feat: Start adding expansions in terms of basis
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jstoobysmith
parent
5a0e52e726
commit
1148234929
5 changed files with 196 additions and 2 deletions
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@ -24,6 +24,23 @@ namespace Lorentz
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def contrMetricVal : (complexContr ⊗ complexContr).V :=
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contrContrToMatrix.symm ((@minkowskiMatrix 3).map ofReal)
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/-- The expansion of `contrMetricVal` into basis vectors. -/
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lemma contrMetricVal_expand_tmul : contrMetricVal =
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complexContrBasis (Sum.inl 0) ⊗ₜ[ℂ] complexContrBasis (Sum.inl 0)
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- complexContrBasis (Sum.inr 0) ⊗ₜ[ℂ] complexContrBasis (Sum.inr 0)
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- complexContrBasis (Sum.inr 1) ⊗ₜ[ℂ] complexContrBasis (Sum.inr 1)
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- complexContrBasis (Sum.inr 2) ⊗ₜ[ℂ] complexContrBasis (Sum.inr 2) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, contrMetricVal, Fin.isValue]
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erw [contrContrToMatrix_symm_expand_tmul]
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simp only [map_apply, ofReal_eq_coe, coe_smul, Fintype.sum_sum_type, Finset.univ_unique,
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Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, Fin.sum_univ_three, ne_eq, reduceCtorEq,
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not_false_eq_true, minkowskiMatrix.off_diag_zero, zero_smul, add_zero, zero_add, Sum.inr.injEq,
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zero_ne_one, Fin.reduceEq, one_ne_zero]
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rw [minkowskiMatrix.inl_0_inl_0, minkowskiMatrix.inr_i_inr_i,
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minkowskiMatrix.inr_i_inr_i, minkowskiMatrix.inr_i_inr_i]
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simp only [Fin.isValue, one_smul, neg_smul]
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rfl
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/-- The metric `ηᵃᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexContr`,
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making its invariance under the action of `SL(2,ℂ)`. -/
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def contrMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexContr where
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@ -51,10 +68,16 @@ def contrMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexContr wh
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apply congrArg
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simp only [LorentzGroup.toComplex_mul_minkowskiMatrix_mul_transpose]
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lemma contrMetric_apply_one : contrMetric.hom (1 : ℂ) = contrMetricVal := by
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change contrMetric.hom.toFun (1 : ℂ) = contrMetricVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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contrMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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/-- The metric `ηᵢᵢ` as an element of `(complexCo ⊗ complexCo).V`. -/
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def coMetricVal : (complexCo ⊗ complexCo).V :=
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coCoToMatrix.symm ((@minkowskiMatrix 3).map ofReal)
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/-- The expansion of `coMetricVal` into basis vectors. -/
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lemma coMetricVal_expand_tmul : coMetricVal =
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complexCoBasis (Sum.inl 0) ⊗ₜ[ℂ] complexCoBasis (Sum.inl 0)
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- complexCoBasis (Sum.inr 0) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 0)
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@ -102,7 +125,8 @@ def coMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo where
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lemma coMetric_apply_one : coMetric.hom (1 : ℂ) = coMetricVal := by
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change coMetric.hom.toFun (1 : ℂ) = coMetricVal
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simp [coMetric]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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coMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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end Lorentz
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end
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@ -27,6 +27,19 @@ def contrContrToMatrix : (complexContr ⊗ complexContr).V ≃ₗ[ℂ]
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Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)
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/-- Expanding `contrContrToMatrix` in terms of the standard basis. -/
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lemma contrContrToMatrix_symm_expand_tmul (M : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ) :
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contrContrToMatrix.symm M =
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∑ i, ∑ j, M i j • (complexContrBasis i ⊗ₜ[ℂ] complexContrBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, contrContrToMatrix, LinearEquiv.trans_symm,
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LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply complexContrBasis complexContrBasis i j]
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rfl
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· simp
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/-- Equivalence of `complexCo ⊗ complexCo` to `4 x 4` complex matrices. -/
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def coCoToMatrix : (complexCo ⊗ complexCo).V ≃ₗ[ℂ]
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Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ :=
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@ -33,60 +33,180 @@ def leftLeftToMatrix : (leftHanded ⊗ leftHanded).V ≃ₗ[ℂ] Matrix (Fin 2)
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `leftLeftToMatrix` in terms of the standard basis. -/
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lemma leftLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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leftLeftToMatrix.symm M = ∑ i, ∑ j, M i j • (leftBasis i ⊗ₜ[ℂ] leftBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, leftLeftToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply leftBasis leftBasis i j]
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rfl
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· simp
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/-- Equivalence of `altLeftHanded ⊗ altLeftHanded` to `2 x 2` complex matrices. -/
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def altLeftaltLeftToMatrix : (altLeftHanded ⊗ altLeftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altLeftBasis altLeftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `altLeftaltLeftToMatrix` in terms of the standard basis. -/
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lemma altLeftaltLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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altLeftaltLeftToMatrix.symm M = ∑ i, ∑ j, M i j • (altLeftBasis i ⊗ₜ[ℂ] altLeftBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altLeftaltLeftToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply altLeftBasis altLeftBasis i j]
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rfl
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· simp
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/-- Equivalence of `leftHanded ⊗ altLeftHanded` to `2 x 2` complex matrices. -/
