3.1 Systematic modified dispersion
Perhaps the simplest kinematic framework for Lorentz violation in particle based experiments is to
propose modified dispersion relations for particles, while keeping the usual energymomentum conservation
laws. This was the approach taken in much of the work using astrophysical phenomena in the late 1990’s. In
a given observer’s frame in flat space, this is done by postulating that the usual Lorentz invariant dispersion
law is replaced by some function . In general the preferred frame is
taken to coincide with the rest frame of the cosmic microwave background. Since we live in
an almost Lorentz invariant world (and are nearly at rest with respect to the CMBR), in the
preferred frame must reduce to the Lorentz invariant dispersion at small energies and
momenta. Hence it is natural to expand about , which yields the expression
where the constant coefficients are dimensionful and arbitrary but presumably such that the
modification is small. The order of the first nonzero term in Equation (13) depends on the underlying
model of quantum gravity taken. Since the underlying motivation for Lorentz violation is quantum gravity,
it is useful to factor out the Planck energy in the coefficients and rewrite Equation (13) as
such that the coefficients are dimensionless.
In most of the literature a simplifying assumption is made that rotation invariance is preserved. In
nature, we cannot have the rotation subgroup of the Lorentz group strongly broken while preserving boost
invariance. Such a scenario leads immediately to broken rotation invariance at every energy which is
unobserved.
Hence, if there is strong rotation breaking there must also be a broken boost subgroup. However, it is
possible to have a broken boost symmetry and unbroken rotation symmetry. Either way, the boost subgroup
must be broken. Phenomenologically, it therefore makes sense to look first at boost Lorentz violation
and neglect any violation of rotational symmetry. If we make this assumption then we have
There is no a priori reason (from a phenomenological point of view) that the coefficients in
Equation (15) are universal (and in fact one would expect the coefficients to be renormalized differently
even if the fundamental Lorentz violation is universal [6]). We will therefore label each as
where represent particle species.

Modified dispersion and effective field theory

Effective field theory (EFT) is not applicable if one wishes to stick to straight kinematics,
however the EFT implications for modified dispersion are so significant that they must be
considered. As will be shown in detail in Section 4.1, universal dispersion relations cannot be
imposed for all from an EFT standpoint. For example, rotationally invariant type
dispersion cannot be imposed universally on photons [90, 230]. The operators that give rise to
dispersion are CPT violating and induce birefringence (the dispersion modifications
change sign based on the photon helicity). Since EFT requires different coefficients for particles
with different properties and there is no underlying reason why all coefficients should be the
same, it is phenomenologically safest when investigating modified dispersion to assume that
each particle has a different dispersion relation. After this general analysis is complete the
universal case can be treated with ease.