Renormalization of lattice-regularized quantum gravity models I. General considerations

Abstract

Lattice regularization is a standard technique for the nonperturbative definition of a quantum theory of fields. Several approaches to the construction of a quantum theory of gravity adopt this technique either explicitly or implicitly. A crucial complement to lattice regularization is the process of renormalization through which a continuous description of the quantum theory arises. I provide a comprehensive conceptual discussion of the renormalization of lattice-regularized quantum gravity models. I begin with a presentation of the renormalization group from the Wilsonian perspective. I then consider the application of the renormalization group in four contexts: quantum field theory on a continuous nondynamical spacetime, quantum field theory on a lattice-regularized nondynamical spacetime, quantum field theory of continuous dynamical spacetime, and quantum field theory of lattice-regularized dynamical spacetime. The first three contexts serve to identify successively the particular issues that arise in the fourth context. These issues originate in the inescability of establishing all scales solely on the basis of the dynamics. While most of this discussion rehearses established knowledge, the attention that I pay to these issues, especially the previously underappreciated role of standard units of measure, is largely novel. I conclude by briefly reviewing past studies of renormalization of lattice-regularized quantum gravity models. In the second paper of this two-part series, I illustrate the ideas presented here by proposing a renormalization group scheme for causal dynamical triangulations.