Quantum Riemannian geometry and particle creation on the integer line

Abstract

We construct noncommutative or 'quantum' Riemannian geometry on the integers Z as a lattice line ⋯ • i−1 − • i − • i+1 ⋯ with its natural 2dimensional differential structure and metric given by arbitrary non-zero edge square-lengths • i a i − • 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 ρ 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 2 ∑ ρ∆ρ where (∆ρ) i = ρ i+1 + ρ i−1 − 2ρ 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 ρ, finding that an initial vacuum state has at later times an occupancy ⟨N ⟩ = (1 − √ ρ) 2 (4 √ ρ) in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically.2000 Mathematics Subject Classification. Primary 81R50, 58B32, 83C57.

Riemannian Geometry of a Discretized Circle and Torus

“…As an immediate consequence we obtain the following two results, which were already formulated in [1] (see also [28] for the case G = Z).…”

“…Proof . Although this result is well-known (for example, the case with G = Z was proved in [28] where also the consequences for the metric compatible with higher-order calculi were studied), for completeness we demonstrate the proof. Since…”

“…In our case of the algebra C(Z N ) since the calculus is inner, we can use As an immediate consequence of the above definition we get the following result: Proof . The argument is exactly the same as in [27,28] for groups Z 2 × Z 2 and Z, respectively. We use shortened notation Ω 1 (A) to denote Ω 1 H (C(Z N )) from Theorem 2.4.…”

“…Geometric aspects of finite groups have been intensively studied by several authors, including [13,17], and also recently in [27] for the infinite cyclic group Z and [28] in case of the group Z 2 × Z 2 . As we consider finite cyclic groups Z N with N ≥ 2 many results are much simpler and therefore we skip the derivation of them, which are mostly adaptations of well-known ones published in the aforementioned literature.…”

“…The existence of large number of possible solutions (i.e., the non-triviality of the moduli space of Levi-Civita connection) without imposing additional conditions was already observed in [27] for the case Z 2 × Z 2 (see also [8,Chapter 8.2]). Some of the arguments we use here were already present in [27,28].…”