Lorentz-violating electrodynamics

Searches for Lorentz violation involving photons provide one possible test of relativity. Examples range from modern versions of the classic Michelson-Morley experiment that utilize highly stable electromagnetic resonant cavities to searches for tiny deviations from c 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 10³⁸.

Minimal Lorentz-violating electrodynamics

The most general framework for studies of relativity violations is an effective field theory called the Standard-Model Extension (SME).[1][2][3] Lorentz-violating operators in the SME are classified by their mass dimension ${\displaystyle d}$. To date, the most widely studied limit of the SME is the minimal SME,[4] which limits attention to operators of renormalizable mass-dimension, ${\displaystyle d=3,4}$, in flat spacetime. Within the minimal SME, photons are governed by the lagrangian density

${\displaystyle {\mathcal {L}}=-\textstyle {{1} \over {4}}\,F_{\mu \nu }F^{\mu \nu }+\textstyle {{1} \over {2}}\,(k_{\mathrm {AF} })^{\kappa }\,\epsilon _{\kappa \lambda \mu \nu }A^{\lambda }F^{\mu \nu }-\textstyle {{1} \over {4}}\,(k_{\mathrm {F} })_{\kappa \lambda \mu \nu }F^{\kappa \lambda }F^{\mu \nu }.}$

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 ${\displaystyle d=3}$ operator and a constant coefficient for Lorentz violation ${\displaystyle (k_{\mathrm {AF} })^{\kappa }}$.[5][6] The second term introduces Lorentz violation, but preserves CPT invariance. It consists of a dimension ${\displaystyle d=4}$ operator contracted with constant coefficients for Lorentz violation ${\displaystyle (k_{\mathrm {F} })_{\kappa \lambda \mu \nu }}$.[7] There are a total of four independent ${\displaystyle (k_{\mathrm {AF} })^{\kappa }}$ coefficients and nineteen ${\displaystyle (k_{\mathrm {F} })_{\kappa \lambda \mu \nu }}$ 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 ${\displaystyle (k_{\mathrm {AF} })^{\kappa }}$ and ${\displaystyle (k_{\mathrm {F} })_{\kappa \lambda \mu \nu }}$ 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.

The mathematics describing Lorentz violation in photons is similar to that of conventional electromagnetism in dielectrics. 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, polarization, and direction of propagation. Consequently, Lorentz violation can introduce dispersion in light propagating in empty space. It can also introduce birefringence, an effect seen in crystals such as calcite. The best constraints on Lorentz violation come from constraints on birefringence in light from astrophysical sources.[8]

Nonminimal Lorentz-violating electrodynamics

The full SME incorporates general relativity and curved spacetimes. It also includes operators of arbitrary (nonrenormalizable) dimension ${\displaystyle d\geq 5}$. The general gauge-invariant photon sector was constructed in 2009 by Kostelecky and Mewes.[9] It was shown that the more general theory could be written in a form similar to the minimal case,

${\displaystyle {\mathcal {L}}=-\textstyle {1 \over 4}F_{\mu \nu }F^{\mu \nu }+\textstyle {1 \over 2}\epsilon ^{\kappa \lambda \mu \nu }A_{\lambda }{({\hat {k}}_{\mathrm {AF} })}_{\kappa }F_{\mu \nu }-\textstyle {1 \over 4}F_{\kappa \lambda }{({\hat {k}}_{\mathrm {F} })}^{\kappa \lambda \mu \nu }F_{\mu \nu }\,,}$

where the constant coefficients are promoted to operators ${\displaystyle {({\hat {k}}_{\mathrm {AF} })}_{\kappa }}$ and ${\displaystyle {({\hat {k}}_{\mathrm {F} })}^{\kappa \lambda \mu \nu }}$, which take the form of power series in spacetime derivatives. The ${\displaystyle {({\hat {k}}_{\mathrm {AF} })}_{\kappa }}$ operator contains all the CPT-odd ${\displaystyle d=3,5,7,\ldots }$ terms, while the CPT-even terms with ${\displaystyle d=4,6,8,\ldots }$ are in ${\displaystyle {({\hat {k}}_{\mathrm {F} })}^{\kappa \lambda \mu \nu }}$. While the nonrenormalizable terms give many of the same types of signatures as the ${\displaystyle d=3,4}$ case, the effects generally grow faster with frequency, due to the additional derivatives. More complex directional dependence typically also arises. Vacuum dispersion of light without birefringence is another feature that is found, which does not arise in the minimal SME.[9]

