Quantum Riemannian geometry of the discrete interval and q-deformation

Abstract

We solve for quantum Riemannian geometries on the finite lattice interval • – • –⋯– • with n nodes (the Dynkin graph of type An) and find that they are necessarily q-deformed with q=eıπn+1. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel “boundary effect” whereby, in order to admit a quantum Levi-Cività connection, the “metric weight” at any edge is forced to be greater pointing toward the bulk compared to toward the boundary, with the ratio given by (i + 1)q/(i)q at node i, where (i)q is a q-integer. The Christoffel symbols are also q-deformed. The limit q → 1 likewise forces the quantum Riemannian geometry of the natural numbers N to have rational metric multiples (i + 1)/i in the direction of increasing i. In both cases, there is a unique Ricci-scalar flat metric up to normalization. Elements of quantum field theory and quantum gravity are exhibited for n = 3 and for the continuum limit of the geometry of N. The Laplacian for the scalar-flat metric becomes the Airy equation operator 1xd2dx2 in so far as a limit exists. Scaling this metric by a conformal factor eψ(i) gives a limiting Ricci scalar curvature proportional to e−ψxd2ψdx2.