refactor: Move dual to minkowskiMatrix
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jstoobysmith
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5 changed files with 60 additions and 54 deletions
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@ -43,7 +43,7 @@ namespace LorentzGroup
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scoped[LorentzGroup] notation (name := lorentzGroup_notation) "𝓛" => LorentzGroup
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open minkowskiMetric
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open minkowskiMatrix
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variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
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/-!
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@ -123,7 +123,7 @@ instance lorentzGroupIsGroup : Group (LorentzGroup d) where
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one := ⟨1, LorentzGroup.one_mem⟩
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one_mul A := Subtype.eq (Matrix.one_mul A.1)
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mul_one A := Subtype.eq (Matrix.mul_one A.1)
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inv A := ⟨minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2⟩
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inv A := ⟨minkowskiMatrix.dual A.1, LorentzGroup.dual_mem A.2⟩
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inv_mul_cancel A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)
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/-- `LorentzGroup` has the subtype topology. -/
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@ -132,6 +132,7 @@ instance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype
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namespace LorentzGroup
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open minkowskiMetric
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open minkowskiMatrix
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variable {Λ Λ' : LorentzGroup d}
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@ -250,7 +251,7 @@ def toProd : LorentzGroup d →* (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ
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(Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)ᵐᵒᵖ :=
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MonoidHom.comp (Units.embedProduct _) toGL
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lemma toProd_eq_transpose_η : toProd Λ = (Λ.1, MulOpposite.op $ minkowskiMetric.dual Λ.1) := rfl
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lemma toProd_eq_transpose_η : toProd Λ = (Λ.1, MulOpposite.op $ minkowskiMatrix.dual Λ.1) := rfl
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lemma toProd_injective : Function.Injective (@toProd d) := by
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intro A B h
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@ -22,6 +22,7 @@ open ComplexConjugate
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namespace LorentzGroup
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open minkowskiMetric
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open minkowskiMatrix
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variable {d : ℕ}
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@ -25,7 +25,7 @@ def SO3ToMatrix (A : SO(3)) : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ :=
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lemma SO3ToMatrix_in_LorentzGroup (A : SO(3)) : SO3ToMatrix A ∈ LorentzGroup 3 := by
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rw [LorentzGroup.mem_iff_dual_mul_self]
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simp only [minkowskiMetric.dual, minkowskiMatrix.as_block, SO3ToMatrix,
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simp only [minkowskiMatrix.dual, minkowskiMatrix.as_block, SO3ToMatrix,
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Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,
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Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mul, Matrix.mul_one,
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neg_mul, one_mul, zero_add, Matrix.mul_neg, neg_zero, mul_neg, neg_neg,
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@ -89,6 +89,58 @@ lemma inr_i_inr_i (i : Fin d) : @minkowskiMatrix d (Sum.inr i) (Sum.inr i) = -1
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simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
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simp_all only [diagonal_apply_eq, Sum.elim_inr]
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variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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/-- The dual of a matrix with respect to the Minkowski metric. -/
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def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
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@[simp]
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lemma dual_id : @dual d 1 = 1 := by
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simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
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@[simp]
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lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
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simp only [dual, transpose_mul]
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trans η * Λ'ᵀ * (η * η) * Λᵀ * η
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· noncomm_ring [minkowskiMatrix.sq]
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· noncomm_ring
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@[simp]
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lemma dual_dual : dual (dual Λ) = Λ := by
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simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
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trans (η * η) * Λ * (η * η)
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· noncomm_ring
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· noncomm_ring [minkowskiMatrix.sq]
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@[simp]
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lemma dual_eta : @dual d η = η := by
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simp only [dual, eq_transpose]
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noncomm_ring [minkowskiMatrix.sq]
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@[simp]
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lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
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simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
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noncomm_ring
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@[simp]
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lemma det_dual : (dual Λ).det = Λ.det := by
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simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
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group
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norm_cast
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simp
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lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
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dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by
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simp only [dual, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
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diagonal_mul, transpose_apply, diagonal_apply_eq]
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lemma dual_apply_minkowskiMatrix (μ ν : Fin 1 ⊕ Fin d) :
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dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by
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rw [dual_apply, mul_assoc]
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simp
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end minkowskiMatrix
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/-!
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@ -238,54 +290,6 @@ section matrices
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variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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/-- The dual of a matrix with respect to the Minkowski metric. -/
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def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
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@[simp]
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lemma dual_id : @dual d 1 = 1 := by
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simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
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@[simp]
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lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
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simp only [dual, transpose_mul]
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trans η * Λ'ᵀ * (η * η) * Λᵀ * η
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· noncomm_ring [minkowskiMatrix.sq]
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· noncomm_ring
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@[simp]
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lemma dual_dual : dual (dual Λ) = Λ := by
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simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
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trans (η * η) * Λ * (η * η)
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· noncomm_ring
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· noncomm_ring [minkowskiMatrix.sq]
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@[simp]
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lemma dual_eta : @dual d η = η := by
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simp only [dual, eq_transpose]
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noncomm_ring [minkowskiMatrix.sq]
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@[simp]
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lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
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simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
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noncomm_ring
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@[simp]
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lemma det_dual : (dual Λ).det = Λ.det := by
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simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
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group
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norm_cast
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simp
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lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
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dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by
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simp only [dual, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
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diagonal_mul, transpose_apply, diagonal_apply_eq]
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lemma dual_apply_minkowskiMatrix (μ ν : Fin 1 ⊕ Fin d) :
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dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by
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rw [dual_apply, mul_assoc]
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simp
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@[simp]
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lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
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simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,
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@ -167,7 +167,7 @@ lemma repSelfAdjointMatrix_basis (i : Fin 1 ⊕ Fin 3) :
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lemma repSelfAdjointMatrix_σSA (i : Fin 1 ⊕ Fin 3) :
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SL2C.repSelfAdjointMatrix M (PauliMatrix.σSA i) =
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∑ j, (toLorentzGroup M⁻¹).1 i j • PauliMatrix.σSA j := by
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have h1 : (toLorentzGroup M⁻¹).1 = minkowskiMetric.dual (toLorentzGroup M).1 := by
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have h1 : (toLorentzGroup M⁻¹).1 = minkowskiMatrix.dual (toLorentzGroup M).1 := by
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simp
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simp only [h1]
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rw [PauliMatrix.σSA_minkowskiMetric_σSAL, _root_.map_smul]
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@ -178,7 +178,7 @@ lemma repSelfAdjointMatrix_σSA (i : Fin 1 ⊕ Fin 3) :
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rw [smul_smul, PauliMatrix.σSA_minkowskiMetric_σSAL, smul_smul]
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apply congrFun
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apply congrArg
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exact Eq.symm (minkowskiMetric.dual_apply_minkowskiMatrix ((toLorentzGroup M).1) i j)
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exact Eq.symm (minkowskiMatrix.dual_apply_minkowskiMatrix ((toLorentzGroup M).1) i j)
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lemma repLorentzVector_stdBasis (i : Fin 1 ⊕ Fin 3) :
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SL2C.repLorentzVector M (LorentzVector.stdBasis i) =
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