Experiments

Vacuum birefringence

Birefringence of light occurs when the solutions to the modified Lorentz-violating Maxwell equations give rise to polarization-dependent speeds.[9][10][11] Light propagates as the combination of two orthogonal polarizations 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 remains fixed when propagating in a vacuum. In the CPT-odd case (d ∈ {odd} ), birefringence causes a simple rotation of the polarization. The CPT-even case (d ∈ {even} ) gives more complicated behavior as linearly polarized light evolves into elliptically polarizations.[9]

The quantity determining the size of the effect is the change in relative phase, ${\displaystyle \Delta \phi =2\pi \Delta v\,t/\lambda }$, where ${\displaystyle \Delta v}$ is the difference in phase speeds, ${\displaystyle t}$ is the propagation time, and ${\displaystyle \lambda }$ is the wavelength. For ${\displaystyle d>3}$, the highest sensitivities are achieved by considering high-energy photons from distant sources, giving large values to the ratio ${\displaystyle t/\lambda }$ that enhance the sensitivity to ${\displaystyle \Delta v}$. The best constraints on vacuum birefringence from ${\displaystyle d>3}$ Lorentz violation come from polarimetry studies of gamma-ray bursts (GRB).[11][12][13][14] For example, sensitivities of 10⁻³⁸ to the ${\displaystyle d=4}$ coefficients for Lorentz violation have been achieved. For ${\displaystyle d=3}$, the velocity difference ${\displaystyle \Delta v}$ is proportional to the wavelength, canceling the ${\displaystyle \lambda }$ 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 cosmic microwave background (CMB). Constraints on ${\displaystyle d=3}$ coefficients for Lorentz violation from the CMB currently stand at around 10⁻⁴³ GeV.[15][16][17][18][19][20][21][22][23][24][25][26][27]

Vacuum dispersion

Lorentz violation with ${\displaystyle d\neq 4}$ can lead to frequency-dependent light speeds.[9] To search for this effect, researchers compare the arrival times of photons from distant sources of pulsed radiation, such as GRB or pulsars. Assuming photons of all energies are produced within a narrow window of time, dispersion 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 ${\displaystyle \Delta t=\Delta vL/c^{2}}$, where ${\displaystyle \Delta v}$ is the difference in the group velocity and ${\displaystyle L}$ is the distance traveled. Sensitivity to Lorentz violation is then increased by considering very distant sources with rapidly changing time profiles. The speed difference ${\displaystyle \Delta v}$ grows as ${\displaystyle E^{d-4}}$, so higher-energy sources provide better sensitivity to effects from ${\displaystyle d>4}$ Lorentz violation, making GRB an ideal source.[9][28][29][30][31][32]

Dispersion may or may not be accompanied by birefringence. Polarization studies typically achieved sensitivities well beyond those achievable through dispersion. As a result, most searches for dispersion focus on Lorentz violation that leads to dispersion but not birefringence. The SME shows that dispersion without birefringence can only arise from operators of even dimension ${\displaystyle d}$. Consequently, the energy dependence in the light speed from nonbirefringent Lorentz violation can be quadratic ${\displaystyle E^{2}}$ or quartic ${\displaystyle E^{4}}$ or any other even power of energy. Odd powers of energy, such as linear ${\displaystyle E}$ and cubic ${\displaystyle E^{3}}$, do not arise in effective field theory.

