Signals of CPT and Lorentz violation are possible in the context of spectroscopy using hydrogen and antihydrogen. We apply the Standard-Model Extension, a broad framework for Lorentz breaking in physics, to various transitions in the hydrogen and antihydrogen spectra. The results show an unsuppressed effect in the transition between the upper two hyperfine sublevels of the ground state of these systems. We also discuss related tests in Penning traps, and recent work on Lorentz violation in curved spacetime.
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