### 6.3 Birefringence

A constraint related to time of flight is birefringence. The dimension five operators in Equation (40) as
well as certain operators in the mSME induce birefringence - different speeds for different photon
polarizations (41).
A number of distant astrophysical objects exhibit strong linear polarization in various low energy bands (see
for example the sources in [178, 127]). Recently, linear polarization at high energies from a GRB has been
reported [85], though this claim has been challenged [248, 274]. Lorentz violating birefringence can erase
linear polarization as a wave propagates, hence measurements of polarization constrain the relevant
operators.
The logic is as follows. We assume for simplicity the framework of Section 4.1.4 and rotation invariance;
the corresponding analysis for the general mSME case can be found in [178]. At the source, assume the
emitted radiation is completely linearly polarized, which will provide the most conservative constraint. To
evolve the wave, we must first decompose the linear polarization into the propagating circularly polarized
states,
If we choose coordinates such that the wave is travelling in the z-direction with initial polarization (0,1,0,0)
then and . Rearranging slightly we have
which describes a wave with a rotating polarization vector. Hence in the presence of birefringence a linearly
polarized wave rotates its direction of polarization during propagation. This fact alone has been used to
constrain the term in the mSME to the level of by analyzing the plane of polarization
of distant galaxies [74].
A variation on this constraint can be derived by considering birefringence when the difference
is a function of . A realistic polarization measurement is an aggregate of the
polarization of received photons in a narrow energy band. If there is significant power across the entire
band, then a polarized signal must have the polarization direction at the top nearly the same
as the direction at the bottom. If the birefringence effect is energy dependent, however, the
polarization vectors across the band rotate differently with energy. This causes polarization “diffusion”
as the photons propagate. Given enough time the spread in angle of the polarization vectors
becomes comparable to and the initial linear polarization is lost. Measurement of linear
polarization from distant sources therefore constrains the size of this effect and hence the Lorentz
violating coefficients. We can easily estimate the constraint from this effect by looking at when the
polarization at two different energies (representing the top and bottom of some experimental
band) is orthogonal, i.e. . Using Equation (77) for the polarization gives

Three main results have been derived using this approach. Birefringence has been applied to the mSME
in [178, 179]. Here, the ten independent components of the two coefficients and (see
Section 5.3) that control birefringence are expressed in terms of a ten-dimensional vector [179]. The
actual bound, calculated from the observed polarization of sixteen astrophysical objects, is
.
A similar energy band was used to constrain in Equation (40) to be [127]. Recently,
the reported polarization of GRB021206 [85] was used to constrain to [156], but since
the polarization claim is uncertain [248, 274] such a figure cannot be treated as an actual
constraint.