Lorentz invariance is the fundamental symmetry of Einstein’s theory of special relativity and has been tested to a great level of detail. However, theories of quantum gravity at the Planck scale indicate that Lorentz symmetry may be broken at that scale, motivating further tests. While the Planck energy is currently unreachable by experiment, tiny residual effects at attainable energies can become measurable when photons propagate over sufficiently large distances. The Standard-Model extension (SME) is an effective field-theory approach to describe low-energy effects of quantum gravity theories. Lorentz- and $CPT$-symmetry-violating effects are introduced by adding additional terms to the Standard-Model Lagrangian. These terms can be ordered by the mass dimension of the corresponding operator, and the leading terms of interest have dimension $d=5$. Effects of these operators are a linear variation of the speed of light with photon energy, and a rotation of the linear polarization of photons quadratic in photon energy, as well as anisotropy. We analyze optical polarization data from 72 active galactic nuclei and GRBs and derive the first set of limits on all 16 coefficients of mass dimension $d=5$ of the SME photon sector. Our constraints imply a lower limit on the energy scale of quantum gravity of $10^6$ times the Planck energy, severely limiting the phase space for any theory that predicts a rotation of the photon polarization quadratic in energy.

• Received 8 January 2017

DOI:https://doi.org/10.1103/PhysRevD.95.083013