Modern tests of Lorentz invariance

Modern tests of the Lorentz invariance serve to check the basic statements of the special theory of relativity or the equivalence principle of the general theory of relativity . The effects related to the Lorentz invariance mainly concern the principle of relativity , the constancy of the speed of light in all inertial systems , CPT symmetry , and the related statements of the Standard Model of particle physics . An essential motivation of these experiments are possible violations of the Lorentz and CPT invariance, which result from various variations of theQuantum gravity or other alternatives to the special and general theory of relativity could follow.

Possible violations of the Lorentz and CPT invariance are demonstrated theoretically by test theories of the special relativity theory or effective field theories (EFT) such as the Standard Model Extension (SME). Here, the standard model is seen as fundamentally valid, but effects that violate Lorentz can gradually be introduced by assuming a preferred reference system. For example, violations of the dispersion relation would lead to a deviation between the speed of light and the limit speed of matter.

Both terrestrial and astronomical experiments (on photons , nucleons , electrons , neutrinos, etc.) were carried out. No violation of the Lorentz invariance has been found in the published work, and exceptional cases in which positive results were reported have not yet been confirmed. For a detailed overview, see Mattingly (2005), and for detailed data, see Kostelecký & Russell (2013). For a current and historical overview, see Liberati (2013). For a general overview see tests of special relativity .

Estimates of violations of Lorentz invariance

Early models that were used to estimate the possibility of small deviations from the Lorentz invariance date back to the 1960s. Most notably, a number of test theories of special relativity and effective field theories have been developed that can be used to analyze experiments:

• The “parametrized post-Newtonian formalism” (PPN) is used as a test theory for general relativity (GTR) and its alternatives, and also contains parameters for Lorentz-violating effects through preferred reference systems or fields.
• The Robertson-Mansouri-Sexl formalism (RMS) contains three parameters with which deviations from the isotropy of the speed of light can be given with respect to a preferred reference system.
• The c ² formalism (a special case of the more general THεμ formalism) uses a modified dispersion relation and describes Lorentz violations in the sense of a discrepancy between the speed of light and the limit speed of matter in the presence of a preferred reference system.
• Doubly special relativity (DSR) leaves the Planck energy or the Planck length invariant as minimal scales without assuming a preferred reference system.
• Very special relativity (very special relativity theory, VSR) contains spacetime symmetries that represent certain subgroups of the Poincaré group . VSR is only equivalent to SRT in the context of local quantum field theory or in the case of CP conservation .
• Lorentz injuries are also discussed in connection with alternative formulations to quantum gravity and GTR, with approaches such as loop quantum gravity, emergent gravity, Einstein's ether theory, Hořava-Lifshitz gravity, or non-commutative geometry.

In many modern experiments, however, possible Lorentz injuries are mainly analyzed using the Standard Model Extension (SME). It was developed by Kostelecky and co-workers in 1997 and thereafter, primarily to assess possible effects of a theory of quantum gravity. It contains all possible Lorentz and CPT- violating coefficients which do not violate the calibration symmetry , whereby these effects are introduced by spontaneous symmetry breaking . It affects not only the SRT, but also the GTR and the standard model . The older RMS and c ² models, the Coleman - Glashow model, where the SME coefficients are limited to dimension 4 operators and rotational invariance, are models whose parameters are related to the SME and can therefore be viewed as SME special cases , and the Gambini - Pullin model or the Meyers-Pospelov model which contains dimension 5 operators from SME or higher.

Speed ​​of Light

Terrestrial

Experiments to measure deviations from the isotropy of the speed of light are carried out with optical resonators . They can be viewed as modern variants of the Michelson-Morley experiment . These experiments are often evaluated with the Robertson-Mansouri-Sexl test theory (RMS), which allows a differentiation between directional and speed-related anisotropies. In Michelson-Morley experiments, the dependence of the speed of light on the orientation of the measuring device or the ratio of the lengths in the longitudinal and transverse direction is checked. In contrast, with variants of the Kennedy-Thorndike experiment , the dependence of the speed of light on the speed of the measuring device or the ratio of the length contraction to the time dilation is checked. The current accuracy, with which an anisotropy of the terrestrially measured speed of light between uniformly moving reflectors can be excluded, is , based on the relative speed between the solar system and the rest system of the cosmic background radiation of approx. 368 km / s. ${\displaystyle 10^{-17}}$

The Standard Model Extension (SME) is also used to limit many isotropy coefficients. For example, even and odd parity coefficients (3x3 matrices) , and are used. These can be interpreted as follows: stands for anisotropic shifts in the two-way (there and back) speed of light, stands for anisotropic one-way speed differences of opposite rays, and stands for isotropic (direction-independent) shifts in the phase speed of light in one direction. With these variations and anisotropies of the speed of light, however, it must be taken into account that they depend on the choice of coordinates, and with a suitable choice of coordinate transformations and field redefinitions can be made to disappear. However, this does not mean that the associated Lorentz violations also disappear, but rather they are only shifted from the photon sector into the matter sector of SME, which does not limit the meaningfulness of the tests as a verification of the Lorentz violation. Ordinary optical resonators are mainly suitable for testing the effects with even parity, and have only little significance for effects with odd parity. In order to test the latter more precisely, asymmetrical resonators are used. Further coefficients in the photon sector, which cannot be redefined in the matter sector, since the light propagation of different rays is compared directly with one another, lead to vacuum dispersion and vacuum birefringence . ${\displaystyle {\tilde {\kappa }}_{e-}}$ ${\displaystyle {\tilde {\kappa }}_{o+}}$ ${\displaystyle {\tilde {\kappa }}_{tr}}$ ${\displaystyle {\tilde {\kappa }}_{e-}}$ ${\displaystyle {\tilde {\kappa }}_{o+}}$ ${\displaystyle {\tilde {\kappa }}_{tr}}$

