Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure: Part I

Schrödinger quantum mechanics is formulated as an infinite-dimensional Hamiltonian system whose phase space carries an additional structure (uncertainty structure) to account for the probabilistic character of the theory. The algebra of observables is described as an algebra of smooth functions on the quantal phase space, with a product naturally induced by the geometrical structures living on that manifold. The possibility of generalizing Schrödinger mechanics along these lines is discussed.