Spin Projection Operators

We go to the rest frame and try to find a projection operator in a covariant form. A candidate for a spin-up particle is $\frac{1+\sigma_z}{2}$, where $\sigma_z$ is the third Pauli-spin matrix. Removing the explicit $z$ dependence we can write

$$\frac{1+\sigma\cdot\hat{u}}{2},$$ (5.212)

where $\hat{u}$ is a unit 3-vector. Extending the operator to 4-dimensions in the rest frame we have

$$\frac{1+\Sigma_3u_3}{2} = \frac{1+i\gamma_1\gamma_2u_3}{2} =... ...}{2} \rightarrow \frac{1+\gamma_5\not{\;\!\!\!u}\gamma_0}{2} .$$ (5.213)

Because we are in the rest frame $\gamma_0$ acting upon the Dirac spinor becomes $\pm 1$ and we can deal with this overall sign later. The covariant Dirac spin projection operator has the form

$$\Sigma(u_3) = \frac{1+\gamma_5\not{\;\!\!\!u}_3}{2}.$$ (5.214)

Notice that $\Sigma(u_3)\ne\Sigma_3$ and that they are different operators. For a general spin vector $s^\mu$ we have

$$\fbox{$\displaystyle \Sigma(s) = \frac{1+\gamma_5\not{\;\!\!\!s}}{2} $}\ .$$ (5.215)

In the reset frame

$$\Sigma( u_3)\omega^1(0) = \frac{1+\gamma_5\gamma_3}{2} \omega^1(0) = \frac{1+\Sigma_3}{2} \omega^1(0) = \omega^1(0) ,$$ (5.216)

$$\Sigma(-u_3)\omega^2(0) = \frac{1-\gamma_5\gamma_3}{2} \omega^2(0) = \frac{1-\Sigma_3}{2} \omega^2(0) = \omega^2(0) ,$$ (5.217)

$$\Sigma(-u_3)\omega^3(0) = \frac{1-\gamma_5\gamma_3}{2} \omega^3(0) = \frac{1-\Sigma_3}{2} \omega^3(0) = \omega^3(0) ,$$ (5.218)

$$\Sigma( u_3)\omega^4(0) = \frac{1+\gamma_5\gamma_3}{2} \omega^4(0) = \frac{1+\Sigma_3}{2} \omega^4(0) = \omega^4(0) .$$ (5.219)

The physical motivation for this apparent backward association of the eigenvalues for the negative-energy solution will appear when we come to the hole theory.

Using the definitions (5.206) we can write

$$\Sigma( u_z)u(p,u_z) = u(p,u_z) ,$$

$$\Sigma( u_z)v(p,u_z) = v(p,u_z) ,$$

$$\Sigma(-u_z)u(p,u_z) = \Sigma(-u_z)v(p,u_z) = 0 .$$ (5.220)

Because of the covariant form of the projection operator, we may write for any polarization vector $s^\mu$

$$\Sigma(s)u(p,s) = u(p,s) ,$$

$$\Sigma(s)v(p,s) = v(p,s) ,$$

$$\Sigma(-s)u(p,s) = \Sigma(-s)v(p,s) = 0 .$$ (5.221)

We now show that the spin projection operators have the projection operator properties:

$$\Sigma(\pm s)\Sigma(\pm s) = \frac{1\pm\gamma_5\not{\;\!\!\!... ...pm 2\gamma_5\not{\;\!\!\!s}-\gamma_5^2 s^2}{4} = \Sigma(\pm s)$$ (5.222)

and

$$\Sigma(\pm s)\Sigma(\mp s) = \frac{1\pm\gamma_5\not{\;\!\!\!... ...p\gamma_5\not{\;\!\!\!s}}{2} = \frac{1+\gamma_5^2 s^2}{4} = 0$$ (5.223)

also

$$\Sigma(\pm s) + \Sigma(\mp s) = \frac{1\pm\gamma_5\not{\;\!\!\!s}}{2} + \frac{1\mp\gamma_5\not{\;\!\!\!s}}{2} = 1.$$ (5.224)

The energy and spin projection operators commute:

$$[\Sigma(s),\Lambda_{\pm}(p)] = \frac{1+\gamma_5\not{\;\!\!\!s}}{2} \frac{\pm\not{\;\!\!\!p}+mc}{2mc} - \Lambda_{\pm}(p)\Sigma(s)$$

$$= \frac{\pm\not{\;\!\!\!p}+mc\pm\gamma_5\gamma^\mu\gamma^\nu s_\mu p_\nu + mc\gamma_5\not{\;\!\!\!s}}{4mc} - \Lambda_{\pm}(p)\Sigma(s)$$

$$= \frac{\pm\not{\;\!\!\!p}+mc\pm\gamma_5(2g^{\mu\nu}-\gamma^\nu\gamma^\mu) s_\mu p_\nu + mc\gamma_5\not{\;\!\!\!s}}{4mc} - \Lambda_{\pm}(p)\Sigma(s)$$

$$= \frac{\pm\not{\;\!\!\!p}+mc\pm(2g^{\mu\nu}\gamma_5 + \gamma^\nu\g... ...^\mu) s_\mu p_\nu + mc\gamma_5\not{\;\!\!\!s}}{4mc} - \Lambda_{\pm}(p)\Sigma(s)$$

$$= \frac{\pm\not{\;\!\!\!p}+mc\pm(2\gamma_5 s\cdot p + \not{\;\!\!\!... ...5\not{\;\!\!\!s}) + mc\gamma_5\not{\;\!\!\!s}}{4mc} - \Lambda_{\pm}(p)\Sigma(s)$$

$$= \frac{\pm\not{\;\!\!\!p}+mc + (\pm\not{\;\!\!\!p}+ mc) \gamma_5\not{\;\!\!\!s}}{4mc} - \Lambda_{\pm}(p)\Sigma(s)$$

$$= 0 .$$ (5.225)