Probing Spacetime Foam with Photons: Suppression of Observable Effects due to Uncertainty in Optical Paths

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It was recently proposed to use extra-galactic point sources to constrain space-time quantum fluctuations in the universe. In these proposals, the fundamental "fuzziness" of distance caused by space-time quantum fluctuations have been directly identified with fluctuations in optical paths. Phase-front corrugations deduced from these optical-path fluctuations are then applied to light from extragalactic point sources, and used to constrain various models of quantum gravity. In particular, the so-called random-walk model has been claimed to be ruled out by existing astrophysical observations from the Hubble Space Telescope. However, when a photon propagates in three spatial dimensions, it does not follow a specific ray, but would rather sample a finite, three-dimensional region around that ray -- thereby averaging over space-time quantum fluctuations all through that region. We use a simple, random-walk type model to demonstrate that, once the appropriate wave optics is applied, the averaging of neighboring space-time fluctuations will cause much less distortions on the phase front. In our model, the extra suppression factor due to diffraction is (Planck Length)/(Wavelength), which is at least 19 orders of magnitude for astronomical observations.