Noncommutative Riemannian geometry on graphs

2019

Class. Quantum Grav.

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We consider functional-integral quantisation of the moduli of all quantum metrics defined as square-lengths a on the edges of a Lorentzian square graph. We determine correlation functions and find a fixed relative uncertainty ∆a ⟨a⟩ = 1 √ 8 for the edge square-lengths relative to their expected value ⟨a⟩. The expected value of the geometry is a rectangle where parallel edges have the same square-length. We compare with the simpler theory of a quantum scalar field on such a rectangular background. We also look at quantum metric fluctuations relative to a rectangular background, a theory which is finite and interpolates between the scalar theory and the fully fluctuating theory.2000 Mathematics Subject Classification. Primary 81R50, 58B32, 83C57.

Section: 2mentioning

confidence: 99%

Quantum gravity on a square graph

2019

Class. Quantum Grav.

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“…where in the graph case θ = ∑ x→y ω x→y and where α ∶ Ω 1 → Ω 1 ⊗ A Ω 1 and σ are bimodule maps (they commute with products by functions from either side). In this case, vanishing torsion and metric compatibility respectively become [22] (2.2) ∧ (id + σ) = 0, ∧α = 0…”

Section: Preliminaries: Quantum Geometric Formalismmentioning

confidence: 99%

Quantum Riemannian geometry and particle creation on the integer line

2019

Class. Quantum Grav.

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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.

“…The mathematics of quantum Riemannian geometry is simply more general than classical Riemannian geometry and includes discrete [27] as well as deformation examples. What is significant in this section is that whatever we find emerges from little else but the axioms applied to a square graph as 'manifold'.…”

Section: Quantum Gravity On a Square Graphmentioning

confidence: 99%

On the emergence of the structure of physics

Phil. Trans. R. Soc. A.

2018

We consider Hilbert's problem of the axioms of physics at a qualitative or conceptual level. This is more pressing than ever as we seek to understand how both general relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a or as a key structure. Here, I outline some of my recent work around the idea of quantum space-time as motivated by this non-standard philosophy, including a new toy model of gravity on a space-time consisting of four points forming a square.This article is part of the theme issue 'Hilbert's sixth problem'.