Modern Tests of Lorentz Invariance

For photon decay with unmodified photon dispersion and electron/positron dispersion relation 88, there are lower and upper thresholds at 14 TeV and 82 TeV, respectively.

“Equation (90) has two positive real roots, at , corresponding to a lower and upper threshold at 14TeV and 82TeV, respectively.”

Most frameworks explicitly break Lorentz invariance, with a preferred set of observers or background field other than the metric.

“Most frameworks explicitly break Lorentz invariance, in that there is a preferred set of observers or background field other than the metric [90, 34].”

• cites [90] Colladay, D., and Kostelecký, V.A., “Lorentz-violating extension of the standard model”, Phys. Rev. D, 58, 116002, (1998). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/9809521. 1, 2, 4.1.1, 5 sourceNot checked yet

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• cites [34] Arkani-Hamed, N., Cheng, H.-C., Luty, M.A., and Mukohyama, S., “Ghost condensation and a consistent infrared modification of gravity”, J. High Energy Phys., 2004(05), 074, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0312099. 1, 4.4, 4.4 sourceNot checked yet

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Microcausality is violated if the group velocity of any field mode is superluminal.

“In this work, the authors find that microcausality is violated if the group velocity of any field mode is superluminal.”

• cites [176] Kostelecký, V.A., and Lehnert, R., “Stability, causality, and Lorentz and CPT violation”, Phys. Rev. D, 63, 065008, (2001). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-th/0012060. 1, 2.7.1, 2.7.2 sourceNot checked yet

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One cannot preserve the equivalence principle with Lorentz violation unless one also modifies Hamiltonian dynamics.

“As a consequence one cannot preserve the equivalence principle with Lorentz violation unless one also modifies Hamiltonian dynamics.”

These latter frameworks lead to only apparent low energy Lorentz violation.

“These latter frameworks lead to only “apparent” low energy Lorentz violation.”

In MSSM with exact SUSY, the only Lorentz violating operators that can appear have mass dimension five or above.

“If, for example, exact SUSY is imposed in the MSSM (minimal supersymmetric standard model), then the only Lorentz violating operators that can appear have mass dimension five or above [137].”

• cites [137] Groot Nibbelink, S., and Pospelov, M., “Lorentz violation in supersymmetric field theories”, (2004). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0404271. 2.3.2, 11Not checked yet

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Strictly enforced, the rule that only kinematic tests are applicable would preclude any use of an experiment that relies on particle interactions.

“Strictly enforced, this rule would preclude any use of an experiment that relies on particle interactions, as these interactions are determined by the dynamics of the theory.”

Energy/momentum conservation requires translation invariance of the underlying spacetime and the Lorentz violating physics.

“Energy/momentum conservation between initial and final particle states requires translation invariance of the underlying spacetime and the Lorentz violating physics.”

Hence a small amount of Lorentz violation should be present at all energies.

“Hence a small amount of Lorentz violation should be present at all energies.”

SUSY can be an exact symmetry even in the presence of Lorentz violation.

“SUSY, while related to Lorentz symmetry, can still be an exact symmetry even in the presence of Lorentz violation.”

Lorentz violation implies a violation of the equivalence principle.

“Lorentz violation implies a violation of the equivalence principle.”

Non-commutative field theory explicitly contains Lorentz violation.

“Other high energy models of spacetime structure, such as non-commutative field theory, do however explicitly contain Lorentz violation [98].”

• cites [98] Douglas, M.R., and Nekrasov, N.A., “Noncommutative field theory”, Rev. Mod. Phys., 73, 977–1029, (2001). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0106048. 1, 4.2 sourceNot checked yet

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DSR evades many constraints on effective field theory formulations of Lorentz violation.

“Due to this unusual (and useful) feature, DSR evades many of the constraints on effective field theory formulations of Lorentz violation.”

The large energy gap between the Planck scale and the highest known energy particles precludes any direct observation of Planck scale Lorentz violation.

“While perhaps theoretically interesting, the large energy gap between the Planck scale and the highest known energy particles, the trans-GZK cosmic rays of 1011 GeV (not to mention accelerator energies of ∼ 1 TeV), precludes any direct observation of Planck scale Lorentz violation.”

The fundamental status of Lorentz symmetry, broken or deformed, as well as the additional symmetries makes a dramatic difference as to which experiments and observations are sensitive.

“The fundamental status of Lorentz symmetry, broken or deformed, as well as the additional symmetries makes a dramatic difference as to which experiments and observations are sensitive.”

Until quantum gravity models become more mature, only kinematic tests of Lorentz invariance are truly applicable.

“Hence until these models become more mature, only kinematic tests of Lorentz invariance are truly applicable.”

Only a select few observations, such as interferometry, birefringence, Doppler shifts, or time of flight, are by construction insensitive to dynamics.

“Only a select few observations, such as interferometry, birefringence, Doppler shifts, or time of flight are by construction insensitive to dynamics.”

CPT violation implies Lorentz violation for local field theories.

“Conversely, however, CPT violation implies Lorentz violation for local field theories [134].”

• cites [134] Greenberg, O.W., “CPT violation implies violation of Lorentz invariance”, Phys. Rev. Lett., 89, 231602, (2002). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0201258. 2.3.1 sourceNot checked yet

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Low energy experiments would see Lorentz violation even if quantum gravity is Lorentz invariant at the Planck scale, in the scenario where tensor fields acquire a VEV.

“Experiments carried out at low energies would therefore see Lorentz violation, even though it is a good symmetry of the theory at the Planck scale.”

Making all Lorentz violating tensors dynamical removes prior geometry.

“The solution is to make all Lorentz violating tensors dynamical [173, 157], thereby removing prior geometry.”

• cites [173] Kostelecký, V.A., “Gravity, Lorentz violation, and the standard model”, Phys. Rev. D, 69, 105009, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0312310. 2.4, 4.4 sourceNot checked yet

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• cites [157] Jacobson, T.A., and Mattingly, D., “Gravity with a dynamical preferred frame”, Phys. Rev. D, 64, 024028, (2001). 2.4, 4.4, 4.4 sourceNot checked yet

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The equivalence principle is violated even when Lorentz violating terms are equal for all particle species.

“We now give a pedagogical example to show that the equivalence principle is violated even in this case.”

In DSR, all reactions forbidden by conservation in ordinary Lorentz invariant physics are also forbidden, despite dispersion relations allowing new reactions.

“This is the situation in DSR: All reactions that are forbidden by conservation in ordinary Lorentz invariant physics are also forbidden in DSR [146], even though particle dispersion relations in DSR would naively allow new reactions.”

• cites [146] Heyman, D., Hinteleitner, F., and Major, S., “On reaction thresholds in doubly special relativity”, Phys. Rev. D, 69, 105016, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0312089. 2.3.3, 6.6.1 sourceNot checked yet

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Lorentz invariance is required for the CPT theorem.

“Lorentz symmetry is intimately tied up with CPT symmetry in that the assumption of Lorentz invariance is required for the CPT theorem [162].”

• cites [162] Jost, R., “Eine Bemerkung zum CPT”, Helv. Phys. Acta, 30, 409, (1957). 2.3.1Not checked yet

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Lorentz violation allows for but does not require CPT violation.

“Lorentz violation therefore allows for (but does not require) CPT violation, even if the other properties of standard quantum field theory are assumed.”

Some frameworks deform the Poincaré algebra, leading to modified transformations between frames but no preferred frame.

“However others try to deform the Poincaré algebra, which would lead to modified transformations between frames but no preferred frame (for a review see [186]).”

• cites [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, 17Not checked yet

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Tests that use particle interactions can be applied to theories where only the kinematics is understood.

“However, the observational situation is often such that tests that use particle interactions can be applied to theories where only the kinematics is understood.”

