It is known that given a symplectic manifold, one can define different Poisson brackets. I am trying to see whether in classical sense ($\hbar=0$) given two different Poisson brackets (i.e. a symplectic manifold can be equipped with non equivalent different Poisson bracket structures), the quantized structures are totally different in both observables and other algebraic structures. And say one can perform quantization by tuning $\hbar=1$. Would the result different due to different Poisson brackets employed? I guess this answer would also apply to field theory content as well. In addition to that question, is the quantization unique up to some equivalence?



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