Bocquet et al. (2010) looked for an anisotropy of the speed of light in a different way . Changes in the triple momentum of photons during the earth's rotation were considered by measuring the Compton scattering of relativistic electrons. ${\displaystyle {\tilde {\kappa }}_{o+}}$

author	year	RMS		SME
		direction	speed	${\displaystyle {\tilde {\kappa }}_{e-}}$	${\displaystyle {\tilde {\kappa }}_{o+}}$	${\displaystyle {\tilde {\kappa }}_{tr}}$
Michimura et al.	2013				${\displaystyle 0{,}7\pm 1\times 10^{-14}}$	${\displaystyle -0{,}4\pm 0{,}9\times 10^{-10}}$
Baynes et al.	2012					${\displaystyle 3\pm 11\times 10^{-10}}$
Baynes et al.	2011				${\displaystyle 0{,}7\pm 1{,}4\times 10^{-12}}$	${\displaystyle 3{,}4\pm 6{,}2\times 10^{-9}}$
Hohensee et al.	2010			${\displaystyle 0{,}8(0{,}6)\times 10^{-16}}$	${\displaystyle -1{,}5(1{,}2)\times 10^{-12}}$	${\displaystyle -1{,}5(0{,}74)\times 10^{-8}}$
Bocquet et al.	2010				${\displaystyle \leq 1{,}6\times 10^{-14}}$
Herrmann et al.	2009	${\displaystyle (4\pm 8)\times 10^{-12}}$		${\displaystyle (-0{,}31\pm 0{,}73)\times 10^{-17}}$	${\displaystyle (-0{,}14\pm 0{,}78)\times 10^{-13}}$
Eisele et al.	2009	${\displaystyle (-1{,}6\pm 6\pm 1{,}2)\times 10^{-12}}$		${\displaystyle (0{,}0\pm 1{,}0\pm 0{,}3)\times 10^{-17}}$	${\displaystyle (1{,}5\pm 1{,}5\pm 0{,}2)\times 10^{-13}}$
Tobar et al.	2009		${\displaystyle -4{,}8(3{,}7)\times 10^{-8}}$
Tobar et al.	2009					${\displaystyle -0{,}3\pm 3\times 10^{-7}}$
Muller et al.	2007			${\displaystyle (7{,}7(4{,}0))\times 10^{-16}}$	${\displaystyle (1{,}7(2{,}0))\times 10^{-12}}$
Carone et al.	2006					${\displaystyle \lesssim 3\times 10^{-8}}$
Stanwix et al.	2006	${\displaystyle 9{,}4(8{,}1)\times 10^{-11}}$		${\displaystyle (-6{,}9(2{,}2))\times 10^{-16}}$	${\displaystyle (-0{,}9(2{,}6))\times 10^{-12}}$
Herrmann et al.	2005	${\displaystyle (-2{,}1\pm 1{,}9)\times 10^{-10}}$		${\displaystyle (-3{,}1(2{,}5))\times 10^{-16}}$	${\displaystyle (-2{,}5(5{,}1))\times 10^{-12}}$
Stanwix et al.	2005	${\displaystyle -0{,}9(2{,}0)\times 10^{-10}}$		${\displaystyle (-0{,}63(0{,}43))\times 10^{-15}}$	${\displaystyle (0{,}20(0{,}21))\times 10^{-11}}$
Antonini et al.	2005	${\displaystyle (+0{,}5\pm 3\pm 0{,}7)\times 10^{-10}}$		${\displaystyle (-2\pm 0{,}2)\times 10^{-14}}$
Wolf et al.	2004			${\displaystyle (-5{,}7\pm 2{,}3)\times 10^{-15}}$	${\displaystyle (-1{,}8\pm 1{,}5)\times 10^{-11}}$
Wolf et al.	2004	${\displaystyle (+1{,}2\pm 2{,}2)\times 10^{-9}}$	${\displaystyle (3{,}7\pm 3{,}0)\times 10^{-7}}$
Muller et al.	2003	${\displaystyle (+2{,}2\pm 1{,}5)\times 10^{-9}}$		${\displaystyle (1{,}7\pm 2{,}6)\times 10^{-15}}$	${\displaystyle (14\pm 14)\times 10^{-11}}$
Lipa et al.	2003			${\displaystyle (1{,}4\pm 1{,}4\times 10^{-13}}$	${\displaystyle \leq 10^{-9}}$
Wolf et al.	2003	${\displaystyle (+1{,}5\pm 4{,}2)\times 10^{-9}}$
Braxmaier et al.	2002		${\displaystyle (1{,}9\pm 2{,}1)\times 10^{-5}}$
Hils and Hall	1990		${\displaystyle 6{,}6\times 10^{-5}}$
Brillet and Hall	1979	${\displaystyle \lesssim 5\times 10^{-9}}$		${\displaystyle \lesssim 10^{-15}}$

Solar system

In addition to terrestrial tests, astrometric tests with Lunar Laser Ranging (LLR), i.e. the laser signal exchange between the earth and the moon, are carried out. Usually these measurements are taken as tests of the general theory of relativity and analyzed by means of the "parametrized post-Newtonian formalism" - but these measurements can also serve as tests of the special theory of relativity, since the constancy of the speed of light is applied and deviations from it are expressed as or orbital changes would show. For example, by analyzing the planetary radar data and LLR , Zoltán Bay and White (1981) were able to show the empirical basis of the Lorentz group and thus the special theory of relativity.

Muller et al. (1995, 1999) carried out a variant of the above-mentioned Kennedy-Thorndike experiment with LLR. A dependence of the speed of light on the speed of the observer relative to a preferred reference system would have to lead to changes in the transit time and thus to variations in the measured earth-moon distance. This can only be balanced out if the length contraction and time dilation assume the exact relativistic values. The result was negative, with a maximum RMS velocity dependence of , which is comparable to the experiments of Hils and Hall (1990, see table above right). ${\displaystyle (-5\pm 12)\times 10^{-5}}$

Vacuum dispersion

In addition to the above-mentioned isotropy measurements, dispersion , i.e. the dependence of the speed of light from distant light sources on its energy or frequency, is examined in astronomical experiments . It is usually assumed that if deviations occur, then these should be  measurable at photon energies starting from the Planck energy of approx. 1.22 × 10 ¹⁹ GeV. In the following works, investigations into gamma-ray flashes and other astronomical sources looked for such deviations. However, with increasing accuracy, no energy dependency or violations of the Lorentz invariance could be determined, with the Fermi group even investigating photon energies of up to 31 GeV. Since there should have been deviations in this area long ago, this class of theories of quantum gravity is practically excluded.

