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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, the authors describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations, and the multiplication in this algebra generates an associative product of functions on the phase space.
Abstract: The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard dp ∧ dq structure on In this paper, we describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations The multiplication in this algebra generates an associative product of functions on the phase space This product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer Zero and constant curvature cases are considered as examples The quantization with both static and time-dependent electromagnetic fields is obtained The expansion of the product by the deformation parameter , written in the covariant form, is compared with the known deformation quantization formulae
41 citations
TL;DR: A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre-Hilbert space (one-boson test function space) are given in this article.
Abstract: A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre‐Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self‐contained fashion.
41 citations
TL;DR: Deformations admitting a unit element of a local associative algebra defined on the space of functions on a manifold as discussed by the authors have been studied in statistical mechanics (KMS conditions) and conformal symplectic geometry.
Abstract: Deformations admitting a unit element of a local associative algebra defined on the space of functions on a manifold Definition and properties of the *
f
-products and conformal symplectic geometry Deformations of a *
f
-products A theorem of rigidity Application to statistical mechanics (KMS conditions)
41 citations
TL;DR: In this paper, a variational principle for simply connected simply connected symplectic manifolds is introduced and the corresponding field equations are studied for two-dimensional compact and non-compact simply connected manifold.
40 citations
TL;DR: In this article, the Segal-Weinless approach to quantization has been extended to real symplectic spaces, and a theorem about unitary equivalence has been proved.
Abstract: We prove a theorem, which, while it fits naturally into the Segal–Weinless approach to quantization seems to have been overlooked in the literature: Let (D,σ) be a symplectic space, and T (t) a one parameter group of symplectics on (D,σ). Let (H, 2Im〈⋅ ‖ ⋅〉) be a complex Hilbert space considered as a real symplectic space, and U(t) a one‐parameter unitary group on H with strictly positive energy. Suppose there is a linear symplectic map K from D to H with dense range, intertwining T (t) and U(t). Then K is unique up to unitary equivalence.
40 citations