Geometry of the $p$-adic special orthogonal group $SO(3)_p$

TL;DR

The paper develops the geometry of $SO(3)_p$ for all primes $p$ by showing every element is a rotation about some axis in ${\mathbb Q}_p^3$ and analyzing the induced planar rotation groups $SO(2)_p^{\kappa}$, which are abelian and parameterized by $P^1({\mathbb Q}_p)$. It classifies the definite 2D forms arising from axis restrictions (3 classes for odd $p$, 7 for $p=2$) and constructs explicit axis-rotation parametrizations, including a p-adic Cardano-type decomposition for odd $p$ and a no-go Euler decomposition in general (with $p=2$ entirely obstructed). The work shows $SO(3)_p$ is compact/profinite as the inverse limit of finite groups modulo $p^k$ and provides a detailed, axis-centered decomposition framework via a general ${\cal R}_{\mathbf{n}}(\sigma)$ parametrization. The results illuminate the structure of p-adic spatial rotations and connect to potential p-adic quantum-theoretic applications and representations.

Abstract

We derive explicitly the structural properties of the $p$-adic special orthogonal groups in dimension three, for all primes $p$, and, along the way, the two-dimensional case. In particular, starting from the unique definite quadratic form in three dimensions (up to linear equivalence and rescaling), we show that every element of $SO(3)_p$ is a rotation around an axis. An important part of the analyis is the classification of all definite forms in two dimensions, yielding a description of the rotation subgroups around any fixed axis, which all turn out to be abelian and parametrised naturally by the projective line. Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$ admits a representation in terms of Cardano angles of rotations around the reference axes, in close analogy to the real orthogonal case. However, this works only for certain orderings of the product of rotations around the coordinate axes, depending on the prime; furthermore, there is no general Euler angle decomposition. For $p=2$, no Euler or Cardano decomposition exists.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• Theorem 6
• proof
• Lemma 7
• proof