Modern searches for Lorentz violation

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Measurements on light from gamma-ray bursts show that the speed of light does not vary with energy

Modern searches for Lorentz violation are scientific studies that look for deviations from Lorentz invariance or symmetry, a set of fundamental frameworks that underpin modern science and fundamental physics in particular. These studies try to determine whether violations or exceptions might exist for well-known physical laws such as special relativity and CPT symmetry, as predicted by some variations of quantum gravity, string theory, and some alternatives to general relativity.

Lorentz violations concern the fundamental predictions of special relativity, such as the principle of relativity, the constancy of the speed of light in all inertial frames of reference, and time dilation, as well as the predictions of the standard model of particle physics. To assess and predict possible violations, test theories of special relativity and effective field theories (EFT) such as the Standard-Model Extension (SME) have been invented. These models introduce Lorentz and CPT violations through spontaneous symmetry breaking caused by hypothetical background fields, resulting in some sort of preferred frame effects. This could lead, for instance, to modifications of the dispersion relation, causing differences between the maximal attainable speed of matter and the speed of light.

Both terrestrial and astronomical experiments have been carried out, and new experimental techniques have been introduced. No Lorentz violations could be measured thus far, and exceptions in which positive results were reported have been refuted or lack further confirmations. For discussions of many experiments, see Mattingly (2005).[1] For a detailed list of results of recent experimental searches, see Kostelecký and Russell (2008–2013).[2] For a recent overview and history of Lorentz violating models, see Liberati (2013).[3] See also the main article Tests of special relativity.

Assessing Lorentz invariance violations[edit]

Early models assessing the possibility of slight deviations from Lorentz invariance have been published between the 1960s and the 1990s.[3] In addition, a series of test theories of special relativity and effective field theories (EFT) for the evaluation and assessment of many experiments have been developed, including:

However, the Standard-Model Extension (SME) in which Lorentz violating effects are introduced by spontaneous symmetry breaking, is used for most modern analyses of experimental results. It was introduced by Kostelecký and coworkers in 1997 and the following years, containing all possible Lorentz and CPT violating coefficients not violating gauge symmetry.[6][7] It includes not only special relativity, but the standard model and general relativity as well. Models whose parameters can be related to SME and thus can be seen as special cases of it, include the older RMS and c2 models,[8] the Coleman-Glashow model confining the SME coefficients to dimension 4 operators and rotation invariance,[9] and the Gambini-Pullin model[10] or the Meyers-Pospelov model[11] corresponding to dimension 5 or higher operators of SME.[12]

Speed of light[edit]

Terrestrial[edit]

Many terrestrial experiments have been conducted, mostly with optical resonators or in particle accelerators, by which deviations from the isotropy of the speed of light are tested. Anisotropy parameters are given, for instance, by the Robertson-Mansouri-Sexl test theory (RMS). This allows to distinguish between the relevant orientation and velocity dependent parameters. In modern variants of the Michelson–Morley experiment, the dependence of light speed on the orientation of the apparatus and the relation of longitudinal and transverse lengths of bodies in motion is analyzed. Also modern variants of the Kennedy–Thorndike experiment, by which the dependence of light speed on the velocity of the apparatus and the relation of time dilation and length contraction is analyzed, have been conducted. The current precision, by which an anisotropy of the speed of light can be excluded, is at the 10−17 level. This is related to the relative velocity between the solar system and the rest frame of the cosmic microwave background radiation of ∼368 km/s (see also Resonator Michelson–Morley experiments).

In addition, the Standard-Model Extension (SME) can be used to obtain a larger number of isotropy coefficients in the photon sector. It uses the even- and odd-parity coefficients (3×3 matrices) \tilde{\kappa}_{e-}, \tilde{\kappa}_{o+} and \tilde{\kappa}_{tr}.[8] They can be interpreted as follows: \tilde{\kappa}_{e-} represent anisotropic shifts in the two-way (forward and backwards) speed of light, \tilde{\kappa}_{o+} represent anisotropic differences in the one-way speed of counterpropagating beams along an axis,[13][14] and \tilde{\kappa}_{tr} represent isotropic (orientation-independent) shifts in the one-way phase velocity of light.[15] It was shown that such variations in the speed of light can be removed by suitable coordinate transformations and field redefinitions, though the corresponding Lorentz violations cannot be removed, because such redefinitions only transfer those violations from the photon sector to the matter sector of SME.[8] While ordinary symmetric optical resonators are suitable for testing even-parity effects and provide only tiny constraints on odd-parity effects, also asymmetric resonators have been built for the detection of odd-parity effects.[15] For additional coefficients in the photon sector leading to birefringence of light in vacuum, which cannot be redefined as the other photon effects, see #Vacuum birefringence.

Another type of test of the \tilde{\kappa}_{o+} related one-way light speed isotropy in combination with the electron sector of the SME was conducted by Bocquet et al. (2010).[16] They searched for fluctuations in the 3-momentum of photons during Earth's rotation, by measuring the Compton scattering of ultrarelativistic electrons on monochromatic laser photons in the frame of the cosmic microwave background radiation, as originally suggested by Vahe Gurzadyan and Amur Margarian [17] (for details on that 'Compton Edge' method and analysis see,[18] discussion e.g.[19]).

Author Year RMS SME
Orientation Velocity \tilde{\kappa}_{e-} \tilde{\kappa}_{o+} \tilde{\kappa}_{tr}
Michimura et al.[20] 2013 \scriptstyle 0.7\pm1\times10^{-14} \scriptstyle -0.4\pm0.9\times10^{-10}
Baynes et al.[21] 2012 \scriptstyle 3\pm11\times10^{-10}
Baynes et al.[22] 2011 \scriptstyle 0.7\pm1.4\times10^{-12} \scriptstyle 3.4\pm6.2\times10^{-9}
Hohensee et al.[13] 2010 \scriptstyle 0.8(0.6)\times10^{-16} \scriptstyle -1.5(1.2)\times10^{-12} \scriptstyle -1.5(0.74)\times10^{-8}
Bocquet et al.[16] 2010 \scriptstyle\leq1.6\times10^{-14} [23]
Herrmann et al.[24] 2009 \scriptstyle(4\pm8)\times10^{-12} \scriptstyle(-0.31\pm0.73)\times10^{-17} \scriptstyle(-0.14\pm0.78)\times10^{-13}
Eisele et al.[25] 2009 \scriptstyle(-1.6\pm6\pm1.2)\times10^{-12} \scriptstyle(0.0\pm1.0\pm0.3)\times10^{-17} \scriptstyle(1.5\pm1.5\pm0.2)\times10^{-13}
Tobar et al.[26] 2009 \scriptstyle -4,8(3,7)\times10^{-8}
Tobar et al.[27] 2009 \scriptstyle -0.3\pm3\times10^{-7}
Müller et al.[28] 2007 \scriptstyle(7.7(4.0))\times10^{-16} \scriptstyle(1.7(2.0))\times10^{-12}
Carone et al.[29] 2006 \scriptstyle \lesssim3\times10^{-8} [30]
Stanwix et al.[31] 2006 \scriptstyle9.4(8.1)\times10^{-11} \scriptstyle(-6.9(2.2))\times10^{-16} \scriptstyle(-0.9(2.6))\times10^{-12}
Herrmann et al.[32] 2005 \scriptstyle(-2.1\pm1.9)\times10^{-10} \scriptstyle(-3.1(2.5))\times10^{-16} \scriptstyle(-2.5(5.1))\times10^{-12}
Stanwix et al.[33] 2005 \scriptstyle-0.9(2.0)\times10^{-10} \scriptstyle(-0.63(0.43))\times10^{-15} \scriptstyle(0.20(0.21))\times10^{-11}
Antonini et al.[34] 2005 \scriptstyle(+0.5\pm3\pm0.7)\times10^{-10} \scriptstyle(-2\pm0{,}2)\times10^{-14}
Wolf et al.[35] 2004 \scriptstyle(-5.7\pm2.3)\times10^{-15} \scriptstyle(-1.8\pm1.5)\times10^{-11}
Wolf et al.[36] 2004 \scriptstyle(+1.2\pm2.2)\times10^{-9} \scriptstyle (3.7\pm3.0)\times10^{-7}
Müller et al.[37] 2003 \scriptstyle(+2.2\pm1.5)\times10^{-9} \scriptstyle(1.7\pm2.6)\times10^{-15} \scriptstyle(14\pm14)\times10^{-11}
Lipa et al.[38] 2003 \scriptstyle(1.4\pm1.4\times10^{-13} \scriptstyle\leq10^{-9}
Wolf et al.[39] 2003 \scriptstyle(+1.5\pm4.2)\times10^{-9}
Braxmaier et al.[40] 2002 \scriptstyle (1.9\pm2.1)\times10^{-5}
Hils and Hall[41] 1990 \scriptstyle 6.6\times10^{-5}
Brillet and Hall[42] 1979 \scriptstyle\lesssim5\times10^{-9} \scriptstyle\lesssim10^{-15}

Solar system[edit]

Besides terrestrial tests also astrometric tests using Lunar Laser Ranging (LLR), i.e. sending laser signals from Earth to Moon and back, have been conducted. They are ordinarily used to test general relativity and are evaluated using the Parameterized post-Newtonian formalism.[43] However, since these measurements are based on the assumption that the speed of light is constant, they can also be used as tests of special relativity by analyzing potential distance and orbit oscillations. For instance, Zoltán Lajos Bay and White (1981) demonstrated the empirical foundations of the Lorentz group and thus special relativity by analyzing the planetary radar and LLR data.[44]

In addition to the terrestrial Kennedy–Thorndike experiments mentioned above, Müller & Soffel (1995)[45] and Müller et al. (1999)[46] tested the RMS velocity dependence parameter by searching for anomalous distance oscillations using LLR. Since time dilation is already confirmed to high precision, a positive result would prove that light speed depends on the observer's velocity and length contraction is direction dependent (like in the other Kennedy–Thorndike experiments). However, no anomalous distance oscillations have been observed, with a RMS velocity dependence limit of \scriptstyle(-5\pm12)\times10^{-5},[46] comparable to that of Hils and Hall (1990, see table above on the right).

