Quantum Riemannian geometry and particle creation on the integer line

Abstract

We construct noncommutative or `quantum' Riemannian geometry on the integers $\Bbb Z$ as a lattice line $\cdots\bullet_{i-1}-\bullet_i-\bullet_{i+1}\cdots$ with its natural 2-dimensional differential structure and metric given by arbitrary non-zero edge square-lengths $\bullet_i{\buildrel a_i\over -}\bullet_{i+1}$. We find for general metrics a unique $*$-preserving quantum Levi-Civita connection, which is flat if and only if $a_i$ are a geometric progression where the ratios $\rho_i=a_{i+1}/a_i$ are constant. More generally, we compute the Ricci tensor for the natural antisymmetric lift of the volume 2-form and find that the quantum Einstein-Hilbert action up to a total divergence is $-{1\over 2}\sum \rho\Delta \rho$ where $(\Delta\rho)_i=\rho_{i+1}+\rho_{i-1}-2\rho_i$ is the standard discrete Laplacian. We take a first look at some issues for quantum gravity on the lattice line. We also examine $1+0$ dimensional scalar quantum theory with mass $m$ and the lattice line as discrete time. As an application, we compute discrete time cosmological particle creation for a step function jump in the metric by a factor $\rho$, finding that an initial vacuum state has at later times an occupancy $ =(1-\sqrt{\rho})^2/(4\sqrt{\rho})$ in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically.