The Algebraic Way

TL;DR

This paper explores the non-commutative symplectic algebra underlying quantum mechanics, revealing its connections to classical and Bohmian mechanics, and deriving fundamental dynamical equations from a unified algebraic framework.

Contribution

It introduces a comprehensive algebraic structure that unifies quantum, classical, and Bohmian mechanics, extending the mathematical foundation of quantum dynamics.

Findings

The algebra contains both Weyl-von Neumann and Moyal algebras.

The algebraic approach yields equations reducing to classical and quantum limits.

Bohmian mechanics is shown as a fragment of the underlying algebra.

Abstract

In this paper we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. We show that this algebra contains both the Weyl-von Neumann algebra and the Moyal algebra. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schr\"{o}dinger equation and its wave function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.