Scalar fields on $p$AdS

TL;DR

This paper explores scalar fields on a $p$-adic analog of Anti-de Sitter space ($p$AdS), proposing a bulk action, deriving Green's functions, and analyzing their behavior and convergence properties.

Contribution

It introduces a subgroup of the isometry group for $p$AdS, proposes a scalar bulk action and equations, and derives analytical Green's functions for this $p$-adic setting.

Findings

Derived analytical Green's functions for $p$AdS

Analyzed the limiting behavior of Green's functions

Studied convergence of small loops in $p$AdS geometry

Abstract

We obtain a subgroup of the isometry group of $p$AdS (a $p$-adic version of AdS alternative to the Bruhat-Tits tree). We propose a candidate for the scalar bulk action and equation of motion on $p$AdS, and work out analytical expressions of the Green's functions for a particular choice of parameter together with an ansatz for general cases. The limiting behaviors of the Green's function are also studied. With their help, the convergence of small loops (whose radii are smaller than AdS length scale of $p$AdS) is analyzed.