2201:
violation, the resonant frequencies may depend on the orientation of the cavity. So, two cavities with different orientations may give different frequencies even if they are otherwise identical. A typical experiment compares the frequencies of two identical cavities oriented at right angles in the laboratory. To distinguish between frequency differences of more conventional origins, such as small defects in the cavities, and
Lorentz violation, the cavities are typically placed on a turntable and rotated in the laboratory. The orientation dependence from Lorentz violation would cause the frequency difference to change as the cavities rotate.
2316:, the effects of Lorentz violation grow with frequency, so optical cavities provide better sensitivity to nonrenormalizable violations, all else being equal. The geometrical symmetries of the cavity also affect the sensitivity since parity symmetric cavities are only directly sensitive to parity-even coefficients for Lorentz violation. Ring resonators provide a complementary class of cavity experiment that can test parity-odd violations. In a ring resonator, two modes propagating in opposites directions in the same ring are compared, rather than modes in two different cavities.
1027:
355:
778:
113:
1022:{\displaystyle {\mathcal {L}}=-\textstyle {1 \over 4}F_{\mu \nu }F^{\mu \nu }+\textstyle {1 \over 2}\epsilon ^{\kappa \lambda \mu \nu }A_{\lambda }{({\hat {k}}_{\mathrm {AF} })}_{\kappa }F_{\mu \nu }-\textstyle {1 \over 4}F_{\kappa \lambda }{({\hat {k}}_{\mathrm {F} })}^{\kappa \lambda \mu \nu }F_{\mu \nu }\,,}
350:{\displaystyle {\mathcal {L}}=-\textstyle {{1} \over {4}}\,F_{\mu \nu }F^{\mu \nu }+\textstyle {{1} \over {2}}\,(k_{\mathrm {AF} })^{\kappa }\,\epsilon _{\kappa \lambda \mu \nu }A^{\lambda }F^{\mu \nu }-\textstyle {{1} \over {4}}\,(k_{\mathrm {F} })_{\kappa \lambda \mu \nu }F^{\kappa \lambda }F^{\mu \nu }.}
2200:
and the geometry of the cavity. The
Lorentz-violating modifications to the Maxwell equations lead to tiny shifts in the resonant frequencies. Experimenters search for these tiny shifts by comparing two or more cavities at different orientations. Since rotation-symmetry violation is a form of Lorentz
1425:
that propagate at slightly different phase velocities. A gradual change in the relative phase results as one of the polarizations outpaces the other. The total polarization (the sum of the two) evolves as the light propagates, in contrast to the
Lorentz-invariant case where the polarization of light
702:
are outside the control of experimenters and can be viewed as constant background fields that fill the entire
Universe, introducing directionality to the otherwise isotropic spacetime. Photons interact with these background fields and experience frame-dependent effects, violating Lorentz invariance.
2181:
While extreme sensitivity to
Lorentz violation is achieved in astrophysical studies, most forms of Lorentz violation have little to no effect on light propagating in a vacuum. These types of violations cannot be tested using astrophysical tests, but can be sought in laboratory-based experiments
3502:
Komatsu, E.; Smith, K. M.; Dunkley, J.; Bennett, C.L.; Gold, B.; Hinshaw, G.; Jarosik, N.; Larson, D.; Nolta, M. R.; Page, L.; Spergel, D.N.; Halpern, M.; Hill, R. S.; Kogut, A.; Limon, M.; Meyer, S.S.; Odegard, N.; Tucker, G.S.; Weiland, J.L.; Wollack, E.; Wright, E.L. (11 January 2011).
3672:
Pagano, Luca; de
Bernardis, Paolo; de Troia, Grazia; Gubitosi, Giulia; Masi, Silvia; Melchiorri, Alessandro; Natoli, Paolo; Piacentini, Francesco; Polenta, Gianluca (24 August 2009). "CMB polarization systematics, cosmological birefringence, and the gravitational waves background".
4619:
Hohensee, Michael A.; Stanwix, Paul L.; Tobar, Michael E.; Parker, Stephen R.; Phillips, David F.; Walsworth, Ronald L. (5 October 2010). "Improved constraints on isotropic shift and anisotropies of the speed of light using rotating cryogenic sapphire oscillators".
1842:
would cause higher-energy photons to run ahead or behind lower-energy photons, leading to otherwise unexplained energy dependence in the arrival time. For two photons of two different energies, the difference in arrival times is approximately given by the ratio
4941:
Müller, Holger; Stanwix, Paul Louis; Tobar, Michael Edmund; Ivanov, Eugene; Wolf, Peter; Herrmann, Sven; Senger, Alexander; Kovalchuk, Evgeny; Peters, Achim (30 July 2007). "Tests of
Relativity by Complementary Rotating Michelson-Morley Experiments".
360:
The first term on the right-hand side is the conventional
Maxwell Lagrangian and gives rise to the usual source-free Maxwell equations. The next term violates both Lorentz and CPT invariance and is constructed from a dimension
4307:; Akhperjanian, A.G.; Barres de Almeida, U.; Bazer-Bachi, A.R.; Becherini, Y.; Behera, B.; et al. (22 October 2008). "Limits on an Energy Dependence of the Speed of Light from a Flare of the Active Galaxy PKS 2155-304".
4177:
Vasileiou, V.; Jacholkowska, A.; Piron, F.; Bolmont, J.; Couturier, C.; Granot, J.; et al. (4 June 2013). "Constraints on
Lorentz invariance violation fromFermi-Large Area Telescope observations of gamma-ray bursts".
4376:
Albert, J.; Aliu, E.; Anderhub, H.; Antonelli, L.A.; Antoranz, P.; et al. (2008). "Probing quantum gravity using photons from a flare of the active galactic nucleus
Markarian 501 observed by the MAGIC telescope".
1358:
1148:
5585:
Reinhardt, Sascha; Saathoff, Guido; Buhr, Henrik; Carlson, Lars A.; Wolf, Andreas; et al. (11 November 2007). "Test of relativistic time dilation with fast optical atomic clocks at different velocities".
5011:
Stanwix, Paul L.; Tobar, Michael E.; Wolf, Peter; Locke, Clayton R.; Ivanov, Eugene N. (4 October 2006). "Improved test of Lorentz invariance in electrodynamics using rotating cryogenic sapphire oscillators".
4876:
4115:
Gubitosi, Giulia; Pagano, Luca; Amelino-Camelia, Giovanni; Melchiorri, Alessandro; Cooray, Asantha (17 August 2009). "A constraint on Planck-scale modifications to electrodynamics with CMB polarization data".
605:
coefficients. Both Lorentz-violating terms are invariant under observer Lorentz transformations, implying that the physics in independent of observer or coordinate choice. However, the coefficient tensors
700:
603:
507:
3274:
Toma, Kenji; Mukohyama, Shinji; Yonetoku, Daisuke; Murakami, Toshio; Gunji, Shuichi; Mihara, Tatehiro; et al. (13 December 2012). "Strict limit on CPT violation from polarization of γ-ray bursts".
3863:
Komatsu, E.; Dunkley, J.; Nolta, M. R.; Bennett, C. L.; Gold, B.; Hinshaw, G.; et al. (1 January 2009). "Five-year Wilkinson Microwave Anisotropy Probe observations: Cosmological interpretation".
1208:
1086:
1834:
can lead to frequency-dependent light speeds. To search for this effect, researchers compare the arrival times of photons from distant sources of pulsed radiation, such as GRB or pulsars. Assuming
1499:
1770:
dependence in the phase shift, implying there is no benefit to considering higher energies. As a result, maximum sensitivity is achieved by studying the most distant source available, the
649:
552:
430:
3985:
Cabella, Paolo; Natoli, Paolo; Silk, Joseph (28 December 2007). "Constraints on CPT violation from Wilkinson Microwave Anisotropy Probe three year polarization data: A wavelet analysis".
4814:
Tobar, Michael E.; Ivanov, Eugene N.; Stanwix, Paul L.; le Floch, Jean-Michel G.; Hartnett, John G. (22 December 2009). "Rotating odd-parity Lorentz invariance test in electrodynamics".
