Modern Tests of Lorentz Invariance

In the paper

“we see that we can solve the conservation equation if p > (4m2Epl)1/3, which is approximately 23TeV. At threshold, x = 1/2 so this corresponds to emission of a hard photon with an energy of 11.5TeV.”

In the paper

“From Equation (94) it is clear that when the Phase space is highly suppressed, while for the phase space in k becomes of order p. The momentum acts as an effective threshold, where the reaction is strongly suppressed below this energy.”

In the paper

“We observe 50 TeV photons from the Crab nebula. There must exist then at least one stable photon polarization.”

In the paper

“Placing kth at 50 TeV yields the constraint and hence |ηR| < 0.125. The same procedure applies in the opposite choice of outgoing particle helicity, so ηL obeys this bound as well.”

In the paper

“For example, there is a wavelength dependent speed of light in DSR models.”

In the paper

“The new choice of group action leads to a new invariant energy scale as well as the invariant velocity c (hence the name doubly special).”

In the paper

“Equivalence principle tests are therefore able to also look for Lorentz violation and vice versa (for an explicit example see [13]).”

What the sources say

• Not checked yetcites [13] Alvarez, C., and Mann, R.B., “The equivalence principle and g-2 experiments”, Phys. Lett. B, 409, 83–87, (1997). Related online version (cited on 08 June 2005): http://arXiv.org/abs/gr-qc/9510070. 2.5 source

In the paper

“For example, it is possible that quantum gravity might be Lorentz invariant but contains tensor fields that acquire a vacuum expectation value at low energies [185], thereby spontaneously breaking the symmetry.”

What the sources say

• Not checked yetcites [185] Kostelecký, V.A., and Samuel, S., “Spontaneous Breaking Of Lorentz Symmetry In String Theory”, Phys. Rev. D, 39, 683, (1989). 1, 4.4 source

In the paper

“A partial realization of this idea is provided by so-called “doubly special relativity” (DSR) [15, 186], which will be discussed more thoroughly in Section 3.4.”

What the sources say

• Not checked yetcites [15] Amelino-Camelia, G., “Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale”, Int. J. Mod. Phys. D, 11, 35–60, (2002). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0012051. 2.1.2, 3.4 source
• Not checked yetcites [186] Kowalski-Glikman, J., “Introduction to Doubly Special Relativity”, (2004). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0405273. 1, 2.1.2, 2.2, 3.4, 3, 3, 15, 17

In the paper

“Even if broken at high energies, Lorentz symmetry can still be an attractive infrared fixed point, thereby yielding an approximately Lorentz invariant low energy world [79].”

What the sources say

• Not checked yetcites [79] Chadha, S., and Nielsen, H.B., “Lorentz invariance as a low-Energy phenomenon”, Nucl. Phys. B, 217, 125, (1983). 1 source

In the paper

“This type of violation is really only “apparent” Lorentz violation. The reader should understand that it is a violation only of the usual linear Lorentz group action on physical quantities.”

In the paper

“although no quantum gravity model predicts Lorentz violation conclusively.”

In the paper

“In this scenario there is still Lorentz invariance, but the Lorentz group acts non-linearly on physical quantities.”

In the paper

“Since in order for a physical theory to be well defined the action must be a spacetime scalar, breaking of active Lorentz invariance is the only physically acceptable type of Lorentz violation.”

In the paper

“If we keep our Lorentz violating tensors as fixed objects, we immediately have non-conservation of stress tensors and inconsistent Einstein equations.”

In the paper

“Since we live in a world where Lorentz invariance is at the very least an excellent approximate symmetry, ταβ must be small in our frame.”

In the paper

“One should keep in mind that low energy experiments cannot directly tell us whether or not quantum gravity is Lorentz invariant. Rather, they can only determine if the “state” that we live in is Lorentz violating.”

In the paper

“It is obvious that when we introduce Lorentz violation we have to rethink causality — there is no universal light cone given by the metric that all fields must propagate within.”

