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- W2968479908 abstract "We construct noncommutative or `quantum' Riemannian geometry on the integers $Bbb Z$ as a lattice line $cdotsbullet_{i-1}-bullet_i-bullet_{i+1}cdots$ with its natural 2-dimensional differential structure and metric given by arbitrary non-zero edge square-lengths $bullet_i{buildrel a_iover -}bullet_{i+1}$. 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 $-{1over 2}sum rhoDelta rho$ where $(Deltarho)_i=rho_{i+1}+rho_{i-1}-2rho_i$ 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/(4sqrt{rho})$ in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically." @default.
- W2968479908 created "2019-08-22" @default.
- W2968479908 creator A5074290225 @default.
- W2968479908 date "2018-11-15" @default.
- W2968479908 modified "2023-09-27" @default.
- W2968479908 title "Quantum Riemannian Geometry and Scalar Fields on the Integer Line" @default.
- W2968479908 hasPublicationYear "2018" @default.
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