Lorentz violation can induce a breakdown of microcausality.

“Lorentz violation can induce a breakdown of microcausality, as shown in [176].”

• cites [176] Kostelecký, V.A., and Lehnert, R., “Stability, causality, and Lorentz and CPT violation”, Phys. Rev. D, 63, 065008, (2001). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-th/0012060. 1, 2.7.1, 2.7.2 sourceNot checked yet

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If Lorentz invariance is violated at the Planck scale, there must be an interpolation to the low energy, nearly Lorentz invariant world we live in.

“If Lorentz invariance is violated at the Planck scale, there must be an interpolation to the low energy, (at least nearly) Lorentz invariant world we live in.”

DSR, if it can eventually be made mathematically consistent in its current incarnation, has one phenomenological advantage: it does not have a preferred frame.

“DSR, if it can eventually be made mathematically consistent in its current incarnation, has one phenomenological advantage — it does not have a preferred frame.”

With Lorentz violation, one must still have some ground state.

“With the introduction of Lorentz violation, one must still have some ground state.”

Rotationally invariant n = 1, 3 type dispersion cannot be imposed universally on photons.

“For example, rotationally invariant n = 1, 3 type dispersion cannot be imposed universally on photons [90, 230].”

• cites [90] Colladay, D., and Kostelecký, V.A., “Lorentz-violating extension of the standard model”, Phys. Rev. D, 58, 116002, (1998). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/9809521. 1, 2, 4.1.1, 5 sourceNot checked yet

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• cites [230] Myers, R.C., and Pospelov, M., “Experimental challenges for quantum gravity”, Phys. Rev. Lett., 90, 211601, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0301124. 2, 4.1.4, 5.4, 8 sourceNot checked yet

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The vacuum Čerenkov effect can be understood as an effect of inequivalent vacua.

“In other words, the vacuum Čerenkov effect (discussed in more detail in Section 6.5) can be understood as an effect of inequivalent vacua.”

Many experiments have observables that depend on a, b, d but not on the synchronization procedure.

“Many experiments, such as those that measure the isotropy of the one way speed of light [275] or propagation of light around closed loops, have observables that depend on a, b, d but not on the synchronization procedure.”

• cites [275] Will, C.M., “Clock synchronization and isotropy of the one-way speed of light”, Phys. Rev. D, 45, 403–411, (1992). 3.2 sourceNot checked yet

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In aether theory, the four-vector can twist so local superluminal propagation leads to energy-momentum flowing around closed paths.

“For example in an aether theory, where Lorentz violation is described by a timelike four-vector, the four-vector can twist in such a way that local superluminal propagation can lead to energy-momentum flowing around closed paths [206].”

• cites [206] Lim, E.A., “Can we see Lorentz-violating vector fields in the CMB?”, Phys. Rev. D, 71, 063504-1–16, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/astro-ph/0407437. 2.7.1, 33 sourceNot checked yet

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It is not fully settled yet whether DSR is mathematically consistent or physically meaningful.

“Furthermore, it is not fully settled yet if DSR is mathematically consistent or physically meaningful.”

The c2 framework is the flat space limit of the THεμ framework.

“The c2 framework [277] is the flat space limit of the THϵμ [205] framework.”

• cites [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, 7.3Not checked yet

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• cites [205] Lightman, A.P., and Lee, D.L., “Restricted proof that the Weak Equivalence Principle implies the Einstein Equivalence Principle”, Phys. Rev. D, 8, 364–376, (1973). 2.2, 3.3 sourceNot checked yet

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For models with higher order dispersion relations (n > 2), there is a stability problem for particles with momentum near the Planck energy if f(n) < 0.

“For the models in Section 3.1 with higher order dispersion relations ( with n > 2) there is a stability problem for particles with momentum near the Planck energy if f(n) < 0 as modes do not have positive energy at these high momenta.”

The RMS framework can be incorporated into the standard model extension.

“In fact, it was shown in [179] that the RMS framework can be incorporated into the standard model extension.”

• cites [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 sourceNot checked yet

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In a concordant frame, physics is perfectly local and causal even though microcausality does not hold.

“If we happened to be in such a concordant frame then physics would be perfectly local and causal even though microcausality does not hold.”

Universal dispersion relations cannot be imposed for all n from an EFT standpoint.

“As will be shown in detail in Section 4.1, universal dispersion relations cannot be imposed for all n from an EFT standpoint.”

It is not possible to compare the RMS parameters of two experiments that use physically different clocks and rods.

“Hence it is not possible to compare the RMS parameters of two experiments that use physically different clocks and rods (for example, an experiment that uses a cesium atomic clock versus an experiment that uses a hydrogen one).”

With Lorentz violation, even inertial observers do not necessarily agree on the vacuum.

“With Lorentz violation even inertial observers do not necessarily agree on the vacuum.”

The THεμ and c2 constructions can also be expressed in terms of the standard model extension.

“The THϵμ and c2 constructions can also be expressed in terms on the standard model extension [179].”

• cites [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 sourceNot checked yet

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If the energy for every mode is positive, then the vacuum state is stable.

“If the energy for every mode is positive, then the vacuum state |0S〉 is stable.”

Dynamical Lorentz violating tensors introduce further possible instabilities.

“Dynamical Lorentz violating tensors introduce further possible instabilities.”

The boost subgroup must be broken.

“Either way, the boost subgroup must be broken.”

The coordinate transformation of Equation (16) only has meaning if the coordinates are identified with measurements made by a particular set of clocks and rods.

“In particular, the coordinate transformation of Equation (16) only has meaning if we identify the coordinates with the measurements made by a particular set of clocks and rods.”

The only existing constraints on non-systematic dispersion come from threshold reactions and the phase coherence of light.

“The only existing constraints on non-systematic dispersion come from threshold reactions (see Section 6.6.2) and the phase coherence of light (see Section 6.9).”

In aether theories, the total energy is proportional to the usual ADM energy of general relativity.

“For aether theories, the total energy is proportional to the usual ADM energy of general relativity [104].”

• cites [104] Eling, C., Jacobson, T.A., and Mattingly, D., “Einstein-aether theory”, (2004). URL (cited on 08 June 2005): http://arXiv.org/abs/gr-qc/0410001. 2.7.2, 4.4Not checked yet

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It is possible to have a broken boost symmetry and unbroken rotation symmetry.

“However, it is possible to have a broken boost symmetry and unbroken rotation symmetry.”

In Lorentz violating theory, breakdown of microcausality does not lead to a breakdown of cluster decomposition, unlike Lorentz invariant theory.

“However, the breakdown of microcausality does not lead to a breakdown of cluster decomposition in a Lorentz violating theory, in contrast to a Lorentz invariant theory.”

Supersymmetry is currently the only known symmetry (other than Lorentz symmetry itself) that can protect Lorentz violating operators of dimension four or less.

“Supersymmetry is currently the only known symmetry (other than Lorentz symmetry itself) that can protect Lorentz violating operators of dimension four or less [137, 159, 65], much as SUSY protects some lower dimension operators in non-commutative field theory [216].”

• cites [137] Groot Nibbelink, S., and Pospelov, M., “Lorentz violation in supersymmetric field theories”, (2004). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0404271. 2.3.2, 11Not checked yet

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• cites [159] Jain, P., and Ralston, J.P., “Supersymmetry and the Lorentz fine tuning problem”, Phys. Lett. B, 621, 213–218, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0502106. 11 sourceNot checked yet

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• cites [65] Bolokhov, P.A., Groot Nibbelink, S., and Pospelov, M., “Lorentz violating supersymmetric quantum electrodynamics”, (2005). URL (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0505029. 2.3.2, 11, 11Not checked yet

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• cites [216] Matusis, A., Susskind, L., and Toumbas, N., “The IR/UV connection in the non-commutative gauge theories”, J. High Energy Phys., 2000(12), 002, (2000). Related online version (cited on 22 February 2005): http://arxiv.org/abs/hep-th/0002075. 4.2, 11 sourceNot checked yet

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The dimension five operators in the QED sector violate CPT.

