  We classify the unitary, renormalizable, Lorentz violating quantum field
theories of interacting scalars and fermions, obtained improving the behavior
of Feynman diagrams by means of higher space derivatives. Higher time
derivatives are not generated by renormalization. Renormalizability is ensured
by a "weighted power counting" criterion. The theories contain a dimensionful
parameter, yet a set of models are classically invariant under a weighted scale
transformation, which is anomalous at the quantum level. Formulas for the
weighted trace anomaly are derived. The renormalization-group properties are
studied.
