Phase Space Formulation of S-matrix

TL;DR

This paper establishes a precise connection between the S-matrix and phase space symplectomorphisms, revealing how quantum and classical transformations relate through a phase space formulation and diagrammatic methods.

Contribution

It introduces an exact relation between the S-matrix and S-symplectomorphism using phase space quantum mechanics, including a diagrammatic approach for quantum eikonal calculations.

Findings

Quantum S-matrix induces a fuzzy diffeomorphism on phase space.

Classical limit of the quantum transformation is the S-symplectomorphism.

Diagrammatic methods effectively compute quantum eikonals in different orderings.

Abstract

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 $\hbar$-deformation of each Poisson bracket in the Magnus formula. Diagrammatic computation of quantum eikonal is illustrated for quantizations in both symmetric and normal orderings.