In the paper

“Unfortunately, the aether stress tensor does not necessarily satisfy the dominant energy condition (although it may for certain choices of coefficients), so there is no proof yet that spacetimes with a dynamical aether have positive energy.”

In the paper

“Another common idea about quantum gravity is that spacetime should have a stochastic [99, 255] or “foamy” [231] structure at very small scales.”

What the sources say

• Not checked yetcites [99] Dowker, F., Henson, J., and Sorkin, R.D., “Quantum gravity phenomenology, Lorentz invariance and discreteness”, Mod. Phys. Lett. A, 19, 1829–1840, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0311055. 2.6, 3.5 source
• Not checked yetcites [255] Shiokawa, K., “Mesoscopic fluctuations in stochastic spacetime”, Phys. Rev. D, 62, 024002-1–14, (2000). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0001088. 2.6, 3.5 source
• Not checked yetcites [231] Ng, Y.J., and van Dam, H., “Limit to space-time measurement”, Mod. Phys. Lett. A, 9, 335–340, (1994). 2.6, 3.5 source

In the paper

“In nature, we cannot have the rotation subgroup of the Lorentz group strongly broken while preserving boost invariance. Such a scenario leads immediately to broken rotation invariance at every energy which is unobserved.2 Hence, if there is strong rotation breaking there must also be a broken boost subgroup.”

In the paper

“DSR is not a complete theory as it has no dynamics and generates problems when applied to macroscopic objects (for a discussion see [186]).”

What the sources say

• Not checked yetcites [186] Kowalski-Glikman, J., “Introduction to Doubly Special Relativity”, (2004). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0405273. 1, 2.1.2, 2.2, 3.4, 3, 3, 15, 17

In the paper

“This requires that the Hamiltonian still be bounded from below and that perturbations around the ground state have real frequencies.”

In the paper

“The RMS framework is incomplete, as it says nothing about dynamics or how given clocks and rods relate to fundamental particles.”

In the paper

“In general the preferred frame is taken to coincide with the rest frame of the cosmic microwave background.”

In the paper

“However, it is usually assumed that these modified dispersion relations are only effective — at the Planck scale there is a UV completion that renders the fundamental theory stable.”

In the paper

“In the RMS framework, there is assumed to be a preferred frame Σ where the speed of light is isotropic.”

In the paper

“Perhaps the simplest kinematic framework for Lorentz violation in particle based experiments is to propose modified dispersion relations for particles, while keeping the usual energy-momentum conservation laws.”

In the paper

“Since EFT requires different coefficients for particles with different properties and there is no underlying reason why all coefficients should be the same, it is phenomenologically safest when investigating modified dispersion to assume that each particle has a different dispersion relation.”

In the paper

“Combining these two ideas has lead a number of authors to the idea of stochastic/non-systematic dispersion where the modifications to the dispersion relation fluctuate over time.”

In the paper

“The operators that give rise to n = 1, 3 dispersion are CPT violating and induce birefringence (the dispersion modifications change sign based on the photon helicity).”

In the paper

“The Robertson-Mansouri-Sexl framework [246, 211, 277] is a well known kinematic test theory for parameterizing deviations from Lorentz invariance.”

What the sources say

• Not checked yetcites [246] Robertson, H., “Postulate versus observation in the special theory of relativity”, Rev. Mod. Phys., 21, 378, (1949). 2.2, 3.2 source
• Not checked yetcites [211] Mansouri, R., and Sexl, R.U., “A test theory of special relativity. I — Simultaneity and clock synchronization”, Gen. Relativ. Gravit., 8, 497–513, (1977). 2.2, 3.2 source
• Not checked yetcites [277] Will, C.M., “The Confrontation between General Relativity and Experiment”, Living Rev. Relativity, 4(4), lrr-2001-4, (2001). URL (cited on 22 February 2005): http://www.livingreviews.org/lrr-2001-4. 3.2, 3.3, 4.4, 31, 7.3, 7.3

In the paper

“Therefore it evades most of the threshold constraints from astrophysics as well as any terrestrial experiment that looks for sidereal variations, while still modifying the usual action of the Lorentz group.”