“One should note that these operators violate CPT.”

For electrons and positrons, where g ≈ 2, the state n, s = −1/2 is almost degenerate with the state n − 1, s = +1/2.

“For electrons and positrons, where g ≈ 2, the state n, s = −1/2 is almost degenerate with the state n − 1, s = +1/2.”

A continuation of the 3He/129Xe experiment has recently been able to directly constrain boost violation at the level of 10−27 GeV.

“A continuation of this experiment has recently been able to directly constrain boost violation at the level of 10−27 GeV [71] (sidereal variations look at rotation invariance).”

• cites [71] Canè, F., Bear, D., Phillips, D.F., Rosen, M.S., Smallwood, C.L., Stoner, R.E., and Walsworth, R.L., “Bound on Lorentz- and CPT-Violating Boost Effects for the Neutron”, Phys. Rev. Lett., 93, 230801, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/physics/0309070. 13 sourceNot checked yet

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Only the differences between aμ for various particles are actually observable.

“However, one can always eliminate one of the aμ, i.e. only the differences between aμ for various particles are actually observable.”

Without some custodial symmetry protecting the theory from Lorentz violating dimension ≤ 4 operators, the lower dimension operators will be more important than irrelevant higher mass dimension operators.

“However, without some custodial symmetry protecting the theory from Lorentz violating dimension ≤ 4 operators, the lower dimension operators will be more important than irrelevant higher mass dimension operators (see Section 4.3 for details).”

No evidence for Lorentz violation has been found.

“However, no evidence for such violation has been found.”

Assuming the stated conditions, aether theories possess a set of coupled aether-metric modes which act as new gravitational degrees of freedom that can be searched for with gravitational wave interferometers or by determining energy loss rates from inspiral systems like the binary pulsar.

“Assuming these conditions, aether theories possess a set of coupled aether-metric modes which act as new gravitational degrees of freedom that can be searched for with gravitational wave interferometers or by determining energy loss rates from inspiral systems like the binary pulsar.”

• AssumptionThe authors state this as an assumption: In aether models, all the Lorentz violation is assumed to be provided by a vector field uα.

Without a custodial symmetry, loop effects with the dimension six operators will induce lower dimension operators.

“Other strong constraints can be derived by noting that without a custodial symmetry loop effects with the dimension six operators will induce lower dimension operators.”

The best overall limit is in the neutron sector of the mSME and comes from a 3He/129Xe maser system.

“The best overall limit is in the neutron sector of the mSME and comes from a 3He/129Xe maser system [44, 45].”

• cites [44] Bear, D., Stoner, R.E., Walsworth, R.L., Kostelecký, V.A., and Lane, C.D., “Limit on Lorentz and CPT violation of the neutron using a two-species noble-gas maser”, Phys. Rev. Lett., 85, 5038–5041, (2000). Related online version (cited on 22 February 2005): http://arXiv.org/abs/physics/0007049. 13 sourceNot checked yet

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• cites [45] Bear, D., Stoner, R.E., Walsworth, R.L., Kostelecký, V.A., and Lane, C.D., “Erratum: Limit on Lorentz and CPT Violation of the Neutron Using a Two-Species Noble-Gas Maser”, Phys. Rev. Lett., 89, 209902, (2002). 13 sourceNot checked yet

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Strong limits can be placed on the birefringent effect from astrophysical sources.

“Strong limits can be placed on this birefringent effect from astrophysical sources [179], as detailed in Section 6.3.”

• cites [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 sourceNot checked yet

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A combination of CPT invariance and SUSY would forbid Lorentz violating operators below dimension six.

“However, dimension five SUSY operators are CPT violating, so a combination of CPT invariance and SUSY would forbid Lorentz violating operators below dimension six.”

Since the dimension five operators violate CPT, they give rise to different dispersions for positrons than electrons.

“We note that since the dimension five operators violate CPT, they give rise to different dispersions for positrons than electrons.”

The formulation with prior geometry generically leads to energy-momentum non-conservation.

“The difficulty with this formulation is that it constitutes prior geometry and generically leads to energy-momentum non-conservation, similar to the bimetric model in Section 2.4.”

Colladay and Kostelecky derived the (minimal) Standard Model Extension (mSME) in flat space.

“In [90] Colladay and Kostelecky derived just such a theory in flat space — the so-called (minimal) Standard Model Extension (mSME).”

• cites [90] Colladay, D., and Kostelecký, V.A., “Lorentz-violating extension of the standard model”, Phys. Rev. D, 58, 116002, (1998). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/9809521. 1, 2, 4.1.1, 5 sourceNot checked yet

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In the dynamical Lorentz violating coefficient scenario, the matter stress tensor is automatically conserved if all the fields are on-shell.

“In this scenario, the matter stress tensor is automatically conserved if all the fields are on-shell.”

In order to match a non-commutative theory to low energy observations, the UV/IR mixing problem of non-commutative field theory must be tamed enough to create a well-defined low energy expansion.

“However, in order to match a non-commutative theory to low energy observations, we must have the appropriate low energy theory, which implies that the infamous UV/IR mixing problem of non-commutative field theory must be tamed enough to create a well-defined low energy expansion.”

The observational bound on the difference between ωa for electrons and positrons leads to a bound on bZ of order bZ ≤ 10−21 me.

“The observational bound on the difference between ωa for electrons and positrons is [97]. This leads to a bound on bZ of order bZ ≤ 10−21me.”

• cites [97] Dehmelt, H.G., Mittleman, R.K., van Dyck Jr, R.S., and Schwinberg, P., “Past electron positron g-2 experiments yielded sharpest bound on CPT violation”, Phys. Rev. Lett., 83, 4694–4696, (1999). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9906262. 5.1 sourceNot checked yet

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In multi-particle theories involving fermion interactions, one cannot remove aμ for all fermions.

“However, in multi-particle theories involving fermion interactions one cannot remove aμ for all fermions [89].”

• cites [89] Colladay, D., and Kostelecký, V.A., “CPT violation and the standard model”, Phys. Rev. D, 55, 6760–6774, (1997). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/9703464. 4.1.3 sourceNot checked yet

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The dimension five SUSY operators do not induce n = 3 type dispersion like the operators (40).

“These dimension five operators do not induce n = 3 type dispersion like the operators (40).”

In the mSME, the frequencies ωa and ωc both receive corrections.

“However, the frequencies ωa and ωc both receive corrections [60].”

• cites [60] Bluhm, R., Kostelecký, V.A., and Russell, N., “CPT and Lorentz tests in Penning traps”, Phys. Rev. D, 57, 3932–3943, (1998). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/9809543. 5.1 sourceNot checked yet

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If one imposes exact SUSY on a Lorentz violating theory, the first allowed operators are of dimension five.

“If one imposes exact SUSY on a Lorentz violating theory, the first allowed operators are of dimension five [137].”

• cites [137] Groot Nibbelink, S., and Pospelov, M., “Lorentz violation in supersymmetric field theories”, (2004). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0404271. 2.3.2, 11Not checked yet

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A stronger bound of ΛNC > 5 × 10^14 GeV on the non-commutativity scale can be derived from clock comparison experiments with Cs/Hg magnetometers.

“A stronger bound of ΛNC > 5 × 1014 GeV [218] on the non-commutativity scale can be derived from clock comparison experiments with Cs/Hg magnetometers [46] (see Section 5.2).”

