Physics 570 Assignment #1 January 11, 2006
Due Wednesday, January 18, 2006

1. Free particle spinors. Let $u_\pm(\u p)$ and $v_\pm(-\u p)$ be the eigenvectors of $h(\u p) \equiv \u \alpha \cdot \u p + \beta m$ with positive or negative eigenvalues, respectively, normalized to $2 E(\u p)$. Prove the relations below. (In the following $p^0 \equiv E(\u p) = \sqrt {\u p^2 + m^2}$ and $\bar u \equiv u^@ (i \gamma^0)$.)
   1. $(i \gamma^\mu p_\mu + m) u_s (p) = 0$ and $(-i \gamma^\mu p_\mu + m) v_s (p) = 0$.
   2. $\bar u_s(p) (i \gamma^\mu p_\mu + m) = 0$ and $\bar v_s(p) (-i \gamma^\mu p_\mu + m) = 0$.
   3. $\bar u_s(p) u_{s'}(p) = -\bar v_s(p) v_{s'}(p) = 2m \delta_{ss'}$ and $\bar u_s(p) v_{s'}(p) = \bar v_s(p) u_{s'}(p) = 0$.
   4. $\Lambda_+(p) \equiv \sum_s u_s(p) \bar u_s(p) = (-i \gamma^\mu p_\mu +m)$.
   5. $\Lambda_-(p) \equiv -\sum_s v_s(p) \bar v_s(p) = (i \gamma^\mu p_\mu +m)$.
   6. $\Lambda_\pm (p) / 2m = \left(\Lambda_\pm (p) / 2m \right)^2$, $\Lambda_+(p) \Lambda_-(p) = \Lambda_-(p) \Lambda_+(p) = 0$, and $\Lambda_+(p) + \Lambda_-(p) = 2m$. Explain why this means that $\Lambda_\pm(p) / 2m$ are mutually orthogonal projection operators.
   7. $-i \sigma_2 \chi_\mp^*(\u p) = \pm \chi_\pm(\u p)$ (if the phases of the two-component basis spinors of definite helicity, $\chi_\pm$, are suitably adjusted).
   8. $u_\pm(p)^* = C v_\pm (p)$, where $C = i \alpha_2 \beta = \gamma_2$ (up to an irrelevant overall phase which may be set to one).
   9. $C^T \alpha_i^T C = \alpha_i$ and $C^T \beta^T C = -\beta$.
2. Majorana Fermions. Let $\psi(x)$ be a spinor field built from spin $\half$ fermion creation and annihilation operators via the mode expansion $\psi(x) = \int {d^3p \over (2\pi)^3} \> {1 \over \sqrt {2 E(\u p)}} \left( ... ...u p) u_s(\u p) e^{ipx} + b^@_s(\u p) v_s(\u p) e^{-ipx} \right)$.
   1. Show that $\left(\psi^@(x) \right)^T = C \psi(x)$.
   2. Consider a redefined field operator, $\psi'(x) \equiv S \psi(x)$, for some unitary 4-by-4 matrix $S$. Show that $S$ may be chosen so that $\psi'(x)$ is a real field, $\psi'(x)^@ = \psi'(x)$.
3. Dirac Fermions.
   1. Show that a free theory of two equal mass Majorana spinor fields, with Hamiltonian $\hat H = \half \int d^3x \> \left\{ \psi_1^@(x) (-i \alpha \cdot \nabla ... ...psi_2^@(x) (-i \alpha \cdot \nabla + \beta m) \psi_2(x) \right\} ,$ is equivalent to the free Dirac Hamiltonian, $\hat H = \int d^3x \> \psi^@(x) (-i \alpha \cdot \nabla + \beta m) \psi(x) ,$ when $\psi(x) \equiv (\psi_1(x) + i \psi_2(x)) / \sqrt 2$.
   2. Express the total momentum in terms of $\psi(x)$ and $\psi^@(x)$. What is the momentum density operator?
   3. Using $\psi(x) = \int {d^3p \over (2\pi)^3} \> {1 \over \sqrt {2 E(\u p)}} \left( ... ...u p) u_s(\u p) e^{ipx} + d^@_s(\u p) v_s(\u p) e^{-ipx} \right)$, the standard mode expansion of a Dirac field, express the operator $Q \equiv \int d^3x \> \psi^@(x) \psi(x) - {\rm (const.)}$ in terms of the individual creation and annhilation operators. Choose the additive constant so that $Q \vert> = 0$. What does $Q$ measure?
   4. Show that $Q$ commutes with the free Hamiltonian. What symmetry does $Q$ generate? I.e., what is the effect of the transformation $\psi(x) \to U(\alpha) \psi(x) U(\alpha)^@$, where $U(\alpha) \equiv \exp (-i \alpha Q)$. What is the effect of the transformation on the original Majorana fields $\psi_1(x)$ and $\psi_2(x)$?
   5. Show that the transformation $\psi(x) \to C \psi^@(x)^T$ leaves the free Hamiltonian invariant. What is the effect of this transformation on the mode operators $b_s(\u p)$, $d_s(\u p)$? On $Q$? Explain why this transformation is called ``charge conjugation''.
   6. Show that the free time-ordered propagator $S(x)_{\alpha\beta} \equiv <0\vert {\scr T} \!\left( \psi_\alpha(x) \bar \psi_\beta(0) \right) \vert>$ is a Green's function for $i(\gamma^\mu \partial_\mu + m)$. What is $\tilde S(p)$?
4. Gamma Matrix Identities. For any four-vector $a^\mu$, let $\slash a$ be an abbreviation for $\gamma_\mu a^\mu$. Define $\gamma_5 \equiv \gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3$. Using only the anti-commutation relation satisfied by the gamma matrices:
   1. Prove that $\gamma_5$ anticommutes with all the $\gamma_\mu$.
   2. Prove that $\gamma_5^2 = 1$.
   3. Prove that the trace of any product of an odd number of gamma matrices vanishes.
   4. Evaluate $\tr (\slash a \slash b)$.
   5. Evaluate $\tr (\slash a \slash b \slash c \slash d)$.
   6. How does the number of terms in a trace of $n$ gamma matrices grow with $n$?
   7. Evaluate $\gamma^\mu \gamma_\mu$.
   8. Evaluate $\gamma^\mu \slash a \gamma_\mu$.
   9. Evaluate $\gamma^\mu \slash a \slash b \gamma_\mu$.
   10. Evaluate $\gamma^\mu \slash a \slash b \slash c \gamma_\mu$.
   Hints: When evaluating traces, feel free to insert $1 = \gamma_5^2$, and use the cyclic identity for traces plus the fact that $\gamma_5$ anti-commutes with all (other) gamma matrices.
5. Landau Levels. Consider a single electron propagating in the presence of a background electromagnetic vector potential, $\undertilde A (\undertilde x) = a \hat z + B x \hat y$. Here, $a$ and $B$ are (time independent) constants.
   1. What is the magnetic field? What is the meaning of the constant $a$?
   2. As a warm-up, solve the non-relativistic Schrodinger equation for an electron in this background field. Do not neglect the magnetic moment of the electron, $\u \mu_e = (e \hbar / 2mc) \u \sigma$. Hint: Use translation invariance to reduce the problem from three spatial dimensions to one, and show that the result is equivalent to a (shifted) harmonic oscillator.
   3. Consider the electron confined to a very large box of size $L^3$. (Ignore boundary effects which vanish as $L \to \infty$.) Explain why the degeneracy of each level in the lowest Landau band is $(\vert e B\vert / 2 \pi) L^2$. What is the degeneracy of levels in higher bands?
   4. Now for the big time. Consider the Dirac equation, $\{\gamma^\mu (\partial_\mu - i e A_\mu) + m \} \psi = 0$, in the presence of this background field. The next several parts of this problem sketch one approach for solving this equation. Feel free to use another (as long as it works!).

