In the litterature about quantization schemes, people tend to use Weyl ordering a lot.
Altough it enjoys some desirable properties like sending real functions into self-adjoint operators or sending Schwarz functions into trace class operators, we know that these features are not unique of Weyl ordering.
Is there any deep reason (being mathematical of physical) to prefer Weyl ordered quantizers?
Quantization may roughly be thought as transition from classical to quantum description of a system. What is its precise definition is not generally agreed since many of related questions are still opened. With different motivations, quantization was approached in different manners, see [1] and references therein. Even in this case, the ordering problem is not trivial. The most important problem regarding the quantization of R2n is in that it is not known how should the algebraic and Lie algebraic multiplications of quantum mechanics be realized in unambiguous and consistent way.
[1] Arens, R., and Babit, D., ” Algebraic difficulties of preserving dynamical relations when forming quantum-mechanical operators, ” J. Math. Phys. 6, 1071-1075 (1965).
From a physical point of view, Weyl ordering provides a consistent procedure for quantizing polynomial Hamiltonians, but of course this is not enough to be preferred as a quantization method. From a mathematical point of view, its importance lies in the subsequent developments of Weyl's idea by Wigner and Moyal which, ultimately, led to the idea of star products and deformation quantization. It was proved by Kontsevich in the late 90s (in a work that gave him the Fields medal) that any Poisson manifold can be quantized following these ideas. You can see a somewhat cursory description of this line of reasoning in the introduction to a paper of mine: https://arxiv.org/pdf/1110.5700.pdf
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