### 5.1 Penning traps

A Penning trap is a combination of static magnetic and electric fields that can keep a charged particle
localized within the trap for extremely long periods of time (for a review of Penning traps see [68]). A
trapped particle moves in a number of different ways. The two motions relevant for Lorentz
violation tests are the cyclotron motion in the magnetic field and Larmor precession due to the
spin. The ratio of the precession frequency to the cyclotron frequency is given by
where is the g-factor of the charged particle. The energy levels for a spin particle are given by
where is an integer and . For electrons and positrons, where , the
state is almost degenerate with the state . The degeneracy breaking
is solely due to the anomalous magnetic moment of the electron and is usually denoted by
. By introducing a small oscillating magnetic field into the trap one can induce
transitions between these almost degenerate energy states and very sensitively determine the value of
.
The primary use of measurements of is that they directly give a very accurate value of
. However, due to their precision, these measurements also provide good tests of CPT and
Lorentz invariance. In the mSME, the only framework that has been applied to Penning trap
experiments, the factor for electrons and positrons receive no corrections at lowest order. However,
the frequencies and both receive corrections [60]. At lowest order in the Lorentz
violating coefficients these corrections are (with the trap’s magnetic field in the z-direction)

expressed in a non-rotating frame. The unmodified frequencies are denoted by and the
Lorentz violating parameters are various components of the general set given in Equations (32)
and (33).
The functional form of Equation (50) immediately makes clear that there are two ways to test for
Lorentz violation. The first is to look for instantaneous CPT violation between electrons and positrons
which occurs if the parameter is non-zero. The observational bound on the difference between
for electrons and positrons is [97]. This leads to a bound on
of order . The second approach is to track over time, looking for
sidereal variations as the orientation of the experimental apparatus changes with respect to the
background Lorentz violating tensors. This approach has been used in [217] to place a bound on the
diurnal variation of the anomaly frequency of , which limits a particular
combination of components of , , and at this level. Finally, we note that similar
techniques have been used to measure CPT violations for proton/anti-proton and hydrogen ion
systems [118]. By measuring the cyclotron frequency over time, bounds on the cyclotron frequency
variation (50) for the anti-proton have established a limit at the level of on components of
.