formulated in terms of general four-dimensional quaternions, which are independent of a special representation and rely only on the algebraic properties of the Clifford algebra [8-10]. Such an implementation of the Dirac theory is known to speed up diagonalisation procedures significantly, and has successfully been employed in modem four-component relativistic program packages like Dirac... 

It took some time until it was realized that the Dirac theory describes the spin correctly because it is a spinor-field theory, and not because it is relativistic [16]. In fact, if one takes the nonrelativistic limit of the Dirac equation, spin survives, and this is consistent with the observation that the Galilei group has spinor representations as well. So, without any doubt, spin is not a relativistic effect. 

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. 

The two-component methods, though much simpler than the approaches based on the 4-spinor representation, bring about some new problems in calculations of expectation values of other than energy operators. The unitary transformation U on the Dirac Hamiltonian ho (Eq.4.23 is accompanied by a corresponding reduction of the wave function to the two-component form (Eq.4.26). The expectation value of any physical observable 0 in the Dirac theory is defined as ... 

According to the standard representation of the Dirac theory, the one-electron operators considered in chapters 5 and 6 are (4 x 4)-matrices acting on the one-particle Hilbert space... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

Recently in [6] we constructed effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric but strongly interacting theories with a Dirac fermion either in the two index symmetric or two index antisymmetric representation of the gauge group. These theories are planar equivalent, at N —> oo to SYM [7], In this limit the non-supersymmetric effective Lagrangians coincide with the bosonic part of the VY Lagrangian. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

The technology for solving the Schrddinger equation is so much farther advanced in r space than in p space that it is most practical to obtain the momentum-space from its position-space counterpart The transformation theories of Dirac [118,119] and Jordan [120,121] provide the hnk between these representations ... 

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. 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

M. Stanke, J. Karwowski, Recent advances in the theory of chemical and physical systems, In J.-R Julien, J. Maruani, S. Wilson, G. Delgado-Barrio (Eds.), Non-Standard Representations of the Dirac Equation and the Variation Method, Springer, Dordrecht, The Netherlands, 2006, pp. 217-228. 

Independently of the approximations used for the representation of the spinors (numerical or basis expansion), matrix equations are obtained for Equations (2.4) that must be solved iteratively, as the potential v(r) depends on the solution spinors. The quality of the resulting solutions can be assessed as in the nonrelativistic case by the use of the relativistic virial theorem (Kim 1967 Rutkowski et al. 1993), which has been generalized to allow for finite nuclear models (Matsuoka and Koga 2001). The extensive contributions by I. P. Grant to the development of the relativistic theory of many-electron systems has been paid tribute to recently (Karwowski 2001). The higher-order QED corrections, which need to be considered for heavy atoms in addition to the four-component Dirac description, have been reviewed in great detail (Mohr et al. 1998) and in Chapter 1 of this book. 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

We proceed somewhat similarly as in the theory of effective Hamiltonians (see the Appendix). However, we do, of course, not consider the matrix representation of the Dirac operator in a given basis, but take directly the matrix form of D in terms of the upper and lower spinor components. 

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