Physics 570 Assignment #2 January 19, 2006
Due Friday, January 27, 2006

1. Noether's Theorem. Consider a local field theory with a Lagrange density $\scr L$ which is a functional of some set of fields $\{ \phi_a(x) \}$ and their derivatives. Let $\phi_a(x) \to \phi_a(x) + \epsilon \delta\phi_a(x)$ be an infinitesimal transformation of the fields which leaves the action invariant, $\partial S/\partial\epsilon = 0$. Now consider a space-time dependent transformation $\phi_a(x) \to \phi_a(x) + \epsilon(x) \delta\phi_a(x)$, where $\epsilon$ is an arbitrary (but infinitesimal) function on spacetime. Define the current $j^\mu(x) \equiv {\partial {\scr L} \over \partial [\partial_\mu \phi_a(x)]} \delta\phi_a(x) .$ Prove Noether's theorem, which states that in a local field theory every continuous symmetry transformation is generated by a conserved current. Specifically:
   1. Show that the variation of the action under the infinitesimal transformation with arbitrary $\epsilon(x)$ is $\delta S = -\int d^4x \; \epsilon(x) \, \partial_\mu j^\mu(x)$. Explain why this implies that the current is conserved, $\partial_\mu j^\mu(x) = 0$ when $\phi_a(x)$ satisfies the classical field equations.
   2. Define a ``charge'' $Q$ as the spatial integral of the time component of the current, $Q \equiv \int d^3x \> j^0(x)$. Show that $Q$ is constant in time (given reasonable boundary conditions at infinity) if the current is conserved.
   3. In the quantized theory, show that the operator $Q$ is the generator of the transformation, $i \, [ Q, \phi_a(x)] = \delta\phi_a(x)$.
   4. In a theory of a complex scalar field with Lagrange density $\vert\partial\phi\vert^2 + V(\vert\phi\vert)$, What is the conserved current associated with the symmetry $\phi(x) \to e^{i\alpha} \phi(x)$?
   5. Explain how Noether's theorem shows that space-time translation invariance implies the existence of a conserved energy-momentum tensor satisfying $\partial_\mu T^{\mu\nu}(x) = 0$. What is $T^{\mu\nu}$ in the above scalar field theory?
   6. What is $T^{\mu\nu}$ in a theory of fermions with Lagrange density $\bar\psi(\slash\partial + m) \psi + \lambda (\bar\psi\psi)^2$?
   7. If there exists some other tensor $\Delta T^{\mu\nu}$ which is independently conserved, $\partial_\mu \Delta T^{\mu\nu}(x) = 0$, then one can always redefine the stress-energy tensor, $T^{\mu\nu}_{\rm improved} = T^{\mu\nu} + \Delta T^{\mu\nu}$. In the above examples of scalar and spinor theories, show that one can define a stress-energy tensor which is symmetric, $T^{\mu\nu} = T^{\nu\mu}$.
   8. What conserved currents are associated with Lorentz transformations?
2. Massless Fermions.
   1. Consider a free massless Dirac fermion field, $\psi$. Decompose $\psi$ into two pieces, $\psi = \psi_L + \psi_R$, where $\gamma_5 \psi_L = \psi_L$ and $\gamma_5 \psi_R = -\psi_R$. Show that the massless Dirac equation does not couple $\psi_L$ and $\psi_R$. Solve it. (If you wish, first find a representation of the gamma matrices in which $\gamma_5$ is diagonal, and all the $\gamma_\mu$ are block off-diagonal.)
   2. Show that positive energy solutions for $\psi_L$ have negative helicity (or spin anti-parallel to the direction of motion, $\undertilde \sigma \cdot \hat p \psi_L = - \psi_L)$, while $\psi_R$ has positive helicity. Explain why this justifies calling $\psi_L$ ``left-handed'', and $\psi_R$ ``right-handed''.
   3. Show that both $j^\mu \equiv \bar \psi \gamma^\mu \psi$ and $j_5^\mu \equiv \bar \psi \gamma_5 \gamma^\mu \psi$ are conserved currents when $\psi$ satisfies the massless Dirac equation. Let $N$ and $N_5$ denote the corresponding conserved charges. What are their particle interpretations? Do these currents remain conserved if a (minimally-coupled) background electromagnetic field is added?
3. Lorentz Transformations for Spinors.
   1. Show that if $\psi(x^\mu)$ satisfies the Dirac equation, then so does $S \psi(x^\nu {\Lambda_\nu}^\mu)$ where $x'^\mu \equiv {\Lambda^\mu}_\nu x^\nu$ is a Lorentz transformation and the matrix $S$ (acting on Dirac indices) satisfies $S^{-1} \gamma^\mu S = {\Lambda^\mu}_\nu \gamma^\nu$.
   2. For an infinitesimal Lorentz transformation with ${\Lambda^\mu}_\nu = {\delta^\mu}_\nu + {\omega^\mu}_\nu + O(\omega^2)$, show that $S = 1 + \coeff 18 \omega_{\mu\nu} [ \gamma^\mu, \gamma^\nu ] + O(\omega^2)$.
   3. Find the explicit form of the matrix $S$ for a finite rotation through an angle $\theta$ about the unit vector $\hat n$.
   4. Find the explicit form of the matrix $S$ for a finite boost with rapidity $\chi$ in the direction $\hat n$.
   5. Find (a consistent choice for) the matrix $S$ for spatial inversion (or parity).
   6. Show that if $\psi(x) \to S \psi(x')$ under some Lorentz transformation, then $\bar\psi(x) \to \bar\psi(x') S^{-1}$.
   7. Show that $\bar\psi(x) \psi(x)$ transforms like a scalar field and $\bar\psi(x) \gamma^\mu \psi(x)$ like a vector field under proper Lorentz transformations and under parity.
   8. Show that $\bar\psi(x) \gamma^5 \psi(x)$ transforms like a pseudoscalar field--a scalar under proper Lorentz transformations, but with an additional minus sign under parity.