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

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... tal 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 ∼ 10) transitions.