The homomorphism of SU

The 0(3) group is homomorphic with the SU(2) group, that of 2 x 2 unitary matrices with unit determinant [6]. It is well known that there is a two to one mapping of the elements of SU(2) onto those of 0(3). However, the group space of SU(2) is simply connected in the vacuum, and so it cannot support an Aharonov-Bohm effect or physical potentials. It has to be modified [26] to SU(2)/Z2 SO(3). 

Exercise 6.3 Consider the representation of SU (2) on defined by matrix multiplication. Consider the group homomorphism 4 T —> SU l) defined by... 

Homomorphism of 0(3) and SU(2). There is an important relationship between 0(3) and SU(2). The elements of SU(2) are associated with rotations in 3D space. To make this relationship explicit, new coordinates are defined ... 

In order to establish the homomorphism between SU(2) and SO(3), we will consider first the dual u v) of the spinor basis u v). Note that no special notation, apart from the bra and ket, is used in C2 to distinguish the spinor basis (u v from its dual u v). A general rotation of the column spinor basis in C2 is effected by (see Altmann (1986), Section 6.7)... 

This result amounts to a 1 to 2 homomorphic mapping of the unitary group SU(2) onto the rotation group. From (28) it follows that the two unitary matrices... 

The 0(3) electrodynamics developed by Evans [2], and its homomorph, the SU(2) electrodynamics of Barrett [10], are substructures of the Sachs theory dependent on a particular choice of metric. Both 0(3) and SU(2) electrodynamics are Yang-Mills structures with a Wu-Yang phase factor, as discussed by Evans and others [2,9]. Using the choice of metric (17), the electromagnetic energy density present in the 0(3) curved spacetime is given by the product... 

In this section we classify the finite-dimensional irreducible representations of 50(3). Compared to the work we did classifying the irreducible representations of SU(2) in Section 6.5, the calculation in this section is a piece of cake. However, the reader should note that we use the 51/ (2) classification in this section. So our classification for 50(3) is not inherently easier. Our trick is to use the group homomorphism 4> 51/(2) —> 50(3) (defined in Section 4.3) to show that any representation of 5 O (3) is just a representation of 51/ (2) in disguise. At the end of the section we show how to use weights to identify irreducible representations. 

The reader should check that Definition 8.7 is satisfied and notice that those factors of 1 /2 are necessary. Thus Ti is a Lie algebra homomorphism. To see that it is an isomorphism, note that the matrices on the three right-hand sides of the defining equations for Ti form a basis of su(2). Similarly, defining 72 0Q 5o(3) by... 

Exercise 10.22 Find a group isomorphism between S O (3) and a subgroup of the physical symmetries of the qubit. Use Proposition 10.1 to find a nontrivial group homomorphism from SU (2) into the group of physical symmetries of the qubit. Finally, express the group homomorphism SU(2) —> 50(3) from Section 4.3 in terms of these functions. 

Proof. We shall use the description of (C2) in terms of matrices given in Theorem 1.14. Suppose Z is a T-invariant O-dimensional subscheme in (C2), and corresponds to a triple of matrices (Bi, B2, i). Recall that it is given as follows Define a iV-dimensional vector space V as H°(Oz), and a 1-dimensional vector space W. Then the multiplications of coordinate functions z, z2 6 C define endomorphisms Bi, B2. The natural map Oc2 —> Oz defines a linear map i W V. Prom this construction, V is a T-module, and W is the trivial T-module. The pair (Bi,B2) is T-equivariant, if it is considered as an element in Hom(V, Q V), where Q is 2-dimensional representation given by the inclusion T C SU(2). (This follows from that (Zi,z2) is an element in Q.) And i is also a T-equi variant homomorphism W —> V. 

There has been an unusual amount of debate concerning the development of 0(3) electrodynamics, over a period of 7 years. When the 2 (3) field was first proposed [48], it was not realized that it was part of an 0(3) electrodynamics homomorphic with Barrett s SU(2) invariant electrodynamics [50] and therefore had a solid basis in gauge theory. The first debate published [70,79] was between Barron and Evans. The former proposed that B,3> violates C and CPT symmetry. This incorrect assertion was adequately answered by Evans at the time, but it is now clear that if B<3) violated C and CPT, so would classical gauge theory, a reduction to absurdity. For example, Barrett s SU(2) invariant theory [50] would violate C... 

It can therefore be inferred that 0(3) electrodynamics is a theory of Rieman-nian curved spacetime, as is the homomorphic SU(2) theory of Barrett [50], Both 0(3) and SU(2) electrodynamics are substructures of general relativity as represented by the irreducible representations of the Einstein group, a continuous Lie group [117]. The Ba> field in vector notation is defined in curved spacetime by... 

Both matrices define zero rotation in 3-dimensional space, so we see that this zero rotation in 3D dimensional space corresponds to two different SU(2) elements depending on the value of (3. There is thus a homomorphism, or many-to-one mapping relationship between 0(3) and SU(2)—where many is 2 in this case—but not a one-to-one mapping. 

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). 

Hence is a group homomorphism. In the dehning formula for given in Equation 4.2, every matrix entry is a differentiable function of the real parameters gft(a), (a), and S( ). Because these parameters are differentiable functions on SU (2), the function is differentiable. Hence 4> is a Lie group homomorphism. 

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