Resonant cavities

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 involving electromagnetic fields. The primary examples are the modern Michelson-Morley experiments based on electromagnetic resonant cavities, which have achieved sensitivities on the order of parts in 10¹⁸ to Lorentz violation.[33][34][35][36][37][38][39][40][41][42][43][44][45][46]

Resonant cavities support electromagnetic standing waves that oscillate at well-defined frequencies determined by the Maxwell equations 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 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.

Several classes of cavity experiment exist with different sensitivities to different types of Lorentz violation. Microwave and optical cavities have been used to constrain ${\displaystyle d=4}$ violations. Microwave experiments have also placed some bounds on nonminimal ${\displaystyle d=6}$ and ${\displaystyle d=8}$ violations. However, for ${\displaystyle d>4}$, 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.

Other experiments

A number of other searches for Lorentz violation in photons have been performed that do not fall under the above categories. These include accelerator based experiments,[47][48][36][49] atomic clocks,[50] and threshold analyses.[9][51][52]

The results of experimental searches of Lorentz invariance violation in the photon sector of the SME are summarized in the Data Tables for Lorentz and CPT violation.[53]

See also

• Standard-Model Extension
• Lorentz-violating neutrino oscillations
• Antimatter Tests of Lorentz Violation
• Bumblebee models
• Tests of special relativity
• Test theories of special relativity

External links

• Background information on Lorentz and CPT violation
• Data Tables for Lorentz and CPT Violation