4684:
Bocquet, J.-P.; Moricciani, D.; Bellini, V.; Beretta, M.; Casano, L.; et al. (17 June 2010). "Limits on Light-Speed Anisotropies from Compton Scattering of High-Energy Electrons".
1891:
714:. As a result, many of the effects of Lorentz violation are also seen in light passing through transparent materials. These include changes in the speed that can depend on frequency,
5073:
Wolf, Peter; Bize, Sébastien; Clairon, André; Santarelli, Giorgio; Tobar, Michael E.; Luiten, André N. (15 September 2004). "Improved test of Lorentz invariance in electrodynamics".
28:
in the speed of light emitted by distant astrophysical sources. Due to the extreme distances involved, astrophysical studies have achieved sensitivities on the order of parts in 10.
5455:
Hohensee, Michael A.; Lehnert, Ralf; Phillips, David F.; Walsworth, Ronald L. (1 April 2009). "Particle-Accelerator Constraints on Isotropic Modifications of the Speed of Light".
4753:
Herrmann, S.; Senger, A.; Möhle, K.; Nagel, M.; Kovalchuk, E. V.; Peters, A. (12 November 2009). "Rotating optical cavity experiment testing Lorentz invariance at the 10 level".
3213:
Laurent, P.; Götz, D.; Binétruy, P.; Covino, S.; Fernandez-Soto, A. (28 June 2011). "Constraints on Lorentz invariance violation using integral/IBIS observations of GRB041219A".
5134:
Wolf, Peter; Tobar, Michael E.; Bize, Sébastien; Clairon, André; Luiten, André N.; Santarelli, Giorgio (2004). "Whispering Gallery Resonators and Tests of Lorentz Invariance".
4046:
Feng, Bo; Li, Mingzhe; Xia, Jun-Qing; Chen, Xuelei; Zhang, Xinmin (7 June 2006). "Searching for CPT violation with cosmic microwave background data from WMAP and Boomerang".
1296:
1252:
5256:
Müller, Holger; Herrmann, Sven; Braxmaier, Claus; Schiller, Stephan; Peters, Achim (10 July 2003). "Modern Michelson-Morley Experiment using Cryogenic Optical Resonators".
1936:
is the distance traveled. Sensitivity to Lorentz violation is then increased by considering very distant sources with rapidly changing time profiles. The speed difference
1620:
1957:
1914:
1768:
1748:
1643:
1562:
1522:
5394:
Hohensee, Michael A.; Lehnert, Ralf; Phillips, David F.; Walsworth, Ronald L. (21 August 2009). "Limits on isotropic Lorentz violation in QED from collider physics".
5195:
Müller, Holger; Herrmann, Sven; Saenz, Alejandro; Peters, Achim; Lämmerzahl, Claus (24 December 2003). "Optical cavity tests of Lorentz invariance for the electron".
3810:
Kahniashvili, Tina; Durrer, Ruth; Maravin, Yurii (22 December 2008). "Testing Lorentz invariance violation with Wilkinson Microwave Anisotropy Probe five year data".
1990:
1832:
766:
2314:
2016:
1673:
1588:
2171:
2124:
2097:
1390:
726:, an effect seen in crystals such as calcite. The best constraints on Lorentz violation come from constraints on birefringence in light from astrophysical sources.
101:
2288:
2262:
2236:
1798:
1725:
1699:
456:
385:
2144:
2070:
1934:
1542:
65:
1392:
case, the effects generally grow faster with frequency, due to the additional derivatives. More complex directional dependence typically also arises. Vacuum
772:
sector was constructed in 2009 by Kostelecky and Mewes. It was shown that the more general theory could be written in a form similar to the minimal case,
5325:
Lipa, J. A.; Nissen, J. A.; Wang, S.; Stricker, D. A.; Avaloff, D. (12 February 2003). "New Limit on Signals of Lorentz Violation in Electrodynamics".
4550:
Parker, Stephen R.; Mewes, Matthew; Stanwix, Paul L.; Tobar, Michael E. (3 May 2011). "Cavity Bounds on Higher-Order Lorentz-Violating Coefficients".
2683:
Carroll, Sean M.; Field, George B.; Jackiw, Roman (15 February 1990). "Limits on a Lorentz- and parity-violating modification of electrodynamics".
2353:
5690:
Klinkhamer, F. R.; Schreck, M. (24 October 2008). "New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory".
1417:
Birefringence of light occurs when the solutions to the modified Lorentz-violating Maxwell equations give rise to polarization-dependent speeds.
3736:"Parity violation constraints using cosmic microwave background polarization spectra from 2006 and 2007 observations by the QUaD polarimeter"
2358:
1301:
1091:
16:
Searches for Lorentz violation involving photons provide one possible test of relativity. Examples range from modern versions of the classic
2324:
A number of other searches for Lorentz violation in photons have been performed that do not fall under the above categories. These include
5629:
Klinkhamer, F. R.; Risse, M. (26 June 2008). "Addendum: Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory".
2373:
2028:. Polarization studies typically achieved sensitivities well beyond those achievable through dispersion. As a result, most searches for
4499:
Baynes, Fred N.; Tobar, Michael E.; Luiten, Andre N. (26 June 2012). "Oscillating Test of the Isotropic Shift of the Speed of Light".
3344:
Kostelecký, V. Alan; Mewes, Matthew (2 October 2006). "Sensitive polarimetric search for relativity violations in gamma-ray bursts".
5524:
Altschul, Brett (14 October 2011). "Bounding Lorentz violation at particle colliders by tracking the motion of charged particles".
654:
557:
461:
17:
2956:
Kostelecký, V. Alan; Mewes, Matthew (29 July 2009). "Electrodynamics with Lorentz-violating operators of arbitrary dimension".
1157:
1035:
3423:
Kostelecký, V. Alan; Mewes, Matthew (3 July 2007). "Lorentz-Violating Electrodynamics and the Cosmic Microwave Background".
2805:
Kostelecký, V. Alan; Mewes, Matthew (29 November 2001). "Cosmological Constraints on Lorentz Violation in Electrodynamics".
1454:
4438:
Boggs, Steven E.; Wunderer, C. B.; Hurley, K.; Coburn, W. (20 July 2004). "Testing Lorentz Invariance with GRB 021206".
5822:
2368:
1771:
1675:
Lorentz violation come from polarimetry studies of gamma-ray bursts (GRB). For example, sensitivities of 10 to the
609:
512:
390:
2734:
Jackiw, R.; Kostelecký, V. Alan (3 May 1999). "Radiatively Induced Lorentz and CPT Violation in Electrodynamics".
2204:
Several classes of cavity experiment exist with different sensitivities to different types of Lorentz violation.
2072:. Consequently, the energy dependence in the light speed from nonbirefringent Lorentz violation can be quadratic
5827:
5817:
2348:
2336:
2041:
1401:
735:
104:
68:
41:
37:
1846:
2612:
Kostelecký, V. Alan; Mewes, Matthew (23 September 2002). "Signals for Lorentz violation in electrodynamics".
2470:
Colladay, D.; Kostelecký, V. Alan (26 October 1998). "Lorentz-violating extension of the standard model".
2183:
3152:
Stecker, Floyd W. (2011). "A new limit on Planck scale Lorentz violation from γ-ray burst polarization".
432:. The second term introduces Lorentz violation, but preserves CPT invariance. It consists of a dimension
36:
The most general framework for studies of relativity violations is an effective field theory called the
3924:
Xia, J.-Q.; Li, H.; Wang, X.; Zhang, X. (19 March 2008). "Testing CPT symmetry with CMB measurements".
