When Poisson and Moyal Brackets are equal?

Abstract

In the phase space $\R^{2d}$, let us denote $\{A,B\}$ the Poisson bracket of two smooth classical observables and $\{A, B\}_\circledast $ 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 $\R^{2d}$, with derivatives of moderate growth, satisfies $\{A,H\}= \{A, H\}_\circledast$ 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.