In the paper

“However, for experiments involving a single type of clock/rod and light, the RMS formalism is applicable and can be used to search for violations of Lorentz invariance in that experiment.”

In the paper

“Hence the synchronization is largely irrelevant and we assume Einstein synchronization.”

In the paper

“Therefore it is somewhat premature to talk about robust constraints on DSR from particle threshold interactions or other experiments.”

In the paper

“Hence the instability to production of Planck energy particles is usually ignored.”

In the paper

“In the flat space limit, which is the c2 formalism, the units are chosen such that limiting speed of the test particles is one while the speed of light is given in terms of the THϵμ parameters by c2 = H/(Tϵμ).”

In the paper

“This transformation is the most general one-to-one transformation that preserves rectilinear motion in the absence of forces.”

In the paper

“Such dispersion modifications have been phenomenologically parameterized by three numbers, the usual coefficient f(n) and exponent n of Section 3.1, and a length scale L which determines the length over which the dispersion is roughly constant.”

In the paper

“The k0 term in Equation (25) is usually required to vanish as it makes the theory unstable.”

In the paper

“It has been argued [75] that any realistic non-commutative theory must eventually reduce to part of the mSME.”

What the sources say

• Not checked yetcites [75] Carroll, S.M., Harvey, J.A., Kostelecký, V.A., Lane, C.D., and Okamoto, T., “Noncommutative field theory and Lorentz violation”, Phys. Rev. Lett., 87, 141601, (2001). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0105082. 9, 9 source

In the paper

“The first is to look for instantaneous CPT violation between electrons and positrons which occurs if the bZ parameter is non-zero.”

In the paper

“The functional form of Equation (50) immediately makes clear that there are two ways to test for Lorentz violation.”

In the paper

“The existence of Θαβ manifestly breaks Lorentz invariance and hence the size of ΛNC is constrained by tests of Lorentz violation.”

In the paper

“Barring conspiratorial cancellations among the coefficients, the bound on is 6.4 ± 5.4 × 10−32 GeV, which is the strongest clock comparison limit on mSME parameters.”

In the paper

“If kF ≠ 0, spacetime acts as a anisotropic medium, and different photon polarizations propagate at different speeds.”

In the paper

“The absence of lower dimension operators implies that either there is a fine tuning in the Lorentz violating sector [91], some other symmetry is present that protects the lower dimension operators, or Lorentz invariance is an exact symmetry.”

What the sources say

• Not checked yetcites [91] Collins, J., Perez, A., Sudarsky, D., Urrutia, L.F., and Vucetich, H., “Lorentz invariance: An additional fine-tuning problem”, Phys. Rev. Lett., 93, 191301, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0403053. 10, 11 source

In the paper

“The most conservative approach for a framework in which to test Lorentz violation from quantum gravity is that of effective field theory (EFT).”

In the paper

“The same scenario generically happens for any tensor field that acquires a VEV dynamically (see Section 7.1), which implies that Lorentz violation can be constrained by the gravitational sector as well as by direct matter couplings.”

In the paper

“Both the standard model and relativity can be considered EFT’s, and the EFT framework can easily incorporate Lorentz violation via the introduction of extra tensors.”

In the paper

“Such modifications are completely unobservable in astrophysical processes, although high precision terrestrial experiments can still probe them.”

In the paper

“If instead one takes Λ ≪ ΛNC then A, . The resulting field theory becomes a subset of the standard model extension; specifically the new operators have the form of the (kF)αβγδFαβ Fγδ term in Equation (35).”

In the paper

“If Λ ≫ ΛNC then A, B are (up to loop factors and coupling coefficients), independent of the scale ΛNC. Such strong Lorentz violation is obviously ruled out by current experiment, implying that in this perturbative approach such a limit is observationally not viable.”

