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Symplectic vector space
About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.
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TL;DR: In this article, a canonical model for the Weil representation of the symplectic group Sp$(V ) was obtained for the case of the Heisenberg group, and a proof of a stronger form of the Stone-von Neumann property for the group was obtained.
Abstract: In this paper, we construct a quantization functor, associating a complex vector space $\cal{H}(V)$ to a finite-dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp$(V )$. The main new technical result is a proof of a stronger form of the Stone–von Neumann property for the Heisenberg group $H(V )$. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group over finite field.
22 citations
TL;DR: Symplectic lattice maps as mentioned in this paper approximate the flow map of a Hamiltonian system to arbitrarily high order, and are a viable tool in the study of Hamiltonian systems, which can be thought of as the restriction of a symplectic map to an invariant lattice.
22 citations
TL;DR: In this article, the authors present a review of Hilbert and symplectic space theory for elliptic differential operators with respect to global boundary conditions and modified Laplace operators, as well as a list of symbols and notations.
Abstract: Introduction: Organization of results Review of Hilbert and symplectic space theory GKN-theory for elliptic differential operators Examples of the general theory Global boundary conditions: Modified Laplace operators Appendix A. List of symbols and notations Bibliography Index.
22 citations
TL;DR: The Dirac operator on Riemannian manifolds is canonically defined in this article, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization.
Abstract: Symplectic Dirac operators, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization, are canonically defined in a similar way as the Dirac operator on Riemannian manifolds. These operators depend on a choice of a metaplectic structure as well as on a choice of a symplectic covariant derivative on the tangent bundle of the underlying manifold. This paper performs a complete study of these relations and shows further basic properties of the symplectic Dirac operators. Various examples are given for illustration.
22 citations
22 citations