NRS³ — the uncertainty cone
Take a state ψ and the pair transport T : position P on an axis with
d positions. Their fluctuations are two vectors of ℂd; their Gram matrix is a
Hermitian 2×2 matrix, and a Hermitian 2×2 matrix is a vector of Minkowski space (van der Waerden 1929).
Robertson–Schrödinger says that vector never leaves the light cone. The quantum of NRS³ says more: from four
positions on, in the band, it stays strictly inside. Every number below is computed live from the
matrices Td, Pd; every claim is a Lean theorem (table at the end).
1Two fluctuation vectors
x = (T − ⟨T⟩)ψ, y = (P − ⟨P⟩)ψ in ℂd.
2Their Gram matrix
Hermitian: G = [‖x‖², ⟪x,y⟫; ⟪y,x⟫, ‖y‖²].
3A four-vector
G = v₀1 + v₁σ₁ − v₂σ₂ + v₃σ₃; v₁ covariance, v₂ commutator.
4The interval
det G = v₀² − v₁² − v₂² − v₃² ≥ 0: inside the cone.
The cone, in the frame v₃ = 0
drag to turnThe arrow is the uncertainty vector, rescaled to height 1. The rim is the speed of light β = 1: saturation. Dots: the maximal current state for d = 2 … 16; dashed ring: the limit 1/C∞, never reached.
Units are a boost
A → λA, B → B/λ keeps [A, B] and slides the vector along its
hyperbola v₀² − v₃² = (‖x‖‖y‖)². The interval, the covariance and the commutator do not move: that is
why CNava has no units.
Step 1, seen: the two vectors site by site
Each site j carries xj (blue) and yj (orange) as arrows in the complex plane. β = |⟪x, y⟫| / (‖x‖‖y‖) = cos θNRS: the two vectors are parallel only when β = 1.
The speed of the vector at the maximal current state: β(d) = 1 / CNava(d)
Null at d = 2, 3, timelike from d = 4, decreasing towards 1/C∞ = 0.8805 without reaching it. Dashed: β², the Mandelstam–Tamm ratio ⟨K⟩² / (4 Var T Var P).
Three axes, one determinant: det|NRS³ at the maximal current state of the cube
On the box dx × dy × dz each axis has its own cone. Robertson 1934 for the six observables gives det Σ / |det Ω| = C(dx)² C(dy)² C(dz)², strict as soon as one axis has 4 positions; in the band it is strict for every state, entangled or not (D49t).
What you see, and the theorem behind it
| On the page | Statement | Lean |
|---|---|---|
| Step 2 → 3 | G in the components of v; every Hermitian 2×2 matrix is a four-vector | gramMatrix_eq, herm_coords |
| Step 4 | det G is the Minkowski interval | det_gramMatrix, det_herm |
| never outside | Robertson–Schrödinger is the closed future cone | mem_futureCone, robertsonSchrodinger_future |
| on the rim | saturation is a null vector | interval_eq_zero_iff |
| λ slider | units are a boost; the interval does not move | coneVector_smul, interval_smul, gram_boost |
| frame v₃ = 0 | a rescaling reaches it, with v₀ = ‖x‖‖y‖ | balanced_frame |
| β(d) | at the maximal current state, β = 1 / CNava(d); null iff d = 2, 3; timelike from 4 | speed_maxCurrentState, null_maxCurrentState_iff, timelike_maxCurrentState |
| “in the band” | every axis strictly timelike for every state in Ϙ, entangled included; det|NRS³ strict | timelike_of_band, robertson_det_strict_of_band |
| ℝ only | over ℝ the commutator component v₂ vanishes | stated in the docstrings; coords_gram_two |
The four components are statistics of the pair — variances, covariance, commutator — not coordinates of events. Outside the band the defect can vanish only trivially (Robertson as 0 = 0, nothing moves). Source: nrs3-uncertainty-cone · base theorem: nava-robertson-schrodinger.