The Fermion Algebra

Abstract

Abstract This chapter introduces the Fermion algebra as the main technical tool for studying in Chapter 7 the two dimensional Ising model as the prototype lattice model in statistical mechanics. It will also feature in Chapter 8 as a means of constructing representations of chiral algebras, particularly the Virasoro algebra, in conformal field theories which can appear in critical statistical mechanical models. Ignoring most of the structure, the Fermion algebra is just an infinite tensor product of 2 x 2 matrix algebras. Such infinite tensor products of matrix algebras, UHF algebras, are studied in Sections 6.2 and 6.3, as well as their representations on infinite tensor products of Hilbert spaces. Subtleties about completions of such infinite tensor products of Hilbert spaces lead to inequivalent representations of UHF algebras, and criteria to decide when infinite tensor automorphisms are inner or weakly inner in the trace representation.