A Discrete Analog of General Covariance -- Part 1: Could the world be fundamentally set on a lattice?

Abstract

A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a theory's substantive content from its merely representational artifacts. It is an indispensable tool for a modern understanding of spacetime theories. Motivated by quantum gravity, one may wish to extend these notions to quantum spacetime theories (whatever those are). Relatedly, one might want to extend these notions to discrete spacetime theories (i.e., lattice theories). This paper delivers such an extension with surprising consequences. One's first intuition regarding discrete spacetime theories may be that they introduce a great deal of fixed background structure (i.e., a lattice) and thereby limit our theory's possible symmetries down to those which preserve this fixed structure (i.e., discrete symmetries). However, as I will discuss, these intuitions are doubly wrong and overhasty. Discrete spacetime theories can and do have continuous translation and rotation symmetries. Moreover, the exact same theory can be given a wide variety of lattice structures and can even be described with no lattice at all. As my discrete analog of general covariance will reveal: lattice structure is rather less like a fixed background structure or part of an underlying manifold and rather more like a coordinate system, i.e., merely a representational artifact. Thus, the world cannot be "fundamentally set on a square lattice" (or any other lattice) any more than it could be "fundamentally set in a certain coordinate system". Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly representational. Spacetime cannot be discrete (even when it might be representable as such).