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An ultrametric space $(X, d)$ satisfies the strong triangle inequality:
$$d(x, z) \leq \max\{d(x, y), d(y, z)\}$$This is stronger than the ordinary triangle inequality. Every triangle in an ultrametric space is isosceles with the two equal sides at least as long as the third. The open balls in an ultrametric space are also closed, and every point in a ball is a center of that ball.
The $p$-adic absolute value $|\cdot|_p$ on $\mathbb{Q}$ induces an ultrametric:
$$|x|_p = p^{-v_p(x)}$$where $v_p(x)$ is the $p$-adic valuation. The completion of $\mathbb{Q}$ with respect to this absolute value yields the field $\mathbb{Q}_p$ of $p$-adic numbers.
The distinction between quantum states in a Hilbert space $\mathcal{H}$ can be quantified by the fidelity:
$$F(\rho, \sigma) = \left( \operatorname{Tr} \sqrt{\sqrt{\rho} \, \sigma \sqrt{\rho}} \right)^2$$For pure states $|\psi\rangle$ and $|\phi\rangle$, this reduces to the squared overlap:
$$F(|\psi\rangle, |\phi\rangle) = |\langle\psi|\phi\rangle|^2$$The Bruhat–Tits building for $\operatorname{PGL}(2, \mathbb{Q}_p)$ is a $(p+1)$-regular tree. Its vertices correspond to homothety classes of lattices in $\mathbb{Q}_p^2$, and the boundary at infinity is the projective line $\mathbb{P}^1(\mathbb{Q}_p)$.
The action of $\operatorname{PGL}(2, \mathbb{Q}_p)$ on this tree preserves the hyperbolic distance:
$$d(v, w) = \log_p |\det(g)| \quad \text{where } g \in \operatorname{GL}(2, \mathbb{Q}_p)$$In $p$-adic quantum mechanics, the wavefunction takes values in $\mathbb{C}_p$, the completion of the algebraic closure of $\mathbb{Q}_p$. The Schrödinger equation becomes:
$$i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \hat{H} \Psi(x, t)$$but with $x \in \mathbb{Q}_p^n$ and the kinetic term interpreted via the Vladimirov operator:
$$(D^\alpha \varphi)(x) = \frac{1}{\Gamma_p(-\alpha)} \int_{\mathbb{Q}_p} \frac{\varphi(y) - \varphi(x)}{|x - y|_p^{1+\alpha}} \, dy$$