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]

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]

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

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