Surname	year	Lower limit in GeV
		95% CL	99% CL
Vasileiou et al.	2013	${\displaystyle >7{,}6\times E_{\mathrm {Pl} }}$
Fermi-LAT-GBM group	2009	${\displaystyle >3{,}42\times E_{\mathrm {Pl} }}$	${\displaystyle >1{,}19\times E_{\mathrm {Pl} }}$
HESS group	2008	${\displaystyle \geq 7{,}2\times 10^{17}}$
MAGIC group	2007	${\displaystyle \geq 0{,}21\times 10^{18}}$
Lamon et al.	2008	${\displaystyle \geq 3{,}2\times 10^{11}}$
Martinez et al.	2006	${\displaystyle \geq 0{,}66\times 10^{17}}$
Ellis et al.	2006/8	${\displaystyle \geq 1{,}4\times 10^{16}}$
Boggs et al.	2004	${\displaystyle \geq 1{,}8\times 10^{17}}$
Ellis et al.	2003	${\displaystyle \geq 6{,}9\times 10^{15}}$
Ellis et al.	2000	${\displaystyle \geq 10^{15}}$
Shepherd	1999	${\displaystyle \geq 2{,}7\times 10^{16}}$
Biller	1999	${\displaystyle >4\times 10^{16}}$
Kaaret	1999	${\displaystyle >1{,}8\times 10^{15}}$

Vacuum birefringence

Violations of the Lorentz invariance (such as the presence of a preferred reference system) could also lead to birefringence in a vacuum and parity violations . Research is conducted into the related deviations in the polarization of photons, for example the rotation of the plane of polarization due to differences in speed between right and left polarized photons. Here, gamma-ray bursts, galaxy radiation and the cosmic microwave background radiation are checked. The following works contain some of the coefficients given by the standard model extension or for Lorentz violations due to birefringence, with 3 and 5 representing the mass dimensions used. The latter corresponds to the expression in the Meyers-Pospelov EFT by where the Planck mass is. So far, no violations of the Lorentz invariance have been found. However, the ESO has found the first indications of the existence of vacuum birefringence. ${\displaystyle k_{(V)00}^{(3)}}$ ${\displaystyle k_{(V)00}^{(5)}}$ ${\displaystyle \xi }$ ${\displaystyle \textstyle k_{(V)00}^{(5)}={\frac {3{\sqrt {4\pi }}\xi }{5m_{\mathrm {P} }}}}$ ${\displaystyle m_{\mathrm {P} }}$

Surname	year	SME limits		EFT limits ( ) ${\displaystyle \xi }$
		${\displaystyle k_{(V)00}^{(3)}}$ in GeV	${\displaystyle k_{(V)00}^{(5)}}$in GeV −1
Götz et al.	2013		${\displaystyle \leq 5{,}9\times 10^{-35}}$	${\displaystyle \leq 3{,}4\times 10^{-16}}$
Toma et al.	2012		${\displaystyle \leq 1{,}4\times 10^{-34}}$	${\displaystyle \leq 8\times 10^{-16}}$
Laurent et al.	2011		${\displaystyle \leq 1{,}9\times 10^{-33}}$	${\displaystyle \leq 1{,}1\times 10^{-14}}$
plug	2011		${\displaystyle \leq 4{,}2\times 10^{-34}}$	${\displaystyle \leq 2{,}4\times 10^{-15}}$
Kostelecký et al.	2009		${\displaystyle \leq 1\times 10^{-32}}$	${\displaystyle \leq 9\times 10^{-14}}$
QUaD collaboration	2008	${\displaystyle \leq 2\times 10^{-43}}$
Kostelecký et al.	2008	${\displaystyle =(2{,}3\pm 5{,}4)\times 10^{-43}}$
Maccione et al.	2008		${\displaystyle \leq 1{,}5\times 10^{-28}}$	${\displaystyle \leq 9\times 10^{-10}}$
Komatsu et al.	2008	${\displaystyle =(1{,}2\pm 2{,}2)\times 10^{-43}}$
Kahniashvili et al.	2008	${\displaystyle \leq 2{,}5\times 10^{-43}}$
Xia et al.	2008	${\displaystyle =(2{,}6\pm 1{,}9)\times 10^{-43}}$
Cabella et al.	2007	${\displaystyle =(2{,}5\pm 3{,}0)\times 10^{-43}}$
Fan et al.	2007		${\displaystyle \leq 3{,}4\times 10^{-26}}$	${\displaystyle \leq 2\times 10^{-7}}$
Feng et al.	2006	${\displaystyle =(6{,}0\pm 4{,}0)\times 10^{-43}}$
Gleiser et al.	2001		${\displaystyle \leq 8{,}7\times 10^{-23}}$	${\displaystyle \leq 2\times 10^{-4}}$
Carroll et al.	1990	${\displaystyle \leq 2\times 10^{-42}}$

Limiting speed of matter

Threshold Energy Effects

Changed dispersion relationships due to Lorentz violations can lead to threshold energy effects that would otherwise not be possible. This is the case, for example, when the limit speed of particles with a charge structure (such as protons, electrons, neutrinos) and photons is different. Depending on which of these types of particles reaches the speed of faster than light, the main search is for the following effects:

• Photon decay when the speed of the photons is greater than the limit speed of other particles. The photons decay into different particles in a very short time, which means that high-energy light from very great distances could no longer reach the earth. The mere presence of high-energy light from distant astronomical regions limits possible differences between the speed of light and the limit speed of other particles.
• Vacuum Cherenkov effect when the limit speed of charged particles is greater than the speed of light. This can lead to the emission of bremsstrahlung in various forms until the energy falls below the threshold. This is analogous to the well-known Cherenkov radiation in media, in which particles move faster than the phase speed of light in this medium. Lorentz injuries due to this effect can be restricted when particles such as ultra-high energy cosmic rays (UHECR) arrive on earth from distant astronomical regions. The greater the energy, the less the possibility of deviations of its limit speed from the speed of light.
• Change in synchrotron radiation due to different boundary speeds between charged particles and light.
• The GZK cutoff could also be lifted by corresponding Lorentz violations. However, since the cutoff has been proven with a high degree of probability in more recent measurements, possible Lorentz violations can be considerably restricted.

Together, these effects limit possible deviations between the limit speed and the speed of light on both sides, as can be seen in the following table. However, since astronomical measurements also contain additional assumptions - regarding the often only approximately known conditions for the emission - terrestrial measurements result in greater reliability, but the specific EFT limit values ​​for deviations in the limit speeds are somewhat lower:

