Autonomous geometro-statistical formalism for quantum mechanics I : Noncommutative symplectic geometry and Hamiltonian mechanics

Considering the problem of autonomous development of quantum mechanics in the broader context of solution of Hilbert's sixth problem (which relates to joint axiomatization of physics and probability theory), a formalism is evolved in this two-part work which facilitates the desired autonomous development and satisfactory treatments of quantum-classical correspondence and quantum measurements. This first part contains a detailed development of superderivation based differential calculus and symplectic structures and of noncommutative Hamiltonian mechanics (NHM) which combines elements of noncommutative symplectic geometry and non-commutative probability in an algebraic setting. The treatment of NHM includes, besides its basics, a reasonably detailed treatment of symplectic actions of Lie groups and noncom-mutative analogues of the momentum map, Poincaré-Cartan form and the symplectic version of Noether's theorem. Consideration of interaction between systems in the NHM framework leads to a division of physical systems into two 'worlds' — the 'commutative world' and the 'noncommutative world' [in which the systems have, respectively, (super-)commutative and non-(super-)commutative system algebras] — with no consistent description of interaction allowed between two systems belonging to different 'worlds'; for the 'noncommutative world', the formalism dictates the introduction of a universal Planck type constant as a consistency requirement.