refactor: Lint
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jstoobysmith
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6 changed files with 84 additions and 69 deletions
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@ -10,7 +10,6 @@ import Mathlib.LinearAlgebra.TensorProduct.Matrix
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# Tensor product of two Weyl fermion
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-/
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namespace Fermion
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@ -96,7 +95,7 @@ lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
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((leftBasis.tensorProduct leftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct leftBasis)
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(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) v)
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(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -108,7 +107,7 @@ lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ j : Fin 2, M.1 i j * leftLeftToMatrix v j x) * M.1 j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 j x := by
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= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -138,8 +137,8 @@ lemma altLeftaltLeftToMatrix_ρ (v : (altLeftHanded ⊗ altLeftHanded).V) (M : S
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((altLeftBasis.tensorProduct altLeftBasis).repr v)))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct altLeftBasis)
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(altLeftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) v)
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(altLeftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -178,8 +177,8 @@ lemma leftAltLeftToMatrix_ρ (v : (leftHanded ⊗ altLeftHanded).V) (M : SL(2,
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((leftBasis.tensorProduct altLeftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct altLeftBasis)
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(leftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) v)
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(leftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -219,8 +218,8 @@ lemma altLeftLeftToMatrix_ρ (v : (altLeftHanded ⊗ leftHanded).V) (M : SL(2,
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((altLeftBasis.tensorProduct leftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct leftBasis)
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(altLeftBasis.tensorProduct leftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) v)
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(altLeftBasis.tensorProduct leftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -260,7 +259,8 @@ lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)
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((rightBasis.tensorProduct rightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct rightBasis)
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(rightBasis.tensorProduct rightBasis) (TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) v)
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(rightBasis.tensorProduct rightBasis)
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(TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -271,8 +271,9 @@ lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)
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* rightRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x:= by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) *
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(M.1.map star) j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
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((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x:= by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -300,8 +301,8 @@ lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M
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((altRightBasis.tensorProduct altRightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct altRightBasis)
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(altRightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) v)
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(altRightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -312,8 +313,9 @@ lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M
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* altRightAltRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) *
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(↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
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((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -340,8 +342,8 @@ lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(
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((rightBasis.tensorProduct altRightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct altRightBasis)
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(rightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
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(rightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -352,8 +354,9 @@ lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(
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* rightAltRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x)
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* (↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2,
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((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -381,8 +384,8 @@ lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(
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((altRightBasis.tensorProduct rightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct rightBasis)
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(altRightBasis.tensorProduct rightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
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(altRightBasis.tensorProduct rightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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@ -393,8 +396,10 @@ lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(
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* altRightRightToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x := by
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have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2,
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(↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) *
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(M.1.map star) j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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@ -464,7 +469,7 @@ lemma altRightAltRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(
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lemma rightAltRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
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TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M) (rightAltRightToMatrix.symm v) =
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rightAltRightToMatrix.symm ((M.1.map star) * v * (((M.1⁻¹).conjTranspose)ᵀ) ) := by
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rightAltRightToMatrix.symm ((M.1.map star) * v * (((M.1⁻¹).conjTranspose)ᵀ)) := by
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have h1 := rightAltRightToMatrix_ρ (rightAltRightToMatrix.symm v) M
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
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rw [← h1]
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@ -478,7 +483,5 @@ lemma altRightRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,
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rw [← h1]
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simp
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end
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end Fermion
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