[Paper Review] A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. I : Noncommutative Symplectic Geometry and Hamiltonian Mechanics

Tulsi Dass|arXiv (Cornell University)|Sep 25, 2009

Noncommutative and Quantum Gravity Theories59 references3 citations

TL;DR

This paper proposes a noncommutative Hamiltonian mechanics (NHM) framework based on noncommutative symplectic geometry and topological superalgebras to unify physics and probability theory, addressing Hilbert's Sixth Problem. It establishes that a universal Planck constant emerges naturally from the formalism when combining two non-supercommutative symplectic superalgebras, providing a foundational, autonomous formulation of quantum mechanics within a unified mechanics framework.

ABSTRACT

This series of papers is devoted to an open-ended project aimed at the solution of Hilbert's sixth problem (concerning joint axiomatization of physics and probability theory) proposed to be constructed in the framework of an all-embracing mechanics. In this first paper, the bare skeleton of such a mechanics is constructed in the form of noncommutative Hamiltonian mechanics (NHM) which combines elements of noncommutative symplectic geometry and noncommutative probability in the framework of topological superalgebras; it includes, besides NHM basics, a treatment of Lie group actions in NHM and noncommutative analogues of the momentum map, Poincar$\acute{e}$-Cartan form and the symplectic version of Noether's theorem. Canonically induced symplectic structure on the (skew) tensor product of two symplectic superalgebras (needed in the description of interaction between systems) is shown to exist if and only if either both system superalgebras are supercommutative or both non-supercommutative with a `quantum symplectic structure' characterized by a \emph{universal} Planck type constant; the presence of such a universal constant is, therefore, \emph{dictated} by the formalism. This provides proper foundation for an autonomous development of quantum mechanics as a universal mechanics.

Motivation & Objective

• To develop a unified, axiomatic framework for physics and probability theory, as posed by Hilbert’s Sixth Problem.
• To construct a noncommutative Hamiltonian mechanics (NHM) that autonomously incorporates quantum mechanics as a subdiscipline.
• To unify noncommutative symplectic geometry with noncommutative probability in a topological superalgebra setting.
• To establish the existence of a universal Planck constant as a formal consequence of symplectic structure on interacting systems.
• To provide a foundation for quantum-classical correspondence and measurement within a single, coherent formalism.

Proposed method

• Develops a superderivation-based differential calculus on topological superalgebras to generalize differential forms and mappings.
• Introduces noncommutative symplectic structures via Poisson brackets and symplectic forms on superalgebras, generalizing classical Hamiltonian mechanics.
• Defines noncommutative Hamiltonian dynamics using state spaces and evolution governed by Hamiltonian functionals in the superalgebra framework.
• Constructs canonically induced symplectic structures on the (skew) tensor product of two symplectic superalgebras.
• Derives noncommutative analogues of the momentum map, Poincaré–Cartan form, and Noether’s theorem using Lie group actions on superalgebras.
• Establishes that the symplectic structure on tensor products exists only if both algebras are either supercommutative or non-supercommutative with a universal Planck-type constant.

Experimental results

Research questions

• RQ1 Under what conditions does a symplectic structure exist on the tensor product of two noncommutative symplectic superalgebras?
• RQ2 How can a universal Planck constant be derived from the formalism of noncommutative symplectic geometry rather than imposed ad hoc?
• RQ3 In what way does noncommutative Hamiltonian mechanics unify classical and quantum mechanics as subdisciplines?
• RQ4 How do symmetries and conservation laws emerge in the noncommutative setting via a generalized Noether’s theorem?
• RQ5 Can a consistent description of interacting quantum systems be formulated within a noncommutative Hamiltonian framework without assuming classical limits a priori?

Key findings

• A canonically induced symplectic structure on the (skew) tensor product of two symplectic superalgebras exists if and only if both are supercommutative or both are non-supercommutative with a universal Planck-type constant.
• The presence of a universal Planck constant is formally dictated by the requirement of consistent symplectic structure on interacting systems, emerging naturally from the formalism.
• Noncommutative Hamiltonian mechanics (NHM) provides a unified framework in which both classical and quantum mechanics arise as special subdisciplines.
• The noncommutative momentum map and symplectic version of Noether’s theorem are successfully generalized to the superalgebraic setting.
• The absence of quantum-classical interaction is not a problem in NHM, as classical systems can be treated as semiclassical approximations within the same formalism.
• The formalism allows for a consistent, autonomous development of quantum mechanics without prior classical foundations, with quantum systems defined by mutual compatibility between observables and pure states.