Thick morphisms and Quantum Mechanics

For an arbitrary morphism of (super)manifolds, the pull-back is a linear map of space of functions.
In 2014 Theodore Voronov has introduced thick morphisms of (super)manifolds which define generally non-linear pull-back of functions. This construction was introduced as an adequate tool to describe $L_\infty$ morphisms of algebras of functions provided with the structure of homotopy Poisson algebra.
Voronov introduced the special geometrical object $S$, which defines the thick morphism.
It turns out that if you go down from "heaven to earth", and consider usual (not super!) manifolds, then we come to constructions which has natural interpretation in classical and Quantum mechanics.
In particular in this case the geometrical object $S$ which defines thick diffeomorphism becomes an action of classical mechanics, and pull-back of thick diffeomorphism with quadratic action give spinor representation.
The talk is based on the joint work with Theodore Voronov: https://arxiv.org/abs/1909.00290