Role of the imaginary part in the Moyal quantization

TL;DR

This paper demonstrates that the imaginary part of the Moyal star-genvalue equation uncovers Hamiltonian symmetries, leading to conserved quantities that serve as variables in the Wigner function, exemplified through the Toda lattice.

Contribution

It reveals the significance of the imaginary part in the Moyal quantization for identifying symmetries and conserved quantities, with application to the Toda lattice.

Findings

Imaginary part uncovers Hamiltonian symmetries.

Conserved quantities derived for Toda lattice.

Conserved quantities used as variables in Wigner function.

Abstract

We show that the imaginary part of the $\star$-genvalue equation in the Moyal quantization reveals the symmetries of the Hamiltonian by which we obtain the conserved quantities. Applying to the Toda lattice equation, we derive conserved quantities which are used as the independent variables of Wigner function.