A prismatic approach to $(varphi, hat G)$-modules and $F$-crystals

We give a new construction of $(varphi, hat G)$-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a different proof about the equivalence between the category of prismatic $F$-crystals in finite locally free $mathcal{O}_{Delta}$-modules over $mathrm{Spf}(mathcal{O}_K)$ and the category of lattices in crystalline representations of $G_K$, where $K$ is a complete discretely valued field of mixed characteristic with perfect residue field. We also propose a possible generalization of this result for semi-stable representations using the absolute logarithmic prismatic site.