New essay: what happens at high energy?
A new essay, What happens at high energy?, is now on the essays page. It takes on the standard objection to discrete spacetime — that a lattice must misbehave at high energy, breaking Lorentz invariance or hiding an arbitrary cutoff — and shows the answer is already in the bipartite Dirac update derived in Paper I, not bolted on.
The short version: there is only one amplitude per site, and the rate at which its phase advances, \Delta\phi=\omega+V, decides how much of it propagates versus stays in place. Staying in place is what inertia is here — the fraction \sin^2(\Delta\phi/2) the phase-rate diverts from hopping — so mass and inertia are two readings of one quantity, not a second substance. The same phase-rate read as momentum enters through a structure factor that saturates at the zone boundary, so energy and momentum per mode are bounded: the lattice is its own ultraviolet regulator, for free, and the familiar E^2=m^2+|\mathbf p|^2 is the low-energy limit. Nothing in the substrate bends — the adjacency graph is fixed; it is the derived energy–momentum curve that bends, the same distinction that lets the framework carry gravity without a curved metric.
This maps onto three PASS rows in Paper I’s audit table: Photon dispersion (group velocity vs. wavenumber, with zone-boundary corrections; exp_09), Inertial persistence (exp_01), and the Dirac equation continuum limit. The essay is paired with the birefringence essay: that one is about the direction of the same propagator, this one about its magnitude and what it does to energy.
Pointers:
- Read the essay: What happens at high energy?
- Companion essay: Birefringence in the A=1 lattice
- Substrate paper: Paper I — Geometry First
- All essays: essays page