  It has been argued by Jacobson, Liberati and Mattingly that synchrotron
radiation from the Crab Nebula imposes a stringent constraint on any
modification of the dispersion relations of the electron that might be induced
by quantum gravity. We supplement their analysis by deriving the spectrum of
synchrotron radiation from the coupling of an electrically-charged particle to
an external magnetic fields in the presence of quantum-gravity effects of the
general form $(E/M_{QG})^\alpha$. We find that the synchrotron constraint from
the Crab Nebula practically excludes $\alpha \lsim 1.74$ for $M_{QG} \sim m_P =
1.2 \times 10^{19}$ GeV. On the other hand, this analysis does not constrain
any modification of the dispersion relation of the photon that might be induced
by quantum gravity. We point out that such quantum-gravity effects need not
obey the equivalence principle, a point exemplified by the Liouville-string
D-particle model of space-time foam. This model suggests a linear modification
of the dispersion relation for the photon, but not for the electron, and hence
is compatible with known constraints from the Crab Nebula and elsewhere.