References

1. Colladay, Don; Kostelecký, V. Alan (1 May 1997). "CPT violation and the standard model". Physical Review D. American Physical Society (APS). 55 (11): 6760–6774. arXiv:hep-ph/9703464. doi:10.1103/physrevd.55.6760. ISSN 0556-2821.
2. Colladay, D.; Kostelecký, V. Alan (26 October 1998). "Lorentz-violating extension of the standard model". Physical Review D. American Physical Society (APS). 58 (11): 116002. doi:10.1103/physrevd.58.116002. hdl:2022/18992. ISSN 0556-2821.
3. Kostelecký, V. Alan (17 May 2004). "Gravity, Lorentz violation, and the standard model". Physical Review D. American Physical Society (APS). 69 (10): 105009. doi:10.1103/physrevd.69.105009. hdl:2022/18692. ISSN 1550-7998.
4. Kostelecký, V. Alan; Mewes, Matthew (23 September 2002). "Signals for Lorentz violation in electrodynamics". Physical Review D. American Physical Society (APS). 66 (5): 056005. doi:10.1103/physrevd.66.056005. hdl:2022/19024. ISSN 0556-2821.
5. Carroll, Sean M.; Field, George B.; Jackiw, Roman (15 February 1990). "Limits on a Lorentz- and parity-violating modification of electrodynamics". Physical Review D. American Physical Society (APS). 41 (4): 1231–1240. doi:10.1103/physrevd.41.1231. ISSN 0556-2821.
6. Jackiw, R.; Kostelecký, V. Alan (3 May 1999). "Radiatively Induced Lorentz andCPTViolation in Electrodynamics". Physical Review Letters. American Physical Society (APS). 82 (18): 3572–3575. doi:10.1103/physrevlett.82.3572. hdl:2022/18677. ISSN 0031-9007.
7. Kostelecký, V. Alan; Mewes, Matthew (29 November 2001). "Cosmological Constraints on Lorentz Violation in Electrodynamics". Physical Review Letters. American Physical Society (APS). 87 (25): 251304. doi:10.1103/physrevlett.87.251304. hdl:2022/19023. ISSN 0031-9007.
8. Kostelecký, V. Alan; Mewes, Matthew (13 November 2008). "Astrophysical tests of Lorentz and CPT violation with photons". The Astrophysical Journal. IOP Publishing. 689 (1): L1–L4. doi:10.1086/595815. ISSN 0004-637X.
9. Kostelecký, V. Alan; Mewes, Matthew (29 July 2009). "Electrodynamics with Lorentz-violating operators of arbitrary dimension". Physical Review D. American Physical Society (APS). 80 (1): 015020. arXiv:0905.0031. doi:10.1103/physrevd.80.015020. ISSN 1550-7998.
10. Carroll, Sean M.; Field, George B. (29 September 1997). "Is there evidence for cosmic anisotropy in the polarization of distant radio sources?". Physical Review Letters. American Physical Society (APS). 79 (13): 2394–2397. doi:10.1103/physrevlett.79.2394. ISSN 0031-9007.
11. Kostelecký, V. Alan; Mewes, Matthew (14 May 2013). "Constraints on Relativity Violations from Gamma-Ray Bursts". Physical Review Letters. American Physical Society (APS). 110 (20): 201601. doi:10.1103/physrevlett.110.201601. ISSN 0031-9007.
12. Stecker, Floyd W. (2011). "A new limit on Planck scale Lorentz violation from γ-ray burst polarization". Astroparticle Physics. Elsevier BV. 35 (2): 95–97. arXiv:1102.2784. doi:10.1016/j.astropartphys.2011.06.007. ISSN 0927-6505.
13. 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". Physical Review D. American Physical Society (APS). 83 (12): 121301(R). arXiv:1106.1068. doi:10.1103/physrevd.83.121301. ISSN 1550-7998.
14. 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". Physical Review Letters. American Physical Society (APS). 109 (24): 241104. arXiv:1208.5288. doi:10.1103/physrevlett.109.241104. ISSN 0031-9007.
15. Kostelecký, V. Alan; Mewes, Matthew (2 October 2006). "Sensitive polarimetric search for relativity violations in gamma-ray bursts". Physical Review Letters. American Physical Society (APS). 97 (14): 140401. doi:10.1103/physrevlett.97.140401. hdl:2022/19617. ISSN 0031-9007.
16. Kostelecký, V. Alan; Mewes, Matthew (3 July 2007). "Lorentz-Violating Electrodynamics and the Cosmic Microwave Background". Physical Review Letters. American Physical Society (APS). 99 (1): 011601. doi:10.1103/physrevlett.99.011601. hdl:2022/18696. ISSN 0031-9007.
17. 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). "Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation". The Astrophysical Journal Supplement Series. IOP Publishing. 192 (2): 18. doi:10.1088/0067-0049/192/2/18. ISSN 0067-0049.