5770:
5709:
5648:
5595:
5543:
5474:
5413:
5344:
5275:
5214:
5153:
5092:
5031:
4961:
4891:
4833:
4772:
4703:
4639:
4569:
4508:
4457:
4396:
4326:
4252:
4197:
4135:
4065:
4004:
3943:
3882:
3829:
3760:
3692:
3585:
3526:
3442:
3363:
3294:
3232:
3171:
3108:
3040:
2975:
2907:
2824:
2753:
2692:
2631:
2560:
2489:
2428:
2325:
1445:
1441:
1422:
1257:
1213:
715:
2196:
support electromagnetic standing waves that oscillate at well-defined frequencies determined by the
45:
2045:
2033:
2029:
2021:
1839:
1393:
719:
3505:"Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation"
5794:
5760:
5751:
Kostelecký, V. Alan; Russell, Neil (10 March 2011). "Data tables for Lorentz and CPT violation".
5733:
5699:
5672:
5638:
5567:
5533:
5506:
5464:
5437:
5403:
5376:
5334:
5307:
5265:
5238:
5204:
5177:
5143:
5116:
5082:
5055:
5021:
4993:
4951:
4923:
4857:
4823:
4796:
4762:
4735:
4693:
4663:
4629:
4601:
4559:
4481:
4447:
4420:
4386:
4358:
4316:
4304:
4286:
4221:
4187:
4159:
4125:
4097:
4055:
4028:
3994:
3967:
3933:
3906:
3872:
3819:
3792:
3750:
3734:
Wu, E.Y.S.; Ade, P.; Bock, J.; Bowden, M.; Brown, M.L.; Cahill, G.; et al. (21 April 2009).
3716:
3682:
3654:
3634:
3575:
3516:
3484:
3432:
3405:
3353:
3326:
3284:
3256:
3222:
3195:
3161:
3098:
3064:
3030:
2999:
2965:
2897:
2866:
2814:
2787:
2743:
2665:
2621:
2594:
2550:
2523:
2479:
2452:
2418:
2335:
The results of experimental searches of Lorentz invariance violation in the photon sector of the
739:
1597:
3735:
5786:
5725:
5664:
5611:
5559:
5498:
5490:
5429:
5368:
5360:
5299:
5291:
5230:
5169:
5108:
5047:
4985:
4977:
4915:
4907:
4849:
4788:
4727:
4719:
4655:
4593:
4585:
4532:
4524:
4473:
4412:
4350:
4342:
4278:
4270:
4213:
4151:
4089:
4081:
4020:
3959:
3898:
3845:
3784:
3776:
3708:
3603:
3544:
3476:
3468:
3397:
3389:
3318:
3310:
3248:
3187:
3134:
3126:
3056:
2991:
2925:
2858:
2850:
2779:
2716:
2708:
2657:
2586:
2515:
2444:
2197:
2193:
2187:
1939:
1896:
1753:
1730:
1625:
1547:
1504:
21:
1962:
1811:
745:
742:
and curved spacetimes. It also includes operators of arbitrary (nonrenormalizable) dimension
5778:
5717:
5656:
5603:
5551:
5482:
5421:
5352:
5283:
5222:
5161:
5100:
5039:
4969:
4899:
4841:
4780:
4711:
4647:
4577:
4516:
4465:
4404:
4334:
4260:
4205:
4143:
4073:
4012:
3951:
3947:
3890:
3837:
3768:
3700:
3644:
3593:
3534:
3458:
3450:
3379:
3371:
3302:
3240:
3179:
3116:
3048:
2983:
2915:
2840:
2832:
2769:
2761:
2700:
2647:
2639:
2576:
2568:
2505:
2497:
2436:
2363:
2293:
2209:
2186:. The primary examples are the modern Michelson-Morley experiments based on electromagnetic
1995:
1652:
1567:
2149:
2102:
2075:
1363:
74:
71:
is the minimal SME, which limits attention to operators of renormalizable mass-dimension,
5165:
2541:
Kostelecký, V. Alan (17 May 2004). "Gravity, Lorentz violation, and the standard model".
2409:
Colladay, Don; Kostelecký, V. Alan (1 May 1997). "CPT violation and the standard model".
2267:
2241:
2215:
1777:
1704:
1678:
435:
364:
5774:
5713:
5652:
5599:
5547:
5478:
5417:
5348:
5279:
5218:
5157:
5096:
5035:
4965:
4895:
4837:
4776:
4707:
4643:
4573:
4512:
4461:
4400:
4330:
4256:
4201:
4147:
4139:
4069:
4008:
3886:
3833:
3764:
3696:
3589:
3530:
3446:
3367:
3298:
3236:
3183:
3175:
3112:
3044:
2979:
2911:
2828:
2757:
2696:
2635:
2564:
2493:
2432:
3019:"Is there evidence for cosmic anisotropy in the polarization of distant radio sources?"
2129:
2055:
1919:
1527:
50:
3894:
3649:
3622:
2190:, which have achieved sensitivities on the order of parts in 10 to Lorentz violation.
5811:
5737:
5676:
5571:
4861:
4800:
4225:
4032:
3910:
3720:
3539:
3504:
3199:
3003:
2791:
2329:
2049:
2037:
2025:
1646:
1397:
1360:. While the nonrenormalizable terms give many of the same types of signatures as the
723:
5510:
5380:
5311:
5242:
5120:
4997:
4927:
4739:
4605:
4485:
4362:
4163:
4101:
3796:
3488:
3330:
3260:
3068:
2870:
2669:
2598:
2385:
5798:
5486:
5441:
5181:
5059:
4903:
4715:
4667:
4581:
4520:
4424:
4408:
4338:
4290:
3971:
3772:
3658:
3598:
3563:
3409:
3306:
3121:
3086:
2527:
2456:
1151:
5356:
5287:
4973:
4077:
3955:
3454:
3375:
2836:
1800:
coefficients for Lorentz violation from the CMB currently stand at around 10 GeV.
1433:), birefringence causes a simple rotation of the polarization. The CPT-even case (
1451:
The quantity determining the size of the effect is the change in relative phase,
3623:"Improved measurements of the temperature and polarization of the CMB from QUaD"
3052:
2765:
711:
5721:
5660:
5555:
5425:
5226:
5104:
5043:
4845:
4784:
4651:
4209:
4016:
3841:
3704:
3244:
2987:
2643:
2572:
2501:
718:, and direction of propagation. Consequently, Lorentz violation can introduce
5790:
5782:
5729:
5668:
5615:
5563:
5494:
5433:
5364:
5295:
5234:
5173:
5112:
5051:
4981:
4911:
4853:
4792:
4723:
4659:
4589:
4528:
4477:
4416:
4346:
4217:
4155:
4085:
4024:
3963:
3902:
3849:
3780:
3712:
3607:
3548:
3472:
3393:
3314:
3252:
3191:
3130:
3060:
2995:
2929:
2854:
2783:
2712:
2704:
2661:
2590:
2519:
2448:
2440:
2238:
violations. Microwave experiments have also placed some bounds on nonminimal
4265:
4240:
2205:
5502:
5372:
5303:
4989:
4919:
4731:
4597:
4536:
4354:
4282:
4093:
3788:
3480:
3401:
3322:
3138:
2862:
2720:
4452:
4060:
3437:
3035:
5339:
5270:
2126:
or any other even power of energy. Odd powers of energy, such as linear
5209:
5087:
4241:"Fermi observations of high-energy gamma-ray emission from GRB 080916C"
3358:
2819:
2748:
2626:
2555:
2484:
2423:
1835:
1591:
1353:{\displaystyle {({\hat {k}}_{\mathrm {F} })}^{\kappa \lambda \mu \nu }}
1143:{\displaystyle {({\hat {k}}_{\mathrm {F} })}^{\kappa \lambda \mu \nu }}
707:
5148:
5026:
4877:"Laboratory Test of the Isotropy of Light Propagation at the 10 Level"
4274:
3463:
3384:
2845:
2774:
2652:
2581:
2510:
1992:, so higher-energy sources provide better sensitivity to effects from
1400:
is another feature that is found, which does not arise in the minimal
5607:
769:
458:
operator contracted with constant coefficients for Lorentz violation
3018:
1590:, the highest sensitivities are achieved by considering high-energy
4469:
4239:
Abdo, A.A.; et al. (Fermi LAT and GBM Collaborations) (2009).
