Quantization Yielding Sharp and Unsharp Observables

The so-called spectral decomposition quantization (SDQ) involving spectral decompositions of the identity operator has been proposed. The quantization yields the standard (sharp) quantum observables in the form of Hermitian operators acting in the Hilbert space of square-integrable functions. Here, the quantizable classical observables are limited as in the case of geometric quantization (GQ). It has been shown that this quantization leads to the same results as those yielded by geometric quantization in the position representation as well as in the momentum representation. The method of quantization also has been formulated in the representation concerning any pair of canonical observables. Then, the quantization has been generalized by replacing the spectral measure appearing in the spectral decompositions of the identity operator with positive-operator-valued measure constructed by using suitable smearing function (e.g. confidence distribution) so that the quantization yields unsharp quantum observables. It has been shown that the unsharp quantum observables are yielded from the quantization of the so-called unsharp quantizable classical observables. Some properties of unsharp classical observables have been investigated. The explicit expressions of unsharp quantum observables have been obtained.