Normalization 4-spinors

These relations were derived on the assumption that the normalization of the spinors was u u =1. It is oftentimes useful (for covariance reasons) to normalize the spinors so that uu = constant. The relation between these two normalizations is readily obtained. Upon multiplying Eq. (9-307) by (p E(p),s) on the left, we obtain... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nonrelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that are normally below the relativistic scale, the Berry phase obtained from the Schrodinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

The IRs of G comprise the vector representations, which are the IRs of G, and new representations called the spinor or double group representations, which correspond to half-integral j. The double group G contains twice as many elements as G but not twice as many classes g, and g,- are in different classes in G except when g,- is a proper or improper BB rotation (that is, a rotation about a binary axis that is normal to another binary axis), in which case g, and gt are in the same class and (gj, (xg,) are necessarily zero in spinor... 

The set of PFs [gj gj] is called the factor system. Associativity (a) and the symmetry of [gi gf ] (d) are true for all factor systems. The standardization (b) and normalization (c) properties are conventions chosen by Altmann and Herzig (1994) in their standard work Point Group Theory Tables. Associativity (a) follows from the associativity property of the multiplication of group elements. For a spinor representation T of G, on introducing [/ j] as an abbreviation for [g, g ], ... 

Exercise 12.6-1 D3h= 2C3 3C2 crh 2S3 3crv, where crh = /C2, aV = IC2", C2-L(C2, C2")- Thus the improper binary axis C2 is normal to the 3C2 proper binary axes and the three improper C2" binary axes. There are, therefore, three irregular classes crh, 3 C2, and 3oy. There are six classes in all and therefore six vector representations (Nv = 6). There are three regular classes and therefore three spinor representations, each of which is doubly degenerate since J]3=1 2 = 22 + 22 + 22 = 12 = g. 

When shown that (a p)2 = p2, the spinor form of Schrodinger s equation reduces to the normal complex form with E and p in operator notation ... 

In essentially all practical cases the normal order of (8, 9) demands the Dirac spinor orbitals < >k to be electron orbitals which according to the remark after (10) are orbitals which develop continuously from the free-electron continuum when the external potential A is continuously switched on. 

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