Generalized Heisenberg Dynamics Revisited

TL;DR

This paper revisits the foundations of Heisenberg's matrix mechanics, extending it to systems with discrete variables using Nambu mechanics, and demonstrates that multiple commutators can serve as a discrete Nambu bracket.

Contribution

It introduces an extended matrix mechanics framework for discrete systems based on Nambu mechanics and clarifies the role of multiple commutators as discrete Nambu brackets.

Findings

Multiple commutators act as discrete Nambu brackets.

Extended matrix mechanics describes systems with generalized matrices.

Reconfirmation of the connection between commutators and Nambu brackets.

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.