Euclidean (A)dS spaces over $p$-adic numbers

Abstract

With the help of Wick rotation over $p$-adic numbers $\mathbb{Q}_p$, the $p$-adic version of Euclidean $\textrm{dS}_2$ space(noted as $p\textrm{dS}_2$) is obtained based on $p\textrm{AdS}_2$($p$-adic version of Euclidean $\textrm{AdS}_2$ space), the latter of which is already known. The corresponding embedding equations are also found. The distances $D(X,Y)$'s on $p\textrm{(A)dS}_1$ and $p\textrm{AdS}_2$ have intuitive explanations. On the graph representations of $\mathbb{Q}_p$ and $\mathbb{Q}_{p^2}$, namely Bruhat-Tits trees $\textrm{T}_{p}$ and $\textrm{T}_{p^2}$, $D(X,Y)$ is found to be the inverse of distance between a particular subgraph and the line connecting $X$ and $Y$.