Projection Operators of Energy and Spin

We have found four operators which project from a given plane-wave solution of momentum $\vec{p}$, the four independent solutions corresponding to positive and negative energy, and to spin up and spin down along a given direction. These projection operators are covariant and satisfy

$$P_r(\vec{p}) w^{r^\prime}(\vec{p}) = \delta_{rr^\prime} w^{r^\prime}(\vec{p})$$ (5.226)

or equivalently

$$P_r(\vec{p})P_{r^\prime}(\vec{p}) = \delta_{rr^\prime} P_r(\vec{p}) .$$ (5.227)

Thus the four projection operators are

$$P_1(\vec{p}) = \Lambda_+(p)\Sigma(u_z) ,$$

$$P_2(\vec{p}) = \Lambda_+(p)\Sigma(-u_z),$$

$$P_3(\vec{p}) = \Lambda_-(p)\Sigma(-u_z),$$

$$P_4(\vec{p}) = \Lambda_-(p)\Sigma(u_z) .$$ (5.228)

We shall rely upon these projection operators very frequently in developing rapid and efficient calculation techniques. They permit us to use closure methods, thus avoiding the necessity of writing out matrices and spinor solutions component by component.

The projection operators are specific to the normalization $\overline{u} u = 1$ and $\overline{v} v = -1$ for the spinors.