Vacuum dispersion[edit]

Another effect often discussed in connection with Quantum gravity (QG) is the possibility of Dispersion of light in vacuum (i.e. the dependence of light speed on photon energy), due to Lorentz violating Dispersion relations. This effect should be strong at energy levels comparable to, or beyond the Planck energy \scriptstyle E_{\mathrm{Pl}}\sim1.22\times10^{19} GeV, while being extraordinarily weak at energies accessible in the laboratory or observed in astrophysical objects. In an attempt to observe a weak dependence of speed on energy, light from distant astrophysical sources such as gamma ray bursts and distant galaxies has been examined in many experiments. Especially the Fermi-LAT group was able show that no energy dependence and thus no observable Lorentz violation occurs in the photon sector even beyond the Planck energy,[47] which excludes a large class of Lorentz-violating quantum gravity models.

Name Year QG Bounds in GeV
95% C.L. 99% C.L.
Vasileiou et al.[48] 2013 \scriptstyle >7.6\times E_{\mathrm{Pl}}
Fermi-LAT-GBM[47] 2009 \scriptstyle >3.42\times E_{\mathrm{Pl}} \scriptstyle >1.19\times E_{\mathrm{Pl}}
H.E.S.S.[49] 2008 \scriptstyle \geq7.2\times10^{17}
MAGIC[50] 2007 \scriptstyle \geq0.21\times10^{18}
Ellis et al.[51][52] 2007 \scriptstyle \geq1.4\times10^{16}
Lamon et al.[53] 2007 \scriptstyle \geq3.2\times10^{11}
Martinez et al.[54] 2006 \scriptstyle \geq0.66\times10^{17}
Boggs et al.[55] 2004 \scriptstyle \geq1.8\times10^{17}
Ellis et al.[56] 2003 \scriptstyle \geq6.9\times10^{15}
Ellis et al.[57] 2000 \scriptstyle \geq10^{15}
Kaaret[58] 1999 \scriptstyle >1.8\times10^{15}
Schaefer[59] 1999 \scriptstyle \geq2.7\times10^{16}
Biller[60] 1999 \scriptstyle >4\times10^{16}

Vacuum birefringence[edit]

Lorentz violating dispersion relations due to the presence of an anisotropic space might also lead to vacuum birefringence and parity violations. For instance, the polarization plane of photons might rotate due to velocity differences between left- and right-handed photons. In particular, gamma ray bursts, galactic radiation, and the cosmic microwave background radiation are examined. The SME coefficients \scriptstyle k_{(V)00}^{(3)} and \scriptstyle k_{(V)00}^{(5)} for Lorentz violation are given, 3 and 5 denote the mass dimensions employed. The latter corresponds to \xi in the EFT of Meyers and Pospelov[11] by {\scriptstyle k_{(V)00}^{(5)}=\frac{3\sqrt{4\pi}\xi}{5m_{\mathrm{P}}}}, m_P being the Planck mass.[61]

Name Year SME bounds EFT bound \xi
\scriptstyle k_{(V)00}^{(3)} in GeV \scriptstyle k_{(V)00}^{(5)} in GeV−1
Götz et al.[62] 2013 \scriptstyle \leq5.9\times10^{-35} \scriptstyle \leq3.4\times10^{-16}
Toma et al.[63] 2012 \scriptstyle \leq1.4\times10^{-34} \scriptstyle \leq8\times10^{-16}
Laurent et al.[64] 2011 \scriptstyle \leq1.9\times10^{-33} \scriptstyle \leq1.1\times10^{-14}
Stecker[61] 2011 \scriptstyle \leq4.2\times10^{-34} \scriptstyle \leq2.4\times10^{-15}
Kostelecký et al.[12] 2009 \scriptstyle \leq1\times10^{-32} \scriptstyle \leq9\times10^{-14}
QUaD[65] 2008 \scriptstyle \leq2\times10^{-43}
Kostelecký et al.[66] 2008 \scriptstyle =(2.3\pm5.4)\times10^{-43}
Maccione et al.[67] 2008 \scriptstyle \leq1.5\times10^{-28} \scriptstyle \leq9\times10^{-10}
Komatsu et al.[68] 2008 \scriptstyle =(1.2\pm2.2)\times10^{-43} [12]
Kahniashvili et al.[69] 2008 \scriptstyle \leq2.5\times10^{-43} [12]
Xia et al.[70] 2008 \scriptstyle =(2.6\pm1.9)\times10^{-43} [12]
Cabella et al.[71] 2007 \scriptstyle =(2.5\pm3.0)\times10^{-43} [12]
Fan et al.[72] 2007 \scriptstyle \leq3.4\times10^{-26} \scriptstyle \leq2\times10^{-7} [61]
Feng et al.[73] 2006 \scriptstyle =(6.0\pm4.0)\times10^{-43} [12]
Gleiser et al.[74] 2001 \scriptstyle \leq8.7\times10^{-23} \scriptstyle \leq4\times10^{-4} [61]
Carroll et al.[75] 1990 \scriptstyle \leq2\times10^{-42}

Maximal attainable speed[edit]

Threshold constraints[edit]

Lorentz violations could lead to differences between the speed of light and the limiting or maximal attainable speed (MAS) of any particle, whereas in special relativity the speeds should be the same. One possibility is to investigate otherwise forbidden effects at threshold energy in connection with particles having a charge structure (protons, electrons, neutrinos). This is because the dispersion relation is assumed to be modified in Lorentz violating EFT models such as SME. Depending on which of these particles travels faster or slower than the speed of light, effects such as the following can occur:[76][77]

  • Photon decay at superluminal speed. These (hypothetical) high-energy photons would quickly decay into other particles, which means that high energy light cannot propagate over long distances. So the mere existence of high energy light from astronomic sources constrains possible deviations from the limiting velocity.
  • Vacuum Cherenkov radiation at superluminal speed of any particle (protons, electrons, neutrinos) having a charge structure. In this case, emission of Bremsstrahlung can occur, until the particle falls below threshold and subluminal speed is reached again. This is similar to the known Cherenkov radiation in media, in which particles are traveling faster than the phase velocity of light in that medium. Deviations from the limiting velocity can be constrained by observing high energy particles of distant astronomic sources that reach Earth.
  • The rate of synchrotron radiation could be modified, if the limiting velocity between charged particles and photons is different.
  • The Greisen–Zatsepin–Kuzmin limit could be evaded by Lorentz violating effects. However, recent measurements indicate that this limit really exists.

Since astronomic measurements also contain additional assumptions – like the unknown conditions at the emission or along the path traversed by the particles, or the nature of the particles –, terrestrial measurements provide results of greater clarity, even though the bounds are lower (the following bounds describe maximal deviations between the speed of light and the limiting velocity of matter):

Name Year Bounds Particle Astr./Terr.
Photon decay Cherenkov Synchrotron GZK
Stecker[78] 2014 \scriptstyle \leq5\times10^{-21} Electron Astr.
Stecker & Scully[79] 2009 \scriptstyle \leq4.5\times10^{-23} UHECR Astr.
Altschul[80] 2009 \scriptstyle \leq5\times10^{-15} Electron Terr.
Hohensee et al.[77] 2009 \scriptstyle \leq-5.8\times10^{-12} \scriptstyle \leq1.2\times10^{-11} Electron Terr.
Bi et al.[81] 2008 \scriptstyle \leq3\times10^{-23} UHECR Astr.
Klinkhamer & Schreck[82] 2008 \scriptstyle \leq-9\times10^{-16} \scriptstyle \leq6\times10^{-20} UHECR Astr.
Klinkhamer & Risse[83] 2007 \scriptstyle \leq2\times10^{-19} UHECR Astr.
Kaufhold et al.[84] 2007 \scriptstyle \leq10^{-17} UHECR Astr.
Altschul[85] 2005 \scriptstyle \leq6\times10^{-20} Electron Astr.
Gagnon et al.[86] 2004 \scriptstyle \leq-2\times10^{-21} \scriptstyle \leq5\times10^{-24} UHECR Astr.
Jacobson et al.[87] 2003 \scriptstyle \leq-2\times10^{-16} \scriptstyle \leq5\times10^{-20} Electron Astr.
Coleman & Glashow[9] 1997 \scriptstyle \leq-1.5\times10^{-15} \scriptstyle \leq5\times10^{-23} UHECR Astr.

Clock comparison and spin coupling[edit]

By this kind of spectroscopy experiments – sometimes called Hughes–Drever experiments as well – violations of Lorentz invariance in the interactions of protons and neutrons are tested by studying the energy levels of those nucleons in order to find anisotropies in their frequencies ("clocks"). Using spin-polarized torsion balances, also anisotropies with respect to electrons can be examined. Methods used mostly focus on vector spin interactions and tensor interactions,[88] and are often described in CPT odd/even SME terms (in particular parameters of bμ and cμν).[89] Such experiments are currently the most sensitive terrestrial ones, because the precision by which Lorentz violations can be excluded lies at the 10−33 GeV level.

These tests can be used to constrain deviations between the maximal attainable speed of matter and the speed of light,[5] in particular with respect to the parameters of cμν that are also used in the evaluations of the threshold effects mentioned above.[80]

Author Year SME bounds Parameters
Proton Neutron Electron
Allmendinger et al.[90] 2013 \scriptstyle<6.7\times10^{-34} bμ
Hohensee et al.[91] 2013 \scriptstyle(-9.0\pm11)\times10^{-17} cμν
Peck et al.[92] 2012 \scriptstyle<4\times10^{-30} \scriptstyle<3.7\times10^{-31} bμ
Smiciklas et al.[88] 2011 \scriptstyle(4.8\pm4.4)\times10^{-32} cμν
Gemmel et al.[93] 2010 \scriptstyle<3.7\times10^{-32} bμ
Brown et al.[94] 2010 \scriptstyle<6\times10^{-32} \scriptstyle<3.7\times10^{-33} bμ
Altarev et al.[95] 2009 \scriptstyle<2\times10^{-29} bμ
Heckel et al.[96] 2008 \scriptstyle(4.0\pm3.3)\times10^{-31} bμ
Wolf et al.[97] 2006 \scriptstyle(-1.8(2.8))\times10^{-25} cμν
Canè et al.[98] 2004 \scriptstyle(8.0\pm9.5)\times10^{-32} bμ
Heckel et al.[99] 2006 \scriptstyle<5\times10^{-30} bμ
Humphrey et al.[100] 2003 \scriptstyle<2\times10^{-27} bμ
Hou et al.[101] 2003 \scriptstyle(1.8\pm5.3)\times10^{-30} bμ
Phillips et al.[102] 2001 \scriptstyle<2\times10^{-27} bμ
Bear et al.[103] 2000 \scriptstyle(4.0\pm3.3)\times10^{-31} bμ

Time dilation[edit]

The classic time dilation experiments such as the Ives–Stilwell experiment, the Moessbauer rotor experiments, and the Time dilation of moving particles, have been enhanced by modernized equipment. For example, the Doppler shift of lithium ions traveling at high speeds is evaluated by using saturated spectroscopy in heavy ion storage rings. For more information, see Modern Ives–Stilwell experiments.

