Energy Projection Operators

The positive and negative-energy projection operators can be guessed from the Dirac equation in momentum space, equation 5.187,

$$\fbox{$\displaystyle \Lambda_r(p) = \frac{\epsilon_r\not{\!p... ...or}\quad \Lambda_{\pm}(p) = \frac{\pm\not{\!p}+mc}{2mc} $}\ .$$ (5.206)

Applying the ``trial'' positive-energy projection operator to an arbitrary Dirac spinor and using equation 5.187, we have

$$\Lambda_+(p) w^r(p) = \frac{\not{\;\!\!\!p}+ mc}{2mc} w^r(p)... ...trm{for}\ r=1,2 \\ 0 & \textrm{for}\ r=3,4 \end{array}\right.$$ (5.207)

and similarly for the negative-energy operator, as required.

Applying the operator twice we have

$$\Lambda_r\Lambda_{r^\prime} = \frac{\epsilon_r\epsilon_{r^\p... ...epsilon_r+\epsilon_{r^\prime})\not{\!p}mc + m^2c^2}{4m^2c^2} .$$ (5.208)

Since $\not{\!p}\not{\!p} = p_\mu p_\nu \gamma^\mu\gamma^\nu = 1/2 p_\mu p_\nu (\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu) = p_\mu p_\nu g^{\mu\nu} = p^2 = m^2c^2$ ,

$$\Lambda_r\Lambda_{r^\prime} = \frac{(\epsilon_r+\epsilon_{r^\prime})\not{\!p} + (1+\epsilon_r\epsilon_{r^\prime})mc}{4mc}$$

$$= \left( \frac{1+\epsilon_r\epsilon_{r^\prime}}{2} \right) \Lambda_r .$$ (5.209)

Therefore

$$\Lambda_{\pm}^2(p) = \Lambda_{\pm}(p) \quad\textrm{and}\quad \Lambda_{\pm}(p)\Lambda_\mp(p) = 0$$ (5.210)

as required of a projection operator.

Also

$$\Lambda_+(p) + \Lambda_-(p) = \frac{\not{\!p}+mc}{2mc} + \frac{-\not{\!p}+mc}{2mc} = 1 ,$$ (5.211)

as required.