Special unitary groups

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. 

There is a Lie group isomorphism between unit quaternions and the special unitary group 50 (2). Define a function T from the unit quaternions to 50 (2) by... 

V) special unitary group, i.e., group of unitary operators of determinant one on a Unite-dimensional scalar product space V, 114... 

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 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... 

Scuseria, Janssen, and Schaefer, for example, developed a set of intermediates based on their reformulation of the CCSD amplitude and energy equations in a unitary group formalism designed to offer special efficiency when the refer-... 

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... 

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. 

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). 

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... 

Exercise 4.3 Show that the set ofly. diagonal special unitary matrices is a group and that it is isomorphic to the group T x T. (See Exercise 4.1 for the definition of the Cartesian product of groups.)... 

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