Quantizing the B\"acklund transformations of Painlev\'e equations and the quantum discrete Painlev\'e VI equation

Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as "symmetries" or the B\"acklund transformations in Painlev\'{e} equations. We thereby propose a quantization of the discrete Painlev\'{e} VI equation as a discrete Hamiltonian flow commuting with the action of $W(D_4^{(1)})$.