  In the electromagnetism of loop quantum gravity, two helicities of a photon
have different phase velocities and group velocities, termed as "vacuum
birefringence". Two novel phenomenons, "peak doubling" and "de-polarization",
are expected to appear for a linearly polarized light from astrophysical
sources. We show that the criteria to observe these two phenomenons are the
same. Further, from recently observed $\gamma$-ray polarization from Cygnus
X-1, we obtain an upper limit $\sim 8.7\times10^{-12}$ for Lorentz-violating
parameter $\chi$, which is the most firm constraint from well-known systems. We
also suggest to analyze possible existence of "peak doubling" through Fermi LAT
GRBs.