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def leftAltLeftToMatrix : (leftHanded ⊗ altLeftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct leftBasis altLeftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `leftAltLeftToMatrix` in terms of the standard basis. -/
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lemma leftAltLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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leftAltLeftToMatrix.symm M = ∑ i, ∑ j, M i j • (leftBasis i ⊗ₜ[ℂ] altLeftBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, leftAltLeftToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply leftBasis altLeftBasis i j]
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rfl
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· simp
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/-- Equivalence of `altLeftHanded ⊗ leftHanded` to `2 x 2` complex matrices. -/
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def altLeftLeftToMatrix : (altLeftHanded ⊗ leftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altLeftBasis leftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `altLeftLeftToMatrix` in terms of the standard basis. -/
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lemma altLeftLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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altLeftLeftToMatrix.symm M = ∑ i, ∑ j, M i j • (altLeftBasis i ⊗ₜ[ℂ] leftBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altLeftLeftToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply altLeftBasis leftBasis i j]
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rfl
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· simp
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/-- Equivalence of `rightHanded ⊗ rightHanded` to `2 x 2` complex matrices. -/
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def rightRightToMatrix : (rightHanded ⊗ rightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct rightBasis rightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `rightRightToMatrix` in terms of the standard basis. -/
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lemma rightRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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rightRightToMatrix.symm M = ∑ i, ∑ j, M i j • (rightBasis i ⊗ₜ[ℂ] rightBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, rightRightToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply rightBasis rightBasis i j]
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rfl
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· simp
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/-- Equivalence of `altRightHanded ⊗ altRightHanded` to `2 x 2` complex matrices. -/
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def altRightAltRightToMatrix : (altRightHanded ⊗ altRightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altRightBasis altRightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `altRightAltRightToMatrix` in terms of the standard basis. -/
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lemma altRightAltRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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altRightAltRightToMatrix.symm M = ∑ i, ∑ j, M i j • (altRightBasis i ⊗ₜ[ℂ] altRightBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altRightAltRightToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply altRightBasis altRightBasis i j]
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rfl
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· simp
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/-- Equivalence of `rightHanded ⊗ altRightHanded` to `2 x 2` complex matrices. -/
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def rightAltRightToMatrix : (rightHanded ⊗ altRightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct rightBasis altRightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `rightAltRightToMatrix` in terms of the standard basis. -/
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lemma rightAltRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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rightAltRightToMatrix.symm M = ∑ i, ∑ j, M i j • (rightBasis i ⊗ₜ[ℂ] altRightBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, rightAltRightToMatrix, LinearEquiv.trans_symm,
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LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply rightBasis altRightBasis i j]
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rfl
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· simp
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/-- Equivalence of `altRightHanded ⊗ rightHanded` to `2 x 2` complex matrices. -/
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def altRightRightToMatrix : (altRightHanded ⊗ rightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altRightBasis rightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `altRightRightToMatrix` in terms of the standard basis. -/
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lemma altRightRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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altRightRightToMatrix.symm M = ∑ i, ∑ j, M i j • (altRightBasis i ⊗ₜ[ℂ] rightBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altRightRightToMatrix, LinearEquiv.trans_symm,
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LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply altRightBasis rightBasis i j]
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rfl
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· simp
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/-- Equivalence of `altLeftHanded ⊗ altRightHanded` to `2 x 2` complex matrices. -/
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def altLeftAltRightToMatrix : (altLeftHanded ⊗ altRightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altLeftBasis altRightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `altLeftAltRightToMatrix` in terms of the standard basis. -/
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lemma altLeftAltRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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altLeftAltRightToMatrix.symm M = ∑ i, ∑ j, M i j • (altLeftBasis i ⊗ₜ[ℂ] altRightBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altLeftAltRightToMatrix,
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LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply altLeftBasis altRightBasis i j]
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rfl
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· simp
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/-- Equivalence of `leftHanded ⊗ rightHanded` to `2 x 2` complex matrices. -/
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def leftRightToMatrix : (leftHanded ⊗ rightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct leftBasis rightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Expanding `leftRightToMatrix` in terms of the standard basis. -/
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lemma leftRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
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leftRightToMatrix.symm M = ∑ i, ∑ j, M i j • (leftBasis i ⊗ₜ[ℂ] rightBasis j) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, leftRightToMatrix, LinearEquiv.trans_symm,
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LinearEquiv.trans_apply, Basis.repr_symm_apply]
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rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
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· erw [Finset.sum_product]
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refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
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erw [Basis.tensorProduct_apply leftBasis rightBasis i j]
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rfl
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· simp
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/-!
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## Group actions
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