  The CPT theorem has been tested to very high precision in a variety of
experiments involving particles and antiparticles confined within Penning
traps. These tests include comparisons of anomalous magnetic moments and
charge-to-mass ratios of electrons and positrons, protons and antiprotons, and
hydrogen ions and antiprotons. We present a theoretical analysis of possible
signals for CPT and Lorentz violation in these systems. We use the framework of
Colladay and Kosteleck\'y, which consists of a general extension of the $SU(3)
\times SU(2) \times U(1)$ standard model including possible CPT and Lorentz
violations arising from spontaneous symmetry breaking at a fundamental level,
such as in string theory. We work in the context of an extension of quantum
electrodynamics to examine CPT and Lorentz tests in Penning traps. Our analysis
permits a detailed study of the effectiveness of experimental tests of CPT and
Lorentz symmetry performed in Penning traps. We describe possible signals that
might appear in principle, and estimate bounds on CPT and Lorentz violation
attainable in present and future experiments.
