there is a short, high energy GRBs from some astrophysical source where all the photons are emitted from the same point at the same time. The expected observational signal is then a correlation between the photon time of arrival and energy. The time of arrival is fairly straightforward to measure, but the reconstruction of the initial photon energy is not so easy. GLAST measures the initial photon energy by calorimetry – the photon goes through a conversion foil and converts to an electronpositron pair. The pair then enters a calorimeter, which measures the energy by scintillation. The initial particle energy is then only known by reconstruction from many events.
Energy reconstruction requires addition of the multitude of low energy signals back into the single high energy incoming photon. Usually this addition in energy is linear (with corrections due to systematics/experimental error). However, if we take the DSR energy summation rules as currently postulated the energies of the low energy events add nonlinearly, leading to a modified high energy signal. One might guess that since the initial particle energy is well below the Planck scale, the nonlinear corrections make little difference to the energy reconstruction. However, to concretely answer such a question, the multiparticle sector of DSR must be properly understood (for a discussion of the problems with multiparticle states in DSR see [
186]
).
Finally, while photons are the most commonly used particle in time of flight tests, other particles may also be employed. For example, it has been proposed in [
81]
that neutrino emission from GRBs may also be used to set limits on
$n=3$
dispersion. Observed neutrino energies can be much higher than the TeV scale used for photon measurements, hence one expects that any time delay is greatly magnified. Neutrino time delay might therefore be a very precise probe of even
$n>3$
dispersion corrections. Of course, first an identifiable GRB neutrino flux must be detected, which has not happened yet [
5]
. Assuming that a flux is seen and able to be correlated on the sky with a GRB, one must still disentangle the signal. In a DSR scenario, where time delay scales uniformly with energy this is not problematic, at least theoretically. However, in an EFT scenario there can be independent coefficients for each helicity, thereby possibly masking an energy dependent signal.
For
$n=3$
this complication is irrelevant if one assumes that all the neutrinos are lefthanded (as would be expected if produced from a standard model interaction) as only
${f}_{\nu L}^{\left(3\right)}$
would then apply.
For
$n>3$
the possible operators are not yet known, so it is not clear what bounds would be set by limits on neutrino time of flight delays.
An amplifier is some other scale (such as travel time or particle mass) which combines with the Planck scale to magnify the effect.
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 ).^{
$\text{18}$
}
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,
$$\begin{array}{c}{A}^{\alpha}={\epsilon}_{L}^{\alpha}{e}^{i{k}_{L}\cdot x}+{\epsilon}_{R}^{\alpha}{e}^{i{k}_{R}\cdot x}.\end{array}$$ 
(76)

If we choose coordinates such that the wave is travelling in the zdirection with initial polarization (0,1,0,0) then
${\epsilon}_{L}=(0,1,i,0)$
and
${\epsilon}_{R}=(0,1,i,0)$
. Rearranging slightly we have
$$\begin{array}{c}{A}^{\alpha}={e}^{i({\omega}_{L}t{k}_{z}z)}\left({\epsilon}_{L}^{\alpha}+{\epsilon}_{R}^{\alpha}{e}^{i({\omega}_{R}{\omega}_{L})t}\right),\end{array}$$ 
(77)

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
${k}_{AF}$
term in the mSME to the level of
${10}^{42}GeV$
by analyzing the plane of polarization of distant galaxies [
74]
.
A variation on this constraint can be derived by considering birefringence when the difference
$\Delta \omega ={\omega}_{R}{\omega}_{L}$
is a function of
${k}_{z}$
. 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
$2\pi $
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.
${A}_{\left({E}_{T}\right)}^{\alpha}{A}_{\left({E}_{B}\right)\alpha}=0$
. Using Equation ( 77 ) for the polarization gives
$$\begin{array}{c}\left\Delta {\omega}_{\left({E}_{T}\right)}\Delta {\omega}_{\left({E}_{B}\right)}\rightt\approx \pi .\end{array}$$ 
(78)

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
${\stackrel{~}{\kappa}}_{e+}$
and
${\stackrel{~}{\kappa}}_{o}$
(see Section 5.3 ) that control birefringence are expressed in terms of a tendimensional vector
${k}^{a}$
[
179]
. The actual bound, calculated from the observed polarization of sixteen astrophysical objects, is
$\left{k}^{a}\right\le {10}^{32}$
.^{
$\text{19}$
}
A similar energy band was used to constrain
$\xi $
in Equation ( 40 ) to be
$\left\xi \right<\mathcal{O}\left({10}^{4}\right)$
[
127]
. Recently, the reported polarization of GRB021206 [
85]
was used to constrain
$\xi $
to
$\left\xi \right<\mathcal{O}\left({10}^{14}\right)$
[
156]
, but since the polarization claim is uncertain [
248,
274]
such a figure cannot be treated as an actual constraint.
6.4 Threshold constraints
We now turn our attention from astrophysical tests involving a single particle species to threshold reactions, which often involve many particle types. Before delving into the calculational details of energy thresholds with Lorentz violation, we give a pedagogical example that shows why particle decay processes (which involve rates) give constraints that are only functions of reaction threshold energies. Consider photon decay,
$\gamma \to {e}^{+}{e}^{}$
(see Section 6.5.5 for details). In ordinary Lorentz invariant physics the photon is stable to this decay process. What forbids this reaction is solely energy/momentum conservation – two timelike fourmomenta (the outgoing pair) cannot add up to the null four momentum of the photon. If, however, we break Lorentz invariance and assume a photon obeys a dispersion relation of the form
$$\begin{array}{c}{\omega}^{2}={k}^{2}+{f}_{\gamma}^{\left(3\right)}\frac{{k}^{3}}{{E}_{Pl}},\end{array}$$ 
(79)

