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Determinants, Pfaffians and quasi-free representations of the car algebra. (English) Zbl 0917.46061

Summary: We apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative \(C^*\)-algebraic construction of the determinant and Pfaffian line bundles discussed by Pressley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the determinant line bundle (after restriction from the Hilbert space Grassmannian to the Siegel manifold, or isotropic Grassmannian, consisting of all complex structures on an associated Hilbert space) is derived from a Fock-anti-Fock correspondence and an application of the Powers-Størmer purification procedure. A Borel-Weil type description of the infinite-dimensional \(\text{Spin}^c\)-representation is obtained, via a Shale-Stinespring implementation of Bogolubov transformations.

MSC:

46L55 Noncommutative dynamical systems
53D50 Geometric quantization
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14F25 Classical real and complex (co)homology in algebraic geometry
81R30 Coherent states
46N50 Applications of functional analysis in quantum physics
46L40 Automorphisms of selfadjoint operator algebras
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