• cites [218] Mocioiu, I., Pospelov, M., and Roiban, R., “Low-energy limits on the antisymmetric tensor field background on the brane and on the non-commutative scale”, Phys. Lett. B, 489, 390–396, (2000). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0005191. 9 sourceNot checked yet

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• cites [46] Berglund, C.J. et. al., “New Limits on Local Lorentz Invariance from Hg and Cs Magnetometers”, Phys. Rev. Lett., 75, 1879, (1995). 9, 13 sourceNot checked yet

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Other clock comparison experiments yield constraints on the proton sector in the mSME.

“Other clock comparison experiments have been performed that yield constraints on the proton sector [84, 243, 196, 46, 241] in the mSME.”

• cites [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 sourceNot checked yet

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• cites [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 sourceNot checked yet

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• cites [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 sourceNot checked yet

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The best electron limits come from two torsion balances, one in Washington and one in Taiwan.

“The best limits on (where i is the spatial direction, including that parallel to the earth’s rotation axis) for the electron come from two balances, one in Washington [170, 141] and one in Taiwan [148].”

• cites [170] Kostelecký, V.A., ed., CPT and Lorentz Symmetry, Proceedings of the Meeting held at Indiana University, Bloomington, USA, November 6–8, 1998, (World Scientific, Singapore, 1999). 5.4Not checked yet

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• cites [141] Heckel, B.R. (for the Eöt-Wash Group), “Torsion Balance Tests of Spin Coupled Forces”, project homepage, University of Washington. URL (cited on 22 February 2005): http://www.npl.washington.edu/eotwash/publications/cpt01.pdf. 5.4Not checked yet

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• cites [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 sourceNot checked yet

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A constraint on the dimension-five neutron operators of Equation (40) was derived using limits on the spatial variation of the Be+ hyperfine nuclear spin transition as a function of the angle between the spin axis and an external magnetic field.

“A constraint of the dimension five operators of Equation (40) for neutrons was recently derived in [52] using limits on the spatial variation of the hyperfine nuclear spin transition in Be+ as a function of the angle between the spin axis and an external magnetic field [64].”

• cites [52] Bertolami, O., and Rosa, J.G., “New bounds on cubic Lorentz-violating terms in the fermionic dispersion relation”, Phys. Rev. D, 71, 097901, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0412289. 13 sourceNot checked yet

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• cites [64] Bollinger, J., Heinzen, D.J., Itano, W.M., Gilbert, S.L., and Wineland, D.J., “Test of the linearity of quantum mechanics by rf spectroscopy of the 9Be+ ground state”, Phys. Rev. Lett., 63, 1031–1034, (1989). 13 sourceNot checked yet

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The Washington torsion balance magnets give an estimated net spin polarization equivalent to approximately 10^23 aligned electron spins.

“The magnets give an estimated net spin polarization equivalent to approximately 1023 aligned electron spins.”

In the mSME, CPT violation also implies Lorentz violation.

“In the framework of the mSME, CPT violation also implies Lorentz violation.”

Spin polarized torsion balances place limits on the electron sector of the mSME comparable to clock comparison limits on protons and neutrons.

“Spin polarized torsion balances are able to place comparable limits on the electron sector of the mSME [56].”

• cites [56] Bluhm, R., and Kostelecký, V.A., “Lorentz and CPT tests with spin-polarized solids”, Phys. Rev. Lett., 84, 1381–1384, (2000). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/9912542. 5.4 sourceNot checked yet

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Using a cavity with a medium gives additional electron sensitivity through the dependence of the resonant frequency on the index of refraction.

“Furthermore, by using a cavity with a medium, the dependence of fr on n gives additional electron sensitivity [226].”

• cites [226] Müller, H., “Testing Lorentz invariance by use of vacuum and matter filled cavity resonators”, (2004). URL (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0412385. 5.3, 1Not checked yet

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The relevant observable for this type of CPT violation in the kaon system is the K0 and anti-K0 mass difference ΔK.

“Hence the relevant observable for this type of CPT violation in the kaon system is the K0 and mass difference, .14”

For certain materials, the strain induced on the cavity by Lorentz violation is large, allowing sensitivity to the electron parameters cμν at a level equivalent to the photon parameters.

“In contrast to sapphire, for certain materials the strain induced on the cavity by Lorentz violation is large. This allows sensitivity to the electron parameters cμν at a level equivalent to the photon parameters.”

Similar proton bounds have been estimated from the Berglund et al. experiment using the Schmidt model for nuclear structure, in which an individual nucleon is assumed to carry the entire nuclear angular momentum.

“Similar bounds have been estimated [175] from the experiment of Berglund et al. [46] using the Schmidt model [251] for nuclear structure, where an individual nucleon is assumed to carry the entire nuclear angular momentum.”

• cites [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 sourceNot checked yet

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• cites [46] Berglund, C.J. et. al., “New Limits on Local Lorentz Invariance from Hg and Cs Magnetometers”, Phys. Rev. Lett., 75, 1879, (1995). 9, 13 sourceNot checked yet

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• cites [251] Schmidt, T., Z. Phys., 106, 358, (1937). 13 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: In the Schmidt model, an individual nucleon carries the entire nuclear angular momentum.

In general, all cavities are sensitive to the photon κ parameters.

“In general, all cavities are sensitive to the photon κ parameters.”

If uα is spacelike, the neutron limits are |η1| < 2 × 10−8 and |η2| < 10−8.

“If uα is spacelike one has |η1| < 2 × 10−8 and |η2| < 10−8.”

• cites [52] Bertolami, O., and Rosa, J.G., “New bounds on cubic Lorentz-violating terms in the fermionic dispersion relation”, Phys. Rev. D, 71, 097901, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0412289. 13 sourceNot checked yet

  Not checked in the free run.

If uα is lightlike, both neutron coefficients are bounded at the 10−8 level.

“If uα is lightlike both coefficients are bounded at the 10−8 level.”

• cites [52] Bertolami, O., and Rosa, J.G., “New bounds on cubic Lorentz-violating terms in the fermionic dispersion relation”, Phys. Rev. D, 71, 097901, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0412289. 13 sourceNot checked yet

  Not checked in the free run.

Constraints from cavity experiments are not on the κ parameters themselves, but on linear combinations of them.

“Constraints from cavity experiments are not on the κ parameters themselves, but rather on the linear combinations”

A field redefinition shows that the Lorentz-violating background tensor (kF)μναβ can be thought of as a dielectric medium with no charge or current density.

“This redefinition shows that the Lorentz violating background tensor (kF)μναβ can be thought of as a dielectric medium with no charge or current density.”

• cites [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 sourceNot checked yet

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CPT violation in the kaon system occurs only when the diagonal components of the Hamiltonian H are not equal.

“CPT violation only occurs when the diagonal components of H are not equal [198].”

• cites [198] Lee, T.D., and Wu, C.S., “Weak Interactions (Second section) Chapter 9: Decays of Neutral K Mesons”, Annu. Rev. Nucl. Sci., 16, 511–590, (1966). 5.5 sourceNot checked yet

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If uα is timelike and coincides with the rest frame of the CMBR, the neutron limits are roughly |η1| < 6 × 10−3 and |η2| < 3.

“The limits are roughly |η1| < 6 × 10−3 and |η2| < 3 if uα is timelike and coincides with the rest frame of the CMBR.”

• cites [52] Bertolami, O., and Rosa, J.G., “New bounds on cubic Lorentz-violating terms in the fermionic dispersion relation”, Phys. Rev. D, 71, 097901, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0412289. 13 sourceNot checked yet

  Not checked in the free run.

Assuming all lower dimension operators are absent, the constraint on the dimension-five operators is |ηR − ηL| < 4.

“Assuming that all lower dimension operators are absent, the constraint on the dimension five operators is |ηR − ηL| < 4 [230].”

