Theoretical Statistics and Mathematics Unit, ISI Delhi

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]

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