Quantum gravity on a square graph

We perform functional-integral quantisation of the moduli of all quantum metrics defined as square-lengths a on the edges of a Lorentzian quadrilateral graph. Noting that the partition function factorises into a theory for the spacelike edges and its conjugate for the timelike ones, we determine correlation functions and find a fixed relative uncertainty for the edge square-lengths relative to their expected value . 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 which at large rectangle scales resembles a pair of scalar fields on the vertical and horizontal edges with Planckian mass.