• cites [230] Myers, R.C., and Pospelov, M., “Experimental challenges for quantum gravity”, Phys. Rev. Lett., 90, 211601, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0301124. 2, 4.1.4, 5.4, 8 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: All lower-dimension operators are absent when constraining the dimension-five operators.

The properties of this dielectric are orientation dependent.

“Note that since and depend on the components of (kF)μναβ, the properties of the dielectric are orientation dependent.”

The absence of any extra twist in the Washington experiment limits all components of the electron coefficient to be less than 10−28 GeV.

“The absence of any extra twist limits all components of for the electron to be less than 10−28 GeV.”

• cites [170] Kostelecký, V.A., ed., CPT and Lorentz Symmetry, Proceedings of the Meeting held at Indiana University, Bloomington, USA, November 6–8, 1998, (World Scientific, Singapore, 1999). 5.4Not checked yet

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• cites [141] Heckel, B.R. (for the Eöt-Wash Group), “Torsion Balance Tests of Spin Coupled Forces”, project homepage, University of Washington. URL (cited on 22 February 2005): http://www.npl.washington.edu/eotwash/publications/cpt01.pdf. 5.4Not checked yet

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Cavity experiments constrain the electromagnetic sector of the mSME because one of the clocks involves photons.

“However, since one of the clocks involves photons, cavity experiments constrain the electromagnetic sector of the mSME as well.”

Torsion balance experiments are sensitive enough to also constrain the dimension-five operators in Equation (40).

“To conclude this section, we note that the torsion balance experiments are actually sensitive enough to also constrain the dimension 5 operators in Equation (40).”

The best test of the RMS parameter αRMS to date comes from spectroscopy of lithium ions in a storage ring and gives |αRMS| < 2 × 10−7.

“The best test to date comes from spectroscopy of lithium ions in a storage ring [249]. In this experiment, 7Li+ ions are trapped in a storage ring at a velocity of 0.064 c. The transition frequencies of the boosted ions are then measured and compared to the transition frequencies at rest, providing a bound on the deviation from the special relativistic Doppler shift of |αRMS| < 2 × 10−7 in the RMS”

• cites [249] Saathoff, G., Karpuk, S., Eisenbarth, U., Huber, G., Krohn, S., Horta, R.M., Reinhardt, S., Schwalm, D., Wolf, A., and Gwinner, G., “Improved Test of Time Dilation in Special Relativity”, Phys. Rev. Lett., 91, 190403, (2003). 5.6 sourceNot checked yet

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Combining the Markarian 421 flare observations, assuming the flare was emitted from the same event at the source, yields the limit |f(3)| < 128.

“If we assume that the flare was emitted from the same event at the source, the time of arrival delay between 1TeV and 2TeV photons must be less than 280 s. Combining all these factors yields the limit |f(3)| < 128.”

• cites [53] Biller, S.D., Breslin, A.C., Buckley, J., Catanese, M., Carson, M., Carter-Lewis, D.A., Cawley, M.F., Fegan, D.J., Finley, J.P., Gaidos, J.A., Hillas, A.M., Krennrich, F., Lamb, R.C., Lessard, R., Masterson, C., McEnery, J.E., McKernan, B., Moriarty, P., Quinn, J., Rose, H.J., Samuelson, F., Sembroski, G., Skelton, P., and Weekes, T.C., “Limits to quantum gravity effects from observations of TeV flares in active galaxies”, Phys. Rev. Lett., 83, 2108–2111, (1999). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/9810044. 16 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The Markarian 421 flare was emitted from the same event at the source.

Analyzing the plane of polarization of distant galaxies has been used to constrain the kAF term in the mSME to the level of 10−42 GeV.

“This fact alone has been used to constrain the kAF term in the mSME to the level of 10−42 GeV by analyzing the plane of polarization of distant galaxies [74].”

• cites [74] Carroll, S.M., Field, G.B., and Jackiw, R., “Limits on a Lorentz and parity violating modification of electrodynamics”, Phys. Rev. D, 41, 1231, (1990). 18 sourceNot checked yet

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An analysis of sidereal variations involving only one of the μ−/μ+ at the current sensitivity could bound the coefficient at the level of 10−25 GeV.

“Furthermore, an analysis of sidereal variations involving only one of the μ−/μ+ at the current sensitivity in [72] could bound the coefficient at the level of 10−25 GeV [58].”

• cites [72] Carey, R.M. et al., “New Measurement of the Anomalous Magnetic Moment of the Positive Muon”, Phys. Rev. Lett., 82, 1632–1635, (1999). 5.7 sourceNot checked yet

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• cites [58] Bluhm, R., Kostelecký, V.A., and Lane, C.D., “CPT and Lorentz tests with muons”, Phys. Rev. Lett., 84, 1098–1101, (2000). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/9912451. 5.7 sourceNot checked yet

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A direct comparison between the g − 2 factors for μ− and μ+ limits the CPT-violating coefficient bZ < 10−22 GeV.

“The first is a direct comparison between the g − 2 factors for μ− and μ+, which limits the CPT violating coefficient bZ < 10−22 GeV.”

• cites [58] Bluhm, R., Kostelecký, V.A., and Lane, C.D., “CPT and Lorentz tests with muons”, Phys. Rev. Lett., 84, 1098–1101, (2000). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/9912451. 5.7 sourceNot checked yet

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• cites [39] Bailey, J., Borer, K., Combley, F., Drumm, H., Eck, C., Farley, F.J.M., Field, J.H., Flegel, W., Hattersley, P.M., Krienen, F., Lange, F., Lebée, G., McMillan, E., Petrucci, G., Picasso, E., Rúnolfsson, O., von Rüden, W., Williams, R.W., and Wojcicki, S., “Final report on the CERN muon storage ring including the anomalous magnetic moment and the electric dipole moment of the muon, and a direct test of relativistic time dilation”, Nucl. Phys. B, 150, 1–75, (1979). 5.7 sourceNot checked yet

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• cites [72] Carey, R.M. et al., “New Measurement of the Anomalous Magnetic Moment of the Positive Muon”, Phys. Rev. Lett., 82, 1632–1635, (1999). 5.7 sourceNot checked yet

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Constraints can be derived with current data that are much less than even for n = 4 Lorentz violation.

“However, the main point remains valid with more complicated forms of Lorentz violation: Constraints can be derived with current data that are much less than even for n = 4 Lorentz violation.”

Upper thresholds do not occur in Lorentz invariant physics.

“Upper thresholds do not occur in Lorentz invariant physics.”

In the presence of birefringence, a linearly polarized wave rotates its direction of polarization during propagation.

“Hence in the presence of birefringence a linearly polarized wave rotates its direction of polarization during propagation.”

For mass dimension > 4 operators, unless f(n) is much greater than 1, the turnover momentum pTO is much higher than any relevant observational energy.

“Hence unless f(n) ≫ 1, pTO is much higher than any relevant observational energy, and we can make the assumption of monotonicity without loss of generality.”

Asymmetric thresholds cannot occur in Lorentz invariant reactions.

“This cannot occur in Lorentz invariant reactions.”

There is currently no natural and complete way that astrophysics might observe Lorentz violation while terrestrial experiments confirm Lorentz invariance.

“In summary there is currently no “natural” and complete way that astrophysics might observe Lorentz violation, but terrestrial experiments confirm Lorentz invariance.”

If a photon obeys a modified dispersion relation of the form in Equation (79) while electrons and positrons have their usual Lorentz-invariant dispersion, then a photon can decay to an electron-positron pair above a threshold energy.

