Exercises

1. Demonstrate that the four operators $\partial_\mu\equiv \partial/\partial x^{\,\mu}$ transform under Lorentz transformation as the covariant components of 4-vector, whereas the four operators $\partial^{\,\mu}\equiv \partial/\partial x_{\mu}$ transform as the contravariant components of the same vector.

2. Demonstrate that Equation (11.29) is equivalent to Equations (11.24)-(11.26).

3. Noting that $\alpha_i=-\beta\,\alpha_i\,\beta$ , prove that the $\alpha_i$ and $\beta$ matrices all have zero trace. Hence, deduce that each of these matrices has $n$ eigenvalues $+1$ , and $n$ eigenvalues $-1$ , where $2n$ is the dimension of the matrices.

4. Verify that the matrices (11.30) and (11.31) satisfy the anti-commutation relations (11.29).

5. Verify that the matrices (11.32) and (11.33) satisfy Equations (11.24)-(11.26).

6. Verify that if
   $$\partial_\mu\,j^{\,\mu} = 0,$$
   where $j^{\,\mu}$ is a 4-vector field, then
   $$\int d^{\,3} {\bf x}\,j^{\,0}$$
   is Lorentz invariant, where the integral is over all space, and it is assumed that $j^{\,\mu}\rightarrow 0$ as $\vert{\bf x}\vert\rightarrow\infty$ .

7. Verify that Equation (11.77) is a solution of Equations (11.76).

8. A Lorentz transformation between frames $S$ and $S'$ takes the form
   $$x^{\,\mu'} = a^{\,\mu}_{~\nu}\,x^{\,\nu}.$$
   If
   $$a^{\,\mu}_{~\nu} = g^{\,\mu}_{~\nu} +\delta\omega\,I^{\,\mu}_{~\nu},$$
   where $I^{\,0}_{~1}=I^{\,1}_{~0} =-1$ , and $I^{\,\mu}_{~\nu}=0$ otherwise, then the transformation corresponds to an infinitesimal velocity boost, $\delta v = c\,\delta\omega$ , parallel to the $x_1$ -axis. Show that if a finite boost is built up from a great many such boosts then the transformation matrix becomes
   $$a^{\,\mu}_{~\nu}=\left( \begin{array}{cccc}\cosh\omega, &-\s... ...\\ [0.5ex] 0,&0,&1,&0\\ [0.5ex] 0,&0,&0,&1 \end{array}\right),$$
   where $v=c\,\tanh\omega$ is the velocity of frame $S'$ relative to frame $S$ . Show that the corresponding transformation rule for spinor wavefunctions is $\psi' = A\,\psi$ , where
   $$A = \exp\left(-\frac{\omega}{2}\,\alpha_1\right)= \cosh\left(\frac{\omega}{2}\right)-\sinh\left(\frac{\omega}{2}\right)\alpha_1.$$

9. Show that the transformation rule for spinor wavefunctions associated with a Lorentz transformation from frame $S$ to some frame $S'$ moving with velocity ${\bf v}= v\,{\bf n}$ with respect to $S$ is $\psi' = A\,\psi$ , where
   $$A=\exp\left[-\frac{\omega}{2}\,({\bf n}\cdot\mbox{\boldmath $\alp... ...t)-\sinh\left(\frac{\omega}{2}\right) ({\bf n}\cdot\mbox{\boldmath $\alpha$}),$$
   and $\omega=\tanh^{-1}(v/c)$ .

10. Consider the spinors
    $$\psi^{\,r} = {\rm e}^{-{\rm i}\,\epsilon_r\,(m_e\,c^{\,2}/\hbar)\,t}\,w^{\,r}(0),$$
    for $r=1,2,3,4$ . Here, $\epsilon_r = +1$ for $r=1,2$ , and $\epsilon_r=-1$ for $r=3,4$ . Moreover,

    $$w^{\,1}(0) =\left(\begin{array}{c}1\\ [0.5ex]0\\ [0.5ex]0\\ [0.5ex]0\end{array}\right), w^{\,2}(0) =\left(\begin{array}{c}0\\ [0.5ex]1\\ [0.5ex]0\\ [0.5ex]0\end{array}\right), w^{\,3}(0) =\left(\begin{array}{c}0\\ [0.5ex]0\\ [0.5ex]1\\ [0.5ex]0\end{array}\right), w^{\,4}(0) =\left(\begin{array}{c}0\\ [0.5ex]0\\ [0.5ex]0\\ [0.5ex]1\end{array}\right).$$

    
    
    Verify that the $\psi^{\,r}$ are solutions of the Dirac equation in free space corresponding to electrons of energy $\epsilon_r\,m_e\,c^{\,2}$ , momentum ${\bf p}={\bf0}$ , and spin angular momentum parallel to the $x_3$ -axis $S_3=\zeta_r\,\hbar/2$ , where $\zeta_r=+1$ for $r=1,3$ , and $\zeta_r=-1$ for $r=2,4$ .

11. Show that the four solutions of the Dirac equation corresponding to an electron of energy $E$ and momentum ${\bf p}$ moving in free space take the form
    $$\psi^{\,r} = {\rm e}^{-{\rm i}\,\epsilon_r\,(p_\mu\,x^{\,\mu}/\hbar)}\,w^{\,r}({\bf p}),$$
    where

    $$w^{\,1}({\bf p}) =\sqrt{\frac{E+m_e\,c^{\,2}}{2\,m_e\,c^{\,2}}}\left(\begin{array}... ...z\,c}{E+m_e\,c^{\,2}}\\ [0.5ex]\frac{p_+\,c}{E+m_e\,c^{\,2}}\end{array}\right), w^{\,2}({\bf p}) =\sqrt{\frac{E+m_e\,c^{\,2}}{2\,m_e\,c^{\,2}}}\left(\begin{array}... ...\,c}{E+m_e\,c^{\,2}}\\ [0.5ex]\frac{-p_z\,c}{E+m_e\,c^{\,2}}\end{array}\right),$$

    $$w^{\,3}({\bf p}) =\sqrt{\frac{E+m_e\,c^{\,2}}{2\,m_e\,c^{\,2}}}\left(\begin{array}... ... [0.5ex]\frac{p_+\,c}{E+m_e\,c^{\,2}}\\ [0.5ex]1\\ [0.5ex] 0\end{array}\right), w^{\,4}({\bf p}) =\sqrt{\frac{E+m_e\,c^{\,2}}{2\,m_e\,c^{\,2}}}\left(\begin{array}... ...[0.5ex]\frac{-p_z\,c}{E+m_e\,c^{\,2}}\\ [0.5ex]0\\ [0.5ex] 1\end{array}\right).$$

    
    
    Here, $p_\pm = p_x\pm {\rm i}\,p_y$ . Demonstrate that these spinors become identical to the $\psi^{\,r}$ of the previous exercise in the limit that ${\bf p}\rightarrow {\bf0}$ .

12. Verify that the $4\times 4$ matrices $\Sigma_{\,i}$ , defined in Equation (11.98), satisfy the standard anti-commutation relations for Pauli matrices: that is,
    $$\{\Sigma_i, \Sigma_j\} = 2\,\delta_{ij}.$$