Publications and Preprints

Supmech: the Geometro-statistical Formalism Underlying Quantum Mechanics
by
Tulsi Dass
A scheme of mechanics, called supmech', is developed which aims at providing a base for a solution of Hilbert's sixth problem (seeking a unified axiomatization of physics and probability theory) and serves to develop quantum mechanics autonomously (i.e. without having to \emph{quantize} classical systems). Integrating noncommutative symplectic geometry and noncommutative probability in an algebraic setting, it associates, with every experimentally accessible' system, a symplectic superalgebra and operates essentially as noncommutative Hamiltonian mechanics with an extra condition of compatible completeness' between observables and pure states incorporated. A noncommutative analogue of the Poincar$\acute{e}$-Cartan form is introduced. It is shown that interactions between systems can be consistently described in supmech only if either (i) all system algebras are supercommutative, or (ii) all system algebras are non-supercommutative and have a quantum symplectic structure characterized by a \emph{universal} Planck type constant of the dimension of action. Standard quantum systems', defined algebraically, are shown to have faithful Hilbert space - based realizations; the rigged Hilbert space - based Dirac bra-ket formalism naturally appears. The formalism has a natural place for commutative superselection rules. Treating massive particles as localizable elementary quantum systems, the Schr$\ddot{o}$dinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. Quantum measurements are satisfactorily treated; the unwanted macroscopic superpositions are shown to be suppressed when the observations on the apparatus are restricted to macroscopically distinguishable pointer readings. This treatment automatically incorporates the decohering effects of the internal environment of the apparatus; a trivial extension also serves to include the external environment.

isid/ms/2008/06 [fulltext]