2920:
2885:
5765:
5704:
5643:
5538:
5469:
5408:
4956:
4828:
4767:
4698:
4634:
4564:
4391:
4321:
4192:
4130:
3999:
3938:
3877:
3824:
3755:
3687:
3639:
3580:
3521:
3289:
3227:
3166:
3103:
2970:
2902:
2390:
2339:
are summarized in the Data Tables for Lorentz and CPT violation.
1418:
1426:
remains fixed when propagating in a vacuum. In the CPT-odd case (
2886:"Astrophysical tests of Lorentz and CPT violation with photons"
1838:
of all energies are produced within a narrow window of time,
4875:
Eisele, Ch.; Nevsky, A. Yu.; Schiller, S. (25 August 2009).
3087:"Constraints on Relativity Violations from Gamma-Ray Bursts"
784:
695:{\displaystyle (k_{\mathrm {F} })_{\kappa \lambda \mu \nu }}
598:{\displaystyle (k_{\mathrm {F} })_{\kappa \lambda \mu \nu }}
502:{\displaystyle (k_{\mathrm {F} })_{\kappa \lambda \mu \nu }}
119:
1701:
coefficients for Lorentz violation have been achieved. For
722:
in light propagating in empty space. It can also introduce
3564:"Probing CPT violation with CMB polarization measurements"
1032:
where the constant coefficients are promoted to operators
387:
operator and a constant coefficient for Lorentz violation
5594:(12). Springer Science and Business Media LLC: 861–864.
3017:
Carroll, Sean M.; Field, George B. (29 September 1997).
2884:
Kostelecký, V. Alan; Mewes, Matthew (13 November 2008).
1203:{\displaystyle {({\hat {k}}_{\mathrm {AF} })}_{\kappa }}
1081:{\displaystyle {({\hat {k}}_{\mathrm {AF} })}_{\kappa }}
1594:
from distant sources, giving large values to the ratio
710:
is similar to that of conventional electromagnetism in
3621:
Brown, M.L.; et al. (QUaD Collaboration) (2009).
1494:{\displaystyle \Delta \phi =2\pi \Delta v\,t/\lambda }
930:
835:
795:
265:
175:
130:
2296:
2270:
2244:
2218:
2152:
2132:
2105:
2078:
2058:
1998:
1965:
1942:
1922:
1899:
1849:
1814:
1780:
1756:
1733:
1707:
1681:
1655:
1628:
1600:
1570:
1550:
1530:
1507:
1457:
1366:
1304:
1260:
1216:
1160:
1094:
1038:
781:
748:
657:
612:
560:
515:
464:
438:
393:
367:
116:
77:
53:
3085:Kostelecký, V. Alan; Mewes, Matthew (14 May 2013).
2386:
Background information on Lorentz and CPT violation
3221:(12). American Physical Society (APS): 121301(R).
2308:
2282:
2256:
2230:
2165:
2138:
2118:
2091:
2064:
2010:
1984:
1951:
1928:
1908:
1885:
1826:
1792:
1762:
1742:
1719:
1693:
1667:
1637:
1614:
1582:
1556:
1536:
1516:
1493:
1384:
1352:
1290:
1246:
1202:
1142:
1080:
1021:
760:
694:
643:
597:
546:
501:
450:
424:
379:
349:
95:
59:
2691:(4). American Physical Society (APS): 1231–1240.
1750:is proportional to the wavelength, canceling the
107:, photons are governed by the Lagrangian density
2052:can only arise from operators of even dimension
1421:propagates as the combination of two orthogonal
706:The mathematics describing Lorentz violation in
67:. To date, the most widely studied limit of the
4822:(12). American Physical Society (APS): 125024.
4507:(26). American Physical Society (APS): 260801.
4186:(12). American Physical Society (APS): 122001.
3562:Xia, Jun-Qing; Li, Hong; Zhang, Xinmin (2010).
3283:(24). American Physical Society (APS): 241104.
2018:Lorentz violation, making GRB an ideal source.
5081:(5). American Physical Society (APS): 051902.
4890:(9). American Physical Society (APS): 090401.
4118:Journal of Cosmology and Astroparticle Physics
5759:(1). American Physical Society (APS): 11–31.
3080:
3078:
644:{\displaystyle (k_{\mathrm {AF} })^{\kappa }}
547:{\displaystyle (k_{\mathrm {AF} })^{\kappa }}
425:{\displaystyle (k_{\mathrm {AF} })^{\kappa }}
8:
1916:is the difference in the group velocity and
730:Nonminimal Lorentz-violating electrodynamics
3865:The Astrophysical Journal Supplement Series
3509:The Astrophysical Journal Supplement Series
20:that utilize highly stable electromagnetic
4679:
4677:
2173:, do not arise in effective field theory.
40:(SME). Lorentz-violating operators in the
5764:
5703:
5642:
5537:
5468:
5407:
5338:
5269:
5208:
5147:
5086:
5025:
4955:
4827:
4766:
4697:
4633:
4563:
4451:
4390:
4320:
4264:
4191:
4129:
4059:
3998:
3937:
3876:
3823:
3754:
3686:
3648:
3638:
3597:
3579:
3538:
3520:
3462:
3436:
3383:
3357:
3288:
3226:
3165:
3120:
3102:
3034:
2969:
2919:
2901:
2844:
2818:
2773:
2747:
2651:
2625:
2580:
2554:
2509:
2483:
2422:
2391:Data Tables for Lorentz and CPT Violation
2295:
2269:
2243:
2217:
2157:
2151:
2131:
2110:
2104:
2083:
2077:
2057:
2032:focus on Lorentz violation that leads to
1997:
1970:
1964:
1941:
1921:
1898:
1877:
1868:
1848:
1813:
1779:
1755:
1732:
1706:
1680:
1654:
1627:
1604:
1599:
1569:
1549:
1529:
1506:
1483:
1479:
1456:
1365:
1335:
1324:
1323:
1312:
1311:
1306:
1303:
1259:
1215:
1194:
1180:
1179:
1168:
1167:
1162:
1159:
1125:
1114:
1113:
1102:
1101:
1096:
1093:
1072:
1058:
1057:
1046:
1045:
1040:
1037:
1012:
1003:
984:
973:
972:
961:
960:
955:
945:
931:
918:
908:
894:
893:
882:
881:
876:
869:
850:
836:
823:
810:
796:
783:
782:
780:
747:
677:
666:
665:
656:
635:
621:
620:
611:
580:
569:
568:
559:
538:
524:
523:
514:
509:. There are a total of four independent
484:
473:
472:
463:
437:
416:
402:
401:
392:
366:
332:
319:
300:
289:
288:
280:
273:
268:
266:
253:
243:
224:
219:
213:
199:
198:
190:
183:
178:
176:
163:
150:
145:
138:
133:
131:
118:
117:
115:
76:
52:
32:Minimal Lorentz-violating electrodynamics
2951:
2949:
2947:
2945:
2943:
2941:
2939:
1886:{\displaystyle \Delta t=\Delta vL/c^{2}}
103:, in flat spacetime. Within the minimal
2401:
2354:Lorentz-violating neutrino oscillations
2359:Antimatter tests of Lorentz violation
1440:) gives more complicated behavior as
1254:terms, while the CPT-even terms with
24:to searches for tiny deviations from
7:
3184:10.1016/j.astropartphys.2011.06.007
2374:Test theories of special relativity
1524:is the difference in phase speeds,
5166:10.1023/b:gerg.0000046188.87741.51
5136:General Relativity and Gravitation
1943:
1900:
1859:
1850:
1734:
1629:
1508:
1473:
1458:
1325:
1210:operator contains all the CPT-odd
1184:
1181:
1115:
1062:
1059:
974:
898:
895:
667:
625:
622:
570:
528:
525:
474:
406:
403:
290:
203:
200:
14:
2024:may or may not be accompanied by
1645:. The best constraints on vacuum
1622:that enhance the sensitivity to
1291:{\displaystyle d=4,6,8,\ldots }
1247:{\displaystyle d=3,5,7,\ldots }
5487:10.1103/physrevlett.102.170402
4904:10.1103/physrevlett.103.090401
4716:10.1103/physrevlett.104.241601
4582:10.1103/physrevlett.106.180401
4521:10.1103/physrevlett.108.260801
4446:(2). IOP Publishing: L77–L80.
