Spinor representation

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. 

The requirement of Postulate 3 that the equations of motion be form-invariant (i.e., that fix ) and A u(x ) satisfy the same equation of motion with respect to a as did fx) and Au x) with respect to x demands that the field variables transform under such transformations according to a finite dimensional representation of the Lorentz group. In other words it demands that transform like a spinor... 

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

This non-relativistic equation in terms of four-component spinors has been studied in detail by Levy-Leblond [44,45], who has shown that it results automatically from a study of the irreducible representations of the Gahlei group and that it gives a correct description of spin. It is easy to see that in the absence of an external magnetic field, equation (63) is equivalent to the Schrodinger equation in the sense that after elimination of the small component ... 

Let T be the fundamental representation of SO(4), and be the positive half-spinor representation. Let Mi = T 0ru(D) and M2 = S +0cHom(D, W). If we choose a complex structure on in other words, a reduction of the symmetry group from SO(4) into SU(2), T could be identified with A° L Hence Mi can be identified with A° 0c End(D). More explicitly, choosing a basis for T, we could write the identification as... 

LB. Frenkel, Spinor representations of affine Lie algebras, Proc. Natl. Acad. Sci. USA 77 (1980), 6303-6306. 

Representation of orbital spinors of symmetric molecules in terms of relativistic double groups [6]. 

The G-spinor representation (12), when substituted into (5), results in a time-dependent charge-current density... 

Whilst this demonstrates that calculations using the methods of this paper may prove very useful in studies of molecules containing only low-Z atoms, a major objective has been to study systems containing heavier atoms. So far, only a limited number of molecular calculations have been carried out with BERTHA at the DHF level, mainly in connection with studies of hyperfine and PT-odd effects in heavy polar molecules such as YbF [33] and TIF [13]. The reader is referred to the literature for an assessment of these calculations and for technical details on the construction of basis sets which must not only describe molecular bonding properly but also give a good representation of spinors close to the heavy nuclei to handle the short-range electron-nuclear electroweak interactions. 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

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. 

Thus, to completely solve the problem of symmetry reduction within the framework of the formulated algorithm above, we need to be able to perform steps 3-5 listed above. However, solving these problems for a system of partial differential equations requires enormous amount of computations moreover, these computations cannot be fully automatized with the aid of symbolic computation routines. On the other hand, it is possible to simplify drastically the computations, if one notes that for the majority of physically important realizations of the Euclid, Galileo, and Poincare groups and their extensions, the corresponding invariant solutions admit linear representation. It was this very idea that enabled us to construct broad classes of invariant solutions of a number of nonlinear spinor equations [31-33]. 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

In order to address polarization/rotation modulation—not just static polarization/rotation—an algebra is required which can reduce the ambiguity of a static representation. Such an algebra which is associated with 2, r, and that reduces the ambiguity up to a sign ambiguity, is available in the twistor formalism [28]. In this formalism, polarization/rotation modulation can be accomodated, and a spinor, K, can be represented not only by a null direction indicated by 2, q, or C, but also a real tangent vector L indicated in Fig. 4. 

All representations except S are two-dimensional. Subscripts g and u have the usual meaning, but a superscript + or is used on S representations according to whether x(oy) = l. For L > 0, x(C2), an(f x(°v) are zero. In double groups the spinor representations depend on the total angular momentum quantum number and are labeled... 

The character tables in Appendix A3 include the spinor representations of the common point groups. Double group characters are not given explicitly but, if required, these may be derived very easily. The extra classes in the double group are given by Opechowski s rules. The character of if in these new classes in vector representations is the same as that of R but in spinor representations x(ff) = — (R). The bases of spinor representations will be described in Section 12.8. 

A partial character table of the point group D3h is given in Table 8.4. Find the missing characters of the vector and spinor representations of the double group D3h. Determine whether El transitions E-/ E3/2 and E./2 —> E5/2 are allowed in a weak crystal field of D3h symmetry. State the polarization of allowed transitions. 

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... 

Example 12.6-1 The point group C2v = E C2z perpendicular axes, all operations except E are irregular and there is consequently only one doubly degenerate spinor representation, Ei/2. Contrast C 21, = E C2z / [ in which rrh is az = IC2z and thus an improper binary rotation about... 

Example 12.6-2 The classes of Td are E 4C3 3C2 6S4 6binary rotations are BB rotations. The six dihedral planes occur in three pairs of perpendicular improper BB rotations so both 3C2 and 6rrd are irregular classes. There are therefore Nv 5 vector representations and Ns = 3 spinor representations. 

Exercise 12.6-1 Determine the number of spinor representations in the point group D3h. 

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 ], ... 

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