The symplectic spinor bundle is defined [2] to be the Hilbert space bundle
associated to the metaplectic structure via the metaplectic representation also called the Segal–Shale–Weil[3][4][5] representation of Here, the notation denotes the group of unitary operators acting on a Hilbert space
The Segal–Shale–Weil representation [6] is an infinite dimensional unitary representation
of the metaplectic group on the space of all complex
valued square Lebesgue integrablesquare-integrable functions Because of the infinite dimension,
the Segal–Shale–Weil representation is not so easy to handle.
Notesedit
^Kostant, B. (1974). "Symplectic Spinors". Symposia Mathematica. XIV. Academic Press: 139–152.
^Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0 page 37
^Segal, I.E (1962), Lectures at the 1960 Boulder Summer Seminar, AMS, Providence, RI
^Shale, D. (1962). "Linear symmetries of free boson fields". Trans. Amer. Math. Soc. 103: 149–167. doi:10.1090/s0002-9947-1962-0137504-6.
^Weil, A. (1964). "Sur certains groupes d'opérateurs unitaires". Acta Math. 111: 143–211. doi:10.1007/BF02391012.
^Kashiwara, M; Vergne, M. (1978). "On the Segal–Shale–Weil representation and harmonic polynomials". Inventiones Mathematicae. 44: 1–47. doi:10.1007/BF01389900.
Further readingedit
Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0