A Coding of Real Null Four-Momenta into World-Sheet Co-ordinates

TL;DR

The paper shows that the Cartesian components of real null four-momenta in four-dimensional space-time can be encoded into complex world-sheet coordinates on a simply connected minimal-surface–like world sheet, without introducing tachyons. This identification is demonstrated through multiple, equivalent routes: minimization of the Koba–Nielsen integrand, a Weierstrass-based minimal-surface construction, and an Eisenhart parametrization of minimal surfaces; in each case, a set of points $z_i$ on the world sheet is tied to the momenta, yielding cross-ratio relations that reproduce Mandelstam variables and Regge behavior. The results hold across space-time signatures, with particularly transparent structure in $(2,2)$ where $z_i$ lie on the real line or a circle, and are connected to Lorentz transformations via $SL(2,\mathbb{C})$. The work situates these connections within a broader context of string amplitudes, AdS extensions, and high-energy asymptotics.

Abstract

The results of minimizing the action for string-like systems on a simply-connected world sheet are shown to encode the Cartesian components of real null momentum four-vectors into co-ordinates on the world sheet. This identification arises consistently from different approaches to the problem.