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6.3 Birefringence

A constraint related to time of flight is birefringence. The dimension five operators in Equation (40View Equation) as well as certain operators in the mSME induce birefringence - different speeds for different photon polarizations (41View Equation).18 A number of distant astrophysical objects exhibit strong linear polarization in various low energy bands (see for example the sources in [178Jump To The Next Citation Point127Jump To The Next Citation Point]). Recently, linear polarization at high energies from a GRB has been reported [85Jump To The Next Citation Point], though this claim has been challenged [248Jump To The Next Citation Point274Jump To The Next Citation Point]. 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 [178Jump To The Next Citation Point]. 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,
a a -ik .x a -ik .x A = eLe L + eRe R . (76)
If we choose coordinates such that the wave is travelling in the z-direction with initial polarization (0,1,0,0) then eL = (0,1, -i,0) and eR = (0,1,i,0). Rearranging slightly we have
Aa = e-i(wLt- kzz)(ea + eae-i(wR-wL)t), (77) L R
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 kAF term in the mSME to the level of -42 10 GeV by analyzing the plane of polarization of distant galaxies [74].

A variation on this constraint can be derived by considering birefringence when the difference Dw = wR - wL is a function of kz. 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 2p 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. Aa A(E )a = 0 (ET) B. Using Equation (77View Equation) for the polarization gives

| | |Dw(ET) - Dw(EB) |t ~~ p. (78)

Three main results have been derived using this approach. Birefringence has been applied to the mSME in [178179Jump To The Next Citation Point]. Here, the ten independent components of the two coefficients ~ke+ and ~ko- (see Section 5.3) that control birefringence are expressed in terms of a ten-dimensional vector ka [179]. The actual bound, calculated from the observed polarization of sixteen astrophysical objects, is |ka|< 10-32.19 A similar energy band was used to constrain q in Equation (40View Equation) to be -4 |q| < O(10 ) [127Jump To The Next Citation Point]. Recently, the reported polarization of GRB021206 [85] was used to constrain q to |q|< O(10 -14) [156Jump To The Next Citation Point], but since the polarization claim is uncertain [248274] such a figure cannot be treated as an actual constraint.

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