The current precision with which time dilation is measured (using the RMS test theory), is at the ~10−8 level. It was shown, that Ives-Stilwell type experiments are also sensitive to the \tilde{\kappa}_{tr} isotropic light speed coefficient of the SME, as introduced above.[15] Chou et al. (2010) even managed to measure a frequency shift of ~10−16 due to time dilation, namely at everyday speeds such as 36 km/h.[104]

Author Year Velocity Maximum deviation
from time dilation
Fourth order
RMS bounds
Novotny et al.[105] 2009 0,34c \scriptstyle \leq1.3\times10^{-6} \scriptstyle \leq1.2\times10^{-5}
Reinhardt et al.[106] 2007 0,064c \scriptstyle \leq8.4\times10^{-8}
Saathoff et al.[107] 2003 0,064c \scriptstyle \leq2.2\times10^{-7}
Grieser et al.[108] 1994 0,064c \scriptstyle \leq1\times10^{-6} \scriptstyle \leq2.7\times10^{-4}

CPT and antimatter tests[edit]

Another fundamental symmetry of nature is CPT symmetry. It was shown that CPT violations lead to Lorentz violations in quantum field theory (even though there are nonlocal exceptions).[109][110] CPT symmetry requires, for instance, the equality of mass, and equality of decay rates between matter and antimatter. For classic tests of decay rates, see Accelerator tests of time dilation and CPT symmetry.

Modern tests by which CPT symmetry has been confirmed are mainly conducted in the neutral meson sector. In large particle accelerators, direct measurements of mass differences between top- and antitop-quarks have been conducted as well.

Neutral B mesons
Author Year
Belle[111] 2012
Kostelecký et al.[112] 2010
BaBar[113] 2008
Belle[114] 2003
Neutral D mesons
FOCUS[115] 2003
Neutral kaons
Author Year
KTeV[116] 2011
KLOE[117] 2006
CPLEAR[118] 2003
KTeV[119] 2003
NA31[120] 1990
Top- and antitop quarks
Author Year
CDF[121] 2012
CMS[122] 2012
D0[123] 2011
CDF[124] 2011
D0[125] 2009

Using SME, also additional consequences of CPT violation in the neutral meson sector can be formulated.[112] Other SME related CPT tests have been performed as well:

  • Using Penning traps in which individual charged particles and their counterparts are trapped, Gabrielse et al. (1999) examined cyclotron frequencies in proton-antiproton measurements, and couldn't find any deviation down to 9·10−11.[126]
  • Hans Dehmelt et al. tested the anomaly frequency, which plays a fundamental role in the measurement of the electron's gyromagnetic ratio. They searched for sidereal variations, and differences between electrons and positrons as well. Eventually they found no deviations, thereby establishing bounds of 10−24 GeV.[127]
  • Hughes et al. (2001) examined muons for sidereal signals in the spectrum of muons, and found no Lorentz violation down to 10−23 GeV.[128]
  • The "Muon g-2" collaboration of the Brookhaven National Laboratory searched for deviations in the anomaly frequency of muons and anti-muons, and for sidereal variations under consideration of Earth's orientation. Also here, no Lorentz violations could be found, with a precision of 10−24 GeV.[129]

Other particles and interactions[edit]

Third generation particles have been examined for potential Lorentz violations using SME. For instance, Altschul (2007) placed upper limits on Lorentz violation of the tau of 10−8, by searching for anomalous absorption of high energy astrophysical radiation.[130] In the BaBar experiment (2007) it was searched for sidereal variations during Earth's rotation using B mesons (thus bottom quarks) and their antiparticles. No Lorentz and CPT violating signal was found with an upper limit of \scriptstyle\leq(-3.0\pm2.4)\times10^{-15}.[113] Also top quark pairs have been examined in the D0 experiment (2012). They showed that the cross section production of these pairs doesn't depend on sidereal time during Earth's rotation.[131]

Lorentz violation bounds on Bhabha scattering have been given by Charneski et al. (2012).[132] They showed that differential cross sections for the vector and axial couplings in QED become direction dependent in the presence of Lorentz violation. They found no indication of such an effect, placing upper limits on Lorentz violations of \scriptstyle\leq10^{14}(\text{eV})^{-1}.

Gravitation[edit]

The influence of Lorentz violation on gravitational fields and thus general relativity was analyzed as well. The standard framework for such investigations is the Parameterized post-Newtonian formalism (PPN), in which Lorentz violating preferred frame effects are described by the parameters \alpha_1, \alpha_2, \alpha_3 (see the PPN article on observational bounds on these parameters). Lorentz violations are also discussed in relation to Alternatives to general relativity such as Loop quantum gravity, Emergent gravity, Einstein aether theory or Hořava–Lifshitz gravity.

Also SME is suitable to analyze Lorentz violations in the gravitational sector. Bailey and Kostelecky (2006) constrained Lorentz violations down to \scriptstyle 10^{-9} by analyzing the perihelion shifts of Mercury and Earth, and down to \scriptstyle 10^{-13} in relation to solar spin precession.[133] Battat et al. (2007) examined Lunar Laser Ranging data and found no oscillatory perturbations in the lunar orbit. Their strongest SME bound excluding Lorentz violation was \scriptstyle(6.9\pm4.5)\times10^{-11}.[134] Iorio (2012) obtained bounds at the \scriptstyle 10^{-9} level by examining Keplerian orbital elements of a test particle acted upon by Lorentz-violating gravitomagnetic accelerations.[135] Xie (2012) analyzed the advance of periastron of binary pulsars, setting limits on Lorentz violation at the \scriptstyle 10^{-10} level.[136]

Neutrino tests[edit]

Neutrino oscillations[edit]

Although neutrino oscillations have been experimentally confirmed, the theoretical foundations are still controversial, as it can be seen in the discussion related to sterile neutrinos. This makes predictions of possible Lorentz violations very complicated. It is generally assumed that neutrino oscillations require a certain finite mass. However, oscillations could also occur as a consequence of Lorentz violations, so there are speculations as to how much those violations contribute to the mass of the neutrinos.[137]

Additionally, a series of investigations have been published in which a sidereal dependence of the occurrence of neutrino oscillations was tested, which could arise when there were a preferred background field. This, possible CPT violations, and other coefficients of Lorentz violations in the framework of SME, have been tested. Here, some of the achieved GeV bounds for the validity of Lorentz invariance are stated:

Name Year SME bounds
in GeV
Double Chooz[138] 2012 \scriptstyle \leq10^{-20}
MINOS[139] 2012 \scriptstyle \leq10^{-23}
MiniBooNE[140] 2012 \scriptstyle \leq10^{-20}
IceCube[141] 2010 \scriptstyle \leq10^{-23}
MINOS[142] 2010 \scriptstyle \leq10^{-23}
MINOS[143] 2008 \scriptstyle \leq10^{-20}
LSND[144] 2005 \scriptstyle \leq10^{-19}

Neutrino speed[edit]

Since the discovery of neutrino oscillations, it is assumed that their speed is slightly below the speed of light. Direct velocity measurements indicated an upper limit for relative speed differences between light and neutrinos of \scriptstyle |v-c|/c<10^{-9}, see measurements of neutrino speed.

Also indirect constraints on neutrino velocity, on the basis of effective field theories such as SME, can be achieved by searching for threshold effects such as Vacuum Cherenkov radiation. For example, neutrinos should exhibit Bremsstrahlung in the form of electron-positron pair production.[145] Another possibility in the same framework is the investigation of the decay of pions into muons and neutrinos. Superluminal neutrinos would considerably delay those decay processes. The absence of those effects indicate tight limits for velocity differences between light and neutrinos.[146]

Velocity differences between neutrino flavors can be constrained as well. A comparison between muon- and electron-neutrinos by Coleman & Glashow (1998) gave a negative result, with bounds \scriptstyle <6\times10^{-22}.[9]

Name Year Energy SME bounds for (v-c)/c
Vacuum Cherenkov Pion decay
Stecker et al.[78] 2014 1 PeV \scriptstyle <5.6\times10^{-19}
Borriello et al.[147] 2013 1 PeV \scriptstyle <10^{-18}
Cowsik et al.[148] 2012 100 TeV \scriptstyle <10^{-13}
Huo et al.[149] 2012 400 TeV \scriptstyle <7.8\times10^{-12}
ICARUS[150] 2011 17 GeV \scriptstyle <2.5\times10^{-8}
Cowsik et al.[151] 2011 400 TeV \scriptstyle <10^{-12}
Bi et al.[152] 2011 400 TeV \scriptstyle <10^{-12}
Cohen/Glashow[153] 2011 100 TeV \scriptstyle <1.7\times10^{-11}

Reports of alleged Lorentz violations[edit]

Open reports[edit]

LSND, MiniBooNE

In 2001, the LSND experiment observed a 3.8σ excess of antineutrino interactions in neutrino oscillations, which contradicts the standard model.[154] First results of the more recent MiniBooNE experiment appeared to exclude this data above an energy scale of 450 MeV, but they had checked neutrino interactions, not antineutrino ones.[155] In 2008, however, they reported an excess of electron-like neutrino events between 200–475 MeV.[156] And in 2010, when carried out with antineutrinos (as in LSND), the result was in agreement with the LSND result, that is, an excess at the energy scale from 450–1250 MeV was observed.[157][158] Whether those anomalies can be explained by sterile neutrinos, or whether they indicate Lorentz violations, is still discussed and subject to further theoretical and experimental researches.[159]

Solved reports[edit]

In 2011 the OPERA Collaboration published (in a non-peer reviewed arXiv preprint) the results of neutrino measurements, according to which neutrinos are slightly traveling faster than light.[160] The neutrinos apparently arrived early by ~60 ns. The standard deviation was 6σ, clearly beyond the 5σ limit necessary for a significant result. However, in 2012 it was found that this result was due to measurement errors. The end result was consistent with the speed of light,[161] see Faster-than-light neutrino anomaly.

