feat: Add properties of the Lorentz basis

HepLean/SpaceTime/LorentzAlgebra/Basis.lean

@@ -21,6 +21,55 @@ open Matrix
 def σMat (μ ν : Fin 4) : Matrix (Fin 4) (Fin 4) ℝ := fun ρ δ ↦
   η^[ρ]_[μ] * η_[ν]_[δ] - η_[μ]_[δ] * η^[ρ]_[ν]
 
+lemma σMat_in_lorentzAlgebra (μ ν : Fin 4) : σMat μ ν ∈ lorentzAlgebra := by
+  rw [mem_iff]
+  funext ρ δ
+  rw [Matrix.neg_mul, Matrix.neg_apply, η_mul, mul_η, transpose_apply]
+  apply Eq.trans ?_ (by ring :
+    - ((η^[ρ]_[μ] * η_[ρ]_[ρ]) * η_[ν]_[δ] - η_[μ]_[δ] * (η^[ρ]_[ν] *  η_[ρ]_[ρ])) =
+    -(η_[ρ]_[ρ] * (η^[ρ]_[μ] * η_[ν]_[δ] - η_[μ]_[δ] * η^[ρ]_[ν])))
+  apply Eq.trans (by ring : (η^[δ]_[μ] * η_[ν]_[ρ] - η_[μ]_[ρ] * η^[δ]_[ν]) * η_[δ]_[δ]
+    = ((η^[δ]_[μ] * η_[δ]_[δ]) * η_[ν]_[ρ] - η_[μ]_[ρ] * (η^[δ]_[ν]  * η_[δ]_[δ])))
+  rw [η_mul_self, η_mul_self, η_mul_self, η_mul_self]
+  ring
+
+/-- Elements of the Lorentz algebra which form a basis thereof. -/
+@[simps!]
+def σ (μ ν : Fin 4) : lorentzAlgebra := ⟨σMat μ ν, σMat_in_lorentzAlgebra μ ν⟩
+
+lemma σ_anti_symm (μ ν : Fin 4) : σ μ ν = - σ ν μ := by
+  refine SetCoe.ext ?_
+  funext ρ δ
+  simp only [σ_coe, σMat, NegMemClass.coe_neg, neg_apply, neg_sub]
+  ring
+
+lemma σMat_mul (α β γ δ  a b: Fin 4) :
+    (σMat α β * σMat γ δ) a b =
+    η^[a]_[α] * (η_[δ]_[b] * η_[β]_[γ] - η_[γ]_[b] * η_[β]_[δ])
+    - η^[a]_[β] * (η_[δ]_[b] * η_[α]_[γ]- η_[γ]_[b] * η_[α]_[δ]) := by
+  rw [Matrix.mul_apply]
+  simp only [σMat]
+  trans (η^[a]_[α] * η_[δ]_[b]) * ∑ x, η^[x]_[γ] * η_[β]_[x]
+    - (η^[a]_[α] * η_[γ]_[b]) * ∑ x, η^[x]_[δ] * η_[β]_[x]
+    - (η^[a]_[β] *  η_[δ]_[b]) * ∑ x, η^[x]_[γ] * η_[α]_[x]
+    + (η^[a]_[β] * η_[γ]_[b]) * ∑ x, η^[x]_[δ] * η_[α]_[x]
+  repeat rw [Fin.sum_univ_four]
+  ring
+  rw [η_contract_self', η_contract_self', η_contract_self', η_contract_self']
+  ring
+
+lemma σ_comm (α β γ δ : Fin 4) :
+    ⁅σ α β , σ γ δ⁆ =
+    η_[α]_[δ] • σ β γ + η_[α]_[γ] • σ δ β + η_[β]_[δ] • σ γ α + η_[β]_[γ] • σ α δ := by
+  refine SetCoe.ext ?_
+  change σMat α β * σ γ δ -  σ γ δ * σ α β = _
+  funext a b
+  simp only [σ_coe, sub_apply, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid,
+    Submodule.coe_smul_of_tower, add_apply, smul_apply, σMat, smul_eq_mul]
+  rw [σMat_mul, σMat_mul, η_symmetric α γ, η_symmetric α δ, η_symmetric β γ, η_symmetric β δ]
+  ring
+
+
 end lorentzAlgebra
 
 end spaceTime