Unlabeled Quantum Simple Graphs*

Background Independent formulations of quantum gravity use quantum versions of simple graphs – vertices and edges connecting them – to model the underlying self-referencing relationships that form the microstates of quantum gravity. These quantum versions of the simple graphs replace the classical edge states which take the values of 1 or 0 by a two-level Hilbert space. The edge states represent distinguishable states since they inherit the labels of the vertices which are labeled and distinguishable. In this talk we discuss the generalization of the anti-symmetrization procedure to represent the Hilbert space of quantum simple graphs and more generally quantum finite simplicial complexes where the vertices are indistinguishable. We then explore the implications of the anti-symmetrization requirement on emergent geometry using simple dynamical models inspired by the Ising and Heisenberg models.