4409:10.1016/j.physletb.2008.08.053
4339:10.1103/physrevlett.101.170402
3773:10.1103/physrevlett.102.161302
3599:10.1016/j.physletb.2010.03.038
3307:10.1103/physrevlett.109.241104
3122:10.1103/physrevlett.110.201601
1331:
1317:
1307:
1190:
1173:
1163:
1154:in spacetime derivatives. The
1121:
1107:
1097:
1068:
1051:
1041:
980:
966:
956:
904:
887:
877:
768:. The general gauge-invariant
674:
658:
632:
613:
577:
561:
535:
516:
481:
465:
413:
394:
297:
281:
210:
191:
1:
5357:10.1103/physrevlett.90.060403
5288:10.1103/physrevlett.91.020401
4974:10.1103/physrevlett.99.050401
4148:10.1088/1475-7516/2009/08/021
4078:10.1103/physrevlett.96.221302
3455:10.1103/physrevlett.99.011601
3376:10.1103/physrevlett.97.140401
2837:10.1103/physrevlett.87.251304
1544:is the propagation time, and
3926:Astronomy & Astrophysics
2896:(1). IOP Publishing: L1–L4.
2212:have been used to constrain
3956:10.1051/0004-6361:200809410
3895:10.1088/0067-0049/180/2/330
3650:10.1088/0004-637X/705/1/978
3053:10.1103/physrevlett.79.2394
2766:10.1103/physrevlett.82.3572
2369:Tests of special relativity
1772:cosmic microwave background
18:Michelson–Morley experiment
5844:
5722:10.1103/physrevd.78.085026
5661:10.1103/physrevd.77.117901
5556:10.1103/physrevd.84.076006
5426:10.1103/physrevd.80.036010
5227:10.1103/physrevd.68.116006
5105:10.1103/physrevd.70.051902
5044:10.1103/physrevd.74.081101
4846:10.1103/physrevd.80.125024
4785:10.1103/physrevd.80.105011
4652:10.1103/physrevd.82.076001
4210:10.1103/physrevd.87.122001
4017:10.1103/physrevd.76.123014
3842:10.1103/physrevd.78.123009
3705:10.1103/physrevd.80.043522
3540:10.1088/0067-0049/192/2/18
3245:10.1103/physrevd.83.121301
2988:10.1103/physrevd.80.015020
2644:10.1103/physrevd.66.056005
2573:10.1103/physrevd.69.105009
2502:10.1103/physrevd.58.116002
2332:, and threshold analyses.
1727:, the velocity difference
1615:{\displaystyle t/\lambda }
1446:elliptically polarizations
554:coefficients and nineteen
5753:Reviews of Modern Physics
4440:The Astrophysical Journal
2890:The Astrophysical Journal
2290:violations. However, for
1150:, which take the form of
5783:10.1103/revmodphys.83.11
2705:10.1103/physrevd.41.1231
2441:10.1103/physrevd.55.6760
2349:Standard-Model Extension
1952:{\displaystyle \Delta v}
1909:{\displaystyle \Delta v}
1763:{\displaystyle \lambda }
1743:{\displaystyle \Delta v}
1638:{\displaystyle \Delta v}
1557:{\displaystyle \lambda }
1517:{\displaystyle \Delta v}
1442:linearly polarized light
44:are classified by their
38:Standard-Model Extension
5457:Physical Review Letters
5327:Physical Review Letters
5258:Physical Review Letters
4944:Physical Review Letters
4884:Physical Review Letters
4686:Physical Review Letters
4552:Physical Review Letters
4501:Physical Review Letters
4309:Physical Review Letters
4266:10.1126/science.1169101
4048:Physical Review Letters
3948:2008A&A...483..715X
3743:Physical Review Letters
3425:Physical Review Letters
3346:Physical Review Letters
3277:Physical Review Letters
3091:Physical Review Letters
3023:Physical Review Letters
2807:Physical Review Letters
2736:Physical Review Letters
1985:{\displaystyle E^{d-4}}
1827:{\displaystyle d\neq 4}
1808:Lorentz violation with
1564:is the wavelength. For
761:{\displaystyle d\geq 5}
2310:
2309:{\displaystyle d>4}
2284:
2258:
2232:
2184:electromagnetic fields
2167:
2140:
2120:
2093:
2066:
2012:
2011:{\displaystyle d>4}
1986:
1953:
1930:
1910:
1887:
1828:
1794:
1774:(CMB). Constraints on
1764:
1744:
1721:
1695:
1669:
1668:{\displaystyle d>3}
1639:
1616:
1584:
1583:{\displaystyle d>3}
1558:
1538:
1518:
1495:
1386:
1354:
1292:
1248:
1204:
1144:
1082:
1023:
762:
696:
645:
599:
548:
503:
452:
426:
381:
351:
97:
61:
3154:Astroparticle Physics
2311:
2285:
2259:
2233:
2168:
2166:{\displaystyle E^{3}}
2141:
2121:
2119:{\displaystyle E^{4}}
2094:
2092:{\displaystyle E^{2}}
2067:
2013:
1987:
1954:
1931:
1911:
1888:
1829:
1795:
1765:
1745:
1722:
1696:
1670:
1640:
1617:
1585:
1559:
1539:
1519:
1496:
1387:
1385:{\displaystyle d=3,4}
1355:
1293:
1249:
1205:
1145:
1083:
1024:
763:
697:
646:
600:
549:
504:
453:
427:
382:
352:
98:
96:{\displaystyle d=3,4}
62:
2294:
2268:
2242:
2216:
2150:
2130:
2103:
2076:
2056:
1996:
1963:
1940:
1920:
1897:
1847:
1812:
1778:
1754:
1731:
1705:
1679:
1653:
1626:
1598:
1568:
1548:
1528:
1505:
1455:
1413:Vacuum birefringence
1364:
1302:
1258:
1214:
1158:
1092:
1036:
779:
746:
655:
610:
558:
513:
462:
436:
391:
365:
114:
75:
51:
5775:2011RvMP...83...11K
5714:2008PhRvD..78h5026K
5653:2008PhRvD..77k7901K
5600:2007NatPh...3..861R
5548:2011PhRvD..84g6006A
5479:2009PhRvL.102q0402H
5418:2009PhRvD..80c6010H
5349:2003PhRvL..90f0403L
5280:2003PhRvL..91b0401M
5219:2003PhRvD..68k6006M
5158:2004GReGr..36.2351W
5097:2004PhRvD..70e1902W
5036:2006PhRvD..74h1101S
4966:2007PhRvL..99e0401M
4896:2009PhRvL.103i0401E
4838:2009PhRvD..80l5024T
4777:2009PhRvD..80j5011H
4708:2010PhRvL.104x1601B
4644:2010PhRvD..82g6001H
4574:2011PhRvL.106r0401P
4513:2012PhRvL.108z0801B
4462:2004ApJ...611L..77B
4401:2008PhLB..668..253M
4331:2008PhRvL.101q0402A
4257:2009Sci...323.1688A
4202:2013PhRvD..87l2001V
4140:2009JCAP...08..021G
4070:2006PhRvL..96v1302F
4009:2007PhRvD..76l3014C
3887:2009ApJS..180..330K
3834:2008PhRvD..78l3009K
3765:2009PhRvL.102p1302W
3697:2009PhRvD..80d3522P
3590:2010PhLB..687..129X
3531:2011ApJS..192...18K
3447:2007PhRvL..99a1601K
3368:2006PhRvL..97n0401K
3299:2012PhRvL.109x1104T
3237:2011PhRvD..83l1301L
3176:2011APh....35...95S
3113:2013PhRvL.110t1601K
3045:1997PhRvL..79.2394C
2980:2009PhRvD..80a5020K
2912:2008ApJ...689L...1K
2829:2001PhRvL..87y1304K
2758:1999PhRvL..82.3572J
2697:1990PhRvD..41.1231C
2636:2002PhRvD..66e6005K
2565:2004PhRvD..69j5009K
2494:1998PhRvD..58k6002C
2433:1997PhRvD..55.6760C
2328:based experiments,
2283:{\displaystyle d=8}
2257:{\displaystyle d=6}
2231:{\displaystyle d=4}
1793:{\displaystyle d=3}
1720:{\displaystyle d=3}
1694:{\displaystyle d=4}
451:{\displaystyle d=4}
380:{\displaystyle d=3}
5823:Special relativity
2306:
2280:
2254:
2228:
2163:
2136:
2116:
2089:
2062:
2008:
1982:
1949:
1926:
1906:
1883:
1824:
1790:
1760:
1740:
1717:
1691:
1665:
1635:
1612:
1580:
1554:
1534:
1514:
1491:
1382:
1350:
1288:
1244:
1200:
1140:
1078:
1019:
1018:
1017:
1016:
758:
740:general relativity
692:
641:
595:
544:
499:
448:
422:
377:
347:
346:
345:
344:
93:
57:
5692:Physical Review D
5631:Physical Review D
5526:Physical Review D
5396:Physical Review D
5197:Physical Review D
5142:(10): 2351–2372.
