Dynamical Quantum Graphs

One avenue in consideration for a fully background-independent formulation of quantum gravity is the use of quantum graphs. In this presentation, we describe an approach that uses unlabeled quantum simple graphs. Ordinary finite simple graphs consisting of vertices and edges have been quantized in analogy with the first and second quantization procedures of particulate quantum mechanics. Each pair of vertices is associated with an "edge" Hilbert space (analogous to a single-particle Hilbert space), isomorphic to the spin-1/2 Hilbert space, which is described by the vertices comprising their corresponding edge. The tensor product of all "edge" Hilbert spaces comprise the graph Hilbert space. In our approach, the dynamics of graphs with unlabeled vertices is considered to generalize the quantum indistinguishability of particles. A group-theoretic method to construct first-quantized graph wavefunctions consistent with quantum statistics is discussed. An algebra of second-quantized operators on the graph Hilbert space consistent with quantum statistics is constructed. Finally, the previously developed formalism is applied to quantum graph systems analogous to the free quantum particle and the Ising model of ferromagnetism. We then discuss the thermodynamic properties of these dynamical systems.