List of Footnotes

1 Note that since there are two metrics, jab and jab + tab there can be two different sets of transformations that leave one of the metrics invariant. In this sense there are two Lorentz groups.
2 In a field theory, broken rotation invariance automatically yields broken boost invariance. For example, if rotation invariance is broken by coupling matter to a non-zero spacelike four-vector, the four-vector is also not boost invariant.
3 It may therefore seem that DSR theories are ”empty”, in the sense that the new definitions of E and p are merely mathematical manipulations without any physical meaning, i.e. e and p represent the true energy and momentum that one would measure. For a discussion of this point see [203, 186Jump To The Next Citation Point]. Since this is not a theoretical paper, we simply make the DSR assumption that E and p are the measurable energy and momentum (or more specifically for analysis of particle interactions, the energy and momentum that are assigned to in and out states in a scattering problem).
4 The underlying approach that yields certain choices of n has been sharply criticized in [38]. However, the constraints on these models are so poor that any n is observationally feasible.
5 In the literature the mSME is often referred to as just the SME, although technically it was introduced in [90] as a minimal subset of an extension that involved non-renormalizable operators as well.
6 (aL,R)mAB can be constant because the mSME deals with only Minkowski space. If one wishes to make the mSME diffeomorphism invariant, these and other coefficients would be dynamical (see Section 2.4).
7 Note that many of the fermion terms in the mSME can also be found in the extended Dirac equation framework (cf. [37, 193]). Similarly, parts of the electromagnetic sector were previously known (cf. [235]) in the context of equivalence principle violations.
8 These dispersion relations also arise in some approaches to low energy dynamics from loop quantum gravity [8, 7, 120]. However, the ultimate low-energy status of Lorentz invariance in loop quantum gravity is still far from clear (cf. [63, 208, 257]).
9 Other methods of removing UV/IR mixing exist, for an example see [271].
10 While we are primarily concerned in this section with dimensional transmutation of higher dimension operators, Lorentz violating renormalizable operators for one particle of course also yield radiative corrections to other particle operators. For a specific example see [29].
11 This can easily be seen by the following argument. Consider a kinetic term in Euclidean space for a scalar field f of the form M ab@af@bf. In four dimensions M ab must be a dimensionless tensor that has hypercubic symmetry. The only such tensor is dab, so rotation invariance is automatically preserved. Interaction terms are by their very nature rotation invariant, which implies that the entire action is invariant under the full rotation group.
12 The aether models take as a starting point general relativity plus an external vector field. Recent work [144] has shown that there is an alternate formulation with the same Lorentz violating consequences in terms of a non-zero shear field in metric-affine gravity.
13 Clock comparison experiments have also been used to place bounds on Lorentz violation in a conjectured low energy state for loop quantum gravity [266].
14 For a more thorough discussion of CPT (and CP) tests, see for example [199].
15 In k-Minkowski space there is currently some debate as to whether the standard relation for group velocity is correct [19, 186Jump To The Next Citation Point]. Until this is resolved, v = @E/@p remains an assumption that might be modified in a DSR context. It obviously holds in field theoretic approaches to Lorentz violation.
16 This is the first example of a significant constraint on terms in particle dispersion/effective field theory that are Planck suppressed, which would naively seem impossible. The key feature of this reaction is the interplay between the long travel time and the large Planck energy. In general any experiment that is sensitive to Planck suppressed operators is either extremely precise (as in terrestrial tests of the mSME) or has some sort of “amplifier”. An amplifier is some other scale (such as travel time or particle mass) which combines with the Planck scale to magnify the effect.
17 There currently seems to be some disagreement about this. For example, in [21Jump To The Next Citation Point, 258Jump To The Next Citation Point] an energy dependent speed of light is argued for, whereas in [186Jump To The Next Citation Point] no such modification is found.
18 Gravitational birefringence has also been studied extensively in the context of non-metric theories of gravitation, which also exhibit Lorentz violation. See for example [259, 116, 244] for discussions of these theories and the parallels with the mSME.
19 This severe limit on birefringence provides an interesting limitation to allowable spacetime metrics in the approach of Hehl and others [142]. In this approach, linear constitutive relations for electromagnetism are postulated as fundamental and the metric is derived from the constitutive relation. A lack of birefringence implies that the metric must be Riemannian in this approach [195].
20 This can easily be done by considering a CPT invariant EFT or choosing the appropriate helicity states/coefficients for a CPT violating EFT.
21 Currently, the observed photon spectrum only extends to 50 TeV. Hence this type of lower/upper threshold structure is observationally indistinguishable from a simple lower threshold with current data.
22 For n = 4 no QED particles reach energies high enough to provide constraints. The only particles of the required energy are ultra-high energy cosmic rays or neutrinos. Assuming the cosmic rays are protons, the corresponding reaction time for Čerenkov emission is -17 10 s.
23 The theoretical model for the non-thermal emission of the Crab nebula is called the synchrotron-self-Compton (SSC) model. For recent fits of data to the SSC spectrum see [4].
24 This would change, of course, if a shift in the cutoff was recorded, as then the detailed dependence of the cutoff location on the Lorentz violating physics would be important.
25 The bounds quoted in [86Jump To The Next Citation Point, 119Jump To The Next Citation Point] vary from this bound by roughly an order of magnitude, as the proton energy is taken to be 1020 eV rather than right at the GZK cutoff.
26 This approach requires that one can evaluate the parton distribution functions up to UHECR energies without errors from the uncertainty in low energy parton distribution functions or effects from new physics becoming appreciable. We caution that such effects can adjust the constraints, as the authors of [119Jump To The Next Citation Point] note.
27 Dimension five CPT violating operators yield helicity dependent dispersion. To derive bounds on these operators, Gagnon and Moore [119Jump To The Next Citation Point] assume that the coefficients are roughly equal.
28 For a discussion of synchrotron radiation in other models with Planck suppressed dispersion corrections see [128].
29 It has recently been suggested that Lorentz violation might play a role in the formation of the pulsar itself, specifically in large pulsar anomalous velocities [192].
30 There are other attempts to explain the LSND results with CPT violation [229, 42, 41]. However, these CPT violating models are not directly correlated with Lorentz violation as they may involve non-local field theories. It is also seems likely that the LSND result is simply incorrect.
31 There are exceptions, for example see [151]. Here gravity is modified by a Chern-Simons form, yet there are still only two gravitational wave polarizations. The only modification is that the intensity of the polarizations differs from what would be expected in general relativity.
32 It has also been proposed that laser interferometry may eventually be capable of direct tests of Planck suppressed Lorentz violation dispersion [22].
33 The B-mode polarization of the CMB might also carry an imprint of Lorentz violation due to modifications in the gravitational sector [206].
34 The above approach presumes inflation and speculates about the low energy signature of Lorentz violating physics. Lorentz violation can also be a component in the so-called variable speed of light (VSL) cosmologies (for a review see [209]) which are a possible alternative to inflation. Some bounds on VSL theories are known from Lorentz symmetry tests, but in these cases the VSL model can be equivalently expressed in one of the frameworks of this review.