In 2010, MINOS reported differences between the disappearance (and thus the masses) of neutrinos and antineutrinos at the 2.3 sigma level. This would violate CPT symmetry and Lorentz symmetry.[162][163][164] However, in 2011 MINOS updated their antineutrino results, reporting that the difference is not as great as initially expected, after evaluating further data.[165] In 2012, they published a paper in which they reported that the difference is now removed.[166]

In 2007, the MAGIC Collaboration published a paper, in which they claimed a possible energy dependence of the speed of photons from the galaxy Markarian 501. They admitted, that also a possible energy-dependent emission effect could have cause this result as well.[50][167] However, the MAGIC result was superseded by the substantially more precise measurements of the Fermi-LAT group, which couldn't find any effect even beyond the Planck energy.[47] For details, see section Dispersion.

In 1997, Nodland & Ralston claimed to have found a rotation of the polarization plane of light coming from distant radio galaxies. This would indicate an anisotropy of space.[168][169][170] This attracted some interest in the media. However, some criticisms immediately appeared, which disputed the interpretation of the data, and who alluded to errors in the publication.[171][172][173][174][175][176][177] More recent researches also haven't found any evidence for this effect, see section Birefringence.

In popular culture[edit]

In the Futurama episode "Law and Oracle" (2011), Erwin Schrödinger is pulled over by cops for violating Lorentz invariance, by going 15 miles per hour over the speed of light.

See also[edit]

References[edit]