In the paper

“A linear combination of Δa0 and Δaz is bounded at the level of 10−20 GeV [168] and a combination of Δax and Δay is constrained at the 10−21 GeV [234] level.”

What the sources say

• Not checked yetcites [168] Kostelecký, V.A., “Sensitivity of CPT tests with neutral mesons”, Phys. Rev. Lett., 80, 1818, (1998). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9809572. 14 source
• Not checked yetcites [234] Nguyen, H.H., “CPT results from KTeV”, (2001). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-ex/0112046. 14

In the paper

“The relevant parameter for CPT and Lorentz violation in neutral kaon systems is aμ for the down and strange quarks (since ). As we mentioned previously, one of the aμ can always be absorbed by a field redefinition. Therefore only the difference between the quark aμ’s, controls the amount of CPT violation and is physically measurable.”

What the sources say

• Not checked yetcites [184] Kostelecký, V.A., and Potting, R., “CPT, strings, and meson factories”, Phys. Rev. D, 51, 3923–3935, (1995). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9501341. 5.5, 5.5 source

In the paper

“In the mSME the deviation ΔK is (as usual) orientation dependent. In terms of Δaμ, we have [171] 68”

What the sources say

• Not checked yetcites [171] Kostelecký, V.A., “Signals for CPT and Lorentz violation in neutral-meson oscillations”, Phys. Rev. D, 61, 016002, (2000). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9909554. 14 source

In the paper

“Note that all these bounds are approximate, as they depend on the spatial orientation of the experiment with respect to spatial components of uα in the lab frame.”

In the paper

“For Be+, the nuclear spin can be thought of as being carried by a single neutron, so this experiment limits the neutron Lorentz violating coefficients.”

In the paper

“In the mSME, the lowest order contribution to the diagonal components of H occurs in the mass matrix M, contributions to Γ are higher order [184].”

What the sources say

• Not checked yetcites [184] Kostelecký, V.A., and Potting, R., “CPT, strings, and meson factories”, Phys. Rev. D, 51, 3923–3935, (1995). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9501341. 5.5, 5.5 source

In the paper

“The bounds from this experiment are of order 10−29 GeV for the components of perpendicular to the spin axis and 10−28 GeV for the parallel component.”

What the sources say

• Not checked yetcites [148] Hou, L.-S., Ni, W.-T., and Li, Y.-C.M., “Test of Cosmic Spatial Isotropy for Polarized Electrons Using a Rotatable Torsion Balance”, Phys. Rev. Lett., 90, 201101, (2003). 5.4, 5.4 source

In the paper

“From this expression it is easy to see that corresponds to a rotationally invariant shift in the speed of light. It can be shown that and also yield a shift in the speed of light, although in a direction dependent manner. The coefficients and control birefringence.”

What the sources say

• Not checked yetcites [179] Kostelecký, V.A., and Mewes, M., “Signals for Lorentz violation in electrodynamics”, Phys. Rev. D, 66, 056005, (2002). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0205211. 3.2, 3.3, 4.1.4, 4.1.4, 5.3, 5.3, 18 source

In the paper

“For example, in sapphire cavities the change in L due to Lorentz violation is negligible compared to the change in c. This allows one to isolate the electromagnetic sector.”

In the paper

“As noted in [175], proton bounds would be derivable with a more detailed model of nuclear structure.”

What the sources say

• Not checked yetcites [175] Kostelecký, V.A., and Lane, C.D., “Constraints on Lorentz violation from clock-comparison experiments”, Phys. Rev. D, 60, 116010, (1999). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9908504. 13, 13, 13 source

In the paper

“The experiments of Chupp [84], Prestage [243], and Lamoreaux [196] are insensitive to proton coefficients in this model, so no proton bounds have yet been established from these experiments.”

