When Poisson and Moyal Brackets are equal?

Abstract

In the phase space ${\mathbb{R}}^{2d}$, let us denote $\{A,B\}$ the Poisson bracket of two smooth classical observables and $\{A,B{\}}_{⊛}$ their Moyal bracket, defined as the Weyl symbol of $i[A,B]$, where $\hat{A}$ is the Weyl quantization of $A$ and $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$ (commutator). In this note we prove that if a smooth Hamiltonian $H$ on the phase space ${\mathbb{R}}^{2d}$, with derivatives of moderate growth, satisfies $\{A,H\}=\{A,H{\}}_{⊛}$ for any smooth and bounded observable $A$ then $H$ must be a polynomial of degree at most 2. This is related with the Groenewold-van Hove Theorem \cite{Gotay, Groen, vHove} concerning quantization of polynomial observables.