Magnusian: Relating the Eikonal Phase, the On-Shell Action, and the Scattering Generator

TL;DR

This work clarifies a longstanding ambiguity in amplitudes: the eikonal phase $\delta$ (log of a matrix element) and the eikonal generator or Magnusian $\chi$ (logarithm of the S-matrix acting via Poisson brackets) are distinct in general, with a nontrivial exact classical relation $\mathcal{S} = \frac{e^{\{\chi,\cdot\}}-1}{\{\chi,\cdot\}}(P_i{\chi,Q_i}) + \chi(t_f,t_i,Q_i,P_i)$ linking them. The authors derive this relation classically and quantum mechanically, showing that in integrable scattering $\chi$ and the radial action $I$ coincide (up to a Legendre transform with $\mathcal{S}$), while spin or radiation generally spoils the simple correspondence. They further elucidate the interpretation of $\chi$ as a time-averaged, near-identity generator in the group of canonical transformations, outline perturbative relations between $\mathcal{S}$ and $\chi$, and illustrate the framework with harmonic and anharmonic oscillator examples. The results offer a robust, generic tool for extracting scattering observables from the Magnusian and hint at broader applications to bound orbits and worldline formalisms beyond scattering.

Abstract

Two fundamentally distinct types of quantities are both called "eikonal" in present amplitudes literature. The unitarity of the S-matrix ensures it can be written as the exponential of a Hermitian operator. The eikonal generator or Magnusian, which is the classical limit of the expectation value of that operator, generates all scattering observables. The leading order classical behavior of the phase of an S-matrix element is called the classical eikonal phase, and it coincides with a classical on-shell action. We demonstrate that the eikonal generator (Magnusian) and the eikonal phase (classical on-shell action) are inequivalent and find the exact general relationship between them. That relationship explains the special case of integrable scattering in which the two do coincide up to a Legendre transformation and explains why such a correspondence fails in general when spin or radiation are included.

Figures (2)

• Figure 1: Example of group trajectories for affine transformations due to evolution using the physical Hamiltonian (shown in solid orange) and Magnusian (shown in dashed blue). This is computed for the case of $u(t)=0.1 \sin(t)$ and $v(t)=0.05+0.3 \cos(t)+0.3 \sin(t)$.
• Figure 2: Example of group trajectories for $SU(2)$ transformations due to evolution using the physical Hamiltonian (shown in solid orange) and Magnusian (shown in dashed blue). This is computed for the case of $\Omega=\sqrt{\pi/2}$ and $\omega=1$.