van der Waerden 1929 — SL(2, ℂ) moves the vector, never out of the cone
A Hermitian 2×2 matrix G = v₀1 + v₁σ₁ + v₂σ₂ + v₃σ₃ is a four-vector, and
det G is its Minkowski interval. A complex matrix M with det M = 1 acts by
G ↦ M G M†: that is a Lorentz transformation. Boosts tilt the vector, rotations turn it; the
interval never changes and the future cone is kept. Start from the uncertainty vector of NRS³ and try to push it out.
The speed ball: 𝐯 / v₀ inside the unit sphere
drag to turnThe sphere is the light cone seen at v₀ = 1: its surface is β = 1. Grey: where the vector started; orange: where M sends it; the trail follows the sliders. It can get close to the sphere, never through it.
The hyperbola: v₀² − |𝐯|² = det G
Every M moves the point along the same hyperbola; the light line |𝐯| = v₀ is its asymptote. A null vector (d = 2, 3) lives on the light line and stays there.
G (start)
M ∈ SL(2, ℂ)
M G M†
Lean (nrs3-uncertainty-cone,
VanDerWaerden1929): every Hermitian 2×2 matrix is a four-vector (herm_coords), its
determinant is the interval (det_herm), SL(2, ℂ) keeps the interval
(det_conj) and the future cone (posSemidef_conj), the Gram matrix of two vectors is future
causal (robertsonSchrodinger_future), and A → λA, B → B/λ is the boost
diag(λ, λ⁻¹) (gram_boost) — the slider η₃ with λ = eη₃/2.
In Physlib it is a Lorentz.Vector 3 that causally follows the origin (causallyFollows_uncertaintyVector).