      Use translation invariance to reduce the equation to the one-dimensional problem,
      

      $$\bigl\{ - i \alpha_1 (\partial_\eta + \Sigma_3 e B \eta) + h_\perp - E \bigr\} \chi (\eta) = 0 ,$$
      
      
      where $\eta \equiv x - k_y / eB$, $\Sigma_3 \equiv -i \alpha_1 \alpha_2$, and $h_\perp \equiv \alpha_3 (k_z {-} ea) + \beta m$. Which pairs of the three matrices $\{ \alpha_1, \Sigma_3, h_\perp \}$ commute? Which anticommute?
   5. Explain why one may, without loss of generality, decompose the spinor $\chi$ as
      
      $$\chi (\eta) = \chi_\uparrow (\eta) + \alpha_1 \chi_\downarrow (\eta) ,$$
      
      
      where both $\chi_\uparrow$ and $\chi_\downarrow$ have spin up, $\Sigma_3 \chi_\uparrow = \chi_\uparrow$ and $\Sigma_3 \chi_\downarrow = \chi_\downarrow$ (but $\alpha_1 \chi_\downarrow$ has spin down). In terms of this decomposition, show that the Dirac equation becomes
      
      $$\pmatrix { h_\perp - E & -i (\partial_\eta - eB \eta) ... ...matrix { \chi_\uparrow \cr \chi_\downarrow \cr } = 0 .$$
      
      

   6. Solve this equation. (Assume, if you wish, that $eB \gt 0$.)
   7. What are (all) the energy levels? What is their degeneracy? Draw a picture of the energy spectrum as a function of $k_z$. How do the results depend on $a$? What is the minimum energy in magnitude? Do the energy levels agree with the non-relativistic result when $\vert E-m\vert \ll m$?