Intrinsic Dynamics of Symplectic Manifolds: Membrane Representation and Phase Product

On general symplectic manifolds we describe a correspondence between symplectic transformations and their phase functions. On the quantum level, this is a correspondence between unitary operators and phase functions of the WKB-approximation. We represent generic functions via symplectic area of membranes and consider related geometric properties of the noncommutative phase product. An interpretation of the phase product in terms of symplectic groupoids and the groupoid extension of Lagrangian submanifolds are described. The membrane representations of corresponding Lagrangian phase functions are obtained. This paper uses the intrinsic dynamic approach based on the notion of Ether Hamiltonian which is a generalization of the notion of symplectic connection. We demonstrate that this approach works for torsion case as well.