  The strong equivalence principle, local Lorentz invariance and CPT symmetry
are fundamental ingredients of the quantum field theories used to describe
elementary particle physics. Nevertheless, each may be violated by simple
modifications to the dynamics while apparently preserving the essential
fundamental structure of quantum field theory itself. In this paper, we analyse
the construction of strong equivalence, Lorentz and CPT violating Lagrangians
for QED and review and propose some experimental tests in the fields of
astrophysical polarimetry and precision atomic spectroscopy. In particular,
modifications of the Maxwell action predict a birefringent rotation of the
direction of linearly polarised radiation from synchrotron emission which may
be studied using radio galaxies or, potentially, gamma-ray bursts. In the Dirac
sector, changes in atomic energy levels are predicted which may be probed in
precision spectroscopy of hydrogen and anti-hydrogen atoms, notably in the
Doppler-free, two-photon $1s-2s$ and $2s-nd (n \sim 10)$ transitions.
