Spera, Mauro; Wurzbacher, Tilmann Determinants, Pfaffians and quasi-free representations of the car algebra. (English) Zbl 0917.46061 Rev. Math. Phys. 10, No. 5, 705-721 (1998). 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. Cited in 8 Documents 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 Keywords:quasi-free states; CAR algebras; Bogolubov automorphisms; Pfaffian line bundles; determinant line bundle; Hilbert space Grassmannian; Siegel manifold; isotropic Grassmannian; Borel-Weil type description PDFBibTeX XMLCite \textit{M. Spera} and \textit{T. Wurzbacher}, Rev. Math. Phys. 10, No. 5, 705--721 (1998; Zbl 0917.46061) Full Text: DOI References: [1] DOI: 10.2977/prims/1195193913 · Zbl 0227.46061 · doi:10.2977/prims/1195193913 [2] DOI: 10.1007/BF01646271 · Zbl 0173.29706 · doi:10.1007/BF01646271 [3] DOI: 10.1007/BF02096939 · Zbl 0785.58009 · doi:10.1007/BF02096939 [4] DOI: 10.2307/1969996 · Zbl 0094.35701 · doi:10.2307/1969996 [5] DOI: 10.2307/2371218 · Zbl 0011.24401 · doi:10.2307/2371218 [6] DOI: 10.1016/0393-0440(90)90007-P · Zbl 0736.53056 · doi:10.1016/0393-0440(90)90007-P [7] DOI: 10.2307/1969817 · Zbl 0051.13103 · doi:10.2307/1969817 [8] DOI: 10.1016/0022-1236(89)90024-4 · Zbl 0717.47006 · doi:10.1016/0022-1236(89)90024-4 [9] DOI: 10.1007/BF01645492 · Zbl 0186.28301 · doi:10.1007/BF01645492 [10] DOI: 10.1007/BF02698802 · Zbl 0592.35112 · doi:10.1007/BF02698802 [11] Shale D., J. Math. Mech. 14 pp 315– (1965) [12] DOI: 10.1093/qmath/44.4.497 · Zbl 0788.58027 · doi:10.1093/qmath/44.4.497 [13] Spera M., Russian J. Math. Phys. 2 pp 383– (1994) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.