Lorentz symmetry breaking as a quantum field theory regulator

TL;DR

This work investigates using Lorentz-symmetry breaking as a ultraviolet regulator in a polynomial scalar QFT, showing through power-counting and Anselmi–Halat results that finiteness can be achieved without compromising the theory’s foundations. By introducing anisotropic scaling with a spatial derivative order $z$, the author derives the superficial degree of divergence and demonstrates finiteness in cases including $d=z$ (with normal ordering) and $z>d$ (without normal ordering), and applies the framework to $3+1$ dimensions with $z=3$, where the field is dimensionless and the theory is UV finite. The paper furthermore connects this regulator approach to Horava’s quantum gravity program, arguing that a finite, Lorentz-violating QFT could recover Lorentz invariance at low energies and potentially provide a tractable route to quantum gravity. The analysis also emphasizes the physical motivation and implications, such as the Bogoliubov-like dispersion relations and the idea that Lorentz symmetry need not be fundamental but can emerge as an effective low-energy phenomenon. Overall, the work offers a concrete, self-consistent regulator paradigm with potential applications to finite QFTs and gravity that warrants further exploration.

Abstract

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.