The special unitary group SU

The Casimir operator for so(3) commutes with all generators and is given by J2 = J + J + J3. The unitary irreducible representations are characterized by a single quantum number j = 0,1, ,..., which also includes the spin representations of su(2) corresponding to the special unitary group SU(2) when j = j,, and are given by... 

The set of 2 x 2 unimodular unitary matrices constitutes the special unitary group SU(2). Such matrices can be parametrized by... 

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... 

It is known as the unitary unimodular group, or the special unitary group denoted by SU(2). Because of the extra condition on the determinant, SU(2) is a three-parameter group. 

Finally, we must introduce the special iniitaix group SU (2). The unitary in the name is analogous to the orthogonal in the group 5(9(3). We set... 

It has to be noted that the relation between the elements of 0(3)+ (also called SO(3), the group representing proper rotations in 3D coordinate space) and SU(2) (the special unitary group in two dimensions) is not a one-to-one correspondence. Rather, each R matches two matrices u. Molecular point groups including symmetry operations for spinors therefore exhibit two times as many elements as ordinary point groups and are dubbed double groups. 

The product of two unitary transformations is a new unitary transformation, and it is easy to show that the set of unitary transformations on the spin space defined above forms a group. Because the transformations all have determinant 1, due to the Pauli matrices being traceless, this is called the special unitary group of dimension 2, or SU 2). 

The IBM-1 is able to treat different collective excitations in a uniform framework. Its dynamic symmetries are shown later in Fig. 2.26. The U(5) (five-dimensional unitary group) corresponds to spherical, the SU(3) (three-dimensional special unitary group) to deformed, the 0(6) (six-dimensional orthogonal group) to y-soft nuclei. 

SU(n) Group Algebra. Unitary transformations, U( ), leave the modulus squared of a complex wavefunction invariant. The elements of a U( ) group are represented by n x n unitary matrices with a determinant equal to 1. Special unitary matrices are elements of unitary matrices that leave the determinant equal to +1. There are n2 — 1 independent parameters. SU( ) is a subgroup of U(n) for which the determinant equals +1. 

An elegant description of rotation in spherical mode is provided in terms of a special unitary matrix of order 2, known as SU(2) in Lie-group space (T2.8.2). The matrices that form a basis for the algebra of SU(2) are those already introduced to represent quaternions. The important result is that the group space SU(2) is compact compared to a noncompact group R that characterizes cylindrical rotation about an axis of infinite extent. If an object... 

Being an angular momentum, the spin should be associated with the symmetry of the rotation group. Thus the rotation group can be described by Special (their determinant is 1) Unitary 2x2 matrices and so this group is called SU(2). The determinant of matrix A is... 

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