Physics 515 Problem Set #4 April 22, 2004
Due Thursday, April 29, 2004

1. (10 pts) Understanding the origin of ultra-high energy cosmic rays is a current problem. One hypothetical scenario involves extremely high energy extra-galactic neutrinos which generate observable cosmic ray events by interacting with low energy massive neutrinos in the galactic halo to produce a Z-boson (which subsequently decays to various hadrons and leptons). The rest mass (times $c^2$) of a Z-boson is 91 GeV. The rest mass of neutrinos is now known to be non-zero, but the precise value is unknown. Treat the low energy neutrinos in the galactic halo as massive particles, nearly at rest, with a mass $m_\nu$ somewhere around 1 eV. What energy $E_{\rm Z}$, as a function of ($m_\nu/1$ eV), must an extra-galactic neutrino have if it is to produce a Z-boson after scattering from a galactic halo neutrino?
2. (20 pts) Solve the equation of motion for a relativistic charged particle, ${du^\mu(\tau) \over d\tau} = {q \over mc} \, F^{\mu\nu}\, u_\nu(\tau)$, for the case of a constant, uniform magnetic field $B$ along the $\hat z$ axis. Show that the particle follows a circular helix (whose projection onto the $x$-$y$ plane is a circle). What is the angular frequency $\omega_p$ of this rotation as measured by the particle's proper time? What is the corresponding angular frequency $\omega_c$ in the lab frame if the particle has total energy $E$?
3. (15 pts) At some event in spacetime, an inertial observer A measures electric and magnetic fields $\E = E_0 \, \hat x$ and $\B = B_0 \, \hat y$.
   1. What is the corresponding field strength tensor $\vert\vert F^{\mu\nu} \vert\vert$ in A's reference frame?
   2. Observer B moves in the $\hat z$ direction at rapidity $\chi$ relative to A. What Lorentz transformation matrix $\vert\vert\Lambda^{\mu}_{\;\nu}\vert\vert$ transforms (the components of) vectors in reference frame A to the co-moving frame of B?
   3. What is the field strength tensor $\vert\vert\bar F^{\mu\nu}\vert\vert$ in frame B?
   4. For what values of the rapidity (if any) will observer B measure a purely electric, or a purely magnetic field?
4. (20 pts) The current density for a swarm of particles with arbitrary spacetime trajectories may be written as $j^\mu(x) = \sum_a q_a c \int d\tau_a \> u_a^\mu(\tau_a) \, \delta^4(x{-}z_a(\tau_a))$, where $q_a$ is the charge of the $a$'th particle, $z^\mu(\tau_a)$ is its spacetime location (as a function of its proper time), and $u^\mu_a(\tau_a)$ is its 4-velocity.
   1. Explain why the above construction makes $j^\mu(x)$ transform as a 4-vector field.
   2. Prove that $\partial_\mu \, j^\mu(x) = 0$.
   3. Justify the claim that this is the correct expression for the 4-current of a collection of charged particles, by performing the proper time integral and expressing $j^\mu(x)$ in terms of a three-dimensional spatial delta function. Write the resulting time and space components of $j^\mu(x)$ explicitly in terms of the physical 3-velocity (and position) of the particle. Do you obtain familar results?
5. (10 pts) Given any rank-2 antisymmetric tensor, such as the field strength tensor $F_{\alpha\beta}(x)$, one may define the dual tensor $\,{}^d F^{\mu\nu}(x) \equiv \half \epsilon^{\mu\nu\alpha\beta} \, F_{\alpha\beta}(x)$. Write out the components of the dual field strength ${}^d F^{\mu\nu}$ in terms of the components of $E$ and $B$. Describe the effect of replacing $F^{\mu\nu}$ by its dual (i.e., a duality transform) as a transformation on $E$ and $B$. What is the net effect of a double dual, ${}^d({}^d F)^{\mu\nu}$? Note that this definition allows one to write Maxwell's equations in the very compact and `symmetric' form: $\partial_\nu F^{\mu\nu} = {1 \over c} \, j^\mu$ and $\partial_\nu \, {}^d F^{\mu\nu} = 0$.
6. (10 pts) Prove that $\E^2 - c^2\,\B^2$ and $\E \cdot \B$ are both Lorentz invariant (under proper orthochronous transformatinos), by expressing them in terms of the field strength $F^{\mu\nu}$. How does parity transform these quantities? Is $\E^2 + c^2\, \B^2$ Lorentz invariant?