  1. ^ Mattingly, David (2005). "Modern Tests of Lorentz Invariance". Living Rev. Relativity 8 (5). 
  2. ^ Kostelecky, V.A.; Russell, N. (2011). "Data tables for Lorentz and CPT violation". Reviews of Modern Physics 83 (1): 11–31. arXiv:0801.0287. Bibcode:2011RvMP...83...11K. doi:10.1103/RevModPhys.83.11. 
  3. ^ a b Liberati, S., V.A. (2013). "Tests of Lorentz invariance: a 2013 update". Classical and Quantum Gravity 30 (13): 133001. arXiv:1304.5795. Bibcode:2013CQGra..30m3001L. doi:10.1088/0264-9381/30/13/133001. 
  4. ^ Haugan, Mark P.; Will, Clifford M. (1987). "Modern tests of special relativity". Physics Today 40 (5): 69–86. Bibcode:1987PhT....40e..69H. doi:10.1063/1.881074. 
  5. ^ a b Will, C.M. (2006). "The Confrontation between General Relativity and Experiment". Living Rev. Relativity 9: 12. arXiv:gr-qc/0510072. Bibcode:2006LRR.....9....3W. doi:10.12942/lrr-2006-3. 
  6. ^ Colladay, Don; Kostelecký, V. Alan (1997). "CPT violation and the standard model". Physical Review D 55 (11): 6760–6774. arXiv:hep-ph/9703464. Bibcode:1997PhRvD..55.6760C. doi:10.1103/PhysRevD.55.6760. 
  7. ^ Colladay, Don; Kostelecký, V. Alan (1998). "Lorentz-violating extension of the standard model". Physical Review D 58 (11): 116002. arXiv:hep-ph/9809521. Bibcode:1998PhRvD..58k6002C. doi:10.1103/PhysRevD.58.116002. 
  8. ^ a b c Kostelecký, V. Alan; Mewes, Matthew (2002). "Signals for Lorentz violation in electrodynamics". Physical Review D 66 (5): 056005. arXiv:hep-ph/0205211. Bibcode:2002PhRvD..66e6005K. doi:10.1103/PhysRevD.66.056005. 
  9. ^ a b c Coleman, Sidney; Glashow, Sheldon L. (1998). "High-energy tests of Lorentz invariance". Physical Review D 59 (11): 116008. arXiv:hep-ph/9812418. Bibcode:1999PhRvD..59k6008C. doi:10.1103/PhysRevD.59.116008. 
  10. ^ Gambini, Rodolfo; Pullin, Jorge (1999). "Nonstandard optics from quantum space-time". Physical Review D 59 (12): 124021. arXiv:gr-qc/9809038. Bibcode:1999PhRvD..59l4021G. doi:10.1103/PhysRevD.59.124021. 
  11. ^ a b Myers, Robert C.; Pospelov, Maxim (2003). "Ultraviolet Modifications of Dispersion Relations in Effective Field Theory". Physical Review Letters 90 (21): 211601. arXiv:hep-ph/0301124. Bibcode:2003PhRvL..90u1601M. doi:10.1103/PhysRevLett.90.211601. 
  12. ^ a b c d e f g Kostelecký, V. Alan; Mewes, Matthew (2009). "Electrodynamics with Lorentz-violating operators of arbitrary dimension". Physical Review D 80 (1): 015020. arXiv:0905.0031. Bibcode:2009PhRvD..80a5020K. doi:10.1103/PhysRevD.80.015020. 
  13. ^ a b Hohensee et al. (2010). "Improved constraints on isotropic shift and anisotropies of the speed of light using rotating cryogenic sapphire oscillators". Physical Review D 82 (7): 076001. arXiv:1006.1376. Bibcode:2010PhRvD..82g6001H. doi:10.1103/PhysRevD.82.076001. 
  14. ^ Hohensee et al. (2010). Covariant Quantization of Lorentz-Violating Electromagnetism. arXiv:1210.2683. Bibcode:2012arXiv1210.2683H. ; Standalone version of work included in the Ph.D. Thesis of M.A. Hohensee.
  15. ^ a b c Tobar et al. (2005). "New methods of testing Lorentz violation in electrodynamics". Physical Review D 71 (2): 025004. arXiv:hep-ph/0408006. Bibcode:2005PhRvD..71b5004T. doi:10.1103/PhysRevD.71.025004. 
  16. ^ a b Bocquet et al. (2010). "Limits on Light-Speed Anisotropies from Compton Scattering of High-Energy Electrons". Physical Review Letters 104 (24): 24160. arXiv:1005.5230. Bibcode:2010PhRvL.104x1601B. doi:10.1103/PhysRevLett.104.241601. 
  17. ^ Gurzadyan, V. G.; Margarian, A. T. (1996). "Inverse Compton testing of fundamental physics and the cosmic background radiation". Physica Scripta 53: 513. Bibcode:1996PhyS...53..513G. doi:10.1088/0031-8949/53/5/001. 
  18. ^ Gurzadyan et al. (2012). "A new limit on the light speed isotropy from the GRAAL experiment at the ESRF". Proc. 12th M.Grossmann Meeting on General Relativity B: 1495. arXiv:1004.2867. Bibcode:2010arXiv1004.2867G. 
  19. ^ Zhou, Lingli, Ma, Bo-Qiang (2012). "A theoretical diagnosis on light speed anisotropy from GRAAL experiment". Astroparticle Physics 36 (1): 37–41. arXiv:1009.1675. Bibcode:2012APh....36...37Z. doi:10.1016/j.astropartphys.2012.04.015. 
  20. ^ Michimura et al. (2013). "New Limit on Lorentz Violation Using a Double-Pass Optical Ring Cavity". Physical Review Letters 110 (20): 200401. arXiv:1303.6709. Bibcode:2013PhRvL.110t0401M. doi:10.1103/PhysRevLett.110.200401. 
  21. ^ Baynes et al. (2012). "Oscillating Test of the Isotropic Shift of the Speed of Light". Physical Review Letters 108 (26): 260801. Bibcode:2012PhRvL.108z0801B. doi:10.1103/PhysRevLett.108.260801. 
  22. ^ Baynes et al. (2011). "Testing Lorentz invariance using an odd-parity asymmetric optical resonator". Physical Review D 84 (8): 081101. arXiv:1108.5414. Bibcode:2011PhRvD..84h1101B. doi:10.1103/PhysRevD.84.081101. 
  23. ^ \scriptstyle{\tilde{\kappa}_{o+}} combined with electron coefficients
  24. ^ Herrmann et al. (2009). "Rotating optical cavity experiment testing Lorentz invariance at the 10-17 level". Physical Review D 80 (100): 105011. arXiv:1002.1284. Bibcode:2009PhRvD..80j5011H. doi:10.1103/PhysRevD.80.105011. 
  25. ^ Eisele et al. (2009). "Laboratory Test of the Isotropy of Light Propagation at the 10-17 level". Physical Review Letters 103 (9): 090401. Bibcode:2009PhRvL.103i0401E. doi:10.1103/PhysRevLett.103.090401. PMID 19792767. 
  26. ^ Tobar et al. (2009). "Testing local Lorentz and position invariance and variation of fundamental constants by searching the derivative of the comparison frequency between a cryogenic sapphire oscillator and hydrogen maser". Physical Review D 81 (2): 022003. arXiv:0912.2803. Bibcode:2010PhRvD..81b2003T. doi:10.1103/PhysRevD.81.022003. 
  27. ^ Tobar et al. (2009). "Rotating odd-parity Lorentz invariance test in elecrodynamics". Physical Review D 80 (12): 125024. arXiv:0909.2076. Bibcode:2009PhRvD..80l5024T. doi:10.1103/PhysRevD.80.125024. 
  28. ^ Müller et al. (2007). "Relativity tests by complementary rotating Michelson-Morley experiments". Phys. Rev. Lett. 99 (5): 050401. arXiv:0706.2031. Bibcode:2007PhRvL..99e0401M. doi:10.1103/PhysRevLett.99.050401. PMID 17930733. 
  29. ^ Carone, et al. (2006). "New bounds on isotropic Lorentz violation". Physical Review D 74 (7): 077901. arXiv:hep-ph/0609150. Bibcode:2006PhRvD..74g7901C. doi:10.1103/PhysRevD.74.077901. 
  30. ^ Measured by examining the anomalous magnetic moment of the electron.
  31. ^ Stanwix et al. (2006). "Improved test of Lorentz invariance in electrodynamics using rotating cryogenic sapphire oscillators". Physical Review D 74 (8): 081101. arXiv:gr-qc/0609072. Bibcode:2006PhRvD..74h1101S. doi:10.1103/PhysRevD.74.081101. 
  32. ^ Herrmann et al. (2005). "Test of the Isotropy of the Speed of Light Using a Continuously Rotating Optical Resonator". Phys. Rev. Lett. 95 (15): 150401. arXiv:physics/0508097. Bibcode:2005PhRvL..95o0401H. doi:10.1103/PhysRevLett.95.150401. PMID 16241700. 
  33. ^ Stanwix et al. (2005). "Test of Lorentz Invariance in Electrodynamics Using Rotating Cryogenic Sapphire Microwave Oscillators". Physical Review Letters 95 (4): 040404. arXiv:hep-ph/0506074. Bibcode:2005PhRvL..95d0404S. doi:10.1103/PhysRevLett.95.040404. PMID 16090785. 
  34. ^ Antonini et al. (2005). "Test of constancy of speed of light with rotating cryogenic optical resonators". Physical Review A 71 (5): 050101. arXiv:gr-qc/0504109. Bibcode:2005PhRvA..71e0101A. doi:10.1103/PhysRevA.71.050101. 
  35. ^ Wolf et al. (2004). "Improved test of Lorentz invariance in electrodynamics". Physical Review D 70 (5): 051902. arXiv:hep-ph/0407232. Bibcode:2004PhRvD..70e1902W. doi:10.1103/PhysRevD.70.051902. 
  36. ^ Wolf et al. (2004). "Whispering Gallery Resonators and Tests of Lorentz Invariance". General Relativity and Gravitation 36 (10): 2351–2372. arXiv:gr-qc/0401017. Bibcode:2004GReGr..36.2351W. doi:10.1023/B:GERG.0000046188.87741.51. 
  37. ^ Müller et al. (2003). "Modern Michelson-Morley experiment using cryogenic optical resonators". Physical Review Letters 91 (2): 020401. arXiv:physics/0305117. Bibcode:2003PhRvL..91b0401M. doi:10.1103/PhysRevLett.91.020401. PMID 12906465. 
  38. ^ Lipa et al. (2003). "New Limit on Signals of Lorentz Violation in Electrodynamics". Physical Review Letters 90 (6): 060403. arXiv:physics/0302093. Bibcode:2003PhRvL..90f0403L. doi:10.1103/PhysRevLett.90.060403. 
  39. ^ Wolf et al. (2003). "Tests of Lorentz Invariance using a Microwave Resonator". Physical Review Letters 90 (6): 060402. arXiv:gr-qc/0210049. Bibcode:2003PhRvL..90f0402W. doi:10.1103/PhysRevLett.90.060402. PMID 12633279. 
  40. ^ Braxmaier et al. (2002). "Tests of Relativity Using a Cryogenic Optical Resonator". Phys. Rev. Lett. 88 (1): 010401. Bibcode:2002PhRvL..88a0401B. doi:10.1103/PhysRevLett.88.010401. PMID 11800924. 
  41. ^ Hils, Dieter; Hall, J. L. (1990). "Improved Kennedy-Thorndike experiment to test special relativity". Phys. Rev. Lett. 64 (15): 1697–1700. Bibcode:1990PhRvL..64.1697H. doi:10.1103/PhysRevLett.64.1697. PMID 10041466. 
  42. ^ Brillet, A.; Hall, J. L. (1979). "Improved laser test of the isotropy of space". Phys. Rev. Lett. 42 (9): 549–552. Bibcode:1979PhRvL..42..549B. doi:10.1103/PhysRevLett.42.549. 
  43. ^ Williams, James G.; Turyshev, Slava G.; Boggs, Dale H. (2009). "Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon". International Journal of Modern Physics D 18 (7): 1129–1175. arXiv:gr-qc/0507083. Bibcode:2009IJMPD..18.1129W. doi:10.1142/S021827180901500X. 
  44. ^ Bay, Z.; White, J. A. (1981). "Radar astronomy and the special theory of relativity". Acta physica Academiae Scientiarum Hungaricae 51 (3): 273–297. Bibcode:1981AcPhy..51..273B. doi:10.1007/BF03155586. 
  45. ^ Müller, J.; Soffel, M. H. (1995). "A Kennedy-Thorndike experiment using LLR data". Physics Letters A 198 (2): 71–73. Bibcode:1995PhLA..198...71M. doi:10.1016/0375-9601(94)01001-B. 
