Decomposition of Representations of CAR Induced by Bogoliubov Endomorphisms

TL;DR

This paper analyzes how Bogoliubov transformations with finite corank decompose the CAR representations into irreducible components, revealing the direct link between corank and the number of subrepresentations.

Contribution

It provides a detailed decomposition of CAR representations under non-surjective Bogoliubov transformations with finite corank, clarifying the structure of reducible representations.

Findings

Number of subrepresentations equals the corank of the Bogoliubov operator

Decomposition explicitly characterized for finite corank cases

Reduces complexity in understanding CAR representation structure

Abstract

In a Fock representation, a non-surjective Bogoliubov transformation of CAR leads to a reducible representation. For the case that the corresponding Bogoliubov operator has finite corank, the decomposition into irreducible subrepresentations is clarified. In particular, it turns out that the number of appearing subrepresentations is completely determined by the corank.