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Quantum Riemannian geometry and particle creation on the integer line

We construct noncommutative or ‘quantum’ Riemannian geometry on the integers as a lattice line with its natural 2-dimensional differential structure and metric given by arbitrary non-zero edge square-lengths . We find for general metrics a unique -preserving quantum Levi-Civita connection, which is flat if and only if ai are a geometric progression where the ratios 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 where 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 , finding that an initial vacuum state has at later times an occupancy in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically.