Cayley-Klein parameters

A general rotation R( o n) in ft3 requires the specification of three independent parameters which can be chosen in various ways. The natural and familiar way is to specify the angle of rotation and the direction of the unit vector n. (The normalization condition on n means that there are only three independent parameters.) A second parameterization R(a b) introduced above involves the Cayley-Klein parameters a, b. A third common parameterization is in terms of the three Euler angles a, (3, and 7 (see Section 11.7). Yet another parameterization using the quaternion or Euler-Rodrigues parameters will be introduced in Chapter 12. 

We now have all the necessary machinery for working out the matrix elements l in the MRs of the proper rotations R in any point group for any required value of The l are given in terms of the Cayley-Klein parameters a, b and their CCs by eq. (11.8.43). The parameters a, b may be evaluated from the quaternion parameters X, A for R, using... 

Table 12.4. Rotation parameters n or m, real (A, A), and complex (p, r) quaternion parameters, and the Cayley-Klein parameters a, b for the operators R D3.

Table 14.5. Rotation parameters

Table 16.19. Matrix representatives for elements of the subgroup Gg of M(X) calculated from eqs. (12.8.3) and (12.8.5) using the Cayley—Klein parameters in Table 16.18 for the symmetrized bases.

Table 16.22. Quaternion and Cayley-Klein parameters for the symmetry operators of the point group S4.

Exercise 16.8-3 Quaternion and Cayley-Klein parameters are given in Table 16.22. Using... 

It is easy to check that the rows and columns of this matrix are orthogonal and its determinant equals unity. The independent complex matrix elements in eq. (3.45) are known as Cayley-Klein parameters of the rotation group. Also, one can see that for quaternions connected by the relation r = ri o r2 the corresponding 2x2 matrices are connected by the same relation with replacement of the quaternion multiplication by the usual matrix product. This establishes isomorphism between the SU(2) group and the group of normalized quaternions HP which can be continued to the homomorphism on 50(3). 

To establish the connection between the spinor and the vector, we now need to verify how transformations in the spinor are manifested as transformations in the vector. Consider a finite unitary transformation of the spinor. The transformation belongs to the unitary group, U(2), and, as we have seen, the determinant of this matrix is unimodular. We consider the special case, however, where the determinant is +1. Such matrices form the special unitary group, SU 2). The most general form of an SU(2) matrix involves two complex parameters, say a and b, subject to the condition that their squared norm, a + b, equals unity. These parameters are also known as the Cayley-Klein parameters. (Cf. Problem 2.1.) One has... 

This conservation of length is the property that confirms the previous identification of the interaction matrix elements with a 3-vector and relates it to ordinary space. In fact, by identifying the rotation matrices in Eqs. (7.3) and (7.32), we may determine the Cayley-Klein parameters. Two solutions with opposite signs are possible ... 

R -a, -n) leaves the Cayley-Klein parameters unchanged. By contrast, the combinations R 2tt — a, —n) and R —2n + a, n) change the signs of both Cayley-Klein parameters. 

For the actual construction of the double group as a group of operators, we need a convention to connect the spatial operators to the spinor matrices. As we have seen in Sect. 7.2, the four possible parametric descriptions of a given rotation yield two different choices for the Cayley-Klein parameters. Hence, our convention should define how to characterize unequivocally the parameters of a rotation. It will consist of two criteria the rotation angle must be positive, and the pole from which the rotation is seen as counterclockwise must belong to the positive hemisphere in the nx,tiy, tiz parameter space. This is the hemisphere above the equatorial plane, i.e., with > 0. In the (rix, Wy)-plane, we include the half-circle of points with positive -value, i.e., with = 0, > 0, and also the point with ny = l,nx = 0, and... 

The corresponding Cayley-Klein parameters are then determined as... 

This character can be zero only for a = n and, hence, for binary rotations with n = 2. To examine whether or not the matrix for a binary rotation can be class-conjugated to minus itself, we may limit ourselves to the study of one orientation of the rotation axis, say C. Indeed, in SU 2) any orientation can always be transformed backward to this standard choice by a unitary transformation. The problem thus reduces to finding a spinor operation X represented by a matrix X with Cayley-Klein parameters ax,bx, which transforms (C ) into minus itself ... 

The product of two rotations is a rotation. Obtain an expression for the Cayley-Klein parameters of the product as a function of the parameters of its factors. Is the product commutative The SU 2) matrices may also be identified as normalized quaternions. 

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