Les Graphes dans la Mecenique Quantique

All fundamental particles in the universe come in two types: fermions and bosons. The former category contains electrons, protons, and neutrons, and forbids two particles from occupying the same quantum state (the Pauli exclusion principle); the latter category, which contains photons and other force-carrying particles, has no such restriction. When bosons and fermions are confined in lattices, such as in solid-state systems, the equations of motion can be phrased in terms of the adjacency matrix of an undirected and generally weighted graph. The properties of these quantum many-particle systems can therefore be analyzed in terms of graph theory. I will discuss these relationships, and show that by using graph theory, it is possible to obtain a more efficient determination of the eigenstates (and therefore the properties and dynamics) of interesting physical systems.