Electroweak and Strong Interactions: An Introduction to by Professor Dr. F. Scheck (auth.)

By Professor Dr. F. Scheck (auth.)

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By Professor Dr. F. Scheck (auth.)

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77) This can also be written as follows: ljJ(x) = (

69) is evident. In general, the answers to questions about fundamental properties of the theory are obtained in a more direct and transparent manner than in the four-component formulation. 4 Plane Wave Solutions of the Dirac Equation In this section we derive plane wave solutions of the Dirac equation. 78). 83) where px = pO XO - px with pO = (m 2 + p2)lt 2; u and v are spin or amplitudes characterized by three-momentum p and spin orientation r = I or 2 with respect to some arbitrary quantization axis.

Expressing these matrices in terms of y-matrices one verifies easily that u(O)u(O) = i(~ + yOH(~ v(O)v(O) = -i(~ - - ysn'Y), yOH(~ - ysn·y). 87,88) that u(p) = N{jj + m)u{O) and v{p) = with N = (Ep + m)-1/2. Using this we calculate the sum 30 - u(p)u(p) + v(p)v(p) = 1 --{(p + m)u(O)u(O)(]j + m) + (]j - m)v(O)v(O)(]j - m)} Ep+m 4(E p1+ m) {(]j + m)(~ + yo)(~ - Ys niyi)(]j + m) - (]j - m)(~ - yo)(~ - ysniyi)(]j - m)} 30We use the "slash" notation, I/. '" a"yw N{jj - m)v(O) 46 1 Fermion Fields and Their Properties 1 {[4mp+2py °p+2m 2 y°J 4(Ep+m) - ysni[2m(yip -IN) - 2m 2 yOyi + 2ziyOyipJ}.

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