Theoretical Statistics and Mathematics Unit, ISI Delhi

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

by Tulsi Dass

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 noncommutative probability in an algebraic setting. The
treatment of NHM includes, besides its basics, a reasonably detailed
treatment of symplectic actions of Lie groups and noncommutative
analogues of the momentum map, Poincar$\acute{e}$-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.

isid/ms/2009/03 [fulltext]

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