  46. ^ a b Müller, J., Nordtvedt, K., Schneider, M., Vokrouhlicky, D.: (1999). "Improved Determination of Relativistic Quantities from LLR". Proceedings of the 11th International Workshop on Laser Ranging Instrumentation 10: 216–222. 
  47. ^ a b c Fermi LAT Collaboration (2009). "A limit on the variation of the speed of light arising from quantum gravity effects". Nature 462 (7271): 331–334. arXiv:0908.1832. Bibcode:2009Natur.462..331A. doi:10.1038/nature08574. PMID 19865083. 
  48. ^ Vasileiou et al. (2013). "Constraints on Lorentz invariance violation from Fermi-Large Area Telescope observations of gamma-ray bursts". Physical Review D 87 (12): 122001. arXiv:1305.3463. Bibcode:2013PhRvD..87l2001V. doi:10.1103/PhysRevD.87.122001. 
  49. ^ HESS Collaboration (2008). "Limits on an Energy Dependence of the Speed of Light from a Flare of the Active Galaxy PKS 2155-304". Physics Letters B 101 (17): 170402. arXiv:0810.3475. Bibcode:2008PhRvL.101q0402A. doi:10.1103/PhysRevLett.101.170402. 
  50. ^ a b MAGIC Collaboration (2008). "Probing quantum gravity using photons from a flare of the active galactic nucleus Markarian 501 observed by the MAGIC telescope". Physics Letters B 668 (4): 253–257. arXiv:0708.2889. Bibcode:2008PhLB..668..253M. doi:10.1016/j.physletb.2008.08.053. 
  51. ^ Ellis et al. (2006). "Robust limits on Lorentz violation from gamma-ray bursts". Astroparticle Physics 25 (6): 402–411. arXiv:astro-ph/0510172. Bibcode:2006APh....25..402E. doi:10.1016/j.astropartphys.2006.04.001. 
  52. ^ Ellis et al. (2007). "Corrigendum to "Robust limits on Lorentz violation from gamma-ray bursts"". Astroparticle Physics 29 (2): 158–159. arXiv:0712.2781. Bibcode:2008APh....29..158E. doi:10.1016/j.astropartphys.2007.12.003. 
  53. ^ Lamon et al. (2007). "Study of Lorentz violation in INTEGRAL gamma-ray bursts". General Relativity and Gravitation 40 (8): 1731–1743. arXiv:0706.4039. Bibcode:2008GReGr..40.1731L. doi:10.1007/s10714-007-0580-6. 
  54. ^ Rodríguez Martínez et al. (2006). "GRB 051221A and tests of Lorentz symmetry". Journal of Cosmology and Astroparticle Physics (5): 017. arXiv:astro-ph/0601556. Bibcode:2006JCAP...05..017R. doi:10.1088/1475-7516/2006/05/017. 
  55. ^ Boggs et al. (2004). "Testing Lorentz Invariance with GRB021206". The Astrophysical Journal 611 (2): L77–L80. arXiv:astro-ph/0310307. Bibcode:2004ApJ...611L..77B. doi:10.1086/423933. 
  56. ^ Ellis et al. (2003). "Quantum-gravity analysis of gamma-ray bursts using wavelets". Astronomy and Astrophysics 402: 409–424. arXiv:astro-ph/0210124. Bibcode:2003A&A...402..409E. doi:10.1051/0004-6361:20030263. 
  57. ^ Ellis et al. (2000). "A Search in Gamma-Ray Burst Data for Nonconstancy of the Velocity of Light". The Astrophysical Journal 535 (1): 139–151. arXiv:astro-ph/9907340. Bibcode:2000ApJ...535..139E. doi:10.1086/308825. 
  58. ^ Kaaret, Philip (1999). "Pulsar radiation and quantum gravity". Astronomy and Astrophysics 345: L32–L34. arXiv:astro-ph/9903464. Bibcode:1999A&A...345L..32K. 
  59. ^ Schaefer, Bradley E. (1999). "Severe Limits on Variations of the Speed of Light with Frequency". Physical Review Letters 82 (25): 4964–4966. arXiv:astro-ph/9810479. Bibcode:1999PhRvL..82.4964S. doi:10.1103/PhysRevLett.82.4964. 
  60. ^ Biller et al. (1999). "Limits to Quantum Gravity Effects on Energy Dependence of the Speed of Light from Observations of TeV Flares in Active Galaxies". Physical Review Letters 83 (11): 2108–2111. arXiv:gr-qc/9810044. Bibcode:1999PhRvL..83.2108B. doi:10.1103/PhysRevLett.83.2108. 
  61. ^ a b c d Stecker, Floyd W. (2011). "A new limit on Planck scale Lorentz violation from γ-ray burst polarization". Astroparticle Physics 35 (2): 95–97. arXiv:1102.2784. Bibcode:2011APh....35...95S. doi:10.1016/j.astropartphys.2011.06.007. 
  62. ^ Götz et al. (2013). "The polarized gamma-ray burst GRB 061122". Monthly Notices of the Royal Astronomical Society 431 (4): 3550–3556. arXiv:1303.4186. Bibcode:2013MNRAS.431.3550G. doi:10.1093/mnras/stt439. 
  63. ^ Toma et al. (2012). "Strict Limit on CPT Violation from Polarization of γ-Ray Bursts". Physical Review Letters 109 (24): 241104. arXiv:1208.5288. Bibcode:2012PhRvL.109x1104T. doi:10.1103/PhysRevLett.109.241104. 
  64. ^ Laurent et al. (2011). "Constraints on Lorentz Invariance Violation using integral/IBIS observations of GRB041219A". Physical Review D 83 (12): 121301. arXiv:1106.1068. Bibcode:2011PhRvD..83l1301L. doi:10.1103/PhysRevD.83.121301. 
  65. ^ QUaD Collaboration (2008). "Parity Violation Constraints Using Cosmic Microwave Background Polarization Spectra from 2006 and 2007 Observations by the QUaD Polarimeter". Physical Review Letters 102 (16): 161302. arXiv:0811.0618. Bibcode:2009PhRvL.102p1302W. doi:10.1103/PhysRevLett.102.161302. 
  66. ^ Kostelecký, V. Alan; Mewes, Matthew (2008). "Astrophysical Tests of Lorentz and CPT Violation with Photons". The Astrophysical Journal 689 (1): L1–L4. arXiv:0809.2846. Bibcode:2008ApJ...689L...1K. doi:10.1086/595815. 
  67. ^ Maccione et al. (2008). "γ-ray polarization constraints on Planck scale violations of special relativity". Physical Review D 78 (10): 103003. arXiv:0809.0220. Bibcode:2008PhRvD..78j3003M. doi:10.1103/PhysRevD.78.103003. 
  68. ^ Komatsu et al. (2009). "Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation". The Astrophysical Journal Supplement 180 (2): 330–376. arXiv:0803.0547. Bibcode:2009ApJS..180..330K. doi:10.1088/0067-0049/180/2/330. 
  69. ^ Kahniashvili et al. (2008). "Testing Lorentz invariance violation with Wilkinson Microwave Anisotropy Probe five year data". Physical Review D 78 (12): 123009. arXiv:0807.2593. Bibcode:2008PhRvD..78l3009K. doi:10.1103/PhysRevD.78.123009. 
  70. ^ Xia et al. (2008). "Testing CPT Symmetry with CMB Measurements: Update after WMAP5". The Astrophysical Journal 679 (2): L61–L63. arXiv:0803.2350. Bibcode:2008ApJ...679L..61X. doi:10.1086/589447. 
  71. ^ Cabella et al. (2007). "Constraints on CPT violation from Wilkinson Microwave Anisotropy Probe three year polarization data: A wavelet analysis". Physical Review D 76 (12): 123014. arXiv:0705.0810. Bibcode:2007PhRvD..76l3014C. doi:10.1103/PhysRevD.76.123014. 
  72. ^ Fan et al. (2007). "γ-ray burst ultraviolet/optical afterglow polarimetry as a probe of quantum gravity". Monthly Notices of the Royal Astronomical Society 376 (4): 1857–1860. arXiv:astro-ph/0702006. Bibcode:2007MNRAS.376.1857F. doi:10.1111/j.1365-2966.2007.11576.x. 
  73. ^ Feng et al. (2006). "Searching for CPT Violation with Cosmic Microwave Background Data from WMAP and BOOMERANG". Physical Review Letters 96 (22): 221302. arXiv:astro-ph/0601095. Bibcode:2006PhRvL..96v1302F. doi:10.1103/PhysRevLett.96.221302. 
  74. ^ Gleiser, Reinaldo J.; Kozameh, Carlos N. (2001). "Astrophysical limits on quantum gravity motivated birefringence". Physical Review D 64 (8): 083007. arXiv:gr-qc/0102093. Bibcode:2001PhRvD..64h3007G. doi:10.1103/PhysRevD.64.083007. 
  75. ^ Carroll et al. (1990). "Limits on a Lorentz- and parity-violating modification of electrodynamics". Physical Review D 41 (4): 1231–1240. Bibcode:1990PhRvD..41.1231C. doi:10.1103/PhysRevD.41.1231. 
  76. ^ Jacobson et al. (2002). "Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics". Physical Review D 67 (12): 124011. arXiv:hep-ph/0209264. Bibcode:2003PhRvD..67l4011J. doi:10.1103/PhysRevD.67.124011. 
  77. ^ a b Hohensee et al. (2009). "Particle-Accelerator Constraints on Isotropic Modifications of the Speed of Light". Physical Review Letters 102 (17): 170402. arXiv:0904.2031. Bibcode:2009PhRvL.102q0402H. doi:10.1103/PhysRevLett.102.170402. 
  78. ^ a b Stecker, Floyd W. (2014). "Constraining Superluminal Electron and Neutrino Velocities using the 2010 Crab Nebula Flare and the IceCube PeV Neutrino Events". Astroparticle Physics 56: 16–18. arXiv:1306.6095. Bibcode:2014APh....56...16S. doi:10.1016/j.astropartphys.2014.02.007. 
  79. ^ Stecker, Floyd W.; Scully, Sean T. (2008). "Searching for new physics with ultrahigh energy cosmic rays". New Journal of Physics 11 (8): 085003. arXiv:0906.1735. Bibcode:2009NJPh...11h5003S. doi:10.1088/1367-2630/11/8/085003. 
  80. ^ a b Altschul, Brett (2009). "Bounding isotropic Lorentz violation using synchrotron losses at LEP". Physical Review D 80 (9): 091901. arXiv:0905.4346. Bibcode:2009PhRvD..80i1901A. doi:10.1103/PhysRevD.80.091901. 
  81. ^ Bi, Xiao-Jun; Cao, Zhen; Li, Ye; Yuan, Qiang (2008). "Testing Lorentz invariance with the ultrahigh energy cosmic ray spectrum". Physical Review D 79 (8): 083015. arXiv:0812.0121. Bibcode:2009PhRvD..79h3015B. doi:10.1103/PhysRevD.79.083015. 
  82. ^ Klinkhamer, F. R.; Schreck, M. (2008). "New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory". Physical Review D 78 (8): 085026. arXiv:0809.3217. Bibcode:2008PhRvD..78h5026K. doi:10.1103/PhysRevD.78.085026. 
  83. ^ Klinkhamer, F. R.; Risse, M. (2007). "Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory". Physical Review D 77 (1): 016002. arXiv:0709.2502. Bibcode:2008PhRvD..77a6002K. doi:10.1103/PhysRevD.77.016002. 
  84. ^ Kaufhold, C.; Klinkhamer, F. R. (2007). "Vacuum Cherenkov radiation in spacelike Maxwell-Chern-Simons theory". Physical Review D 76 (2): 025024. arXiv:0704.3255. Bibcode:2007PhRvD..76b5024K. doi:10.1103/PhysRevD.76.025024. 
  85. ^ Altschul, Brett (2005). "Lorentz violation and synchrotron radiation". Physical Review D 72 (8): 085003. arXiv:hep-th/0507258. Bibcode:2005PhRvD..72h5003A. doi:10.1103/PhysRevD.72.085003. 
  86. ^ Gagnon, Olivier; Moore, Guy D. (2004). "Limits on Lorentz violation from the highest energy cosmic rays". Physical Review D 70 (6): 065002. arXiv:hep-ph/0404196. Bibcode:2004PhRvD..70f5002G. doi:10.1103/PhysRevD.70.065002. 
  87. ^ Jacobson et al. (2003). "New Limits on Planck Scale Lorentz Violation in QED". Physical Review Letters 93 (2): 021101. arXiv:astro-ph/0309681. Bibcode:2004PhRvL..93b1101J. doi:10.1103/PhysRevLett.93.021101. 
  88. ^ a b M. Smiciklas et al. (2011). "New Test of Local Lorentz Invariance Using a 21Ne-Rb-K Comagnetometer". Physical Review Letters 107 (17): 171604. arXiv:1106.0738. Bibcode:2011PhRvL.107q1604S. doi:10.1103/PhysRevLett.107.171604. PMID 22107506. 