What the sources say

• Not checked yetcites [84] Chupp, T.E., Hoare, R.J., Loveman, R.A., Oteiza, E.R., Richardson, J.M., Wagshul, M.E., and Thompson, A.K., “Results of a new test of local Lorentz invariance: A search for mass anisotropy in 21Ne”, Phys. Rev. Lett., 63, 1541–1545, (1989). 13 source
• Not checked yetcites [243] Prestage, J.D., Bollinger, J.J., Itano, W.M., and Wineland, D.J., “Limits for spatial anisotrophy by use of nuclear spin polarized Be-9+ ions”, Phys. Rev. Lett., 54, 2387–2390, (1985). 13 source
• Not checked yetcites [196] Lamoreaux, S.K., Jacobs, J.P., Heckel, B.R., Raab, F.J., and Fortson, E.N., “New limits on spatial anisotrophy from optically pumped He-201 and Hg-199”, Phys. Rev. Lett., 57, 3125–3128, (1986). 13 source

In the paper

“Related to the existence of asymmetric thresholds is the hard Čerenkov threshold, which also occurs only when n > 2 with negative coefficients. However, in this case both the outgoing and incoming particles must have negative coefficients.”

In the paper

“If a 50 TeV photon is above threshold, the decay time is then approximately . The travel time of a photon from the Crab is ∼ 1011 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 50 TeV.”

In the paper

“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].”

What the sources say

• Not checked yetcites [86] Coleman, S.R., and Glashow, S.L., “Cosmic ray and neutrino tests of special relativity”, Phys. Lett. B, 405, 249–252, (1997). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9703240. 6.5, 6.5.6, 25, 6.8.1 source
• Not checked yetcites [130] Gonzalez-Mestres, L., “Properties of a possible class of particles able to travel faster than light”, (1995). URL (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/9505117. 6.5, 6.5.6
• Not checked yetcites [154] Jacobson, T.A., Liberati, S., and Mattingly, D., “Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics”, Phys. Rev. D, 67, 124011, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0209264. 6.4, 6.5, 6.5.1, 6.5.1, 20, 21, 6.5.5, 22, 23, 22, 6.5.6, 6.5.6, 25, 6.6.2, 6.8.2 source

In the paper

“Recently, the reported polarization of GRB021206 [85] was used to constrain ξ to [156], but since the polarization claim is uncertain [248, 274] such a figure cannot be treated as an actual constraint.”

What the sources say

• Not checked yetcites [85] Coburn, W., and Boggs, S.E., “Polarization of the prompt gamma-ray emission from the gamma-ray burst of 6 December 2002”, Nature, 423, 415–417, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0305377. 18, 19 source
• Not checked yetcites [156] Jacobson, T.A., Liberati, S., Mattingly, D., and Stecker, F.W., “New limits on Planck scale Lorentz violation in QED”, Phys. Rev. Lett., 93, 021101, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0309681. 19, 6.5.5 source
• Not checked yetcites [248] Rutledge, R.E., and Fox, D.B., “Re-Analysis of Polarization in the Gamma-ray flux of GRB 021206”, Mon. Not. R. Astron. Soc., 350, 1272, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0310385. 18, 19 source

In the paper

“The only general statement that can be made is that asymmetric thresholds are not relevant when the outgoing particles have n = 2 type dispersion modifications (either positive or negative) or for strictly positive coefficients at any n.”

In the paper

“The limit |f(3)| < 128 can be easily applied to the EFT operators in Equation (40). From Equation (41) we trivially see that the constraint on ξ is |ξ| < 64, again comparing the 2TeV peak to the 1TeV.”

In the paper

“The torsion experiments bound the t and z components (where z is parallel to the earth’s rotation axis) at the level of 10−27 GeV and the transverse components at 10−25 GeV.”

In the paper

“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.”

What the sources say

• Not checked yetcites [81] Choubey, S., and King, S.F., “Gamma ray bursts as probes of neutrino mass, quantum gravity and dark energy”, Phys. Rev. D, 67, 073005, (2003). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0207260. 17 source

In the paper

“Theorem 2: At a threshold all outgoing momenta are parallel to p1 and all other incoming momentum are anti-parallel.”

