A Discrete Analog of General Covariance -- Part 2: Despite what you've heard, a perfectly Lorentzian lattice theory

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, especially regarding their background structures and symmetry. 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). In [1] I developed two discrete analogs of general covariance for non-Lorentzian lattice theories. This paper extends these results to a Lorentzian setting.In either setting these discrete analogs of general covariance reveal that lattice structure is rather less like a fixed background structure and rather more like a coordinate system, i.e., merely a representational artifact. These discrete analogs are built upon a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. From this, in [1], I argued that properly understood there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories. It is well-noted by the causal set theory community that no theory on a fixed spacetime lattice is Lorentz invariant, however as I will discuss this is ultimately a problem of representation, not of physics. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Nothing prevents us from using Cartesian coordinates to describe rotationally invariant states/dynamics. As this paper shows, the same is true of lattices in a Lorentzian setting: nothing prevents us from defining a perfectly Lorentzian lattice(-representable) theory.

Entanglement structure of quantum fields through local probes

“…We only expect detectors to be physically able to probe a finite number of modes within a given spatial region, and therefore, they can be used to define an operationally motivated cutoff to the field theory. This suggests that, from the point of view of detectors, even a continuous field theory with infinitely many degrees of freedom may behave as an effectively discretized chain of oscillators, up to the resolution allowed by the detectors [44,45,64].…”

“…( 51) leads to couplings between the amplitudes at different sites, and therefore we see explicitly that the price we pay for imposing a UV cutoff is to have a non-local Hamiltonian, i.e., we need to give up locality. However, if we additionally perform an approximation of the derivative to nearest neighbors (see, for instance, [45]), we get that the field Hamiltonian reduces to…”

“…This is not a problem because the continuum limit of this approximation is still the continuous derivative and (16) indeed becomes (6) in the limit ε → 0. For a deeper discussion on this subtlety see, e.g.,[45].…”