Lorentz-violating electrodynamics

Searches for Lorentz violation involving photons are among the best tests 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. Searches for Lorentz violation involving photons are among the best tests 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. The most general framework for studies of relativity violations is an effective field theory called the Standard-Model Extension (SME). Lorentz-violating operators in the SME are classified by their mass dimension d {displaystyle d} . To date, the most widely studied limit of the SME is the minimal SME, which limits attention to operators of renormalizable mass-dimension, d = 3 , 4 {displaystyle d=3,4} , in flat spacetime. Within the minimal SME, photons are governed by the lagrangian density 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 d = 3 {displaystyle d=3} operator and a constant coefficient for Lorentz violation ( k A F ) κ {displaystyle (k_{AF})^{kappa }} . The second term introduces Lorentz violation, but preserves CPT invariance. It consists of a dimension d = 4 {displaystyle d=4} operator contracted with constant coefficients for Lorentz violation ( k F ) κ λ μ ν {displaystyle (k_{F})_{kappa lambda mu u }} . There are a total of four independent ( k A F ) κ {displaystyle (k_{AF})^{kappa }} coefficients and nineteen ( k F ) κ λ μ ν {displaystyle (k_{F})_{kappa lambda mu u }} 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 ( k A F ) κ {displaystyle (k_{AF})^{kappa }} and ( k F ) κ λ μ ν {displaystyle (k_{F})_{kappa lambda mu u }} 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. The full SME incorporates general relativity and curved spacetimes. It also includes operators of arbitrary (nonrenormalizable) dimension d ≥ 5 {displaystyle dgeq 5} . The general gauge-invariant photon 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, where the constant coefficients are promoted to operators ( k ^ A F ) κ {displaystyle {({hat {k}}_{AF})}_{kappa }} and ( k ^ F ) κ λ μ ν {displaystyle {({hat {k}}_{F})}^{kappa lambda mu u }} , which take the form of power series in spacetime derivatives. The ( k ^ A F ) κ {displaystyle {({hat {k}}_{AF})}_{kappa }} operator contains all the CPT-odd d = 3 , 5 , 7 , … {displaystyle d=3,5,7,ldots } terms, while the CPT-even terms with d = 4 , 6 , 8 , … {displaystyle d=4,6,8,ldots } are in ( k ^ F ) κ λ μ ν {displaystyle {({hat {k}}_{F})}^{kappa lambda mu u }} . While the nonrenormalizable terms give many of the same types of signatures as the d = 3 , 4 {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. Birefringence of light occurs when the solutions to the modified Lorentz-violating Maxwell equations give rise to polarization-dependent speeds. 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. The quantity determining the size of the effect is the change in relative phase, Δ ϕ = 2 π Δ v t / λ {displaystyle Delta phi =2pi Delta v,t/lambda } , where Δ v {displaystyle Delta v} is the difference in phase speeds, t {displaystyle t} is the propagation time, and λ {displaystyle lambda } is the wavelength. For d > 3 {displaystyle d>3} , the highest sensitivities are achieved by considering high-energy photons from distant sources, giving large values to the ratio t / λ {displaystyle t/lambda } that enhance the sensitivity to Δ v {displaystyle Delta v} . The best constraints on vacuum birefringence from d > 3 {displaystyle d>3} Lorentz violation come from polarimetry studies of gamma-ray bursts (GRB). For example, sensitivities of 10−38 to the d = 4 {displaystyle d=4} coefficients for Lorentz violation have been achieved. For d = 3 {displaystyle d=3} , the velocity difference Δ v {displaystyle Delta v} is proportional to the wavelength, canceling the λ {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 d = 3 {displaystyle d=3} coefficients for Lorentz violation from the CMB currently stand at around 10−43 GeV. Lorentz violation with d ≠ 4 {displaystyle d eq 4} 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 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 Δ t = Δ v L / c 2 {displaystyle Delta t=Delta vL/c^{2}} , where Δ v {displaystyle Delta v} is the difference in the group velocity and L {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 Δ v {displaystyle Delta v} grows as E d − 4 {displaystyle E^{d-4}} , so higher-energy sources provide better sensitivity to effects from d > 4 {displaystyle d>4} Lorentz violation, making GRB an ideal source.