Dequantization Techniques for Weyl Quantization

Abstract

By quantization we mean the linear bijection from ${\mathcal{S}}^{\prime }(\Pi )$ to $\mathcal{L}(\mathcal{S}(R)),{\mathcal{S}}^{\prime }(R))$ due to Weyl, and by dequantization we mean its inverse. We propose two new, but related, dequantization schemes. The first is adapted to knowledge of the matrix elements (with respect to the Hermite–Gaussian functions) of the operator to be dequantized, while the second is adapted to its integral kernel. Our dequantization schemes are completely general. We apply these methods to the case where the operators in question are Toeplitz operators related to functions of angle on phase space. This enables us to compare the symbols of these Toeplitz operators with the functions of angle themselves.