  Violations of Lorentz invariance that appear via operators of dimension four
or less are completely parameterized in the Standard Model Extension (SME). In
the pure photonic sector of the SME, there are nineteen dimensionless,
Lorentz-violating parameters. Eighteen of these have experimental upper bounds
ranging between 10^{-11} and 10^{-32}; the remaining parameter, k_tr, is
isotropic and has a much weaker bound of order 10^{-4}. In this Brief Report,
we point out that k_tr gives a significant contribution to the anomalous
magnetic moment of the electron and find a new upper bound of order 10^{-8}.
With reasonable assumptions, we further show that this bound may be improved to
10^{-14} by considering the renormalization of other Lorentz-violating
parameters that are more tightly constrained. Using similar renormalization
arguments, we also estimate bounds on Lorentz violating parameters in the pure
gluonic sector of QCD.
