Projective representations

The double group G was introduced in Chapter 8 in order to deal with irreducible representations (IRs) that correspond to half-integral values of j. Because 

R 2n z) / for j equal to a half-integer. Bethe (1929) therefore introduced the new operator is = R(2n z) / E, thus doubling the size of G = g, by forming the double group 

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 rx oy, where ax /C2x and ry = IC2y. Because x, y, z are mutually 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 6 rd. The three binary 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. 

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Conventional representation of a carbon atom (e.g. C-2 of D-glucose) in the Fischer projection. Representation (e) will be used in general in the present document. 

In this section we define irreducible projective representations and find the irreducible projective representations of 50(3). These turn out to correspond to the different kinds of spin elementary particles can have, namely, 0, 1/2, 1, 3/2. 

Notice that the representation pi is reminiscent of a push-forward representation (see Section 5.6). It is inherently problematic to push forward along a two-to-one function however, because of the projective equivalence, the push-forward turns out to be well defined in this case. We can use this trick to define a whole family of projective representations of SO(3). These representations arise in the study of spin angidar momentum. 

All the irreducible linear Lie group representations of 5 U (2) correspond to spin representations of particles, i.e., to irreducible projective representations. The definition is quite natural. 

Our next task is to identify the projective representation of 50(3) on the state space. This representation is determined by the representations on the factors, but the projection must be handled carefully. The spin-1/2 projective representation of SO (3) on (C ) descends from the linear representation on C". The natural representation of SO (3) on (Section 4.4) descends... 

Proof of the Correspondence between Irreducible Linear Representations of 5f/(2) and Irreducible Projective Representations of 5<9(3)... 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

We have shown that the projective representation cr is the pushforward of the representation p, completing the proof. ... 

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