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Preprints, Working Papers, ... Year : 2022

When Poisson and Moyal Brackets are equal?

Didier Robert
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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.
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Dates and versions

hal-03726775 , version 1 (18-07-2022)
hal-03726775 , version 2 (27-02-2023)
hal-03726775 , version 3 (01-04-2023)
hal-03726775 , version 4 (23-05-2023)


  • HAL Id : hal-03726775 , version 1


Didier Robert. When Poisson and Moyal Brackets are equal?. 2022. ⟨hal-03726775v1⟩
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