next up previous
Next: Physical Constants Up: Relativistic Electron Theory Previous: Positron Theory


  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

    $\displaystyle \partial_\mu\,j^{\,\mu} = 0,$

    where $ j^{\,\mu}$ is a 4-vector field, then

    $\displaystyle \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

    $\displaystyle x^{\,\mu'} = a^{\,\mu}_{~\nu}\,x^{\,\nu}.


    $\displaystyle 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

\begin{array}{cccc}\cosh\omega, &-\s...
...\\ [0.5ex]
0,&0,&1,&0\\ [0.5ex]

    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

    $\displaystyle 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

    $\displaystyle A=\exp\left[-\frac{\omega}{2}\,({\bf n}\cdot\mbox{\boldmath $\alp...
({\bf n}\cdot\mbox{\boldmath $\alpha$}),

    and $ \omega=\tanh^{-1}(v/c)$ .

  10. Consider the spinors

    $\displaystyle \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,

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

    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

    $\displaystyle \psi^{\,r} = {\rm e}^{-{\rm i}\,\epsilon_r\,(p_\mu\,x^{\,\mu}/\hbar)}\,w^{\,r}({\bf p}),


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

    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,

    $\displaystyle \{\Sigma_i, \Sigma_j\} = 2\,\delta_{ij}.

next up previous
Next: Physical Constants Up: Relativistic Electron Theory Previous: Positron Theory
Richard Fitzpatrick 2016-01-22