What the sources say

• Not checked yetcites [88] Coleman, S.R., and Glashow, S.L., “High-energy tests of Lorentz invariance”, Phys. Rev. D, 59, 116008, (1999). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/9812418. 6.5.3, 6.5.6, 6.5.6, 30 source
• Not checked yetcites [215] Mattingly, D., Jacobson, T.A., and Liberati, S., “Threshold configurations in the presence of Lorentz violating dispersion relations”, Phys. Rev. D, 67, 124012, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0211466. 6.5.3 source

In the paper

“As mentioned in Section 4.3, exact SUSY, which is the only known mechanism to completely protect lower dimension operators, yields dispersion modifications (the primary method used in astrophysics) that are unobservable.”

In the paper

“The orders of magnitude of constraints on estimated from the threshold equation alone (i.e. we have neglected the possibility that the matrix elements are small) for various particles are given in Table 2 [154].”

What the sources say

• Not checked yetcites [154] Jacobson, T.A., Liberati, S., and Mattingly, D., “Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics”, Phys. Rev. D, 67, 124011, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0209264. 6.4, 6.5, 6.5.1, 6.5.1, 20, 21, 6.5.5, 22, 23, 22, 6.5.6, 6.5.6, 25, 6.6.2, 6.8.2 source

In the paper

“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 ka [179]. The actual bound, calculated from the observed polarization of sixteen astrophysical objects, is |ka| ≤ 10−32.19”

What the sources say

• Not checked yetcites [178] Kostelecký, V.A., and Mewes, M., “Cosmological constraints on Lorentz violation in electrodynamics”, Phys. Rev. Lett., 87, 251304, (2001). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0111026. 18, 18 source
• Not checked yetcites [179] Kostelecký, V.A., and Mewes, M., “Signals for Lorentz violation in electrodynamics”, Phys. Rev. D, 66, 056005, (2002). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0205211. 3.2, 3.3, 4.1.4, 4.1.4, 5.3, 5.3, 18 source

In the paper

“Terrestrial experiments are invariably concerned with low energy processes. They are therefore best suited for looking at the mSME, which involves lower dimension operators. Astrophysics is more suited for directly constraining higher dimension operators as the Lorentz violating effects scale with energy.”

In the paper

“The group velocity of A, v = dE/dp, is equal to one at a momentum 81 and so we see that the threshold momenta can actually be far below the Planck energy, as it is controlled by the particle mass as well. For example, electrons would be unstable with n = 3 and of at 10TeV, well below the maximum electron energies in”

In the paper

“We see 50 TeV 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 ωth is above 50 TeV, constraining and limiting this type of Lorentz violation.”

What the sources say

• Not checked yetcites [268] Tanimori, T., Sakurazawa, K., Dazeley, S.A., Edwards, P.G., Hara, T., Hayami, Y., Kamei, S., Kifune, T., Konishi, T., Matsubara, Y., Matsuoka, T., Mizumoto, Y., Masaike, A., Mori, M., Muraishi, H., Muraki, Y., Naito, T., Oda, S., Ogio, S., Osaki, T., Patterson, J.R., Roberts, M.D., Rowell, G.P., Suzuki, A., Suzuki, R., Sako, T., Tamura, T., Thornton, G.J., Susukita, R., Yanagita, S., Yoshida, T., and Yoshikoshi, T., “Detection of Gamma Rays of up to 50 TeV from the Crab Nebula”, Astrophys. J. Lett., 492, L33–L36, (1998). Related online version (cited on 22 February 2005): http://arXiv.org/abs/ source
• Not checked yetcites [152] Jacobson, T.A., Liberati, S., and Mattingly, D., “TeV astrophysics constraints on Planck scale Lorentz violation”, Phys. Rev. D, 66, 081302, (2002). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0112207. 6.4, 22 source

In the paper

“Lorentz violating birefringence can erase linear polarization as a wave propagates, hence measurements of polarization constrain the relevant operators.”

