High Energy Astrophysics Tests of Lorentz Invariance and Quantum Gravity Models

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 of approximately 10(exp -35)m. I will discuss the possible signatures of Lorentz invariance violation (LIV) that can be manifested by observing of the spectra, polarization, and timing of gamma-rays from active galactic nuclei and y-ray bursts. Other sensitive tests are provided by observations of the spectra of ultrahigh energy cosmic rays and neutrinos. Using the latest data from the Pierre Auger Observatory one can already derive an upper limit of 4.5 x 10(exp -23) on the fraction of LIV at a Lorentz factor of approximately 2 x 10(exp 11). This result has fundamental implications for quantum gravity models. I will also discuss the possibilities of using more sensitive space-based detection techniques to improve searches for LIV in the future.