18. Xia, Jun-Qing; Li, Hong; Zhang, Xinmin (2010). "Probing CPT violation with CMB polarization measurements". Physics Letters B. Elsevier BV. 687 (2–3): 129–132. doi:10.1016/j.physletb.2010.03.038. ISSN 0370-2693.
19. Brown, M.L.; et al. (QUaD Collaboration) (2009). "Improved measurements of the temperature and polarization of the CMB from QUaD". Astrophys. J. 705: 978.
20. 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". Physical Review D. American Physical Society (APS). 80 (4): 043522. arXiv:0905.1651. doi:10.1103/physrevd.80.043522. ISSN 1550-7998.
21. Wu, E.Y.S.; Ade, P.; Bock, J.; Bowden, M.; Brown, M.L.; Cahill, G.; et al. (21 April 2009). "Parity violation constraints using cosmic microwave background polarization spectra from 2006 and 2007 observations by the QUaD polarimeter". Physical Review Letters. American Physical Society (APS). 102 (16): 161302. doi:10.1103/physrevlett.102.161302. ISSN 0031-9007.
22. Kahniashvili, Tina; Durrer, Ruth; Maravin, Yurii (22 December 2008). "Testing Lorentz invariance violation with Wilkinson Microwave Anisotropy Probe five year data". Physical Review D. American Physical Society (APS). 78 (12): 123009. arXiv:0807.2593. doi:10.1103/physrevd.78.123009. ISSN 1550-7998.
23. 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". The Astrophysical Journal Supplement Series. IOP Publishing. 180 (2): 330–376. arXiv:0803.0547. doi:10.1088/0067-0049/180/2/330. ISSN 0067-0049.
24. Xia, J.-Q.; Li, H.; Wang, X.; Zhang, X. (19 March 2008). "Testing CPT symmetry with CMB measurements". Astronomy & Astrophysics. EDP Sciences. 483 (3): 715–718. arXiv:0710.3325. doi:10.1051/0004-6361:200809410. ISSN 0004-6361.
25. Cabella, Paolo; Natoli, Paolo; Silk, Joseph (28 December 2007). "Constraints onCPTviolation from Wilkinson Microwave Anisotropy Probe three year polarization data: A wavelet analysis". Physical Review D. American Physical Society (APS). 76 (12): 123014. arXiv:0705.0810. doi:10.1103/physrevd.76.123014. ISSN 1550-7998.
26. 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". Physical Review Letters. American Physical Society (APS). 96 (22): 221302. arXiv:astro-ph/0601095. doi:10.1103/physrevlett.96.221302. ISSN 0031-9007.
27. 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". Journal of Cosmology and Astroparticle Physics. IOP Publishing. 2009 (08): 021–021. arXiv:0904.3201. doi:10.1088/1475-7516/2009/08/021. ISSN 1475-7516.
28. 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". Physical Review D. American Physical Society (APS). 87 (12): 122001. arXiv:1305.3463. doi:10.1103/physrevd.87.122001. ISSN 1550-7998.
29. Abdo, A.A.; et al. (Fermi LAT and GBM Collaborations) (2009). "Fermi observations of high-energy gamma-ray emission from GRB 080916C". Science. 323: 1688.
30. Aharonian, F.; 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". Physical Review Letters. American Physical Society (APS). 101 (17): 170402. arXiv:0810.3475. doi:10.1103/physrevlett.101.170402. ISSN 0031-9007.
31. 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". Physics Letters B. Elsevier BV. 668 (4): 253–257. arXiv:0708.2889. doi:10.1016/j.physletb.2008.08.053. ISSN 0370-2693.
32. Boggs, Steven E.; Wunderer, C. B.; Hurley, K.; Coburn, W. (20 July 2004). "Testing Lorentz Invariance with GRB 021206". The Astrophysical Journal. IOP Publishing. 611 (2): L77–L80. arXiv:astro-ph/0310307. doi:10.1086/423933. ISSN 0004-637X.
33. Baynes, Fred N.; Tobar, Michael E.; Luiten, Andre N. (26 June 2012). "Oscillating Test of the Isotropic Shift of the Speed of Light". Physical Review Letters. American Physical Society (APS). 108 (26): 260801. doi:10.1103/physrevlett.108.260801. ISSN 0031-9007.
34. Parker, Stephen R.; Mewes, Matthew; Stanwix, Paul L.; Tobar, Michael E. (3 May 2011). "Cavity Bounds on Higher-Order Lorentz-Violating Coefficients". Physical Review Letters. American Physical Society (APS). 106 (18): 180401. arXiv:1102.0081. doi:10.1103/physrevlett.106.180401. ISSN 0031-9007.
35. 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". Physical Review D. American Physical Society (APS). 82 (7): 076001. arXiv:1006.1376. doi:10.1103/physrevd.82.076001. ISSN 1550-7998.
36. 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". Physical Review Letters. American Physical Society (APS). 104 (24): 241601. arXiv:1005.5230. doi:10.1103/physrevlett.104.241601. ISSN 0031-9007.
