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

Abstract

With the help of Wick rotation over p-adic numbers Q p , the p-adic version of Euclidean dS 2 space(noted as pdS 2 ) is obtained based on pAdS 2 (p-adic version of Euclidean 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(A)dS 1 and pAdS 2 have intuitive explanations. On the graph representations of Q p and Q p 2 , namely Bruhat-Tits trees T p and 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 .