Publication Date

Summer 2023

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Physics and Astronomy

Advisor

Kassahun Betre; Christopher Smallwood; Curtis Asplund

Abstract

This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.

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