Moreover, Groenewold first understood and demonstrated that the Moyal bracket is isomorphic to the quantum commutator, and thus that the latter cannot be made to faithfully correspond to the Poisson bracket, as had been envisioned by Paul Dirac. This observation and his counterexamples contrasting Poisson brackets to commutators have been generalized and codified to what is now known as the Groenewold– Van Hove theorem. See Groenewold's theorem for one version.