A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems

TL;DR

This paper introduces supmech, a noncommutative Hamiltonian mechanics framework that unifies classical and quantum systems, providing a natural derivation of quantum features and Schrödinger's equation without classical assumptions.

Contribution

It develops supmech as a universal mechanics framework, deriving quantum mechanics features and Schrödinger's equation from a purely algebraic, noncommutative formalism.

Findings

Supmech provides a universal mechanics covering all physical phenomena.

Quantum systems naturally admit Hilbert space realizations and superselection rules.

Schrödinger's equation and wave functions emerge naturally without classical Hamiltonian assumptions.

Abstract

Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM) (developed in paper I) with two extra ingredients : positive observable valued measures (PObVMs) [which serve to connect state-induced expectation values and classical probabilities] and the `CC condition' [which stipulates that the sets of observables and pure states be mutually separating] is proposed as a universal mechanics potentially covering all physical phenomena. It facilitates development of an autonomous formalism for quantum mechanics. Quantum systems, defined algebraically as supmech Hamiltonian systems with non-supercommutative system algebras, are shown to inevitably have Hilbert space based realizations (so as to accommodate rigged Hilbert space based Dirac bra-ket formalism), generally admitting commutative superselection rules. Traditional features of quantum mechanics of finite particle systems appear naturally. A treatment of localizability much simpler and more general than the traditional one is given. Treating massive particles as localizable elementary quantum systems, the Schr$\ddot{o}$dinger wave functions with traditional Born interpretation appear as natural objects for the description of their pure states and the Schr$\ddot{o}$dinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. A provisional set of axioms for the supmech program is given.