“If, however, we break Lorentz invariance and assume a photon obeys a dispersion relation of the form 79 while electrons/positrons have their usual Lorentz invariant dispersion, then it is possible to satisfy energy conservation equation if (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”

Hard photon emission can occur even when the velocity condition is never satisfied, if the photon energy-momentum vector is spacelike with n > 2 dispersion.

“Hard photon emission can occur even when the velocity condition is never satisfied, if the photon energy-momentum vector is spacelike with n > 2 dispersion.”

Spin transitions in muonium have been used to place a bound on the muon coefficient (Equation 73).

“First, spin transitions in muonium (μ+e−) have been used to place a bound on for the muon (see Equation (52) for the definition of ) [149].”

• cites [149] Hughes, V.W., Grosse Perdekamp, M., Kawall, D., Liu, W., Jungmann, K., and zu Pulitz, G., “Test of CPT and Lorentz Invariance from Muonium Spectroscopy”, Phys. Rev. Lett., 87, 111804-1–4, (2001). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ex/0106103. 5.7Not checked yet

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Constraints on the antisymmetric part of (kφφ)μν and on (kφB)μν and (kφW)μν come from birefringence constraints on photon propagation and yield a constraint on all three coefficients of order 10−16.

“Constraints on the antisymmetric part of (kϕϕ)μν, which we denote , and (kϕB)μν, (kϕW)μν come from the birefringence constraints on photon propagation (see Section 6.3). Here the loop corrections to the photon propagator induce a non-zero (kF)αβμν, which can be directly constrained. This yields a constraint on all three coefficients of order 10−16.”

The lithium storage-ring results have been reinterpreted in the mSME to give approximate bounds for the electron/proton sector.

“Recently, the results of [249] have been reinterpreted in the context of the mSME. For the electron/proton sector the approximate bounds are [197] 69 70 71 72”

• cites [249] Saathoff, G., Karpuk, S., Eisenbarth, U., Huber, G., Krohn, S., Horta, R.M., Reinhardt, S., Schwalm, D., Wolf, A., and Gwinner, G., “Improved Test of Time Dilation in Special Relativity”, Phys. Rev. Lett., 91, 190403, (2003). 5.6 sourceNot checked yet

  Not checked in the free run.

• cites [197] Lane, C.D., “Probing Lorentz violation with Doppler-shift experiments”, (2005). URL (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0505130. 5.6Not checked yet

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Asymmetric thresholds occur because the minimum energy configuration is not necessarily the symmetric configuration.

“Asymmetric thresholds occur because the minimum energy configuration is not necessarily the symmetric configuration.”

The existence of Lorentz-violating higher-dimensional operators would generically generate lower-dimension operators.

“As mentioned in Section 4.3, the existence of Lorentz violating higher dimensional operators would generically generate lower dimension ones.”

Such a threshold structure would produce a deficit in the observed photon spectrum in this energy band.

“Such a threshold structure would produce a deficit in the observed photon spectrum in this energy band.”

Loop corrections from Lorentz violation in the Higgs sector can be used to derive indirect constraints on unmeasured sectors.

“Such an approach has been recently taken in [28], where loop corrections to mSME coefficients from Lorentz violation in the Higgs sector are considered.”

• cites [28] Anderson, D.L., Sher, M., and Turan, I., “Lorentz and CPT violation in the Higgs sector”, Phys. Rev. D, 70, 016001, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0403116. 5.8 sourceNot checked yet

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Under the assumption that the dynamics is not drastically different from Lorentz-invariant effective field theory, particle reaction constraints can be applied to kinematic theories because decay times are extremely short above threshold.

“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.”

• AssumptionThe authors state this as an assumption: The dynamics is not drastically different from that of Lorentz-invariant effective field theory.

Upper thresholds are possible with Lorentz violation.

“It is easy to see that they are possible with Lorentz violation, however.”

At a threshold for a particle with momentum p1, the configuration is the minimum energy configuration of all other particles that conserves momentum.

“Theorem 1: The configuration at a threshold for a particle with momentum p1 is the minimum energy configuration of all other particles that conserves momentum.”

• cites [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 sourceNot checked yet

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• cites [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 sourceNot checked yet

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The He/Xe maser system gives a better, although less clean, bound on the transverse components of order 10−31 GeV.

“The He/Xe maser system gives a better, although less clean, bound on the transverse components of order 10−31 GeV.”

The parton approach yields two-sided bounds on dimension six operators of order a certain value for all constituent particles, depending on assumptions about equal parton dispersion corrections.

“The parton approach yields two-sided bounds on dimension six operators of order for all constituent particles, depending on the assumptions made about equal parton dispersion corrections.”

• cites [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 sourceNot checked yet

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AGASA found trans-GZK events inconsistent with the GZK cutoff at 2.5σ.

“AGASA found trans-GZK events inconsistent with the GZK cutoff at 2.5σ [96]”

• cites [96] De Marco, D., Blasi, P., and Olinto, A.V., “On the statistical significance of the GZK feature in the spectrum of ultra high energy cosmic rays”, Astropart. Phys., 20, 53–65, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0301497. 6.5.6 sourceNot checked yet

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For higher dimension operators that yield energy dependent dispersion, assigning electrons and protons equal coefficients f(n) does yield comparable constraints.

“However, for higher dimension operators that yield energy dependent dispersion, simply assigning electrons and protons equal coefficients f(n) does yield comparable constraints.”

Lorentz violation can become important when the modification to the dispersion relation is of the same order of magnitude as the proton mass.

“From this heuristic threshold analysis, however, it is clear that Lorentz violation can become important when the modification to the dispersion relation is of the same order of magnitude as the proton mass.”

For n = 2 vacuum Čerenkov, the constraint in mSME parameters is c − |d| − kF/2 ≤ 10−16.

“For n = 2 the constraint is [154], where is the coefficient for one of the electron helicities. In terms of the mSME parameters this condition can be written as c − |d| − kF/2 ≤ 10−16.”

• cites [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 sourceNot checked yet

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If one treated electrons, positrons, and protons as fundamental constituents with only n = 2 dispersion and assigned each a common speed 1 − ε, one would obtain no constraints.

“Note that if one treated electrons, positrons, and protons as the fundamental constituents with only n = 2 dispersion and assigned each a common speed 1 − ϵ, one would obtain no constraints.”

The high energy photon spectrum above 10 TeV from astrophysical sources such as Markarian 501 and 421 has been observed to show signs of absorption due to scattering off the IR background.

“The high energy photon spectrum (above 10TeV) from astrophysical sources such as Markarian 501 and 421 has been observed to show signs of absorption due to scattering off the IR background.”

For n = 2 with CPT preserved, the photon decay constraint translates to kF/2 + c ≤ 4 × 10−16.

“For n = 2 with CPT preserved we have [154]. If we set d = 0 in Equation (39) so that there is no helicity dependence, this translates to the constraint kF/2 + c ≤ 4 × 10−16.”

• cites [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 sourceNot checked yet

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Hi-Res has found evidence for the GZK cutoff, although at a lower confidence level.

“while Hi-Res has found evidence for the GZK cutoff (although at a lower confidence level; for a discussion see [263]).”

• cites [263] Stecker, F.W., “Cosmic physics: The high energy frontier”, J. Phys. G, G29, R47–R88, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0309027. 22, 6.5.6 sourceNot checked yet

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Current data from blazars suggest that both polarizations must be absorbed to some degree.

“although current data from blazars suggest that both polarizations must be absorbed to some degree [263].”

• cites [263] Stecker, F.W., “Cosmic physics: The high energy frontier”, J. Phys. G, G29, R47–R88, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0309027. 22, 6.5.6 sourceNot checked yet

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If d ≠ 0, both helicities of electrons/positrons must satisfy kF/2 + (c ± d) ≤ 4 · 10−16.

