DETERMINANTS, PFAFFIANS AND QUASI-FREE REPRESENTATIONS OF THE CAR ALGEBRA

In this paper we apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative C-algebraic const ruction of the Determinant and Pfaffian line bundles discussed by Pres sley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the Determinant line bundle (aft er restriction from the Hilbert space Grassmannian to the Siegel manif old, or isotropic Grassmannian, consisting of all complex structures o n an associated Hilbert space) is derived from a Fock-anti-Fock corres pondence and an application of the Powers-Stormer purification procedu re. A Borel-Weil type description of the infinite dimensional Spin(c)- representation is obtained, via a Shale-Stinespring implementation of Bogolubov transformations.