Algebraical structures that characterize different statistics for describing a system of particles are represented by equivalence classes of bilinear maps. Theorems that connect these structures are presented.
We show that the Hubbard model with infinite-range Coulomb coupling is equivalent to an ideal gas of three species of particles obeying fractional exclusion statistics. A full appreciation of this mapping requires an extension of the pertinent formalism. This very simple, but rather peculiar model is exactly solvable in any dimension and exhibits a Mott metal-insulator transition, whose universality class is shown to be that of a free spinless Fermi gas. A modified version of the Luttinger theorem is shown to apply in any dimension. We also characterize the metallic and insulating phases by obtaining the electronic band structure as well as the interacting density of states. The fractional statistics manifests itself on the amplitudes of several thermodynamic quantities and, in particular, the Pauli spin susceptibility is subdominant in all metallic phases, and a Curie-type of response appears.
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 In the previous sections we treated the irreducible representation space associated with the vacuum state. In this chapter we treat other representation spaces necessary to describe particle systems with the vacuum as a background. Such representation spaces are characterized as transportable localized excitations on the vacuum background. If we assume Haag’s duality of the vacuum representation, such a localized excitation is described by a localized endomorphism of the observables and its equivalence classes correspond to different representations called sectors(eigensubspaces of the superselection rules in each of which unrestricted superposition principle holds). From the discussion about composition and conjugation of localized endomorphisms, we define the statistics (of a sector) which represents the behaviour of more than two equivalent localized excitations under permutation, and we derive the connection of spin and statistics. We also introduce the lattice of local field algebras as operators intertwining different sectors, and the gauge group acting on them. Then the local observable algebras are the gauge invariant part of the local field algebras and sectors are inequivalent representations of the gauge group. The local field algebras can be chosen to satisfy the standard commutation or anticommutation relations between space-like separated regions and we can derive the connection of spin and statistics of sectors. In this way we obtain a deep understanding of the reason why Fermi fields in spatially separated regions anticommute.