Quantum gravity on a square graph

Abstract

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.

Quantum Riemannian geometry and particle creation on the integer line

“…Although the formalism has been around for a few years now, a downside of the bimodule approach is that the construction of a quantum Levi-Civita connection (QLC) for a chosen quantum metric involves a nonlinear (quadratic) condition which can be solved in individual cases but is hard to solve uniformly across a significant moduli of metrics. This was achieved recently for the square graph, cf [24], and in the present paper our main result at a geometric level is in Section 3 to achieve the same for the integers regarded as a line graph. Quantum metrics themselves are easy to describe for a graph calculus, namely the data is just a 'square length' associated to each edge much as in most ideas for lattice approximations.…”

“…The first and most striking conclusion is that the integer lattice admits a full moduli of quantum Riemannian geometries defined by 'square lengths' attached to the edges, with most of them having curvature for the uniquely associated QLC. Key to this was to modify a previous 'quantum symmetry' condition in favour of an 'edge-symmetric' condition, which had also proven useful to impose in the only other known full moduli graph example [24]. This suggests that the edge-symmetric condition is more relevant and should be adopted for any graph.…”

“…An introduction to the formalism of the constructive 'bottom up' approach to quantum Riemannian geometry [5,6,7,8,22,25] that we will use is in [23] and in our recent work [24], so here we will say just enough to be self-contained. We will only need the case of graphs from [22] and moreover only the discrete group case.…”

“…The idea that our concept of spacetime should be modified as we approach the Planck scale is now widely accepted, though how precisely to do it is not clear. One popular idea is that spacetime coordinates should become noncommutative as an expression of quantum gravity corrections, see [29,19,20,13] and our recent work [24] for some of the background to this 'quantum spacetime hypothesis'. By now there are also several approaches as to how this might be done, such as the 'Dirac operator' (spectral triple) approach of Connes [11] and, which is the one we use, a constructive approach starting with an abstractly defined 'bimodule' Ω 1 of 1forms on the possibly noncommutative coordinate algebra, e.g.…”

“…There are also more sophisticated methods such as dynamical triangulations [2] and causal set models [1]. At least the basic graph approach can be seen as quantum Riemannian geometry [22,24], as we explain briefly in the preliminary Section 2.…”