Showing papers on "Symplectic vector space published in 1979"

A uniqueness result in the Segal–Weinless approach to linear Bose fields

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.

Existence and invariance of twisted products on a symplectic manifold

Abstract: It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.

Reduction of symplectic manifolds through constants of the motion

Abstract: We describe the reduction of a dynamical system on a symplectic manifold by the use of constants of the motion. A constant of the motion together with a symplectic structure defines a distribution, from which one obtains a foliation. The Hamiltonian dynamical system is reduced to another of lower dimension on a certain quotient manifold defined by the foliation. The role of the dynamics remaining on the leaves is discussed.

Weil's representations of the symplectic groups over finite fields

Abstract: $Sp(2m)$ as connected semisimple algebraic groups defined over $F(q)$ endowed with the Frobenius map $F$. Let $M_{2n.m}(F(q))$ be the set of all $2n\\times m$ matrices with entries in $F(q)$ and $S(M_{2n,m}(F(q)))$ be the space of all complex valued functions on $M_{2n,m}(F(q))$ . Then we can construct, associated with $S$ , so called Weil’s representation $\\pi_{S,m}$ of $Sp(2m)^{F}$ realized on $S(M_{2n.m}(F(q)))$ . The representation $\\pi_{S,m}$ can be decomposed naturally according to representations of

Equivalence and slice theory for symplectic forms on closed manifolds

Abstract: ABsmAcr. In this paper, a study is made of the pullback action of the diffeomorphism group on the totality of symplectic forms on a compact manifold. For this action, the orbit is shown to be a smooth (Banach) manifold consisting of a denumerable union of submanifolds, each lying in a fixed cohomology class. In addition, a precise characterization is given of those symplectic manifolds for which there is a local factorization of the pullback action in the sense of a transverse "slice" of closed 2-forms, invariant under the group of symplectic diffeomorphisms.

The reflection character of the finite symplectic and odd-dimensional orthogonal groups

Abstract: (1979). The reflection character of the finite symplectic and odd-dimensional orthogonal groups. Communications in Algebra: Vol. 7, No. 16, pp. 1747-1757.