5075:Physical Review D
5014:Physical Review D
4816:Physical Review D
4755:Physical Review D
4622:Physical Review D
4379:Physics Letters B
4251:(5922): 1688–93.
4180:Physical Review D
3987:Physical Review D
3812:Physical Review D
3675:Physical Review D
3568:Physics Letters B
3215:Physical Review D
3029:(13): 2394–2397.
2958:Physical Review D
2742:(18): 3572–3575.
2685:Physical Review D
2614:Physical Review D
2543:Physical Review D
2472:Physical Review D
2417:(11): 6760–6774.
2411:Physical Review D
2320:Other experiments
2198:Maxwell equations
2194:Resonant cavities
2188:resonant cavities
2177:Resonant cavities
2139:{\displaystyle E}
2065:{\displaystyle d}
1929:{\displaystyle L}
1804:Vacuum dispersion
1537:{\displaystyle t}
1396:of light without
1320:
1176:
1110:
1054:
969:
939:
890:
844:
804:
278:
188:
143:
60:{\displaystyle d}
22:resonant cavities
5835:
5803:
5802:
5768:
5748:
5742:
5741:
5707:
5687:
5681:
5680:
5646:
5626:
5620:
5619:
5608:10.1038/nphys778
5582:
5576:
5575:
5541:
5521:
5515:
5514:
5472:
5452:
5446:
5445:
5411:
5391:
5385:
5384:
5342:
5322:
5316:
5315:
5273:
5253:
5247:
5246:
5212:
5192:
5186:
5185:
5151:
5131:
5125:
5124:
5090:
5070:
5064:
5063:
5029:
5020:(8): 081101(R).
5008:
5002:
5001:
4959:
4938:
4932:
4931:
4881:
4872:
4866:
4865:
4831:
4811:
4805:
4804:
4770:
4750:
4744:
4743:
4701:
4681:
4672:
4671:
4637:
4616:
4610:
4609:
4567:
4547:
4541:
4540:
4496:
4490:
4489:
4455:
4453:astro-ph/0310307
4435:
4429:
4428:
4394:
4373:
4367:
4366:
4324:
4301:
4295:
4294:
4268:
4236:
4230:
4229:
4195:
4174:
4168:
4167:
4133:
4112:
4106:
4105:
4063:
4061:astro-ph/0601095
4043:
4037:
4036:
4002:
3982:
3976:
3975:
3941:
3921:
3915:
3914:
3880:
3860:
3854:
3853:
3827:
3807:
3801:
3800:
3758:
3740:
3731:
3725:
3724:
3690:
3669:
3663:
3662:
3652:
3642:
3618:
3612:
3611:
3601:
3583:
3574:(2–3): 129–132.
3559:
3553:
3552:
3542:
3524:
3499:
3493:
3492:
3466:
3440:
3438:astro-ph/0702379
3420:
3414:
3413:
3387:
3361:
3341:
3335:
3334:
3292:
3271:
3265:
3264:
3230:
3210:
3204:
3203:
3169:
3149:
3143:
3142:
3124:
3106:
3082:
3073:
3072:
3038:
3036:astro-ph/9704263
3014:
3008:
3007:
2973:
2953:
2934:
2933:
2923:
2905:
2881:
2875:
2874:
2848:
2822:
2802:
2796:
2795:
2777:
2751:
2731:
2725:
2724:
2680:
2674:
2673:
2655:
2629:
2609:
2603:
2602:
2584:
2558:
2538:
2532:
2531:
2513:
2487:
2467:
2461:
2460:
2426:
2406:
2364:Bumblebee models
2315:
2313:
2312:
2307:
2289:
2287:
2286:
2281:
2263:
2261:
2260:
2255:
2237:
2235:
2234:
2229:
2210:optical cavities
2172:
2170:
2169:
2164:
2162:
2161:
2145:
2143:
2142:
2137:
2125:
2123:
2122:
2117:
2115:
2114:
2098:
2096:
2095:
2090:
2088:
2087:
2071:
2069:
2068:
2063:
2017:
2015:
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386:
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356:
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327:
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314:
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295:
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279:
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63:
58:
5843:
5842:
5838:
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5832:
5828:Hendrik Lorentz
5818:Electrodynamics
5808:
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5750:
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5689:
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5340:physics/0302093
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5271:physics/0305117
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3207:
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3146:
3084:
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3076:
3016:
3015:
3011:
2955:
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2937:
2883:
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2878:
2804:
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2799:
2733:
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2728:
2682:
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2611:
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2606:
2540:
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2266:
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2240:
2239:
2214:
2213:
2179:
2153:
2148:
2147:
2128:
2127:
2106:
2101:
2100:
2079:
2074:
2073:
2054:
2053:
1994:
1993:
1966:
1961:
1960:
1938:
1937:
1918:
1917:
1895:
1894:
1873:
1845:
1844:
1810:
1809:
1806:
1776:
1775:
1752:
1751:
1729:
1728:
1703:
1702:
1677:
1676:
1651:
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1624:
1623:
1596:
1595:
1566:
1565:
1546:
1545:
1526:
1525:
1503:
1502:
1453:
1452:
1434:
1427:
1415:
1410:
1362:
1361:
1310:
1305:
1300:
1299:
1256:
1255:
1212:
1211:
1166:
1161:
1156:
1155:
1100:
1095:
1090:
1089:
1044:
1039:
1034:
1033:
999:
959:
954:
941:
914:
880:
875:
865:
846:
819:
806:
777:
776:
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743:
732:
673:
661:
653:
652:
631:
616:
608:
607:
576:
564:
556:
555:
534:
519:
511:
510:
480:
468:
460:
459:
434:
433:
412:
397:
389:
388:
363:
362:
328:
315:
296:
284:
249:
239:
220:
209:
194:
159:
146:
112:
111:
73:
72:
49:
48:
34:
12:
11:
5:
5841:
5839:
5831:
5830:
5825:
5820:
5810:
5809:
5805:
5804:
5743:
5682:
5637:(11): 117901.
5621:
5588:Nature Physics
5577:
5516:
5463:(17): 170402.
5447:
5386:
5317:
5248:
5210:hep-ph/0401016
5203:(11): 116006.
5187:
5126:
5088:hep-ph/0407232
5065:
5003:
4933:
4867:
4806:
4761:(10): 105011.
4745:
4692:(24): 241601.
4673:
4611:
4558:(18): 180401.
4542:
4491:
4470:10.1086/423933
4430:
4385:(4): 253–257.
4368:
4315:(17): 170402.
4296:
4231:
4169:
4107:
4054:(22): 221302.
4038:
3993:(12): 123014.
3977:
3932:(3): 715–718.
3916:
3871:(2): 330–376.
3855:
3818:(12): 123009.
3802:
3749:(16): 161302.
