Tests of Lorentz Invariance Using High Energy Astrophysics Observations

High-energy astrophysics observations provide the best possibilities to detect a very small violation of Lorentz invariance, such as may be related to the structure of space-time near the Planck scale. I discuss the possible signatures of Lorentz invariance violation that can be manifested by observing the spectra, polarization, and timing of gamma-rays from active galactic nuclei and gamma-ray bursts. Other sensitive tests are provided by observations of the spectra of ultrahigh-energy cosmic rays and very high-energy neutrinos. I also discuss a new time-of-flight analysis of observations of GRB 090510 by the Fermi gamma-ray Space Telescope. These results, based on high-energy astrophysical observations, have fundamental implications for space-time physics and quantum gravity models.