37. 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 10⁻¹⁷ level". Physical Review D. American Physical Society (APS). 80 (10): 105011. arXiv:1002.1284. doi:10.1103/physrevd.80.105011. ISSN 1550-7998.
38. 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". Physical Review D. American Physical Society (APS). 80 (12): 125024. arXiv:0909.2076. doi:10.1103/physrevd.80.125024. ISSN 1550-7998.
39. Eisele, Ch.; Nevsky, A. Yu.; Schiller, S. (25 August 2009). "Laboratory Test of the Isotropy of Light Propagation at the 10⁻¹⁷ Level" (PDF). Physical Review Letters. American Physical Society (APS). 103 (9): 090401. doi:10.1103/physrevlett.103.090401. ISSN 0031-9007.
40. 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". Physical Review Letters. American Physical Society (APS). 99 (5): 050401. arXiv:0706.2031. doi:10.1103/physrevlett.99.050401. ISSN 0031-9007.
41. 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". Physical Review D. American Physical Society (APS). 74 (8): 081101(R). arXiv:gr-qc/0609072. doi:10.1103/physrevd.74.081101. ISSN 1550-7998.
42. Wolf, Peter; Bize, Sébastien; Clairon, André; Santarelli, Giorgio; Tobar, Michael E.; Luiten, André N. (15 September 2004). "Improved test of Lorentz invariance in electrodynamics". Physical Review D. American Physical Society (APS). 70 (5): 051902. arXiv:hep-ph/0407232. doi:10.1103/physrevd.70.051902. ISSN 1550-7998.
43. Wolf, Peter; Tobar, Michael E.; Bize, Sébastien; Clairon, André; Luiten, André N.; Santarelli, Giorgio (2004). "Whispering Gallery Resonators and Tests of Lorentz Invariance". General Relativity and Gravitation. Springer Nature. 36 (10): 2351–2372. arXiv:gr-qc/0401017. doi:10.1023/b:gerg.0000046188.87741.51. ISSN 0001-7701.
44. Müller, Holger; Herrmann, Sven; Saenz, Alejandro; Peters, Achim; Lämmerzahl, Claus (24 December 2003). "Optical cavity tests of Lorentz invariance for the electron". Physical Review D. American Physical Society (APS). 68 (11): 116006. arXiv:hep-ph/0401016. doi:10.1103/physrevd.68.116006. ISSN 0556-2821.
45. Müller, Holger; Herrmann, Sven; Braxmaier, Claus; Schiller, Stephan; Peters, Achim (10 July 2003). "Modern Michelson-Morley Experiment using Cryogenic Optical Resonators". Physical Review Letters. American Physical Society (APS). 91 (2): 020401. arXiv:physics/0305117. doi:10.1103/physrevlett.91.020401. ISSN 0031-9007.
46. 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". Physical Review Letters. American Physical Society (APS). 90 (6): 060403. arXiv:physics/0302093. doi:10.1103/physrevlett.90.060403. ISSN 0031-9007.
47. Hohensee, Michael A.; Lehnert, Ralf; Phillips, David F.; Walsworth, Ronald L. (21 August 2009). "Limits on isotropic Lorentz violation in QED from collider physics". Physical Review D. American Physical Society (APS). 80 (3): 036010. arXiv:0809.3442. doi:10.1103/physrevd.80.036010. ISSN 1550-7998.
48. 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". Physical Review Letters. American Physical Society (APS). 102 (17): 170402. arXiv:0904.2031. doi:10.1103/physrevlett.102.170402. ISSN 0031-9007.
49. Altschul, Brett (14 October 2011). "Bounding Lorentz violation at particle colliders by tracking the motion of charged particles". Physical Review D. American Physical Society (APS). 84 (7): 076006. arXiv:1108.3827. doi:10.1103/physrevd.84.076006. ISSN 1550-7998.
50. 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". Nature Physics. Springer Science and Business Media LLC. 3 (12): 861–864. doi:10.1038/nphys778. ISSN 1745-2473.
51. Klinkhamer, F. R.; Risse, M. (26 June 2008). "Addendum: Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory". Physical Review D. American Physical Society (APS). 77 (11): 117901. arXiv:0806.4351. doi:10.1103/physrevd.77.117901. ISSN 1550-7998.
52. Klinkhamer, F. R.; Schreck, M. (24 October 2008). "New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory". Physical Review D. American Physical Society (APS). 78 (8): 085026. arXiv:0809.3217. doi:10.1103/physrevd.78.085026. ISSN 1550-7998.
53. Kostelecký, V. Alan; Russell, Neil (10 March 2011). "Data tables for Lorentz andCPTviolation". Reviews of Modern Physics. American Physical Society (APS). 83 (1): 11–31. arXiv:0801.0287. doi:10.1103/revmodphys.83.11. ISSN 0034-6861.