Surname	year	EFT limits				Particle	Astr./Terr.
		Photon decay	Cherenkov	Synchrotron	GZK
plug	2014		${\displaystyle \leq 5\times 10^{-21}}$			Electron	Astr.
Plug & Scully	2009				${\displaystyle \leq 4{,}5\times 10^{-23}}$	UHECR	Astr.
Altschul	2009			${\displaystyle \leq 5\times 10^{-15}}$		Electron	Terr.
Hohensee et al.	2009	${\displaystyle \leq -5{,}8\times 10^{-12}}$	${\displaystyle \leq 1{,}2\times 10^{-11}}$			Electron	Terr.
Bi et al.	2008				${\displaystyle \leq 3\times 10^{-23}}$	UHECR	Astr.
Klinkhamer & Schreck	2008	${\displaystyle \leq -9\times 10^{-16}}$	${\displaystyle \leq 6\times 10^{-20}}$			UHECR	Astr.
Klinkhamer & Risse	2007		${\displaystyle \leq 2\times 10^{-19}}$			UHECR	Astr.
Kaufhold et al.	2007		${\displaystyle \leq 10^{-17}}$			UHECR	Astr.
Altschul	2005			${\displaystyle \leq 6\times 10^{-20}}$		Electron	Astr.
Gagnon et al.	2004	${\displaystyle \leq -2\times 10^{-21}}$	${\displaystyle \leq 5\times 10^{-24}}$			UHECR	Astr.
Jacobson et al.	2003	${\displaystyle \leq -2\times 10^{-16}}$	${\displaystyle \leq 5\times 10^{-20}}$			Electron	Astr.
Coleman & Glass Show	1997	${\displaystyle \leq -1{,}5\times 10^{-15}}$	${\displaystyle \leq 5\times 10^{-23}}$			UHECR	Astr.

Clock comparison and spin measurements

These spectroscopic tests, often referred to as Hughes-Drever or “Clock comparison experiments”, examine deviations from the Lorentz invariance of protons and neutrons . Investigations of the energy level are carried out, whereby anisotropies in the frequencies ("clocks") can be determined, which should occur, for example, due to the presence of a preferred reference system according to SME. Vector spin and tensor interactions are investigated and are often described using CPT-odd / even terms of the SME, especially with regard to the parameters of and . These experiments are among the most precise of all, since deviations of up to GeV could be determined. By using spin-polarized torsion balances , corresponding limits could also be reached for electrons. ${\displaystyle b_{\mu }}$ ${\displaystyle c_{\mu \nu }}$ ${\displaystyle 10^{-33}}$

The limit velocity of matter can also be investigated with these experiments, especially in connection with the parameters of analogous to the threshold energy effects. ${\displaystyle c_{\mu \nu }}$

author	year	SME limits			parameter
		proton	neutron	electron
Allmendinger et al.	2013		${\displaystyle <6{,}7\times 10^{-34}}$		${\displaystyle b_{\mu }}$
Hohensee et al.	2013			${\displaystyle (-9{,}0\pm 11)\times 10^{-17}}$	${\displaystyle c_{\mu \nu }}$
Peck et al.	2012	${\displaystyle <4\times 10^{-30}}$	${\displaystyle <3{,}7\times 10^{-31}}$		${\displaystyle b_{\mu }}$
Smiciklas, et al.	2011		${\displaystyle (4{,}8\pm 4{,}4)\times 10^{-32}}$		${\displaystyle c_{\mu \nu }}$
Gemmel et al.	2010		${\displaystyle <3{,}7\times 10^{-32}}$		${\displaystyle b_{\mu }}$
Brown et al.	2010	${\displaystyle <6\times 10^{-32}}$	${\displaystyle <3{,}7\times 10^{-33}}$		${\displaystyle b_{\mu }}$
Altarev et al.	2009		${\displaystyle <2\times 10^{-29}}$		${\displaystyle b_{\mu }}$
Heckel et al.	2008			${\displaystyle (4{,}0\pm 3{,}3)\times 10^{-31}}$	${\displaystyle b_{\mu }}$
Wolf et al.	2006	${\displaystyle (-1{,}8(2{,}8))\times 10^{-25}}$			${\displaystyle c_{\mu \nu }}$
Canè et al.	2004		${\displaystyle (8{,}0\pm 9{,}5)\times 10^{-32}}$		${\displaystyle b_{\mu }}$
Heckel et al.	2006			${\displaystyle <5\times 10^{-30}}$	${\displaystyle b_{\mu }}$
Humphrey et al.	2003			${\displaystyle <2\times 10^{-27}}$	${\displaystyle b_{\mu }}$
Hou et al.	2003			${\displaystyle (1{,}8\pm 5{,}3)\times 10^{-30}}$	${\displaystyle b_{\mu }}$
Phillips et al.	2001	${\displaystyle <2\times 10^{-27}}$			${\displaystyle b_{\mu }}$
Bear et al.	2000		${\displaystyle (4{,}0\pm 3{,}3)\times 10^{-31}}$		${\displaystyle b_{\mu }}$

Time dilation

The classic experiments to prove the time dilation or the relativistic Doppler effect , such as the Ives-Stilwell experiment , the Mössbauer-rotor experiments , and the time dilation of moving particles , are also repeated. For example, lithium ions are used in storage rings. Even at everyday speeds of 36 km / h, Chou et al. (2010) measured a corresponding frequency shift of around 10 ⁻¹⁶ due to time dilation.

author	year	speed	Max. Deviation from the time dilation	Fourth order RMS limits
Novotny et al.	2009	0.34c	${\displaystyle \leq 1{,}3\times 10^{-6}}$	${\displaystyle \leq 1{,}2\times 10^{-5}}$
Reinhardt et al.	2007	0.064c	${\displaystyle \leq 8{,}4\times 10^{-8}}$
Saathoff et al.	2003	0.064c	${\displaystyle \leq 2{,}2\times 10^{-7}}$
Grieser et al.	1994	0.064c	${\displaystyle \leq 1\times 10^{-6}}$	${\displaystyle \leq 2{,}7\times 10^{-4}}$

CPT and antimatter tests

The Lorentz invariance is closely related to the CPT symmetry in local quantum field theories (with non-local exceptions). From this it follows u. a. a strict symmetry in the mass of particles and their antiparticles as well as equal decay times (for older tests of this type see time dilation of moving particles ). In modern tests, mainly neutral mesons are examined. In addition, mass differences between top and antitop quarks are investigated in large particle accelerators .

