Fractional Klein-Gordon Equation on AdS$_{2+1}$

TL;DR

This paper introduces a covariant fractional Klein-Gordon equation on AdS$_{2+1}$, deriving explicit kernel representations and analyzing the conditions for the existence of propagators, with potential applications in AdS/CFT correspondence.

Contribution

It provides a metric-independent, covariant definition of the fractional Klein-Gordon equation on AdS space and computes explicit kernel and propagator representations.

Findings

Propagator exists only for small mass relative to inverse AdS radius

Explicit kernel representation of fractional Laplace-Beltrami operator derived

Results applicable to AdS/CFT in statistical mechanics and quantum information

Abstract

We propose a covariant definition of the fractional Klein-Gordon equation with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional Klein-Gordon equation on AdS$_{2+1}$, computing the explicit kernel representation of the fractional Laplace-Beltrami operator as well as the two-point propagator of the fractional Klein-Gordon equation. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information.