hydrogen atom. In fact, conventional physics predicts that the spectrum of antihydrogen is identical to that of regular hydrogen. In the presence of the background fields of the SME, the spectra of hydrogen and antihydrogen are expected to show tiny differences in some lines, and no differences in others. Calculations of these SME effects in antihydrogen and hydrogen were published in Physical Review Letters in 1999. One of the main results found is that hyperfine transitions are sensitive to Lorentz breaking effects. Several experimental groups at CERN are working on producing antihydrogen: AEGIS, ALPHA, ASACUSA, ATRAP, and GBAR. Creating trapped antihydrogen in sufficient quantities to do spectroscopy is an enormous experimental challenge. Signatures of Lorentz violation are similar to those expected in Penning traps. There would be sidereal effects causing variations in the spectral frequencies as the experimental laboratory turns with the Earth. There would also be the possibility of finding instantaneous Lorentz breaking signals when antihydrogen spectra are compared directly with conventional hydrogen spectra In October 2017, the BASE experiment at CERN reported a measurement of the antiproton magnetic moment to a precision of 1.5 parts per billion. It is consistent with the most precise measurement of the proton magnetic moment (also made by BASE in 2014), which supports the hypothesis of CPT symmetry. This measurement represents the first time that a property of antimatter is known more precisely than the equivalent property in matter. Lorentz violation with muons The muon and its positively charged antiparticle have been used to perform tests of Lorentz symmetry. Since the lifetime of the muon is only a few microseconds, the experiments are quite different from ones with electrons and positrons. Calculations for muon experiments aimed at probing Lorentz violation in the SME were first published in the year 2000. In the year 2001, Hughes and collaborators published their results from a search for sidereal signals in the spectrum of muonium, an atom consisting of an electron bound to a negatively charged muon. Their data, taken over a two-year period, showed no evidence for Lorentz violation. This placed a stringent constraint on a combination of -type coefficients in the SME, published in Physical Review Letters. In 2008, the Muon Collaboration at the Brookhaven National Laboratory published results after searching for signals of Lorentz violation with muons and antimuons. In one type of analysis, they compared the anomaly frequencies for the muon and its