Nonuniqueness of Green's functions at special points

TL;DR

Using AdS/CFT, the paper reveals that retarded Green's functions can be nonunique at special points in complex momentum space, with these points located at $\omega_* = -i\,2\pi T$ in the examples and tied to horizon regularity rather than to conventional chaos. Across scalar, Maxwell (vector and scalar modes), and shear gravitational perturbations, the authors show that the near-horizon regularity yields two regular solutions and that the incoming boundary condition becomes slope-dependent, producing a $G^R$ that depends on the approach in $(\omega,q)$. They demonstrate universality of the mechanism beyond AdS black holes, discuss phenomenological implications, and distinguish three distinct senses of a special point, clarifying when pole-skipping is linked to hydrodynamic poles or chaos. The results provide a robust framework for understanding nonunique holographic response functions and hint at broad implications for the late-time decay of perturbations in finite-temperature field theories.

Abstract

We investigate a new property of retarded Green's functions using AdS/CFT. The Green's functions are not unique at special points in complex momentum space. This arises because there is no unique incoming mode at the horizon and is similar to the "pole-skipping" phenomenon in holographic chaos. Our examples include the bulk scalar field, the bulk Maxwell vector and scalar modes, and the shear mode of gravitational perturbations. In these examples, the special points are always located at $ω_\star = -i(2πT)$ with appropriate values of complex wave number.