`BookProof.ChapterWignerLittleGroupOrbits.orbit_iff_massSq_eq` {p q : Fin 4 → ℝ} (hp0 : 0 < p 0) (hq0 : 0 < q 0) (hmass : 0 ≤ p 0 ^ 2 - p 1 ^ 2 - p 2 ^ 2 - p 3 ^ 2) : (∃ A : Matrix

Proved

BookProof.ChapterWignerLittleGroupOrbits.orbit_iff_massSq_eq

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWignerLittleGroupOrbits.

BookProof.ChapterWignerLittleGroupOrbits.orbit_iff_massSq_eq {p q : Fin 4 → ℝ} (hp0 : 0 < p 0) (hq0 : 0 < q 0) (hmass : 0 ≤ p 0 ^ 2 - p 1 ^ 2 - p 2 ^ 2 - p 3 ^ 2) : (∃ A : Matrix (Fin 2) (Fin 2) ℂ, A.det = 1 ∧ act A (hermOfMom p) = hermOfMom q) ↔ q 0 ^ 2 - q 1 ^ 2 - q 2 ^ 2 - q 3 ^ 2 = p 0 ^ 2 - p 1 ^ 2 - p 2 ^ 2 - p 3 ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterWignerLittleGroupOrbits.orbit_iff_massSq_eq.

Preamble

-- Generated from ChapterWignerLittleGroupOrbits.lean — theorem BookProof.ChapterWignerLittleGroupOrbits.orbit_iff_massSq_eq
import Mathlib
import Definitions.Def_ChapterWignerLittleGroupOrbits
import Definitions.Def_ChapterWignerLittleGroup
open BookProof.ChapterWignerLittleGroup
open BookProof.ChapterWignerLittleGroupOrbits


open Matrix Complex
open scoped ComplexOrder


open BookProof.ChapterWignerLittleGroup

Formal statement

theorem BookProof.ChapterWignerLittleGroupOrbits.orbit_iff_massSq_eq {p q : Fin 4 → ℝ} (hp0 : 0 < p 0) (hq0 : 0 < q 0)
    (hmass : 0 ≤ p 0 ^ 2 - p 1 ^ 2 - p 2 ^ 2 - p 3 ^ 2) :
    (∃ A : Matrix (Fin 2) (Fin 2) ℂ, A.det = 1 ∧ act A (hermOfMom p) = hermOfMom q) ↔
      q 0 ^ 2 - q 1 ^ 2 - q 2 ^ 2 - q 3 ^ 2 = p 0 ^ 2 - p 1 ^ 2 - p 2 ^ 2 - p 3 ^ 2 := by sorry

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