  Perturbative expansions of relativistic quantum field theories typically
contain ultraviolet divergences requiring regularization and renormalization.
Many different regularization techniques have been developed over the years,
but most regularizations require severe mutilation of the logical foundations
of the theory. In contrast, breaking Lorentz invariance, while it is certainly
a radical step, at least does not damage the logical foundations of the theory.
We shall explore the features of a Lorentz symmetry breaking regulator in a
simple polynomial scalar field theory, and discuss its implications. We shall
quantify just "how much" Lorentz symmetry breaking is required to fully
regulate the theory and render it finite. This scalar field theory provides a
simple way of understanding many of the key features of Horava's recent article
[arXiv:0901.3775 [hep-th]] on 3+1 dimensional quantum gravity.
