(a) An electromagnetic field is specified by a four-vector potential

$$A^{\mu}(\mathbf{x}, t)=(\phi(\mathbf{x}, t) / c, \mathbf{A}(\mathbf{x}, t))$$

Define the corresponding field-strength tensor $F^{\mu \nu}$ and state its transformation property under a general Lorentz transformation.

(b) Write down two independent Lorentz scalars that are quadratic in the field strength and express them in terms of the electric and magnetic fields, $\mathbf{E}=-\boldsymbol{\nabla} \phi-\partial \mathbf{A} / \partial t$ and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that both these scalars vanish when evaluated on an electromagnetic plane-wave solution of Maxwell's equations of arbitrary wavevector and polarisation.

(c) Find (non-zero) constant, homogeneous background fields $\mathbf{E}(\mathbf{x}, t)=\mathbf{E}_{0}$ and $\mathbf{B}(\mathbf{x}, t)=\mathbf{B}_{0}$ such that both the Lorentz scalars vanish. Show that, for any such background, the field-strength tensor obeys

$$F_{\rho}^{\mu} F_{\sigma}^{\rho} F_{\nu}^{\sigma}=0$$

(d) Hence find the trajectory of a relativistic particle of mass $m$ and charge $q$ in this background. You should work in an inertial frame where the particle is at rest at the origin at $t=0$ and in which $\mathbf{B}_{0}=\left(0,0, B_{0}\right)$.