Showing papers on "Symplectic representation published in 1979"

A Symplectic Framework for Field Theories

Abstract: An intuitive derivation of symplectic concepts in mechanics and field theory- Nonrelativistic particle dynamics- Field theory- Examples

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.

Completely integrable systems and symplectic actions

Abstract: We study results on a class of completely integrable systems, for instance, with Hamiltonian H (x,y) = (1/2) Jni=1y2i +Ji

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

New expressions for the eigenvalues of the invariant operators of O(N) and Sp(2n)

Abstract: In the spirit of the recent work of Popov for u(n), we derive a direct expansion of the eigenvalues of the invariant operators Cp for the orthogonal and symplectic groups in terms of the power sums with completely specified coefficients βp(ν), which are easy to compute. The resulting expression, which is a complete analog of the u(n) results, is closed, simple and manifests the general structure of the Cp. It is now possible to say for what value of p a particular combination of the Sk’s begin to appear. Explicit applications of these results in computing the Cp for p<8 illustrate fully their simplicity. Thus our work simplifies and unifies the treatment of this aspect of the problem for the semisimple Lie groups.

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.