In the paper

“If we assume the scale is particle mass, then muon based experiments would have a signal at least 102 larger than equivalent electron experiments due to the larger mass of the muon.”

In the paper

“DSR theories may also predict a time of flight signal, where the speed of light is effectively given by the group velocity of an n = 3 type dispersion relation.17 If there is such a frequency dependence, it is not expected that DSR also yields birefringence as in the EFT case. An n = 3 type dispersion for photons without birefringence would hence be a strong signal for DSR or something similar.”

In the paper

“Hence by itself, this IC Čerenkov constraint can always be satisfied in the Crab and gives no limits at all.”

In the paper

“A complicated lower/upper threshold structure has been applied to the trans-GZK cosmic ray events, with the lower threshold mimicking the GZK-cutoff at 5 × 1019 GeV and the upper entering below the highest events at 3 × 1020 GeV [154].”

What the sources say

• Not checked yetcites [154] Jacobson, T.A., Liberati, S., and Mattingly, D., “Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics”, Phys. Rev. D, 67, 124011, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0209264. 6.4, 6.5, 6.5.1, 6.5.1, 20, 21, 6.5.5, 22, 23, 22, 6.5.6, 6.5.6, 25, 6.6.2, 6.8.2 source

In the paper

“The numerical values of these constraints should not be taken too literally. While the order of magnitude is correct, simply moving the value of the threshold for the proton that interacts with a CMBR photon at some energy does not give accurate numbers.”

In the paper

“Helicity decay has no threshold in the traditional sense; the reaction happens at all energies. However, below a certain energy the phase space is highly suppressed, so we have an effective threshold that practically speaking is indistinguishable from a real threshold.”

In the paper

“If one further assumes that the charged particles are electrons, it can then be inferred that 50 TeV electrons must propagate. However, only one of the electron helicities may be propagating, so we can only constrain one of the helicities.”

In the paper

“The direct limits from photon emission, treating a 5 × 1019 eV proton as a single constituent are [154, 86, 119] for n = 225, for n = 3 [154], and for n = 4 [154].”

What the sources say

• Not checked yetcites [154] Jacobson, T.A., Liberati, S., and Mattingly, D., “Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics”, Phys. Rev. D, 67, 124011, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0209264. 6.4, 6.5, 6.5.1, 6.5.1, 20, 21, 6.5.5, 22, 23, 22, 6.5.6, 6.5.6, 25, 6.6.2, 6.8.2 source
• Not checked yetcites [86] Coleman, S.R., and Glashow, S.L., “Cosmic ray and neutrino tests of special relativity”, Phys. Lett. B, 405, 249–252, (1997). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9703240. 6.5, 6.5.6, 25, 6.8.1 source
• Not checked yetcites [119] Gagnon, O., and Moore, G.D., “Limits on Lorentz violation from the highest energy cosmic rays”, Phys. Rev. D, 70, 065002, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0404196. 6.5.6, 25, 27, 26 source

In the paper

“If we demand that the GZK cutoff is between 2 × 1019 eV and 7 × 1019 eV then for we have . If then there is a wedge shaped region in the parameter space that is allowed [154].”

What the sources say

• Not checked yetcites [154] Jacobson, T.A., Liberati, S., and Mattingly, D., “Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics”, Phys. Rev. D, 67, 124011, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0209264. 6.4, 6.5, 6.5.1, 6.5.1, 20, 21, 6.5.5, 22, 23, 22, 6.5.6, 6.5.6, 25, 6.6.2, 6.8.2 source

In the paper

“The corresponding bounds for n = 3, 4 type dispersion are not known, but one can easily estimate their size. The particle speed is approximately . For a proton at an energy of 1020 eV (10−8EPl) the constraint on the coefficient f(3) is then of . Note though, that in this case only one of the UHECR protons must satisfy this bound due to helicity dependence. Similarly, the n = 4 bound is of .”