3726:
3664:
3613:
3554:
3494:
3415:
3359:hep-ph/0607084
3352:(14): 140401.
3336:
3266:
3205:
3144:
3097:(20): 201601.
3074:
3009:
2935:
2921:10.1086/595815
2876:
2820:hep-ph/0111026
2813:(25): 251304.
2797:
2749:hep-ph/9901358
2726:
2675:
2627:hep-ph/0205211
2604:
2556:hep-th/0312310
2549:(10): 105009.
2533:
2485:hep-ph/9809521
2478:(11): 116002.
2462:
2424:hep-ph/9703464
2400:
2398:
2395:
2394:
2393:
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2380:External links
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156:
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136:
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126:
121:
92:
89:
86:
83:
80:
56:
46:mass dimension
33:
30:
13:
10:
9:
6:
4:
3:
2:
5840:
5829:
5826:
5824:
5821:
5819:
5816:
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5800:
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5698:(8): 085026.
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5402:(3): 036010.
5401:
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5333:(6): 060403.
5332:
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5301:
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5293:
5289:
5285:
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5264:(2): 020401.
5263:
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5240:
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5232:
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5191:
5188:
5183:
5179:
5175:
5171:
5167:
5163:
5159:
5155:
5150:
5149:gr-qc/0401017
5145:
5141:
5137:
5130:
5127:
5122:
5118:
5114:
5110:
5106:
5102:
5098:
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5084:
5080:
5076:
5069:
5066:
5061:
5057:
5053:
5049:
5045:
5041:
5037:
5033:
5028:
5027:gr-qc/0609072
5023:
5019:
5015:
5007:
5004:
4999:
4995:
4991:
4987:
4983:
4979:
4975:
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4967:
4963:
4958:
4953:
4950:(5): 050401.
4949:
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4925:
4921:
4917:
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4628:(7): 076001.
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4306:
4305:Aharonian, F.
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3681:(4): 043522.
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3431:(1): 011601.
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3369:
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3360:
3355:
3351:
3347:
3340:
3337:
3332:
3328:
3324:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3291:
3286:
3282:
3278:
3270:
3267:
3262:
3258:
3254:
3250:
3246:
3242:
3238:
3234:
3229:
3224:
3220:
3216:
3209:
3206:
3201:
3197:
3193:
3189:
3185:
3181:
3177:
3173:
3168:
3163:
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3148:
3145:
3140:
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3132:
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3123:
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3114:
3110:
3105:
3100:
3096:
3092:
3088:
3081:
3079:
3075:
3070:
3066:
3062:
3058:
3054:
3050:
3046:
3042:
3037:
3032:
3028:
3024:
3020:
3013:
3010:
3005:
3001:
2997:
2993:
2989:
2985:
2981:
2977:
2972:
2967:
2964:(1): 015020.
2963:
2959:
2952:
2950:
2948:
2946:
2944:
2942:
2940:
2936:
2931:
2927:
2922:
2917:
2913:
2909:
2904:
2899:
2895:
2891:
2887:
2880:
2877:
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2868:
2864:
2860:
2856:
2852:
2847:
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2838:
2834:
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2808:
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2793:
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2737:
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2727:
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2667:
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2633:
2628:
2623:
2620:(5): 056005.
2619:
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2608:
2605:
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2592:
2588:
2583:
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2360:
2357:
2355:
2352:
2350:
2347:
2346:
2342:
2340:
2338:
2333:
2331:
2330:atomic clocks
2327:
2319:
2317:
2303:
2300:
2297:
2277:
2274:
2271:
2251:
2248:
2245:
2225:
2222:
2219:
2211:
2207:
2202:
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2195:
2191:
2189:
2185:
2176:
2174:
2158:
2154:
2133:
2111:
2107:
2084:
2080:
2059:
2051:
2050:birefringence
2047:
2043:
2039:
2038:birefringence
2035:
2031:
2027:
2026:birefringence
2023:
2019:
2005:
2002:
1999:
1977:
1974:
1971:
1967:
1946:
1923:
1903:
1878:
1874:
1869:
1865:
1862:
1856:
1853:
1841:
1837:
1821:
1818:
1815:
1803:
1801:
1787:
1784:
1781:
1773:
1757:
1737:
1714:
1711:
1708:
1688:
1685:
1682:
1662:
1659:
1656:
1648:
1647:birefringence
1632:
1609:
1605:
1601:
1593:
1577:
1574:
1571:
1551:
1531:
1511:
1488:
1484:
1480:
1476:
1470:
1467:
1464:
1461:
1449:
1447:
1444:evolves into
1443:
1437:
1430:
1424:
1423:polarizations
1420:
1412:
1407:
1405:
1403:
1399:
1398:birefringence
1395:
1379:
1376:
1373:
1370:
1367:
1345:
1342:
1339:
1336:
1314:
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1282:
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1153:
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1129:
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1013:
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1000:
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919:
915:
909:
884:
870:
866:
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847:
841:
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827:
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814:
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738:incorporates
737:
729:
727:
725:
724:birefringence
721:
717:
713:
709:
704:
687:
684:
681:
678:
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636:
617:
590:
587:
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581:
565:
539:
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494:
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310:
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274:
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250:
244:
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231:
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225:
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195:
184:
179:
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164:
160:
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110:
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108:
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3805:
3746:
3742:
3729:
3678:
3674:
3667:
3630:
3627:Astrophys. J
3626:
3616:
3571:
3567:
3557:
3512:
3508:
3497:
3428:
3424:
3418:
3349:
3345:
3339:
3280:
3276:
3269:
3218:
3214:
3208:
3160:(2): 95–97.
3157:
3153:
3147:
3094:
3090:
3026:
3022:
3012:
2961:
2957:
2893:
2889:
2879:
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2806:
2800:
2739:
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2334:
2323:
2203:
2192:
2180:
2020:
1807:
1450:
1435:
1428:
1416:
1152:power series
1031:
733:
716:polarization
705:
359:
35:
25:
15:
2326:accelerator
2099:or quartic
2044:shows that
1408:Experiments
712:dielectrics
5812:Categories
4124:(8): 021.
3464:2022/18696
3385:2022/19617
2846:2022/19023
2775:2022/18677
2653:2022/19024
2582:2022/18692
2511:2022/18992
2397:References
2182:involving
2146:and cubic
2046:dispersion
2034:dispersion
2030:dispersion
2022:Dispersion
1840:dispersion
1394:dispersion
720:dispersion
5791:0034-6861
5766:0801.0287
5738:119293488
5730:1550-7998
5705:0809.3217
5677:118461658
5669:1550-7998
5644:0806.4351
5616:1745-2473
5572:118502052
5564:1550-7998
5539:1108.3827
5495:0031-9007
5470:0904.2031
5434:1550-7998
5409:0809.3442
5365:0031-9007
5296:0031-9007
5235:0556-2821
5174:0001-7701
5113:1550-7998
5052:1550-7998
4982:0031-9007
4957:0706.2031
4912:0031-9007
4862:119175604
4854:1550-7998
4829:0909.2076
4801:118346408
4793:1550-7998
4768:1002.1284
4724:0031-9007
4699:1005.5230
4660:1550-7998
4635:1006.1376
4590:0031-9007
4565:1102.0081
4529:0031-9007
4478:0004-637X
4417:0370-2693
4392:0708.2889
4347:0031-9007
4322:0810.3475
4226:119222087
4218:1550-7998
4193:1305.3463
4156:1475-7516
4131:0904.3201
4086:0031-9007
4033:118717161
4025:1550-7998
4000:0705.0810
3964:0004-6361
3939:0710.3325
3911:119290314
3903:0067-0049
3878:0803.0547
3850:1550-7998
3825:0807.2593
3781:0031-9007
3756:0811.0618
3721:118421845
3713:1550-7998
3688:0905.1651
3640:0906.1003
3608:0370-2693
3581:0908.1876
3549:0067-0049
3522:1001.4538
3515:(2): 18.
