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This result is exactly what we expect for electrons. An analogous calculation determines the spin of a zero-momentum antifermion. But in this case, since the order of the b and by terms in Jz is reversed, we get an extra minus sign from evaluating bp byp by0] = ; bypbp by0]. Thus for positrons, the association between the spinors s and the ; ; spin angular 1 momentum is reversed: 0 corresponds to spin ;1=2, while 01 corresponds to spin +1=2. This reversal of sign agrees with the prediction of Dirac hole theory.

Generalize these results to the case of n identical complex scalar elds. 3 Evaluate the function Z d3p 1 h0j (x) (y) j0i = D(x ; y) = (2 )3 2Ep e;ip (x;y) for (x ; y) spacelike so that (x ; y)2 = ;r2 , explicitly in terms of Bessel functions. Chapter 3 The Dirac Field Having exhaustively treated the simplest relativistic eld equation, we now move on to the second simplest, the Dirac equation. * In this chapter our viewpoint will be quite di erent. First we will rederive the Dirac equation as a classical relativistic eld equation, with special emphasis on its relativistic invariance.

But we haven't proved yet that this \spin" has anything to do with angular momentum. To do this, we must write down the angular momentum operator. 2 by looking for the conserved quantity associated with translational invariance. We can nd the angular momentum operator in a similar way as a consequence of rotational invariance. Under a rotation (or any Lorentz transformation), the Dirac eld transforms (in our original convention) according to (x) ! 0 (x) = 12 ( 1 x): To apply Noether's theorem we must compute the change in the eld at a xed point, that is, = 0 (x) ; (x) = 12 ( 1 x) ; (x): Consider for de niteness an in nitesimal rotation of coordinates by an angle about the z -axis.

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