while electrons/positrons have their usual Lorentz invariant dispersion, then it is possible to satisfy energy conservation equation if
${f}_{\gamma}^{\left(3\right)}>0$
(to see this intuitively, note that the extra term at high energies acts as a large effective mass for a photon). Therefore a photon can decay to an electron positron pair.
This type of reaction is called a
threshold reaction as it can happen only above some threshold energy
${\omega}_{th}\sim ({m}_{e}^{2}{E}_{Pl}/{f}_{\gamma}^{\left(3\right)}{)}^{1/3}$
where
${m}_{e}$
is the electron mass. The threshold energy is translated into a constraint on
${f}_{\gamma}^{\left(3\right)}$
in the following manner. We see
$50TeV$
photons from the Crab nebula [
268]
, hence this reaction must not occur for photons up to this energy as they travel to us from the Crab. If the decay rate is high enough, one could demand that
${\omega}_{th}$
is above
$50TeV$
, constraining
${f}_{\gamma}^{\left(3\right)}$
and limiting this type of Lorentz violation. For
$\mathcal{O}\left(1\right)$
${f}_{\gamma}^{\left(3\right)}$
,
${\omega}_{th}\sim 10TeV$
, and so we can get a slightly better than
$\mathcal{O}\left(1\right)$
constraint on
${f}_{\gamma}^{\left(3\right)}$
from
$50TeV$
photons [
152]
. If, however, the rate is very small then even though a photon is above threshold it could still reach us from the Crab. Using the Lorentz invariant expression for the matrix element
$\mathcal{\mathcal{M}}$
(i.e. just looking at the kinematical aspect of Lorentz violation) one finds that as
$\omega $
increases above
${\omega}_{th}$
the rate very rapidly becomes proportional to
${f}_{\gamma}^{\left(3\right)}{\omega}^{2}/{E}_{Pl}$
. If a
$50TeV$
photon is above threshold, the decay time is then approximately
${10}^{11}/{f}_{\gamma}^{\left(3\right)}s$
. The travel time of a photon from the Crab is
$\sim {10}^{11}$
seconds. Hence if a photon is at all above threshold it will decay almost instantly relative to the observationally required lifetime. Therefore we can neglect the actual rate and derive constraints simply by requiring that the threshold itself is above
$50TeV$
.
It has been argued that technically, threshold constraints can't truly be applicable to a kinematic model where just modified dispersion is postulated and the dynamics/matrix elements are not known. This isn't actually a concern for most threshold constraints. For example, if we wish to constrain
${f}_{\gamma}^{\left(3\right)}$
at
$\mathcal{O}\left(1\right)$
by photon decay, then we can do so as long as
$\mathcal{\mathcal{M}}$
is within 11 orders of magnitude of its Lorentz invariant value (since the decay rate goes as
$\mathcal{\mathcal{M}}{}^{2}$
). Hence for rapid reactions, even an enormous change in the dynamics is irrelevant for deriving a kinematic constraint. Since kinematic estimates of reaction rates are usually fairly accurate (for an example see [
202,
201]
) one can derive constraints using only kinematic models. In general, under the assumption that the dynamics is not drastically different from that of Lorentz invariant effective field theory, one can effectively apply particle reaction constraints to kinematic theories since the decay times are extremely short above threshold.
There are a few exceptions where the rate is important, as the decay time is closer to the travel time of the observed particle. Any type of reaction involving a weakly interacting particle such as a neutrino or graviton will be far more sensitive to changes in the rate. For these particles, the decay time of observed particles can be comparable to their travel time. As well, any process involving scattering, such as the GZK reaction (
$p+{\gamma}_{CMBR}\u27f6p+{\pi}^{0}$
) or photon annihilation (
$2\gamma \u27f6{e}^{+}+{e}^{}$
) is more susceptible to changes in
$\mathcal{\mathcal{M}}$
as the interaction time is again closer to the particle travel time. Even for scattering reactions, however,
$\mathcal{\mathcal{M}}$
would need to change significantly to have any effect. Finally,
$\mathcal{\mathcal{M}}$
is important in reactions like (
$\gamma \u27f63\gamma $
), which are not observed in nature but do not have thresholds [
154,
183,
3,
2,
124]
. In these situations, the small reaction rate is what may prevent the reaction from happening on the relevant timescales. For all of these cases, kinematics only models should be applied with extreme care. We now turn to the calculation of threshold constraints assuming EFT.
6.5 Particle threshold interactions in EFT
When Lorentz invariance is broken there are a number of changes that can occur with threshold reactions. These changes include shifting existing reaction thresholds in energy, adding additional thresholds to existing reactions, introducing new reactions entirely, and changing the kinematic configuration at threshold [
86,
130,
154,
200]
. By demanding that the energy of these thresholds is inside or outside a certain range (so as to be compatible with observation) one can derive stringent constraints on Lorentz violation.
In this section we will describe various threshold phenomena introduced by Lorentz violation in EFT and the constraints that result from high energy astrophysics. Thresholds in other models are discussed in Section
6.6 . We will use rotationally invariant QED as the prime example when analyzing new threshold behavior. The same methodology can easily be transferred to other particles and interactions. A diagram of the necessary elements for threshold constraints and the appropriate sections of this review is shown in Figure 1 .