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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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01 Jan 2008
TL;DR: In this article, the dual complex numbers dened as dual quaternions have been considered as a generalization of complex numbers and the dual unitary matrices that are more general form than unitary matrix were obtained.
Abstract: In this study, the dual complex numbers dened as the dual quaternions have been considered as a generalization of complex numbers. In addition, the dual unitary matrices that are more general form than unitary matrices were obtained. Finally, the group of the dual symplectic matrices was attained by using symplectic structure upon dual quater- nions. In particular, the group of symplectic matrices that are isomorphic to dual unitary matrices was studied. M.S.C. 2000: 15A33.
13 citations
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TL;DR: In this article, it was shown that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space and that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields.
Abstract: We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semigroups (3,4,5), (3,5,7), (3,7,8).
13 citations
TL;DR: In this article, a privileged basis of primitive forms in a symplectic space (V2n, ω) is defined recursively and applied in the setting of Heisenberg groups ℍn, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin's complex of differential forms.
Abstract: This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space (V2n, ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups ℍn, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin’s complex of differential forms in ℍn.
13 citations
01 Jan 1986
TL;DR: In this paper, the authors discuss geometrical optics and its scalar wave counterpart, Fourier optics, as a natural outgrowth of the group theoretical machinery emanating from the symplectic structure of (optical) phase space.
Abstract: We discuss (linear) geometrical optics and its scalar wave counterpart, Fourier optics, as a natural outgrowth of the group theoretical machinery emanating from the symplectic structure of (optical) phase space, as it derives from the Fermat principle We do, in fact, quantize geometrical optics The symplectic structure is preserved by the (symplectic) group of linear canonical transformations Every element of the symplectic group can be realized as a part of an image system in geometrical optics On the other hand, one may associate a (nilpotent) Lie algebra to the (symplectic) phase space on which the symplectic group Sp (4, ℜ) acts in the natural way Looking further on the irreducible unitary representations of the underlying simply connected Lie group, the Heisenberg group, one finds that the symplectic action on the Heisenberg group induces a group of unitary transformations in the space of each (infinite-dimensional) irreducible representation This group, called the metaplectic group, describes the evolution of the (scalar) wave field (Fourier optics) corresponding to the evolution of rays (geometrical optics), and thereby connects the two views on optics We put emphasis on the interplay of concepts, and for this reason, we also invoke intuition and terminology from classical and quantum mechanics
13 citations
TL;DR: In this article, the authors review the prequantization procedure in the context of symplectic supermanifolds with a symplectic form which is not necessarily homogeneous, and show that the Poisson algebra no longer is the set of smooth functions on the manifold, but a subset of functions with values in a super vector space of dimension 1 | 1.
13 citations