  89. ^ Kostelecký, V. Alan; Lane, Charles D. (1999). "Constraints on Lorentz violation from clock-comparison experiments". Physical Review D 60 (11): 116010. arXiv:hep-ph/9908504. Bibcode:1999PhRvD..60k6010K. doi:10.1103/PhysRevD.60.116010. 
  90. ^ Allmendinger et al. (2013). "New limit on Lorentz and CPT violating neutron spin interactions using a free precession 3He-129Xe co-magnetometer". Physical Review Letters 112 (11): 110801. arXiv:1312.3225. Bibcode:2014PhRvL.112k0801A. doi:10.1103/PhysRevLett.112.110801. 
  91. ^ Hohensee et al. (2013). "Limits on violations of Lorentz symmetry and the Einstein equivalence principle using radio-frequency spectroscopy of atomic dysprosium". Physical Review Letters 111 (5): 050401. arXiv:1303.2747. Bibcode:2013PhRvL.111e0401H. doi:10.1103/PhysRevLett.111.050401. 
  92. ^ Peck et al. (2012). "New Limits on Local Lorentz Invariance in Mercury and Cesium". Physical Review A 86 (1): 012109. arXiv:1205.5022. Bibcode:2012PhRvA..86a2109P. doi:10.1103/PhysRevA.86.012109. 
  93. ^ Gemmel et al. (2010). "Limit on Lorentz and CPT violation of the bound neutron using a free precession He3/Xe129 comagnetometer". Physical Review D 82 (11): 111901. arXiv:1011.2143. Bibcode:2010PhRvD..82k1901G. doi:10.1103/PhysRevD.82.111901. 
  94. ^ Brown et al. (2010). "New Limit on Lorentz- and CPT-Violating Neutron Spin Interactions". Physical Review Letters 105 (15): 151604. arXiv:1006.5425. Bibcode:2010PhRvL.105o1604B. doi:10.1103/PhysRevLett.105.151604. PMID 21230893. 
  95. ^ Altarev, I. et al. (2009). "Test of Lorentz Invariance with Spin Precession of Ultracold Neutrons". Physical Review Letters 103 (8): 081602. arXiv:0905.3221. Bibcode:2009PhRvL.103h1602A. doi:10.1103/PhysRevLett.103.081602. PMID 19792714. 
  96. ^ Heckel et al. (2008). "Preferred-frame and CP-violation tests with polarized electrons". Physical Review D 78 (9): 092006. arXiv:0808.2673. Bibcode:2008PhRvD..78i2006H. doi:10.1103/PhysRevD.78.092006. 
  97. ^ Wolf et al. (2006). "Cold Atom Clock Test of Lorentz Invariance in the Matter Sector". Physical Review Letters 96 (6): 060801. arXiv:hep-ph/0601024. Bibcode:2006PhRvL..96f0801W. doi:10.1103/PhysRevLett.96.060801. PMID 16605978. 
  98. ^ Canè et al. (2004). "Bound on Lorentz and CPT Violating Boost Effects for the Neutron". Physical Review Letters 93 (23): 230801. arXiv:physics/0309070. Bibcode:2004PhRvL..93w0801C. doi:10.1103/PhysRevLett.93.230801. PMID 15601138. 
  99. ^ Heckel et al. (2006). "New CP-Violation and Preferred-Frame Tests with Polarized Electrons". Physical Review Letters 97 (2): 021603. arXiv:hep-ph/0606218. Bibcode:2006PhRvL..97b1603H. doi:10.1103/PhysRevLett.97.021603. PMID 16907432. 
  100. ^ Humphrey et al. (2003). "Testing CPT and Lorentz symmetry with hydrogen masers". Physical Review A 68 (6): 063807. arXiv:physics/0103068. Bibcode:2003PhRvA..68f3807H. doi:10.1103/PhysRevA.68.063807. 
  101. ^ Hou et al. (2003). "Test of Cosmic Spatial Isotropy for Polarized Electrons Using a Rotatable Torsion Balance". Physical Review Letters 90 (20): 201101. arXiv:physics/0009012. Bibcode:2003PhRvL..90t1101H. doi:10.1103/PhysRevLett.90.201101. PMID 12785879. 
  102. ^ Phillips et al. (2000). "Limit on Lorentz and CPT violation of the proton using a hydrogen maser". Physical Review D 63 (11): 111101. arXiv:physics/0008230. Bibcode:2001PhRvD..63k1101P. doi:10.1103/PhysRevD.63.111101. 
  103. ^ Bear et al. (2000). "Limit on Lorentz and CPT Violation of the Neutron Using a Two-Species Noble-Gas Maser". Physical Review Letters 85 (24): 5038–5041. arXiv:physics/0007049. Bibcode:2000PhRvL..85.5038B. doi:10.1103/PhysRevLett.85.5038. PMID 11102181. 
  104. ^ Chou et al. (2010). "Optical Clocks and Relativity". Science 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843. 
  105. ^ Novotny, C. et al. (2009). "Sub-Doppler laser spectroscopy on relativistic beams and tests of Lorentz invariance". Physical Review A 80 (2): 022107. Bibcode:2009PhRvA..80b2107N. doi:10.1103/PhysRevA.80.022107. 
  106. ^ Reinhardt et al. (2007). "Test of relativistic time dilation with fast optical atomic clocks at different velocities". Nature Physics 3 (12): 861–864. Bibcode:2007NatPh...3..861R. doi:10.1038/nphys778. 
  107. ^ Saathoff et al. (2003). "Improved Test of Time Dilation in Special Relativity". Phys. Rev. Lett. 91 (19): 190403. Bibcode:2003PhRvL..91s0403S. doi:10.1103/PhysRevLett.91.190403. 
  108. ^ Grieser et al. (1994). "A test of special relativity with stored lithium ions". Applied Physics B Lasers and Optics 59 (2): 127–133. Bibcode:1994ApPhB..59..127G. doi:10.1007/BF01081163. 
  109. ^ Greenberg, O. W. (2002). "CPT Violation Implies Violation of Lorentz Invariance". Physical Review Letters 89 (23): 231602. arXiv:hep-ph/0201258. Bibcode:2002PhRvL..89w1602G. doi:10.1103/PhysRevLett.89.231602. 
  110. ^ Greenberg, O. W. Remarks on a challenge to the relation between CPT and Lorentz violation. arXiv:1105.0927. Bibcode:2011arXiv1105.0927G. 
  111. ^ Belle Collaboration (2012). "Search for time-dependent CPT violation in hadronic and semileptonic B decays". Physical Review D 85 (7): 071105. arXiv:1203.0930. Bibcode:2012PhRvD..85g1105H. doi:10.1103/PhysRevD.85.071105. 
  112. ^ a b Kostelecký, V. Alan; van Kooten, Richard J. (2010). "CPT violation and B-meson oscillations". Physical Review D 82 (10): 101702. arXiv:1007.5312. Bibcode:2010PhRvD..82j1702K. doi:10.1103/PhysRevD.82.101702. 
  113. ^ a b BaBar Collaboration (2008). "Search for CPT and Lorentz Violation in B0-Bmacr0 Oscillations with Dilepton Events". Physical Review Letters 100 (3): 131802. arXiv:0711.2713. Bibcode:2008PhRvL.100m1802A. doi:10.1103/PhysRevLett.100.131802. 
  114. ^ Belle Collaboration (2003). "Studies of B0-B0 mixing properties with inclusive dilepton events". Physical Review D 67 (5): 052004. arXiv:hep-ex/0212033. Bibcode:2003PhRvD..67e2004H. doi:10.1103/PhysRevD.67.052004. 
  115. ^ FOCUS Collaboration (2003). "Charm system tests of CPT and Lorentz invariance with FOCUS". Physics Letters B 556 (1-2): 7–13. arXiv:hep-ex/0208034. Bibcode:2003PhLB..556....7F. doi:10.1016/S0370-2693(03)00103-5. 
  116. ^ KTeV Collaboration (2011). "Precise measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system". Physical Review D 83 (9): 092001. arXiv:1011.0127. Bibcode:2011PhRvD..83i2001A. doi:10.1103/PhysRevD.83.092001. 
  117. ^ KLOE Collaboration (2006). "First observation of quantum interference in the process ϕ→KK→ππππ: A test of quantum mechanics and CPT symmetry". Physics Letters B 642 (4): 315–321. arXiv:hep-ex/0607027. Bibcode:2006PhLB..642..315K. doi:10.1016/j.physletb.2006.09.046. 
  118. ^ CPLEAR Collaboration (2003). "Physics at CPLEAR". Physics Reports 374 (3): 165–270. Bibcode:2003PhR...374..165A. doi:10.1016/S0370-1573(02)00367-8. 
  119. ^ KTeV Collaboration (2003). "Measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system". Physical Review D 67 (1): 012005. arXiv:hep-ex/0208007. Bibcode:2003PhRvD..67a2005A. doi:10.1103/PhysRevD.67.012005. 
  120. ^ NA31 Collaboration (1990). "A measurement of the phases of the CP-violating amplitudes in K0-->2π decays and a test of CPT invariance". Physics Letters B 237 (2): 303–312. Bibcode:1990PhLB..237..303C. doi:10.1016/0370-2693(90)91448-K. 
  121. ^ CDF Collaboration (2012). "Measurement of the Mass Difference Between Top and Anti-top Quarks". Physical Review D 87 (5): 052013. arXiv:1210.6131. Bibcode:2013PhRvD..87e2013A. doi:10.1103/PhysRevD.87.052013. 
  122. ^ CMS Collaboration (2012). "Measurement of the Mass Difference between Top and Antitop Quarks". Journal of High Energy Physics: 109. arXiv:1204.2807. Bibcode:2012JHEP...06..109C. doi:10.1007/JHEP06(2012)109. 
  123. ^ D0 Collaboration (2011). "Direct Measurement of the Mass Difference between Top and Antitop Quarks". Physical Review D 84 (5): 052005. arXiv:1106.2063. Bibcode:2011PhRvD..84e2005A. doi:10.1103/PhysRevD.84.052005. 
  124. ^ CDF Collaboration (2011). "Measurement of the Mass Difference between t and t¯ Quarks". Physical Review Letters 106 (15): 152001. arXiv:1103.2782. Bibcode:2011PhRvL.106o2001A. doi:10.1103/PhysRevLett.106.152001. 
  125. ^ D0 Collaboration (2009). "Direct Measurement of the Mass Difference between Top and Antitop Quarks". Physical Review Letters 103 (13): 132001. arXiv:0906.1172. Bibcode:2009PhRvL.103m2001A. doi:10.1103/PhysRevLett.103.132001. 
  126. ^ Gabrielse et al. (1999). "Precision Mass Spectroscopy of the Antiproton and Proton Using Simultaneously Trapped Particles". Physical Review Letters 82 (16): 3198–3201. Bibcode:1999PhRvL..82.3198G. doi:10.1103/PhysRevLett.82.3198. 
  127. ^ Dehmelt et al. (1999). "Past Electron-Positron g-2 Experiments Yielded Sharpest Bound on CPT Violation for Point Particles". Physical Review Letters 83 (23): 4694–4696. arXiv:hep-ph/9906262. Bibcode:1999PhRvL..83.4694D. doi:10.1103/PhysRevLett.83.4694. 
  128. ^ Hughes et al. (2001). "Test of CPT and Lorentz Invariance from Muonium Spectroscopy". Physical Review Letters 87 (11): 111804. arXiv:hep-ex/0106103. Bibcode:2001PhRvL..87k1804H. doi:10.1103/PhysRevLett.87.111804. 
  129. ^ Bennett et al. (2008). "Search for Lorentz and CPT Violation Effects in Muon Spin Precession". Physical Review Letters 100 (9): 091602. arXiv:0709.4670. Bibcode:2008PhRvL.100i1602B. doi:10.1103/PhysRevLett.100.091602. 
  130. ^ Altschul, Brett (2007). "Astrophysical limits on Lorentz violation for all charged species". Astroparticle Physics 28 (3): 380–384. arXiv:hep-ph/0610324. Bibcode:2007APh....28..380A. doi:10.1016/j.astropartphys.2007.08.003. 
  131. ^ D0 Collaboration (2012). "Search for violation of Lorentz invariance in top quark pair production and decay". Physical Review Letters 108 (26): 261603. arXiv:1203.6106. Bibcode:2012PhRvL.108z1603A. doi:10.1103/PhysRevLett.108.261603. 
