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THE METAPLECTIC REPRESENTATION
APPENDIX TO CHAPTER 5
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algebra g bijective classical mechanics Clearly computation connected Lie group conservation laws curv(L.V define Definition denote diffeomorphism differential equations dim g dynamical system equivalent exists exp tX f e g f e S(Kn finite dimensional Fourier function G acts g e G group of diffeomorphisms group with Lie Hamiltonian G-space Heisenberg group Hence Hermann Hermitian structure Hilbert space implies integral irreducible unitary representation isomorphic Kostant Lemma l.2 Let G Let H line bundle linear Math mathematical metaplectic representation nilpotent Lie group observables one-parameter group orbit orthonormal basis p(exp physics Poisson bracket polarization of f Proof prove quantization quantum mechanics representation of G satisfies Section simply connected Stone-Von Neumann theorem subalgebra subgroup subspace Suppose symplectic manifold symplectic structure trace class unitary representation vector field vector space