On the Statistical Properties of Linear Quantization and Inverse Operator

The paper uses an approach to the formal description of linear quantization as a statistical mixture of operators of the so-called tau quantization. This allows us to consider arbitrary Hermitian linear quantization as a linear integral operator acting on functions from the space of canonical variables and translating them into matrix elements of the quantum operator. In such a scheme, various quantization rules are uniformly studied: Weyl, Born, Jordan, and a number of others. The main result relates to the proof of the uniqueness of Weyl quantization as an operator, which is answered by the inverse operator generating a positive definite probability measure. For any other quantization, the inverse operator (dequantization) generates a measure that is not positive definite.