# Hidden Nambu mechanics II: Quantum/semiclassical dynamics

## TL;DR

This paper demonstrates that Nambu mechanics, a generalized Hamiltonian framework, is also applicable to certain quantum and semiclassical dynamics, revealing hidden structures in expectation value evolutions.

## Contribution

The paper introduces a method to identify hidden Nambu structures in quantum and semiclassical systems with one degree of freedom, extending the applicability of Nambu mechanics.

## Key findings

- Nambu structure found in quantum harmonic oscillator dynamics.
- Semiclassical wave packet dynamics exhibit Nambu mechanics.
- Numerical results support the formalism's validity in simple models.

## Abstract

Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon--Heiles model of two interacting oscillators.