The Algebraic Way

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 Schrödinger 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.

26 pages. No figures

References in corpus (5)

• Quantum optics in the phase space - A tutorial on Gaussian states
• Algebras of distributions suitable for phase-space quantum mechanics. I
• Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
• The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach
• The Clifford Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles