Emergent Matter of Quantum Geometry
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This thesis studies matter emergent as topological excitations of quantum geometry in quantum gravity models. In these models, states are framed four-valent spin networks embedded in a topological three manifold, and the local evolution moves are dual Pachner moves. We first formulate our theory of embedded framed four-valent spin networks by proposing a new graphic calculus of these networks. With this graphic calculus, we study the equivalence classes and the evolution of these networks, and find what we call 3-strand braids, as topological excitations of embedded four-valent spin networks. Each 3-strand braid consists of two nodes that share three edges that may or may not be braided and twisted. The twists happen to be in units of 1/3. Under certain stability condition, some 3-strand braids are stable. Stable braids have rich dynamics encoded in our theory by dual Pachner moves. Firstly, all stable braids can propagate as induced by the expansion and contraction of other regions of their host spin network under evolution. Some braids can also propagate actively, in the sense that they can exchange places with substructures adjacent to them in the graph under the local evolution moves. Secondly, two adjacent braids may have a direct interaction: they merge under the evolution moves to form a new braid if one of them falls into a class called actively interacting braids. The reverse of a direct interaction may happen too, through which a braid decays to another braid by emitting an actively interacting braid. Thirdly, two neighboring braids may exchange a virtual actively interacting braid and become two different braids, in what is called an exchange interaction. Braid dynamics implies an analogue between actively interacting braids and bosons. We also invent a novel algebraic formalism for stable braids. With this new tool, we derive conservation laws from interactions of the braid excitations of spin networks. We show that actively interacting braids form a noncommutative algebra under direction interaction. Each actively interacting braid also behaves like a morphism on non-actively interacting braids. These findings reinforce the analogue between actively interacting braids and bosons. Another important discovery is that stable braids admit seven, and only seven, discrete transformations that uniquely correspond to analogues of C, P, T, and their products. Along with this finding, a braid's electric charge appears to be a function of a conserved quantity, effective twist, of the braids, and thus is quantized in units of 1/3. In addition, each $CPT$-multiplet of actively interacting braids has a unique, characteristic non-negative integer. Braid interactions turn out to be invariant under C, P, and T. Finally, we present an effective description, based on Feynman diagrams, of braid dynamics. This language manifests the analogue between actively interacting braids and bosons, as the topological conservation laws permit them to be singly created and destroyed and as exchanges of these excitations give rise to interactions between braids that are charged under the topological conservation rules. Additionally, we find a constraint on probability amplitudes of braid interactions. We discuss some subtleties, open issues, future directions, and work in progress at the end.