Neutral B mesons author year Belle 2012 Kostelecký et al. 2010 BaBar 2008 Belle 2003 Neutral D mesons FOCUS 2003

Neutral kaons author year KTeV 2011 KLOE 2006 CPLEAR 2003 KTeV 2003 NA31 1990

Top and antitop quarks author year CDF 2012 CMS 2012 D0 experiment 2011 CDF 2011 D0 2009

According to the standard model extension, the following factors are also of importance: the electromagnetic, gravitational and nuclear fields that determine the system must be taken into account, as well as the rotation and orbit of the earth. In addition to the above experiments, such effects can be determined using Penning traps , for example. Individual, charged particles and their antimaterial counterparts are captured. A strong magnetic field is used to keep the particles close to the central axis, and an electric field that aligns the particles accordingly if they are too far along the axis. The movement frequencies of the captured particles can be monitored and measured with considerable precision. Gabrielse et al. (1999) carried out proton-antiproton measurements. The cyclotron frequencies of the captured particles were compared, with an accuracy of . Hans Dehmelt et al. (1999) used Penning traps to check the anomaly frequencies, which play an important role in the measurement of the gyromagnetic ratio of the electron, whereby the maximum deviation could be limited to 10 ⁻²⁴ GeV. ${\displaystyle 9\cdot 10^{-11}}$

Tests on muons are also carried out. Since the lifetime of muons is only a few milliseconds, these experiments differ significantly from those with electrons and positrons. Hughes et al. (2001) published data on their search for sidereal signals in the spectrum of muonium . No deviations from the Lorentz invariance could be found, although the maximum limit for this could be narrowed down to. The “Muon g − 2” group at Brookhaven National Laboratory looked for deviations in the anomaly frequencies of muons and antimuons. They also looked for sidereal variations, taking into account the orientation of the earth relative to a sun-centered inertial system. Here, too, no deviations from the Lorentz invariance could be found. ${\displaystyle 2\cdot 10^{-23}}$

Other particles and interactions

Third generation particles were also examined for possible Lorentz injuries and analyzed as part of the SME. Altschul (2007) researched the anomalous absorption of cosmic high-energy rays and found an upper limit for Lorentz injuries for the τ lepton of 10 ⁻⁸ . During the BaBar experiment (2007), research was carried out using B mesons (and thus bottom quarks ) for sidereal variations during the Earth's rotation. Lorentz and CPT injuries could be excluded with an upper limit of . Also top quark pairs were used during the D0 experiment investigated (2012), the generation of these pairs and has been shown that it is not dependent on the sidereal time during the rotation of the earth. ${\displaystyle \leq (-3{,}0\pm 2{,}4)\times 10^{-15}}$

Limits for Lorentz violations in Bhabha scattering (the quantum electrodynamic electron-positron scattering) were determined by Charneski et al . (2012) determined. They showed that the differential cross-sections for the vectorial and axial couplings in quantum field theory become direction-dependent in the presence of Lorentz violations. The absence of this effect gave an upper limit of . ${\displaystyle \leq 10^{14}({\text{eV}})^{-1}}$

Gravity

Since the SRT and thus also the Lorentz invariance have to be fulfilled locally within the framework of the general theory of relativity (GTR), violations of the Lorentz invariance can also be examined by analyzing the effects of gravitational fields. In general, the parameterized post-Newtonian formalism (PPN) was developed to analyze deviations from the ART . This also includes the parameter , and for the description of effects of a "preferred reference system", which correspond to Lorentz injury. A number of tests have been carried out for this purpose, see Tests of general relativity . Lorentz injuries are also discussed in the context of various alternatives to GTR such as loop quantum gravity, emergent gravity, Einstein's ether theory, or Hořava-Lifshitz gravity. ${\displaystyle \alpha _{1}}$ ${\displaystyle \alpha _{2}}$ ${\displaystyle \alpha _{3}}$

The influence of Lorentz injuries on gravitational fields was also examined in the context of SME. Bailey and Kostelecky (2006) narrowed the possibility of Lorentz injuries to up to 10 ⁻⁹ by examining the apsidal rotation of Mercury, and up to 10 ⁻¹³ by examining solar spin precession. Battat et al . (2007) studied lunar laser ranging data and found no oscillatory changes in the lunar orbit. They reached an upper limit for Lorentz injuries of up to . Iorio (2012) determined limits on the 10 ⁻⁹ level by checking the Keplerian orbital elements of a test particle under the possible influence of Lorentz-violating gravitomagnetic accelerations. Xie (2012) analyzes the advancement of the periastron of double pulsars , with upper limits for Lorentz injuries at the 10 ⁻¹⁰ level. ${\displaystyle (6{,}9\pm 4{,}5)\times 10^{-11}}$

Neutrino tests

Neutrino oscillations

Although neutrino oscillations have been proven experimentally, the theoretical basis is still controversial, which makes it difficult to predict possible deviations from the Lorentz invariance or to interpret it should experimental findings suggest such (such as the problem of sterile neutrinos ). While it is commonly believed that these oscillations prove the presence of a neutrino mass, there are also assumptions that the neutrinos are massless and the oscillations are the product of violations of Lorentz invariance. That is to say, the occurrence of oscillations should already be understood as a Lorentz violation. However, there are already satisfactory Lorentz variant explanatory models that complement the standard model , so that any Lorentz violations are only to be understood as deviations from these models.

Lorentz and CPT violations, for example due to anisotropy of space in the presence of a preferred background, could lead to a sidereal dependence of the probability of the occurrence of neutrino oscillations. Anisotropy measurements could significantly limit the scope for such anisotropies:

Surname	year	SME limits in GeV
Double chooz	2012	${\displaystyle \leq 10^{-20}}$
MINOS	2012	${\displaystyle \leq 10^{-23}}$
MiniBooNE	2012	${\displaystyle \leq 10^{-20}}$
IceCube	2010	${\displaystyle \leq 10^{-23}}$
MINOS	2010	${\displaystyle \leq 10^{-23}}$
MINOS	2008	${\displaystyle \leq 10^{-20}}$
LSND |	2005	${\displaystyle \leq 10^{-19}}$

speed

A series of measurements of the speed of the neutrinos was made. If it is assumed that neutrinos are massless, they would have to move at the speed of light. Since the discovery of neutrino oscillations , however, it has been assumed that they have mass and are therefore (insignificantly) slower than the speed of light. Previous tests show no significant deviations from the speed of light, with an upper limit of . For details see measurements of neutrino velocity . ${\displaystyle (v-c)/c<10^{-9}}$

According to effective field theories such as SME, there are also indirect methods of limiting the speed of neutrinos faster than light, such as the application of the vacuum Cherenkov effect. So here electron-positron generation comes into question, whereby faster than light neutrinos would lose a large part of their energy in a short time. Another effect according to the same model would be the lengthening of the decay times of pions in muons and neutrinos. The absence of these effects considerably limits possible speed differences between light and neutrinos.

