Projective Representations and Spin

Somewhere in the east early morning set off at dawn, travel round in front of the sun, steal a day s march on him. Keep it up for ever never grow a day older technically. 

It is a bit of a lie to say, as we did in previous chapters, that complex scalar product spaces are state spaces for quantum mechanical systems. Certainly every nonzero vector in a complex scalar product space determines a quantum mechanical state however, the converse is not true. If two vectors differ only by a phase factor, or if two vectors normaUze to the same vector, then they will determine the same physical state. This is one of the fundamental assumptions of quantum mechanics. The quantum model we used in Chapters 2 through 9 ignored this subtlety. However, to understand spin we must face this issue. 

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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. 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

The zero rest mass case, m = 0, also leads to physically important representations. In this case Cj = 0 when C2 = 0, both and are null vectors with = 0, so that we can take = oP, where c — P. J/Pq. Since Pq = P for a null vector, this identifies anti-parallel. In this chapter, the most important case is the photon, for which a = 1. Particles with helicity of opposite signs are related by space inversion. 

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