“If d ≠ 0 then both helicities of electrons/positrons must satisfy this bound since the photon has a decay channel into every possible combination of electron/positron helicity. The corresponding limit is kF/2 + (c ± d) ≤ 4 · 10−16.”

• cites [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 sourceNot checked yet

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The GZK cutoff could vanish or be shifted due to source effects as well.

“Hence the GZK cutoff could vanish or be shifted due to source effects as well.”

• AssumptionThe authors state this as an assumption: The source spectrum follows the same power law distribution as at lower energies.

The level of constraints from threshold reactions at 50 TeV is around 10−2.

“The level of constraints from threshold reactions at 50 TeV is around 10−2 [152, 167].”

• cites [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 sourceNot checked yet

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• cites [167] Konopka, T.J., and Major, S.A., “Observational limits on quantum geometry effects”, New J. Phys., 4, 57, (2002). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0201184. 20, 22, 6.5.6 sourceNot checked yet

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The corresponding constraint from the observation of high energy cosmic rays is cp − 1 ≤ 2 × 10−15.

“The corresponding constraint from the observation of high energy cosmic rays is cp − 1 ≤ 2 × 10−15.”

• cites [224] Moore, G.D., and Nelson, A.E., “Lower bound on the propagation speed of gravity from gravitational Cherenkov radiation”, J. High Energy Phys., 2001(09), 023, (2001). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0106220. 6.5.5, 6.5.7 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The cosmic rays used for the gravitational Čerenkov bound are protons.
• AssumptionThe authors state this as an assumption: All cosmic ray protons travel at the same velocity.

Ultra-high energy cosmic rays, if they are protons, will interact strongly with the cosmic microwave background and produce pions, losing energy.

“Ultra-high energy cosmic rays (UHECR), if they are protons, will interact strongly with the cosmic microwave background and produce pions, p + γ ⟶ p + π0, losing energy in the process.”

The region of parameter space where such a lower/upper threshold scenario might happen is extremely small.

“The region of parameter space where such a scenario might happen is extremely small, however.”

High energy particles travelling faster than the speed of graviton modes will emit graviton Čerenkov radiation.

“High energy particles travelling faster than the speed of graviton modes will emit graviton Čerenkov radiation.”

• cites [224] Moore, G.D., and Nelson, A.E., “Lower bound on the propagation speed of gravity from gravitational Cherenkov radiation”, J. High Energy Phys., 2001(09), 023, (2001). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ph/0106220. 6.5.5, 6.5.7 sourceNot checked yet

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When the fermion dispersion is helicity dependent, helicity decay occurs: one helicity is unstable and decays into the other as a particle propagates, emitting radiation.

“When the fermion dispersion is helicity dependent, the phenomenon of helicity decay occurs. One of the helicities is unstable and will decay into the other as a particle propagates, emitting some sort of radiation depending on the exact process considered.”

The 50 TeV photons observed from the Crab nebula are believed to be produced via inverse Compton scattering of charged particles off the ambient soft photon background.

“The 50 TeV photons observed from the Crab nebula are believed to be produced via inverse Compton (IC) scattering of charged particles off the ambient soft photon background.”

Lorentz violation shifts the location of the GZK cutoff.

“Since Lorentz violation shifts the location of the GZK cutoff, significant information about Lorentz violation (even for n = 4 type dispersion) can be gleaned from the UHECR spectrum.”

The birefringence constraint on ξ is |ξ| ≤ 10−4.

“We can simplify the problem dramatically by noting that the birefringence constraint on ξ in Equation (40) is |ξ| ≤ 10−4 [127].”

• cites [127] Gleiser, R.J., and Kozameh, C.N., “Astrophysical limits on quantum gravity motivated birefringence”, Phys. Rev. D, 64, 083007, (2001). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0102093. 18, 19, 22 sourceNot checked yet

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Lorentz violating terms can be chosen such that photons become unstable to decay into electron-positron pairs.

“Lorentz violating terms can be chosen such that photons become unstable to decay into electron-positron pairs [152].”

• cites [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 sourceNot checked yet

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None of the analyses of photon annihilation take into account the EFT requirement for n = 3 that opposite photon polarizations have opposite Lorentz violating terms.

“However, none of the analyses take into account the EFT requirement for n = 3 that opposite photon polarization have opposite Lorentz violating terms.”

Bounds on the coefficients of CPT violating dimension five operators are of the order 10−15.

“Bounds on the coefficients of CPT violating dimension five operators are of the order 10−15.”

• cites [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 sourceNot checked yet

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The lifetime of a high energy particle in QED above threshold is short enough that constraints can be established simply by looking at threshold conditions.

“The lifetime of a high energy particle in QED above threshold is therefore short enough that we can establish constraints simply by looking at threshold conditions.”

A vacuum Čerenkov effect must be forbidden up to the highest observed UHECR energy.

“A complimentary constraint to the GZK analysis can be derived by recognizing that 1019 − 1020 eV protons reach us — a vacuum Čerenkov effect must be forbidden up to the highest observed UHECR energy [88, 154, 119].”

• cites [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 sourceNot checked yet

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• cites [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 sourceNot checked yet

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• cites [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 sourceNot checked yet

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In the aether theory there are three new modes, corresponding to the three new degrees of freedom introduced by the constrained aether vector.

“In the aether theory (see Section 4.4) there are three new modes, corresponding to the three new degrees of freedom introduced by the constrained aether vector.”

Lorentz violation affects the amount of absorption in photon annihilation.

“While this process occurs in Lorentz invariant physics, the amount of absorption is affected by Lorentz violation.”

If no gravity-aether mode can be superluminal, then these bounds imply that every coefficient is generically bounded by |ci| < 10−15.

“If, as the authors of [105] argue, no gravity-aether mode can be superluminal, then these bounds imply that every coefficient is generically bounded by |ci| < 10−15.”

• cites [105] Elliott, J.W., Moore, G.D., and Stoica, H., “Constraining the New Aether: Gravitational Cherenkov Radiation”, J. High Energy Phys., 2005(08), 066, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0505211. 12, 6.5.7, 6.5.7, 31 sourceNot checked yet

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The possible importance of positrons in the Crab implies that the synchrotron constraint is always satisfied when considered by itself.

“The possible importance of positrons in the Crab implies that the synchrotron constraint is always satisfied when considered by itself.”

It seems unlikely that Lorentz violation could be the sole source of neutrino oscillations.

“Current data support neutrino oscillations that vary as a function of L/E [35, 125], so it seems unlikely that Lorentz violation could be the sole source of neutrino oscillations.”

• cites [35] Ashie, Y. et al. (Super-Kamiokande Collaboration), “Evidence for an Oscillatory Signature in Atmospheric Neutrino Oscillation”, Phys. Rev. Lett., 93, 101801, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ex/0404034. 6.8.1 sourceNot checked yet

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• cites [125] Giacomelli, G., and Margiotta, A., “MACRO results on atmospheric neutrino oscillations”, Eur. Phys. J. C, 33, s826–s828, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-ex/0406037. 6.8.1 sourceNot checked yet

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In the presence of dynamical Lorentz violation, one generically expects new gravitational wave polarizations.

“In the presence of dynamical Lorentz violation, where the entire action is diffeomorphism invariant, one generically expects new gravitational wave polarizations.”

Lorentz violation can raise or lower the Bahcall-Waxman bound on the expected flux of ultra-high energy neutrinos significantly.

“It has been shown [18] that Lorentz violation can in fact raise (or lower) this bound significantly.”

• cites [18] Amelino-Camelia, G., Arzano, M., Ng, Y.J., Piran, T., and van Dam, H., “Implications of spacetime quantization for the Bahcall-Waxman neutrino bound”, J. Cosmol. Astropart. Phys., 0402, 009, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0307027. 6.8.1, 30 sourceNot checked yet

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Currently, there is no experimental evidence that Lorentz symmetry is not an exact symmetry in nature.

