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refactor: Lint

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jstoobysmith 2024-10-15 11:39:40 +00:00
parent 89d1b1a50b
commit f72d69e2ba
6 changed files with 84 additions and 69 deletions
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11
HepLean/SpaceTime/WeylFermion/Contraction.lean
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@ -21,7 +21,6 @@ open MatrixGroups
open Complex
open TensorProduct
/-!
## Contraction of Weyl fermions.
@ -76,7 +75,6 @@ def altLeftBi : altLeftHanded →ₗ[ℂ] leftHanded →ₗ[ℂ] ℂ where
simp only [_root_.map_smul, smul_dotProduct, vec2_dotProduct, Fin.isValue, smul_eq_mul,
LinearMap.coe_mk, AddHom.coe_mk, RingHom.id_apply, LinearMap.smul_apply]
/-- The bi-linear map corresponding to contraction of a right-handed Weyl fermion with a
alt-right-handed Weyl fermion. -/
def rightAltBi : rightHanded →ₗ[ℂ] altRightHanded →ₗ[ℂ] ℂ where
@ -170,7 +168,8 @@ The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift rightAltBi
comm M := TensorProduct.ext' fun ψ φ => by
change (M.1.map star *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
change (M.1.map star *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) =
ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
have h1 : (M.1)⁻¹ᴴ = ((M.1)⁻¹.map star)ᵀ := by rfl
rw [dotProduct_mulVec, h1, vecMul_transpose, mulVec_mulVec]
have h2 : ((M.1)⁻¹.map star * (M.1).map star) = 1 := by
@ -193,8 +192,9 @@ def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2
-/
def altRightContraction : altRightHanded ⊗ rightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift altRightBi
comm M := TensorProduct.ext' fun φ ψ => by
change (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1.map star *ᵥ ψ.toFin2ℂ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
comm M := TensorProduct.ext' fun φ ψ => by
change (M.1⁻¹.conjTranspose *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1.map star *ᵥ ψ.toFin2ℂ) =
φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
have h1 : (M.1)⁻¹ᴴ = ((M.1)⁻¹.map star)ᵀ := by rfl
rw [dotProduct_mulVec, h1, mulVec_transpose, vecMul_vecMul]
have h2 : ((M.1)⁻¹.map star * (M.1).map star) = 1 := by
@ -251,7 +251,6 @@ informal_lemma altLeftWeylContraction_invariant where
the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
deps :≈ [``altLeftContraction]
informal_lemma rightAltWeylContraction_invariant where
math :≈ "The contraction rightAltWeylContraction is invariant with respect to
the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."