Quantizing the discrete Painlev\'e VI equation : The Lax formalism

TL;DR

This paper explores two methods for quantizing the discrete Painlevé VI equation, demonstrating that the Lax formalism approach yields the same quantization as the affine Weyl group symmetry method, thus confirming their equivalence.

Contribution

It shows that the Lax formalism approach to quantizing the discrete Painlevé VI equation is successful and aligns with the affine Weyl group symmetry method.

Findings

Lax formalism provides a valid quantization method.

Both approaches produce the same quantization results.

Confirms the equivalence of two quantization methods.

Abstract

A discretization of Painlev\'e VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point of view. It turns out that the second approach is also successful and gives the same quantization as in the first approach.