Weyl quantization of fractional derivatives

The quantum analogs of the derivatives with respect to coordinates q(k) and momenta p(k) are commutators with operators P(k) and Q(k). We consider quantum analogs of fractional Riemann-Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann-Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier transformation. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.3009533]