Limits on Spacetime Foam
  Plausibly spacetime is "foamy" on small distance scales, due to quantum
fluctuations. We elaborate on the proposal to detect spacetime foam by looking
for seeing disks in the images of distant quasars and AGNs. This is a null test
in the sense that the continued presence of unresolved "point" sources at the
milli-arc second level in samples of distant compact sources puts severe
constraints on theories of quantized spacetime foam at the Planckian level. We
discuss the geometry of foamy spacetime, and the appropriate distance measure
for calculating the expected angular broadening. We then deal with recent data
and the constraints they put on spacetime foam models. While time lags from
distant pulsed sources such as GRBs have been posited as a possible test of
spacetime foam models, we demonstrate that the time-lag effect is rather
smaller than has been calculated, due to the equal probability of positive and
negative fluctuations in the speed of light inherent in such models. Thus far,
images of high-redshift quasars from the Hubble Ultra-Deep Field (UDF) provide
the most stringent test of spacetime foam theories. While random walk models
($\alpha = 1/2$) have already been ruled out, the holographic ($\alpha=2/3$)
model remains viable. Here $\alpha \sim 1$ parametrizes the different spacetime
foam models according to which the fluctuation of a distance $l$ is given by
$\sim l^{1 - \alpha} l_P^{\alpha}$ with $l_P$ being the Planck length. Indeed,
we see a slight wavelength-dependent blurring in the UDF images selected for
this study. Using existing data in the {\it Hubble Space Telescope (HST)}
archive we find it is impossible to rule out the $\alpha=2/3$ model, but
exclude all models with $\alpha<0.65$. By comparison, current GRB time lag
observations only exclude models with $\alpha<0.3$.