  132. ^ Charneski et al (2012). "Lorentz violation bounds on Bhabha scattering". Physical Review D 86 (4): 045003. arXiv:1204.0755. Bibcode:2012PhRvD..86d5003C. doi:10.1103/PhysRevD.86.045003. 
  133. ^ Bailey, Quentin G.; Kostelecký, V. Alan (2006). "Signals for Lorentz violation in post-Newtonian gravity". Physical Review D 74 (4): 045001. arXiv:gr-qc/0603030. Bibcode:2006PhRvD..74d5001B. doi:10.1103/PhysRevD.74.045001. 
  134. ^ Battat, James B. R.; Chandler, John F.; Stubbs, Christopher W. (2007). "Testing for Lorentz Violation: Constraints on Standard-Model-Extension Parameters via Lunar Laser Ranging". Physical Review Letters 99 (24): 241103. arXiv:0710.0702. Bibcode:2007PhRvL..99x1103B. doi:10.1103/PhysRevLett.99.241103. 
  135. ^ Iorio, L. (2012). "Orbital effects of Lorentz-violating standard model extension gravitomagnetism around a static body: a sensitivity analysis". Classical and Quantum Gravity 29 (17): 175007. arXiv:1203.1859. Bibcode:2012CQGra..29q5007I. doi:10.1088/0264-9381/29/17/175007. 
  136. ^ Xie, Yi (2012). "Testing Lorentz violation with binary pulsars: constraints on standard model extension". Research in Astronomy and Astrophysics 13 (1): 1–4. arXiv:1208.0736. Bibcode:2013RAA....13....1X. doi:10.1088/1674-4527/13/1/001. 
  137. ^ Díaz, Jorge S.; Kostelecký, V. Alan (2012). "Lorentz- and CPT-violating models for neutrino oscillations". Physical Review D 85 (1): 016013. arXiv:1108.1799. Bibcode:2012PhRvD..85a6013D. doi:10.1103/PhysRevD.85.016013. 
  138. ^ Double Chooz collaboration (2012). "First test of Lorentz violation with a reactor-based antineutrino experiment". Physical Review D 86 (11): 112009. arXiv:1209.5810. Bibcode:2012PhRvD..86k2009A. doi:10.1103/PhysRevD.86.112009. 
  139. ^ MINOS collaboration (2012). "Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector". Physical Review D 85 (3): 031101. arXiv:1201.2631. Bibcode:2012PhRvD..85c1101A. doi:10.1103/PhysRevD.85.031101. 
  140. ^ MiniBooNE Collaboration (2012). "Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses". Physics Letters B 718 (4): 1303–1308. arXiv:1109.3480. Bibcode:2013PhLB..718.1303M. doi:10.1016/j.physletb.2012.12.020. 
  141. ^ IceCube Collaboration (2010). "Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube". Physical Review D 82 (11): 112003. arXiv:1010.4096. Bibcode:2010PhRvD..82k2003A. doi:10.1103/PhysRevD.82.112003. 
  142. ^ MINOS collaboration (2010). "Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector". Physical Review Letters 105 (15): 151601. arXiv:1007.2791. Bibcode:2010PhRvL.105o1601A. doi:10.1103/PhysRevLett.105.151601. 
  143. ^ MINOS collaboration (2008). "Testing Lorentz Invariance and CPT Conservation with NuMI Neutrinos in the MINOS Near Detector". Physical Review Letters 101 (15): 151601. arXiv:0806.4945. Bibcode:2008PhRvL.101o1601A. doi:10.1103/PhysRevLett.101.151601. 
  144. ^ LSND collaboration (2005). "Tests of Lorentz violation in ν¯μ→ν¯e oscillations". Physical Review D 72 (7): 076004. arXiv:hep-ex/0506067. Bibcode:2005PhRvD..72g6004A. doi:10.1103/PhysRevD.72.076004. 
  145. ^ Mattingly et al. (2010). "Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation". Journal of Cosmology and Astroparticle Physics (02): 007. arXiv:0911.0521. Bibcode:2010JCAP...02..007M. doi:10.1088/1475-7516/2010/02/007. 
  146. ^ Kostelecky, Alan; Mewes, Matthew (May 25, 2012). "Neutrinos with Lorentz-violating operators of arbitrary dimension". Physical Review D 85 (9). 096005. arXiv:1112.6395. Bibcode:2012PhRvD..85i6005K. doi:10.1103/PhysRevD.85.096005. 
  147. ^ Borriello et al. (2013). "Stringent constraint on neutrino Lorentz invariance violation from the two IceCube PeV neutrinos". Physical Review D 87 (11): 116009. arXiv:1303.5843. Bibcode:2013PhRvD..87k6009B. doi:10.1103/PhysRevD.87.116009. 
  148. ^ Cowsik et al. (2012). "Testing violations of Lorentz invariance with cosmic rays". Physical Review D 86 (4): 045024. arXiv:1206.0713. Bibcode:2012PhRvD..86d5024C. doi:10.1103/PhysRevD.86.045024. 
  149. ^ Huo, Yunjie; Li, Tianjun; Liao, Yi; Nanopoulos, Dimitri V.; Qi, Yonghui (2012). "Constraints on neutrino velocities revisited". Physical Review D 85 (3): 034022. arXiv:1112.0264. Bibcode:2012PhRvD..85c4022H. doi:10.1103/PhysRevD.85.034022. 
  150. ^ ICARUS Collaboration (2012). "A search for the analogue to Cherenkov radiation by high energy neutrinos at superluminal speeds in ICARUS". Physics Letters B 711 (3-4): 270–275. arXiv:1110.3763. Bibcode:2012PhLB..711..270A. doi:10.1016/j.physletb.2012.04.014. 
  151. ^ Cowsik, R.; Nussinov, S.; Sarkar, U. (2011). "Superluminal neutrinos at OPERA confront pion decay kinematics". Physical Review Letters 107 (25): 251801. arXiv:1110.0241. Bibcode:2011PhRvL.107y1801C. doi:10.1103/PhysRevLett.107.251801. 
  152. ^ Bi, Xiao-Jun; Yin, Peng-Fei; Yu, Zhao-Huan; Yuan, Qiang (2011). "Constraints and tests of the OPERA superluminal neutrinos". Physical Review Letters 107 (24): 241802. arXiv:1109.6667. Bibcode:2011PhRvL.107x1802B. doi:10.1103/PhysRevLett.107.241802. 
  153. ^ Cohen, Andrew G.; Glashow, Sheldon L. (2011). "Pair Creation Constrains Superluminal Neutrino Propagation". Physical Review Letters 107 (18): 181803. arXiv:1109.6562. Bibcode:2011PhRvL.107r1803C. doi:10.1103/PhysRevLett.107.181803. 
  154. ^ LSND Collaboration (2001). "Evidence for neutrino oscillations from the observation of ν¯e appearance in a ν¯μ beam". Physical Review D 64 (11): 112007. arXiv:hep-ex/0104049. Bibcode:2001PhRvD..64k2007A. doi:10.1103/PhysRevD.64.112007. 
  155. ^ MiniBooNE Collaboration (2007). "Search for Electron Neutrino Appearance at the Δm2˜1eV2 Scale". Physical Review Letters 98 (23): 231801. arXiv:0704.1500. Bibcode:2007PhRvL..98w1801A. doi:10.1103/PhysRevLett.98.231801. 
  156. ^ MiniBooNE Collaboration (2008). "Unexplained Excess of Electronlike Events from a 1-GeV Neutrino Beam". Physical Review Letters 102 (10): 101802. arXiv:0812.2243. Bibcode:2009PhRvL.102j1802A. doi:10.1103/PhysRevLett.102.101802. 
  157. ^ "MiniBooNE results suggest antineutrinos act differently". Fermilab today. June 18, 2010. Retrieved 14 December 2011. 
  158. ^ MiniBooNE Collaboration (2010). "Event Excess in the MiniBooNE Search for ν¯μ→ν¯e Oscillations". Physical Review Letters 105 (18): 181801. arXiv:1007.1150. Bibcode:2010PhRvL.105r1801A. doi:10.1103/PhysRevLett.105.181801. 
  159. ^ Diaz, Jorge S. (2011). "Overview of Lorentz Violation in Neutrinos". Proceedings of the DPF-2011 Conference. arXiv:1109.4620. Bibcode:2011arXiv1109.4620D. 
  160. ^ OPERA collaboration (2011). Measurement of the neutrino velocity with the OPERA detector in the CNGS beam. arXiv:1109.4897. Bibcode:2012JHEP...10..093A. doi:10.1007/JHEP10(2012)093. 
  161. ^ OPERA collaboration (2012). Measurement of the neutrino velocity with the OPERA detector in the CNGS beam. arXiv:1109.4897v4. 
  162. ^ "New measurements from Fermilab’s MINOS experiment suggest a difference in a key property of neutrinos and antineutrinos". Fermilab press release. June 14, 2010. Retrieved 14 December 2011. 
  163. ^ MINOS Collaboration (2011). "First Direct Observation of Muon Antineutrino Disappearance". Physical Review Letters 107 (2): 021801. arXiv:1104.0344. Bibcode:2011PhRvL.107b1801A. doi:10.1103/PhysRevLett.107.021801. 
  164. ^ MINOS Collaboration (2011). "Search for the disappearance of muon antineutrinos in the NuMI neutrino beam". Physical Review D 84 (7): 071103. arXiv:1108.1509. Bibcode:2011PhRvD..84g1103A. doi:10.1103/PhysRevD.84.071103. 
  165. ^ "Surprise difference in neutrino and antineutrino mass lessening with new measurements from a Fermilab experiment". Fermilab press release. August 25, 2011. Retrieved 14 December 2011. 
  166. ^ MINOS Collaboration (2012). "An improved measurement of muon antineutrino disappearance in MINOS". Physical Review Letters 108 (19): 191801. arXiv:1202.2772. Bibcode:2012PhRvL.108s1801A. doi:10.1103/PhysRevLett.108.191801. 
  167. ^ George Musser (22 August 2007). "Hints of a breakdown of relativity theory?". Scientific American. Retrieved 15 October 2011. 
  168. ^ Nodland, Borge; Ralston, John P. (1997). "Indication of Anisotropy in Electromagnetic Propagation over Cosmological Distances". Physical Review Letters 78 (16): 3043–3046. arXiv:astro-ph/9704196. Bibcode:1997PhRvL..78.3043N. doi:10.1103/PhysRevLett.78.3043. 
  169. ^ Nodland, Borge; Ralston, John P. (1997). "Nodland and Ralston Reply:". Physical Review Letters 79 (10): 1958. arXiv:astro-ph/9705190. Bibcode:1997PhRvL..79.1958N. doi:10.1103/PhysRevLett.79.1958. 
  170. ^ Borge Nodland, John P. Ralston (1997), Response to Leahy's Comment on the Data's Indication of Cosmological Birefringence, arXiv:astro-ph/9706126
  171. ^ J.P. Leahy: http://www.jb.man.ac.uk/~jpl/screwy.html
  172. ^ Ted Bunn: https://facultystaff.richmond.edu/~ebunn/biref/
  173. ^ Eisenstein, Daniel J.; Bunn, Emory F. (1997). "Appropriate Null Hypothesis for Cosmological Birefringence". Physical Review Letters 79 (10): 1957. arXiv:astro-ph/9704247. Bibcode:1997PhRvL..79.1957E. doi:10.1103/PhysRevLett.79.1957. 
  174. ^ Carroll, Sean M.; Field, George B. (1997). "Is There Evidence for Cosmic Anisotropy in the Polarization of Distant Radio Sources?". Physical Review Letters 79 (13): 2394–2397. arXiv:astro-ph/9704263. Bibcode:1997PhRvL..79.2394C. doi:10.1103/PhysRevLett.79.2394. 
  175. ^ J. P. Leahy: (1997) Comment on the Measurement of Cosmological Birefringence, arXiv:astro-ph/9704285
  176. ^ Wardle et al. (1997). "Observational Evidence against Birefringence over Cosmological Distances". Physical Review Letters 79 (10): 1801–1804. arXiv:astro-ph/9705142. Bibcode:1997PhRvL..79.1801W. doi:10.1103/PhysRevLett.79.1801. 
  177. ^ Loredo et al. (1997). "Bayesian analysis of the polarization of distant radio sources: Limits on cosmological birefringence". Physical Review D 56 (12): 7507–7512. arXiv:astro-ph/9706258. Bibcode:1997PhRvD..56.7507L. doi:10.1103/PhysRevD.56.7507. 

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