Phase Space Formulation of S-matrix

TL;DR

The paper establishes an exact link between the S-matrix and classical S-symplectomorphisms within a phase-space formulation of quantum mechanics, revealing that the S-matrix acts as a fuzzy diffeomorphism whose classical limit reproduces the S-symplectomorphism. It introduces a deformation-quantization framework in which the quantum eikonal $\chi^{\star}$ is obtained by $\hbar$-deforming every Poisson bracket in the classical Magnus-expansion formula for the eikonal $\chi$, enabling diagrammatic computation of quantum corrections. Two ordering schemes are treated—Moyal (symmetric) and Wick (normal)—with explicit Magnus graph formalisms and coefficients ($\omega_1$, $\omega_2$) for up to three vertices, and a clear connection to standard KMOC-type impulse calculations via impulse operators. The construction provides a principled, all-orders diagrammatic method to obtain quantum eikonals and unifies classical and quantum scattering within a single phase-space language, with potential extensions to Kontsevich-type quantization and quantum field theory. This framework clarifies how classical time evolution and unitarity emerge as limits of a noncommutative, fuzzy phase-space evolution, and offers concrete computational tools for classical observables derived from quantum data.

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.

Figures (5)

• Figure 1: Translation is a symplectomorphism, whose generator is the momentum $p$.
• Figure 2: Scattering is a symplectomorphism, whose generator is the eikonal $\chi$.
• Figure 3: The bichrome notation for doubly directed graphs stands as a technique of visualization that aims to efficiently display two orientation structures within a single diagram. In this figure, the same $\mathcal{G} \in \mathop{\mathrm{Mag}}\nolimits^2(7,1)$ is drawn, while emphasizing the time ordering ${\to_\mathrm{T}}$ (first row) or the operator ordering ${\to_\mathrm{O}}$ (second row).
• Figure 4: A graphical representation of Eqs. (\ref{['Fp2']}) and (\ref{['Fm2']}). Ideally, we would have drawn two arrows per each edge to visualize a doubly directed graph.
• Figure 5: Summary of exact relations in classical and quantum scattering theory. Here, we pursue mathematical precision. Classical observables in the phase space formulation are identified as elements of the Poisson algebra $\mathcal{A}_{{\{\space{\,\,},{\,\,}\space\}}}$. Quantum observables in the phase space formulation are identified as formal power series in $\hbar$ whose coefficients are elements of the noncommutative ring $\mathcal{A}_\star$. Also, $\mathop{\mathrm{Aut}}\nolimits$ and $\mathop{\mathrm{Der}}\nolimits$ are notations for the sets of automorphisms and derivations, respectively.