In the paper

“At very high energies (5 × 1019 eV), the interaction length (a function of the power spectrum of interacting background photons coupled with the reaction cross section) becomes of order 50 Mpc. Since cosmic ray sources are probably at further distances than this, the spectrum of high energy protons should show a cutoff around 5 × 1019 eV [135, 281].”

What the sources say

• Not checked yetcites [135] Greisen, K., “End to the cosmic ray spectrum?”, Phys. Rev. Lett., 16, 748–750, (1966). 6.5.6 source
• Not checked yetcites [281] Zatsepin, G.T., and Kuzmin, V.A., “Upper limit of the spectrum of cosmic rays”, J. Exp. Theor. Phys. Lett., 4, 78–80, (1966). 6.5.6

In the paper

“Note that for the range of ξ allowed by birefringence, the relevant constraint is ηR < 0.012 or ηL < 0.012.”

What the sources say

• Not checked yetcites [154] Jacobson, T.A., Liberati, S., and Mattingly, D., “Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics”, Phys. Rev. D, 67, 124011, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0209264. 6.4, 6.5, 6.5.1, 6.5.1, 20, 21, 6.5.5, 22, 23, 22, 6.5.6, 6.5.6, 25, 6.6.2, 6.8.2 source

In the paper

“If this model is correct, as seems highly probable, then the observed radiation from the Crab at 100 MeV is due to synchrotron emission from high energy electrons and/or positrons.”

In the paper

“Cosmology also provides a way to test Lorentz violation. The most obvious connection is via inflation.”

In the paper

“Assuming the source particles are electrons, if then there is a maximum electron velocity for any energy particle and hence a maximum possible value for ωc.”

In the paper

“In short, since we see high energy cosmic rays at energies of 1020 eV, the results of [160, 12, 11, 24] imply that we could not have n = 3 non-systematic dispersion unless the coefficients are , while for n = 4 the coefficients would only have to be an order of magnitude or two below .”

What the sources say

• Not checked yetcites [160] Jankiewicz, M., Buniy, R.V., Kephart, T.W., and Weiler, T.J., “Space-time foam and cosmicray interactions”, Astropart. Phys., 21, 651–666, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0312221. 6.6.2 source
• Not checked yetcites [12] Aloisio, R., Blasi, P., Ghia, P.L., and Grillo, A.F., “Probing the structure of space-time with cosmic rays”, Phys. Rev. D, 62, 053010, (2000). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0001258. 2.6, 6.5.6, 6.6.2 source
• Not checked yetcites [11] Aloisio, R., Blasi, P., Galante, A., Ghia, P.L., and Grillo, A.F., “Space time fluctuations and ultra high energy cosmic ray interactions”, Astropart. Phys., 19, 127–133, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0205271. 6.6.2 source

In the paper

“Performing threshold analyses on non-systematic dispersion assumes that energy-momentum conservation can be applied with a single fluctuation (i.e. the reaction effectively happens at a point). It further assumes that the matrix element is roughly unchanged.”

In the paper

“Note, finally, that these arguments only concern the rate of decay — the conclusion that high energy particles would decay in this framework is unchanged.”

In the paper

“Observation of Airy rings implies that photons are in phase and hence limits the fluctuations in the dispersion relation [204, 245, 233].”

What the sources say

• Not checked yetcites [204] Lieu, R., and Hillman, L.W., “The phase coherence of light from extragalactic sources: Direct evidence against first order quantum gravity fluctuations in time and space”, Astrophys. J. Lett., 585, L77–L80, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0301184. 6.9, 6.9 source
• Not checked yetcites [245] Ragazzoni, R., Turatto, M., and Gaessler, W., “Lack of observational evidence for quantum structure of space-time at Plank scales”, Astrophys. J. Lett., 587, L1–L4, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0303043. 6.9 source
• Not checked yetcites [233] Ng, Y.J., van Dam, H., and Christiansen, W.A., “Probing Planck-scale physics with extra-galactic sources”, Astrophys. J. Lett., 591, L87–L90, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0302372. 6.9 source