A No—Go Theorem for Nonlinear Canonical Quantization

We want to point out thc following strcngthcning of the classical theorem of Grocnewold and van Hove:There exists no ,napping Op from polynomial observables f (p, q) on the phase space R2n into linear operators on L2(Rn)which would map Poisson brackets into commutators, the position and momentum obscrvablcs p and q into the usual(Schrodinger) position and momentum operators, and would obey the yon Neumann rifle 0p(cfk) = c 0p(f)k for k = 1,2, 3and c R. The point is that neither linearity, nor continuity etc. of Op are assumed.