When Poisson and Moyal Brackets are equal? (original) (raw)

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[Â,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 given smooth Hamiltonian H on the phase space R 2d , with derivatives of moderate growth, satisfies {A, H} = {A, H} ⊛ for any observable A in the Schwartz space S(R 2d), then, as it is expected, H must be a polynomial of degree at most 2 in R 2d. A related answer to this question is given in the Groenewold-van Hove Theorem [4, 5, 8] concerning quantization of polynomial observables. We consider here more general classes of Hamiltonians.