Surname	year	energy	SME limits for (vc) / c
			Vacuum Cherenkov	Pioneer disintegration
Steck et al.	2014	1 PeV	${\displaystyle <5{,}6\times 10^{-19}}$
Borriello et al.	2013	1 PeV	${\displaystyle <10^{-18}}$
Cowsik et al.	2012	100 TeV	${\displaystyle <10^{-13}}$
Huo et al.	2012	400 TeV	${\displaystyle <7{,}8\times 10^{-12}}$
ICARUS	2011	17 GeV	${\displaystyle <2{,}5\times 10^{-8}}$
Cowsik et al.	2011	400 TeV		${\displaystyle <10^{-12}}$
Bi et al.	2011	400 TeV		${\displaystyle <10^{-12}}$
Cohen / Glass Show	2011	100 TeV	${\displaystyle <1{,}7\times 10^{-11}}$

In addition, there could also be speed differences between different types of neutrinos. A comparison between muon and electron neutrinos by Sidney Coleman & Sheldon Lee Glashow (1998) gave a negative result, with an upper limit of . ${\displaystyle <6\times 10^{-22}}$

Controversial measurements

LSND, MiniBooNE

In 2001, the LSND group reported a 3.8σ excess of antineutrino interactions in neutrino oscillations that contradict the Standard Model. The first results of the more recent MiniBooNE experiment seemed to refute this, as no excess was measured from neutrino energies of 450 MeV. However, in 2008 an excess of electron-like neutrino events between 200 and 475 MeV was measured. And in 2010, when the experiment was carried out with antineutrinos (as with LSND), an excess of energies of 450 to 1250 MeV could be measured in agreement with the LSND result. Whether these anomalies can be explained by sterile neutrinos or another hypothesis, or whether there is a Lorentz injury, is the subject of further experimental and theoretical investigations.

Solved

In 2011, the OPERA group published in an arXiv preprint measurements of neutrinos that allegedly move faster than light (with 6σ a high significance was given). In the meantime, however, the result has been attributed to measurement errors by the OPERA group and a result that corresponds to the speed of light has been presented. See Neutrino Velocity Measurements for more details .

In 2010 MINOS reported results that there was a 40% difference between the masses of neutrinos and antineutrinos. This would contradict the CPT and Lorentz symmetry. However, in 2011 they corrected their analysis and announced that the effect is not as great as previously assumed. In 2012 a work was finally published in which the difference and thus the anomaly had disappeared.

In 2007 the MAGIC group published a paper according to which they measured a possible energy dependence of the speed of photons from Markarjan 501 . However, they added that energy-dependent effects during emission would be a possible alternative explanation. However, this became irrelevant due to more recent measurements, in particular by the Fermi-LAT-GBM group, which could not determine any effect with far greater accuracy and higher photon energies. See section Vacuum Dispersion .

In 1997, Nodland and Ralston claimed to have measured a birefringence-related rotation of the plane of polarization of light from distant radio galaxies . This indicates an anisotropy of space. This caused a sensation in some media, however, a number of reviews appeared, which rejected this interpretation and pointed to errors in the evaluation. More recent work could not find any signs of such an effect either , see section vacuum birefringence .