3473:0031-9007
3394:0031-9007
3315:0031-9007
3290:1208.5288
3253:1550-7998
3228:1106.1068
3200:119280055
3192:0927-6505
3167:1102.2784
3131:0031-9007
3104:1301.5367
3061:0031-9007
3004:119241509
2996:1550-7998
2971:0905.0031
2930:0004-637X
2903:0809.2846
2855:0031-9007
2792:119471418
2784:0031-9007
2713:0556-2821
2662:0556-2821
2591:1550-7998
2520:0556-2821
2449:0556-2821
2206:Microwave
1975:−
1959:grows as
1944:Δ
1901:Δ
1860:Δ
1851:Δ
1819:≠
1758:λ
1735:Δ
1630:Δ
1610:λ
1552:λ
1509:Δ
1489:λ
1474:Δ
1471:π
1462:ϕ
1459:Δ
1438:∈ {even}
1346:ν
1343:μ
1340:λ
1337:κ
1318:^
1286:…
1242:…
1196:κ
1174:^
1136:ν
1133:μ
1130:λ
1127:κ
1108:^
1074:κ
1052:^
1008:ν
1005:μ
995:ν
992:μ
989:λ
986:κ
967:^
950:λ
947:κ
928:−
923:ν
920:μ
910:κ
888:^
871:λ
861:ν
858:μ
855:λ
852:κ
848:ϵ
828:ν
825:μ
815:ν
812:μ
793:−
753:≥
734:The full
688:ν
685:μ
682:λ
679:κ
637:κ
591:ν
588:μ
585:λ
582:κ
540:κ
495:ν
492:μ
489:λ
486:κ
418:κ
337:ν
334:μ
324:λ
321:κ
311:ν
308:μ
305:λ
302:κ
263:−
258:ν
255:μ
245:λ
235:ν
232:μ
229:λ
226:κ
222:ϵ
215:κ
168:ν
165:μ
155:ν
152:μ
128:−
5511:13682668
5503:19518765
5381:38353693
5373:12633280
5312:15770750
5304:12906465
5243:51302132
5121:19178203
4998:33003084
4990:17930733
4928:33875626
4920:19792767
4740:20890367
4732:20867292
4606:23180659
4598:21635069
4537:23004951
4486:15649601
4363:15789937
4355:18999724
4283:19228997
4164:18811259
4102:29494306
4094:16803298
3797:84181915
3789:19518694
3489:30064523
3481:17678146
3402:17155222
3331:42198517
3323:23368301
3261:53603505
3139:25167393
3069:13943605
2871:11401195
2863:11736558
2721:10012457
2670:21309077
2599:55185765
2343:See also
2048:without
2036:but not
1893:, where
1501:, where
1431:∈ {odd}
5799:3236027
5771:Bibcode
5710:Bibcode
5649:Bibcode
5596:Bibcode
5544:Bibcode
5475:Bibcode
5442:3723253
5414:Bibcode
5345:Bibcode
5276:Bibcode
5215:Bibcode
5182:8799879
5154:Bibcode
5093:Bibcode
5060:3222284
5032:Bibcode
4962:Bibcode
4892:Bibcode
4834:Bibcode
4773:Bibcode
4704:Bibcode
4668:2612817
4640:Bibcode
4570:Bibcode
4509:Bibcode
4458:Bibcode
4425:5103618
4397:Bibcode
4327:Bibcode
4291:7821247
4275:1357451
4253:Bibcode
4245:Science
4198:Bibcode
4136:Bibcode
4066:Bibcode
4005:Bibcode
3972:6795044
3944:Bibcode
3883:Bibcode
3830:Bibcode
3761:Bibcode
3693:Bibcode
3659:1918381
3633:: 978.
3586:Bibcode
3527:Bibcode
3443:Bibcode
3410:1451493
3364:Bibcode
3295:Bibcode
3233:Bibcode
3172:Bibcode
3109:Bibcode
3041:Bibcode
2976:Bibcode
2908:Bibcode
2825:Bibcode
2754:Bibcode
2693:Bibcode
2632:Bibcode
2561:Bibcode
2528:4013391
2490:Bibcode
2457:7651433
2429:Bibcode
1836:photons
1592:photons
1298:are in
708:photons
5797:
5789:
5736:
5728:
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5667:
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2526:
2518:
2455:
2447:
2040:. The
770:photon
5795:S2CID
5761:arXiv
5734:S2CID
5700:arXiv
5673:S2CID
5639:arXiv
5568:S2CID
5534:arXiv
5507:S2CID
5465:arXiv
5438:S2CID
5404:arXiv
5377:S2CID
5335:arXiv
5308:S2CID
5266:arXiv
5239:S2CID
5205:arXiv
5178:S2CID
5144:arXiv
5117:S2CID
5083:arXiv
5056:S2CID
5022:arXiv
4994:S2CID
4952:arXiv
4924:S2CID
4880:(PDF)
4858:S2CID
4824:arXiv
4797:S2CID
4763:arXiv
4736:S2CID
4694:arXiv
4664:S2CID
4630:arXiv
4602:S2CID
4560:arXiv
4482:S2CID
4448:arXiv
4421:S2CID
4387:arXiv
4359:S2CID
4317:arXiv
4287:S2CID
4222:S2CID
4188:arXiv
4160:S2CID
4126:arXiv
4098:S2CID
4056:arXiv
4029:S2CID
3995:arXiv
3968:S2CID
3934:arXiv
3907:S2CID
3873:arXiv
3820:arXiv
3793:S2CID
3751:arXiv
3739:(PDF)
3717:S2CID
3683:arXiv
3655:S2CID
3635:arXiv
3576:arXiv
3517:arXiv
3485:S2CID
3433:arXiv
3406:S2CID
3354:arXiv
3327:S2CID
3285:arXiv
3257:S2CID
3223:arXiv
3196:S2CID
3162:arXiv
3099:arXiv
3065:S2CID
3031:arXiv
3000:S2CID
2966:arXiv
2898:arXiv
2867:S2CID
2815:arXiv
2788:S2CID
2744:arXiv
2666:S2CID
2622:arXiv
2595:S2CID
2551:arXiv
2524:S2CID
2480:arXiv
2453:S2CID
2419:arXiv
1649:from
1419:Light
5787:ISSN
5726:ISSN
5665:ISSN
5612:ISSN
5560:ISSN
5499:PMID
5491:ISSN
5430:ISSN
5369:PMID
5361:ISSN
5300:PMID
5292:ISSN
5231:ISSN
5170:ISSN
5109:ISSN
5048:ISSN
4986:PMID
4978:ISSN
4916:PMID
4908:ISSN
4850:ISSN
4789:ISSN
4728:PMID
4720:ISSN
4656:ISSN
4594:PMID
4586:ISSN
4533:PMID
4525:ISSN
4474:ISSN
4413:ISSN
4351:PMID
4343:ISSN
4279:PMID
4271:OSTI
4214:ISSN
4152:ISSN
4122:2009
4090:PMID
4082:ISSN
4021:ISSN
3960:ISSN
3899:ISSN
3846:ISSN
3785:PMID
3777:ISSN
3709:ISSN
3604:ISSN
3545:ISSN
3477:PMID
3469:ISSN
3398:PMID
3390:ISSN
3319:PMID
3311:ISSN
3249:ISSN
3188:ISSN
3135:PMID
3127:ISSN
3057:ISSN
2992:ISSN
2926:ISSN
2859:PMID
2851:ISSN
2780:ISSN
2717:PMID
2709:ISSN
2658:ISSN
2587:ISSN
2516:ISSN
2445:ISSN
2301:>
2264:and
2208:and
2003:>
1660:>
1575:>
1088:and
651:and
5779:doi
5718:doi
5657:doi
5604:doi
5552:doi
5483:doi
5461:102
5422:doi
5353:doi
5284:doi
5223:doi
5162:doi
5101:doi
5040:doi
4970:doi
4900:doi
4888:103
4842:doi
4781:doi
4712:doi
4690:104
4648:doi
4578:doi
4556:106
4517:doi
4505:108
4466:doi
4444:611
4405:doi
4383:668
4335:doi
4313:101
4261:doi
4249:323
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