Spinors and bispinors

The last equation needs a comment. Because the matrices a have dimension 4, 

How to operate the V-component spinor (for V = 4 we have called them bispinors) Let us construct the proper Hilbert space for the Af-component spinors. As usual (p. 895), first, we will define the sum of two spinors in the following way  

we check that the spinors form an Abelian group with respect to the above defined addition (cf. Appendix C, p. 903) and, that the conditions for the vector space are fulfilled (Appendix B). Then, we define the scalar product of two spinors 

An operator acting on a spinor means a spinor with each component resulting from action on the corresponding component 

Sometimes we will use the notation, in which a matrix of operators acts on a spinor. In this case the result corresponds to multiplication of the matrix (of operators) and the vector (spinor) 

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Using matrix multiplication rules, the Dirac equation [Eq. (3.54)] with bispinors can be rewritten in the form of two equations with spinors and 0 ... 

Non-relativistic quantum theory of atoms and molecules is built upon wave-functions constructed from antisymmetrized products of single particle wave-functions. The same scheme has been adopted for relativistic theories, the main difference now being that the single particle functions are 4-component spinors (bispinors). The finite matrix method approximates such 4-spinors by writing... 

The basic reason for the divergences observed at the end of the previous subsection is that there is a fundamental difference between the upper large) component g> of the Dirac bispinor, and the 2-component FW spinor... 

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

Taking into account the particular structure of the bispinors i l and we obtain the same equations expressed in (two component) spinors ... 

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