Phase Space Formulation of S-matrix

We establish an exact relation between the S-symplectomorphism and the S-matrix by means of the phase space formulation of quantum mechanics. The adjoint action of the S-matrix defines a fuzzy diffeomorphism on phase space whose classical limit is the S-symplectomorphism. The relation between classical and quantum eikonals is immediate via $\mathrm{\hslash }$-deformation of each Poisson bracket in the Magnus formula. Diagrammatic computation of quantum eikonal is illustrated for quantizations in both symmetric and normal orderings.