Generalized Heisenberg Dynamics Revisited

Abstract

Taking as a model the fact that Heisenberg's matrix mechanics was derived from Hamiltonian mechanics using the correspondence principle, we explore a class of dynamical systems involving discrete variables, with Nambu mechanics as the starting point. Specifically, we reconstruct an extended version of matrix mechanics that describes dynamical systems possessing physical quantities expressed through generalized matrices. Furthermore, we reconfirm that a multiple commutator involving generalized matrices can serve as a discrete (quantized) version of the Nambu bracket or the Jacobian.

Figures (3)

• Figure 1: Extensions of Hamiltonian mechanics and matrix mechanics.
• Figure 2: Strategy for exploring new dynamics.
• Figure 3: Classical theory and its quantum (or quantum analog of) theory.