It is shown how many problems involving aspects of spin-1 matrix algebra may be handled in a way analogous to the spin-1/2 algebra of the Pauli matrices. A number of examples of the main point are given in illustration.
The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations. Since gamma matrices are the direct products of two Pauli spin matrices, they provide an appropriate way to describe a system of two spin-1/2 particles. Such multiparticle spin states are intimately connected with the theorems of John Bell. The graph is helpful in analyzing an important example of the Bell–Kochen–Specker theorem.
Abstract. We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter is described by a map between modules over the translation-group algebra, so homological methods are applicable. In any dimension every point-like charge appears as a vertex of a fractal operator, and can be isolated with energy barrier at most logarithmic in the separation distance. For a topologically ordered system in three dimensions, there must exist a point-like nontrivial charge. A connection between the ground state degeneracy and the number of points on an algebraic set is discussed. Tools to handle local Clifford unitary transformations are given.