Quantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation

Abstract

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 Bäcklund transformations in Painlevé equations. We thereby propose a quantization of discrete Painlevé VI equation as a discrete Hamiltonian flow commuting with the action of $W (D_4^{(1)})$.