Following a suggestion of Kostelecký et al. we have evaluated a test of CPT and Lorentz invariance from the microwave spectroscopy of muonium. Hamiltonian terms beyond the standard model violating CPT and Lorentz invariance would contribute frequency shifts δν12 and δν34 to ν12 and ν34, the two transitions involving muon spin flip, which were precisely measured in ground state muonium in a strong magnetic field of 1.7 T. The shifts would be indicated by anti-correlated oscillations in ν12 and ν34 at the earth's sidereal frequency. No time dependence was found in ν12 or ν34 at the level of 20 Hz, limiting the size of some CPT and Lorentz violating parameters at the level of 2 × 10 −23 GeV, representing Planck scale sensitivity and an order of magnitude improvement in sensitivity over previous limits for the muon. 11.30.Er, 11.30.Cp, 12.20.Fv, 36.10.Dr Much current theoretical work is devoted to finding a more fundamental and general underlying theory from which the standard model of particle physics could be deduced as the low energy limit. String theory is a central candidate, with the feature that the assumptions underlying the CPT theorem are invalid for strings, which are extended objects. CPT violation then becomes a possible signature of more fundamental underlying theories, which can be probed by sensitive experimental tests of CPT invariance.Some years ago Kostelecký and coworkers [1-3] developed a plausible extension of the standard model based on spontaneous breaking of Lorentz and CPT symmetry in an underlying theory without gravity. Their low energy effective theory provides a theoretical basis for establishing quantitative bounds on CPT invariance. The analysis was done in the context of conventional relativistic quantum mechanics and quantum field theory in four dimensions, retaining the usual gauge structure and renormalizability. The Lorentz and CPT violating additions to the standard model Lagrangian are highly suppressed to remain compatible with established experimental bounds.As applied to muonium (µ + e − bound state), the most relevant terms in the extension to the standard model are those in the QED limit, involving only muons, electrons and photons. The additional terms in the Lorentzviolating Lagrangian lead to a modified Dirac equation and are given by [2]

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