When are Poisson and Moyal Brackets equal?

Quand les crochets de Poisson et de Moyal sont-ils égaux?

Didier Robert (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

In the phase space 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 A is the Weyl quantization of A and [ A, B] = A B − B 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} ⊛ 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 [3, 4, 6] concerning quantization of polynomial observables.

Domaines: Physique mathématique [math-ph]

Didier Robert. When are Poisson and Moyal Brackets equal?. 2023. ⟨hal-03726775v4⟩