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
.19
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.
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