2006/06/20 by Abe, Mitsuo, Kawamura, Katsunori
#47L55 #81T05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.math-ph/0606047
Previously, we have shown that the CAR algebra for fermions is embedded in the Cuntz algebra \cal O2 in such a way that the generators are expressed in terms of polynomials in the canonical generators of the latter, and it coincides with the U(1)-fixed point subalgebra \cal A≡ \cal O2U(1) of \cal O2 for the canonical gauge action. Based on this embedding formula, some properties of \cal A are studied in detail by restricting those of \cal O2. Various endomorphisms of \cal O2, which are defined by polynomials in the canonical generators, are explicitly constructed, and transcribed into those of \cal A. Especially, we investigate branching laws for a certain family of such endomorphisms with respect to four important representations, i.e., the Fock representation, the infinite wedge representation and their duals. These endomorphisms are completely classified by their branching laws. As an application, we show that the reinterpretation of the Fock vacuum as the Dirac vacuum is described in representation theory through a mixture of fermions.