Individual evidence

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20. ↑ combined with electron coefficients. ${\displaystyle {\tilde {\kappa }}_{o+}}$
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107. ↑ Greenberg, OW: Remarks on a challenge to the relation between CPT and Lorentz violation . In: Cornell University . August. arxiv : 1105.0927 .
108. ↑ Belle Collaboration: Search for time-dependent CPT violation in hadronic and semileptonic B decays . In: Physical Review D . 85, No. 7, 2012, p. 071105. arxiv : 1203.0930 . doi : 10.1103 / PhysRevD.85.071105 .
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110. ↑ ^{a b} BaBar Collaboration: Search for CPT and Lorentz Violation in B0-Bmacr0 Oscillations with Dilepton Events . In: Physical Review Letters . 100, No. 3, 2008, p. 131802. arxiv : 0711.2713 . doi : 10.1103 / PhysRevLett.100.131802 .
111. ^ Belle Collaboration: Studies of B0-B0 mixing properties with inclusive dilepton events . In: Physical Review D . 67, No. 5, 2003, p. 052004. arxiv : hep-ex / 0212033 . doi : 10.1103 / PhysRevD.67.052004 .
112. ^ FOCUS Collaboration: Charm system tests of CPT and Lorentz invariance with FOCUS . In: Physics Letters B . 556, No. 1-2, 2003, pp. 7-13. arxiv : hep-ex / 0208034 . doi : 10.1016 / S0370-2693 (03) 00103-5 .
113. ↑ KTeV Collaboration: Precise measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system . In: Physical Review D . 83, No. 9, 2011, p. 092001. arxiv : 1011.0127 . doi : 10.1103 / PhysRevD.83.092001 .
114. ↑ KLOE Collaboration: First observation of quantum interference in the process ϕ → KK → ππππ: A test of quantum mechanics and CPT symmetry . In: Physics Letters B . 642, No. 4, 2006, pp. 315-321. arxiv : hep-ex / 0607027 . doi : 10.1016 / j.physletb.2006.09.046 .
115. ↑ CPLEAR Collaboration: Physics at CPLEAR . In: Physics Reports . 374, No. 3, 2003, pp. 165-270. doi : 10.1016 / S0370-1573 (02) 00367-8 .
116. ↑ KTeV Collaboration: Measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system . In: Physical Review D . 67, No. 1, 2003, p. 012005. arxiv : hep-ex / 0208007 . doi : 10.1103 / PhysRevD.67.012005 .
117. ↑ NA31 Collaboration: A measurement of the phases of the CP-violating amplitudes in K0 -> 2π decays and a test of CPT invariance . In: Physics Letters B . 237, No. 2, 1990, pp. 303-312. doi : 10.1016 / 0370-2693 (90) 91448-K .
118. ↑ CDF Collaboration: Measurement of the Mass Difference Between Top and Anti-top Quarks . In: Physical Review D . 87, No. 5, 2012, p. 052013. arxiv : 1210.6131 . doi : 10.1103 / PhysRevD.87.052013 .
119. ^ CMS Collaboration: Measurement of the Mass Difference between Top and Antitop Quarks . In: Journal of High Energy Physics . 2012, p. 109. arxiv : 1204.2807 . doi : 10.1007 / JHEP06 (2012) 109 .
120. ^ D0 Collaboration: Direct Measurement of the Mass Difference between Top and Antitop Quarks . In: Physical Review D . 84, No. 5, 2011, p. 052005. arxiv : 1106.2063 . doi : 10.1103 / PhysRevD.84.052005 .
121. ↑ CDF Collaboration: Measurement of the Mass Difference between t and t¯ Quarks . In: Physical Review Letters . 106, No. 15, 2011, p. 152001. arxiv : 1103.2782 . doi : 10.1103 / PhysRevLett.106.152001 .
122. ^ D0 Collaboration: Direct Measurement of the Mass Difference between Top and Antitop Quarks . In: Physical Review Letters . 103, No. 13, 2009, p. 132001. arxiv : 0906.1172 . doi : 10.1103 / PhysRevLett.103.132001 .
123. ↑ Gabrielse, G. et al. : Precision Mass Spectroscopy of the Antiproton and Proton Using Simultaneously Trapped Particles . In: Physical Review Letters . 82, No. 16, 1999, pp. 3198-3201. doi : 10.1103 / PhysRevLett.82.3198 .
124. ↑ Mittleman, RK; Ioannou, II; Dehmelt, HG; Russell, Neil: Bound on CPT and Lorentz Symmetry with a Trapped Electron . In: Physical Review Letters . 83, No. 1, 1999, pp. 2116-2119. doi : 10.1103 / PhysRevLett.83.2116 .
125. ↑ Dehmelt, H .; Mittleman, R .; van Dyck, RS, Jr .; Schwinberg, P .: Past Electron-Positron g-2 Experiments Yielded Sharpest Bound on CPT Violation for Point Particles . In: Physical Review Letters . 83, No. 23, 1999, pp. 4694-4696. arxiv : hep-ph / 9906262 . doi : 10.1103 / PhysRevLett.83.4694 .
126. ↑ Hughes, VW et al. : Test of CPT and Lorentz Invariance from Muonium Spectroscopy . In: Physical Review Letters . 87, No. 11, 2001, p. 111804. arxiv : hep-ex / 0106103 . doi : 10.1103 / PhysRevLett.87.111804 .
127. ↑ Bennett et al. : Search for Lorentz and CPT Violation Effects in Muon Spin Precession . In: Physical Review Letters . 100, No. 9, 2008, p. 091602. arxiv : 0709.4670 . doi : 10.1103 / PhysRevLett.100.091602 .
128. ↑ Altschul, Brett: Astrophysical limits on Lorentz violation for all charged species . In: Astroparticle Physics . 28, No. 3, 2007, pp. 380-384. arxiv : hep-ph / 0610324 . doi : 10.1016 / j.astropartphys.2007.08.003 .
129. ^ D0 Collaboration: Search for violation of Lorentz invariance in top quark pair production and decay . In: Physical Review Letters . 108, No. 26, 2012, p. 261603. arxiv : 1203.6106 . doi : 10.1103 / PhysRevLett.108.261603 .
130. ↑ Charneski et al : Lorentz violation bounds on Bhabha scattering . In: Physical Review D . 86, No. 4, 2012, p. 045003. arxiv : 1204.0755 . doi : 10.1103 / PhysRevD.86.045003 .
131. ↑ Bailey, Quentin G .; Kostelecký, V. Alan: Signals for Lorentz violation in post-Newtonian gravity . In: Physical Review D . 74, No. 4, 2006, p. 045001. arxiv : gr-qc / 0603030 . doi : 10.1103 / PhysRevD.74.045001 .
132. ↑ Battat, James BR; Chandler, John F .; Stubbs, Christopher W .: Testing for Lorentz Violation: Constraints on Standard-Model-Extension Parameters via Lunar Laser Ranging . In: Physical Review Letters . 99, No. 24, 2007, p. 241103. arxiv : 0710.0702 . bibcode : 2007PhRvL..99x1103B . doi : 10.1103 / PhysRevLett.99.241103 .
133. ^ Iorio, L .: Orbital effects of Lorentz-violating standard model extension gravitomagnetism around a static body: a sensitivity analysis . In: Classical and Quantum Gravity . 29, No. 17, 2012, p. 175007. arxiv : 1203.1859 . doi : 10.1088 / 0264-9381 / 29/17/175007 .
134. ↑ Xie, Yi: Testing Lorentz violation with binary pulsars: constraints on standard model extension . In: Research in Astronomy and Astrophysics . 13, No. 1, 2012, pp. 1-4. arxiv : 1208.0736 . doi : 10.1088 / 1674-4527 / 13/1/001 .
135. ↑ Díaz, Jorge S .; Kostelecký, V. Alan: Lorentz- and CPT-violating models for neutrino oscillations . In: Physical Review D . 85, No. 1, 2012, p. 016013. arxiv : 1108.1799 . doi : 10.1103 / PhysRevD.85.016013 .
136. ^ Double Chooz collaboration: First test of Lorentz violation with a reactor-based antineutrino experiment . In: Physical Review D . 86, No. 11, 2012, p. 112009. arxiv : 1209.5810 . doi : 10.1103 / PhysRevD.86.112009 .
137. ^ MINOS collaboration: Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector . In: Physical Review D . 85, No. 3, 2012, p. 031101. arxiv : 1201.2631 . doi : 10.1103 / PhysRevD.85.031101 .
