Distance between quantum field theories as a measure of Lorentz violation

We study the distance between symmetry-violating quantum field theories and the surface of symmetric theories. We use this notion to quantify how precise Lorentz symmetry is today, according to experimental data. The metric in parameter space is defined a la Zamolodchikov, from the two-point function of the Lagrangian perturbation. The distance is obtained minimizing the length of paths connecting the Lorentz-violating theory to the Lorentz surface. This definition depends on the Lagrangian used to formulate the theory, including total derivatives and the choice of coordinate frame. We eliminate such dependencies minimizing with respect to them. We derive a number of general formulas and evaluate the distance in the CPT-invariant, QED subsectors of the standard model extension and the renormalizable high-energy-Lorentz-violating standard model. We study the properties of the distance and address a number of applications.