Quantum Riemannian geometry and particle creation on the integer line

Abstract

We construct noncommutative or `quantum' Riemannian geometry on the integers $\mathbb{Z}$ 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 $\ast$-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 $-\frac{1}{2}\sum \rho \Delta \rho$ 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 $<N>=(1-\sqrt{\rho }{)}^2/(4\sqrt{\rho })$ in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically.

Change in title as per the accepted version, improvements to the referencing; 22 pages latex, no figures

References in corpus (2)

• Loop quantum cosmology of k=1 FRW models
• The Vacuum State of Primordial Fluctuations in Hybrid Loop Quantum Cosmology

Cited by in corpus (7)

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• Fuzzy and discrete black hole models
• Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$
• Quantum Riemannian geometry of the discrete interval and q-deformation
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model