Euclid Levi Civita symbol contract one
euclidLeviCivita_symbol_contract_one
Plain-language statement
Triple Euclidean Levi-Civita contraction ∑_h (ε4)_{a,h} · (ε4)_{b,h} = 6 · δ[a,b] at the symbol level: contracting three of the four Fin 4 component slots of ε4 with the naive Kronecker pairing leaves one free pair a, b and the factor 3! = 6. The Lorentz form carries an extra det η = -1.
Exact Lean statement
lemma euclidLeviCivita_symbol_contract_one (a b : Fin 4) :
∑ h : Fin 3 → Fin 4, euclidLeviCivita (Fin.cons a h) * euclidLeviCivita (Fin.cons b h)
= 6 * ((kroneckerDelta a b : ℕ) : ℝ)Formal artifact
Lean source
lemma euclidLeviCivita_symbol_contract_one (a b : Fin 4) : ∑ h : Fin 3 → Fin 4, euclidLeviCivita (Fin.cons a h) * euclidLeviCivita (Fin.cons b h) = 6 * ((kroneckerDelta a b : ℕ) : ℝ) := by have hcast : ∀ h' : Fin 3 → Fin 4, (generalizedKroneckerDelta (Fin.cons a h') id : ℝ) * (generalizedKroneckerDelta (Fin.cons b h') id : ℝ) = ((generalizedKroneckerDelta (Fin.cons a h') id * generalizedKroneckerDelta (Fin.cons b h') id : ℤ) : ℝ) := fun h' => by push_cast; ring erw [Finset.sum_congr rfl fun h' _ => hcast h', ← Int.cast_sum, sum_generalizedKroneckerDelta_mul_cons] push_cast; ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/LeviCivita/Contractions.lean:78-89
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