138. ↑ MiniBooNE Collaboration: Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses . In: Physics Letters B . 718, No. 4, 2012, pp. 1303-1308. arxiv : 1109.3480 . doi : 10.1016 / j.physletb.2012.12.020 .
139. ↑ IceCube Collaboration: Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube . In: Physical Review D . 82, No. 11, 2010, p. 112003. arxiv : 1010.4096 . doi : 10.1103 / PhysRevD.82.112003 .
140. ^ MINOS collaboration: Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector . In: Physical Review Letters . 105, No. 15, 2010, p. 151601. arxiv : 1007.2791 . doi : 10.1103 / PhysRevLett.105.151601 .
141. ↑ MINOS collaboration: Testing Lorentz Invariance and CPT Conservation with NuMI Neutrinos in the MINOS Near Detector . In: Physical Review Letters . 101, No. 15, 2008, p. 151601. arxiv : 0806.4945 . doi : 10.1103 / PhysRevLett.101.151601 .
142. ↑ LSND collaboration: Tests of Lorentz violation in ν¯μ → ν¯e oscillations . In: Physical Review D . 72, No. 7, 2005, p. 076004. arxiv : hep-ex / 0506067 . doi : 10.1103 / PhysRevD.72.076004 .
143. ↑ Mattingly et al. : Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation . In: Journal of Cosmology and Astroparticle Physics . No. 02, 2010, p. 007. arxiv : 0911.0521 . doi : 10.1088 / 1475-7516 / 2010/02/007 .
144. ↑ Kostelecky, Alan; Mewes, Matthew: Neutrinos with Lorentz-violating operators of arbitrary dimension . In: Physical Review D . 85, No. 9, 2012. arxiv : 1112.6395 . doi : 10.1103 / PhysRevD.85.096005 .
145. ↑ Borriello et al. : Stringent constraint on neutrino Lorentz invariance violation from the two IceCube PeV neutrinos . In: Physical Review D . 87, No. 11, 2013, p. 116009. arxiv : 1303.5843 . doi : 10.1103 / PhysRevD.87.116009 .
146. ↑ Cowsik et al. : Testing violations of Lorentz invariance with cosmic rays . In: Physical Review D . 86, No. 4, 2012, p. 045024. arxiv : 1206.0713 . doi : 10.1103 / PhysRevD.86.045024 .
147. ↑ Huo, Yunjie; Li, Tianjun; Liao, Yi; Nanopoulos, Dimitri V .; Qi, Yonghui: Constraints on neutrino velocities revisited . In: Physical Review D . 85, No. 3, 2012, p. 034022. arxiv : 1112.0264 . doi : 10.1103 / PhysRevD.85.034022 .
148. ↑ ICARUS Collaboration: A search for the analogue to Cherenkov radiation by high energy neutrinos at superluminal speeds in ICARUS . In: Physics Letters B . 711, No. 3-4, 2012, pp. 270-275. arxiv : 1110.3763 . doi : 10.1016 / j.physletb.2012.04.014 .
149. ↑ Cowsik, R .; Nussinov, S .; Sarkar, U .: Superluminal neutrinos at OPERA confront pion decay kinematics . In: Physical Review Letters . 107, No. 25, 2011, p. 251801. arxiv : 1110.0241 . bibcode : 2011PhRvL.107y1801C . doi : 10.1103 / PhysRevLett.107.251801 .
150. ↑ Bi, Xiao-Jun; Yin, peng-fei; Yu, Zhao-Huan; Yuan, Qiang: Constraints and tests of the OPERA superluminal neutrinos . In: Physical Review Letters . 107, No. 24, 2011, p. 241802. arxiv : 1109.6667 . bibcode : 2011PhRvL.107x1802B . doi : 10.1103 / PhysRevLett.107.241802 .
151. ↑ Cohen, Andrew G .; Glashow, Sheldon L .: Pair Creation Constrains Superluminal Neutrino Propagation . In: Physical Review Letters . 107, No. 18, 2011, p. 181803. arxiv : 1109.6562 . bibcode : 2011PhRvL.107r1803C . doi : 10.1103 / PhysRevLett.107.181803 .
152. ↑ LSND Collaboration: Evidence for neutrino oscillations from the observation of ν¯e appearance in a ν¯μ beam . In: Physical Review D . 64, No. 11, 2001, p. 112007. arxiv : hep-ex / 0104049 . bibcode : 2001PhRvD..64k2007A . doi : 10.1103 / PhysRevD.64.112007 .
153. ↑ MiniBooNE Collaboration: Search for Electron Neutrino Appearance at the Δm2˜1eV2 Scale . In: Physical Review Letters . 98, No. 23, 2007, p. 231801. arxiv : 0704.1500 . bibcode : 2007PhRvL..98w1801A . doi : 10.1103 / PhysRevLett.98.231801 .
154. ↑ MiniBooNE Collaboration: Unexplained Excess of Electronlike Events from a 1-GeV Neutrino Beam . In: Physical Review Letters . 102, No. 10, 2008, p. 101802. arxiv : 0812.2243 . bibcode : 2009PhRvL.102j1802A . doi : 10.1103 / PhysRevLett.102.101802 .
155. ↑ MiniBooNE results suggest antineutrinos act differently . Fermilab today. June 18, 2010. Retrieved December 14, 2011.
156. ↑ MiniBooNE Collaboration: Event Excess in the MiniBooNE Search for ν¯μ → ν¯e Oscillations . In: Physical Review Letters . 105, No. 18, 2010, p. 181801. arxiv : 1007.1150 . bibcode : 2010PhRvL.105r1801A . doi : 10.1103 / PhysRevLett.105.181801 .
157. ^ Diaz, Jorge S .: Overview of Lorentz Violation in Neutrinos . In: Proceedings of the DPF-2011 Conference . 2011. arxiv : 1109.4620 . bibcode : 2011arXiv1109.4620D .
158. ↑ 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 December 14, 2011.
159. ^ MINOS Collaboration: First Direct Observation of Muon Antineutrino Disappearance . In: Physical Review Letters . 107, No. 2, 2011, p. 021801. arxiv : 1104.0344 . bibcode : 2011PhRvL.107b1801A . doi : 10.1103 / PhysRevLett.107.021801 .
160. ^ MINOS Collaboration: Search for the disappearance of muon antineutrinos in the NuMI neutrino beam . In: Physical Review D . 84, No. 7, 2011, p. 071103. arxiv : 1108.1509 . bibcode : 2011PhRvD..84g1103A . doi : 10.1103 / PhysRevD.84.071103 .
161. ↑ Surprise difference in neutrino and antineutrino mass lessening with new measurements from a Fermilab experiment . Fermilab press release. August 25, 2011. Retrieved December 14, 2011.
162. ↑ MINOS Collaboration: An improved measurement of muon antineutrino disappearance in MINOS . In: Physical Review Letters . 108, No. 19, 2012, p. 191801. arxiv : 1202.2772 . doi : 10.1103 / PhysRevLett.108.191801 .
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165. ↑ Nodland, Borge; Ralston, John P .: Nodland and Ralston Reply: . In: Physical Review Letters . 79, No. 10, 1997, p. 1958. arxiv : astro-ph / 9705190 . doi : 10.1103 / PhysRevLett.79.1958 .
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167. ↑ JP Leahy: Is the Universe Screwy?
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170. ↑ Carroll, Sean M .; Field, George B .: Is There Evidence for Cosmic Anisotropy in the Polarization of Distant Radio Sources? . In: Physical Review Letters . 79, No. 13, 1997, pp. 2394-2397. arxiv : astro-ph / 9704263 . doi : 10.1103 / PhysRevLett.79.2394 .
171. ↑ JP Leahy: (1997) Comment on the Measurement of Cosmological Birefringence , arxiv : astro-ph / 9704285
172. ↑ Wardle et al. : Observational Evidence against Birefringence over Cosmological Distances . In: Physical Review Letters . 79, No. 10, 1997, pp. 1801-1804. arxiv : astro-ph / 9705142 . doi : 10.1103 / PhysRevLett.79.1801 .
173. ↑ Loredo et al. : Bayesian analysis of the polarization of distant radio sources: Limits on cosmological birefringence . In: Physical Review D . 56, No. 12, 1997, pp. 7507-7512. arxiv : astro-ph / 9706258 . doi : 10.1103 / PhysRevD.56.7507 .

Web links

• Roberts, Schleif (2006); Relativity FAQ: What is the experimental basis of special relativity?
• Kostelecký: Background information on Lorentz and CPT violation