“Currently, we have no experimental evidence that Lorentz symmetry is not an exact symmetry in nature.”

Neutrino oscillations are sensitive enough to directly probe non-renormalizable Lorentz violating terms.

“Neutrino oscillations are sensitive enough to directly probe non-renormalizable Lorentz violating terms.”

The above approach, modified conservation plus unchanged matrix element/rate when the reaction is allowed, is not correct when a low energy particle is involved.

“The above approach, modified conservation plus unchanged matrix element/rate when the reaction is allowed, is not correct when a low energy particle is involved.”

Lorentz violation can cause neutrino oscillation even if the mass is zero.

“The first is simply that neutrino oscillation still occurs even if the mass is zero.”

Barring cancellations, the observational limit on α2 translates to a very strong bound of order 10−7 on the coefficients ci in the aether action.

“Barring cancellations this translates to a very strong bound of order 10−7 on the coefficients ci in the aether action.”

• cites [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, 7.3Not checked yet

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DSR cannot be ruled out or confirmed by any threshold type analysis we currently have.

“Hence DSR cannot be ruled out or confirmed by any threshold type analysis we currently have.”

Carroll and Lim constrain the size of aether-model coefficients c1, c2, and c3 by comparing the effect on big bang nucleosynthesis with observed nucleosynthesis limits.

“By comparing this effect with observed nucleosynthesis limits, Carroll and Lim are able to constrain the size of c1, c2, and c3.”

• cites [76] Carroll, S.M., and Lim, E.A., “Lorentz-violating vector fields slow the universe down”, Phys. Rev. D, 70, 123525, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0407149. 34 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The aether-model coefficients satisfy positive definite Hamiltonian, non-tachyonic, and subluminal propagation conditions.

In the case of the Crab nebula, Equation (103) for the maximum frequency of synchrotron radiation in the presence of Lorentz violation is correct.

“More detailed calculations [222, 111, 221] show that in the case of the Crab nebula Equation (103) is correct”

• cites [222] Montemayor, R., and Urrutia, L.F., “Synchrotron radiation in Myers-Pospelov effective electrodynamics”, Phys. Lett. B, 606, 86–94, (2005). Related online version (cited on 22 February 2005): http://arXiv.org/abs/hep-th/0410143. 6.7 sourceNot checked yet

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• cites [111] Ellis, J.R., Mavromatos, N.E., and Sakharov, A.S., “Synchrotron radiation from the Crab Nebula discriminates between models of space-time foam”, Astropart. Phys., 20, 669–682, (2004). Related online version (cited on 08 June 2005): http://arXiv.org/abs/astro-ph/0308403. 6.7 sourceNot checked yet

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• cites [221] Montemayor, R., and Urrutia, L.F., “Synchrotron radiation in Lorentz-violating electrodynamics: The Myers-Pospelov model”, (2005). URL (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0505135. 6.7Not checked yet

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Neutrinos can provide excellent probes of Lorentz violation because their mass is much smaller than any other known particle.

“Neutrinos can provide excellent probes of Lorentz violation, as their mass is much smaller than any other known particle.”

Few constraints currently exist on dynamical Lorentz violation from gravitational wave observations.

“Unfortunately, few constraints currently exist on dynamical Lorentz violation from gravitational wave observations as the spectrum is only part of the story.”

For n = 3 non-systematic dispersion with normally distributed coefficients, all cosmic ray protons would undergo photo-pion production at energies above 10^15 eV.

“Looking solely at the kinematical threshold condition, they find that all cosmic ray protons would undergo photo-pion production at energies above 1015 eV.”

• cites [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 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The net effect of spacetime foam can be derived by considering energy conservation and non-systematic dispersions at a point.

The actual shift of threshold energies due to DSR is negligible at the level of sensitivity of current astrophysical observations.

“However, the actual shift of threshold energies due to DSR is negligible at the level of sensitivity we have with astrophysical observations [21].”

• cites [21] Amelino-Camelia, G., Kowalski-Glikman, J., Mandanici, G., and Procaccini, A., “Phenomenology of doubly special relativity”, (2003). URL (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0312124. 3, 17, 6.6.1Not checked yet

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The synchrotron constraint can be combined with Čerenkov constraints to create a two-sided bound that is a true constraint.

“However, the synchrotron constraint can be combined with Čerenkov constraints to create a two sided bound [155].”

• cites [155] Jacobson, T.A., Liberati, S., and Mattingly, D., “Lorentz violation at high energy: Concepts, phenomena and astrophysical constraints”, (2005). URL (cited on 08 June 2005): http://arXiv.org/abs/astro-ph/0505267. 1, 6.5.4, 6.5.5, 29Not checked yet

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Synchrotron emission from the Crab nebula (and other astrophysical objects) is very sensitive to n = 3 modified dispersion relations.

“Jacobson et al. [153] noticed that the synchrotron emission from the Crab nebula (and other astrophysical objects) is very sensitive to n = 3 modified dispersion relations.”

• cites [153] Jacobson, T.A., Liberati, S., and Mattingly, D., “Lorentz violation and Crab synchrotron emission: A new constraint far beyond the Planck scale”, Nature, 424, 1019–1021, (2003). Related online version (cited on 22 February 2005): http://arXiv.org/abs/astro-ph/0212190. 6.7, 6.7, 28 sourceNot checked yet

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Current bounds on Lorentz violation are strong enough that there is no easy way to put Lorentz violating operators of dimension ≤ 6 coming solely from Planck scale physics into our field theories.

“We currently have bounds on Lorentz violation strong enough that there is no easy way to put Lorentz violating operators of dimension ≤ 6 coming solely from Planck scale physics into our field theories.”

If a reaction does not occur in ordinary Lorentz invariant physics, it does not occur in DSR.

“By using the pseudo-momentum π and and energy ϵ one can show that if a reaction does not occur in ordinary Lorentz invariant physics, it does not occur in DSR [146].”

• cites [146] Heyman, D., Hinteleitner, F., and Major, S., “On reaction thresholds in doubly special relativity”, Phys. Rev. D, 69, 105016, (2004). Related online version (cited on 22 February 2005): http://arXiv.org/abs/gr-qc/0312089. 2.3.3, 6.6.1 sourceNot checked yet

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At this point there is no method by which the ERS parameters can be used to constrain Lorentz violating theories.

“Therefore, at this point there is no method by which the ERS parameters can be used to constrain Lorentz violating theories.”

DSR evades all of the photon decay and vacuum Čerenkov constraints that give strong limits on EFT Lorentz violation.

“The first difference between DSR and EFT is that DSR evades all of the photon decay and vacuum Čerenkov constraints that give strong limits on EFT Lorentz violation.”

The same type of argument extended to n = 4 dispersion yields a cutoff in the spectrum at 10^18 eV.

“This same type of argument can be rapidly extended to n = 4 dispersion, yielding a cutoff in the spectrum at 1018 eV.”

Demanding that high energy cosmic rays not emit the extra aether modes and assuming no significant Lorentz violation for cosmic rays yields bounds on the coefficients in Equation (48).

“Demanding that high energy cosmic rays not emit these extra modes and assuming no significant Lorentz violation for cosmic rays yields the bounds 102 on the coefficents in Equation (48).”

• cites [105] Elliott, J.W., Moore, G.D., and Stoica, H., “Constraining the New Aether: Gravitational Cherenkov Radiation”, J. High Energy Phys., 2005(08), 066, (2005). Related online version (cited on 08 June 2005): http://arXiv.org/abs/hep-ph/0505211. 12, 6.5.7, 6.5.7, 31 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: There is no significant Lorentz violation for cosmic rays.