Dirac spinors

Transformation properties of Dirac spinors in particular under inversions Marshak, R. E., and Sudarshan, E. C. G., Introduction to Elementary Particle Physics, Interscience Publishers, Inc., New York, 1961. 

Note that the projection operator P = (1 a, vF)/2 projects out the particle state, wF, and the anti-particle state, ip- (or more preciselyip-), from the Dirac spinor field k. The quasi-quarks in a patch carries the residual momentum l1 and is given as... 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

As we can see, the FW two-component wave function is not the large component of the Dirac spinor, but it is related to it by an expression involving X. Consider a similarity transformation based on U parameterized as... 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

The orbital angular momentum quantum numbers, = I and corresponding, respectively, to the large and to the small components of the Dirac spinor are equal to... 

In another physics application, the Dirac equation for states of an electron in relativistic space-time requires wave functions taking values in the complex vector space C" = (ci, C2, c, C4) ci, C2, C3, C4 e C. These wave functions are called Dirac spinors. 

To take this theory further would be to embed it into an 517(4) gauge theory. The gauge potentials are described by 4 x 4 traceless Hermitian matrices and the Dirac spinor has 16 components. The neutrality of the photon is then given by the sum over charges, which vanishes by the tracelessness of the theory. The Higgs field is described by a 4 x 4 matrix of entries. 

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