193 lines
9.2 KiB
Text
193 lines
9.2 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzVector.Complex.Two
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import HepLean.SpaceTime.MinkowskiMetric
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import HepLean.SpaceTime.LorentzVector.Complex.Contraction
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import HepLean.SpaceTime.LorentzVector.Complex.Unit
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/-!
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# Metric for complex Lorentz vectors
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-/
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noncomputable section
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open SpaceTime
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open CategoryTheory.MonoidalCategory
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namespace Lorentz
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/-- The metric `ηᵃᵃ` as an element of `(complexContr ⊗ complexContr).V`. -/
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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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hom := {
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toFun := fun a =>
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let a' : ℂ := a
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a' • contrMetricVal,
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map_add' := fun x y => by
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simp only [add_smul],
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map_smul' := fun m x => by
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simp only [smul_smul]
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rfl}
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comm M := by
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ext x : 2
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
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Function.comp_apply]
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let x' : ℂ := x
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change x' • contrMetricVal =
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(TensorProduct.map (complexContr.ρ M) (complexContr.ρ M)) (x' • contrMetricVal)
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simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
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apply congrArg
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simp only [Action.instMonoidalCategory_tensorObj_V, contrMetricVal]
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erw [contrContrToMatrix_ρ_symm]
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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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- complexCoBasis (Sum.inr 1) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 1)
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- complexCoBasis (Sum.inr 2) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 2) := by
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simp only [Action.instMonoidalCategory_tensorObj_V, coMetricVal, Fin.isValue]
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erw [coCoToMatrix_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 `ηᵢᵢ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo`,
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making its invariance under the action of `SL(2,ℂ)`. -/
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def coMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo where
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hom := {
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toFun := fun a =>
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let a' : ℂ := a
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a' • coMetricVal,
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map_add' := fun x y => by
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simp only [add_smul],
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map_smul' := fun m x => by
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simp only [smul_smul]
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rfl}
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comm M := by
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ext x : 2
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
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Function.comp_apply]
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let x' : ℂ := x
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change x' • coMetricVal =
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(TensorProduct.map (complexCo.ρ M) (complexCo.ρ M)) (x' • coMetricVal)
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simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
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apply congrArg
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simp only [Action.instMonoidalCategory_tensorObj_V, coMetricVal]
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erw [coCoToMatrix_ρ_symm]
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apply congrArg
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rw [LorentzGroup.toComplex_inv]
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simp only [lorentzGroupIsGroup_inv, SL2C.toLorentzGroup_apply_coe,
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LorentzGroup.toComplex_transpose_mul_minkowskiMatrix_mul_self]
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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 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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/-!
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## Contraction of metrics
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-/
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lemma contrCoContraction_apply_metric : (β_ complexContr complexCo).hom.hom
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((complexContr.V ◁ (λ_ complexCo.V).hom)
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((complexContr.V ◁ contrCoContraction.hom ▷ complexCo.V)
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((complexContr.V ◁ (α_ complexContr.V complexCo.V complexCo.V).inv)
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((α_ complexContr.V complexContr.V (complexCo.V ⊗ complexCo.V)).hom
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(contrMetric.hom (1 : ℂ) ⊗ₜ[ℂ] coMetric.hom (1 : ℂ)))))) =
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coContrUnit.hom (1 : ℂ) := by
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rw [contrMetric_apply_one, coMetric_apply_one]
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rw [contrMetricVal_expand_tmul, coMetricVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Fin.isValue, tmul_sub, add_tmul, neg_tmul, map_sub, map_add, map_neg, tmul_sub, sub_tmul]
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have h1 (x1 x2 : complexContr) (y1 y2 :complexCo) :
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(complexContr.V ◁ (λ_ complexCo.V).hom)
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((complexContr.V ◁ contrCoContraction.hom ▷ complexCo.V) (((complexContr.V ◁
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(α_ complexContr.V complexCo.V complexCo.V).inv)
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((α_ complexContr.V complexContr.V (complexCo.V ⊗ complexCo.V)).hom
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((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
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= x1 ⊗ₜ[ℂ] ((λ_ complexCo.V).hom ((contrCoContraction.hom (x2 ⊗ₜ[ℂ] y1)) ⊗ₜ[ℂ] y2)) := rfl
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repeat rw (config := { transparency := .instances }) [h1]
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repeat rw [contrCoContraction_basis']
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simp only [Fin.isValue, ↓reduceIte, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul,
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reduceCtorEq, zero_tmul, map_zero, tmul_zero, sub_zero, zero_sub, Sum.inr.injEq, one_ne_zero,
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Fin.reduceEq, sub_neg_eq_add, zero_ne_one, sub_self]
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erw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]
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rfl
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lemma coContrContraction_apply_metric : (β_ complexCo complexContr).hom.hom
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((complexCo.V ◁ (λ_ complexContr.V).hom)
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((complexCo.V ◁ coContrContraction.hom ▷ complexContr.V)
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((complexCo.V ◁ (α_ complexCo.V complexContr.V complexContr.V).inv)
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((α_ complexCo.V complexCo.V (complexContr.V ⊗ complexContr.V)).hom
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(coMetric.hom (1 : ℂ) ⊗ₜ[ℂ] contrMetric.hom (1 : ℂ)))))) =
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contrCoUnit.hom (1 : ℂ) := by
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rw [coMetric_apply_one, contrMetric_apply_one]
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rw [coMetricVal_expand_tmul, contrMetricVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Fin.isValue, tmul_sub, add_tmul, neg_tmul, map_sub, map_add, map_neg, tmul_sub, sub_tmul]
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have h1 (x1 x2 : complexCo) (y1 y2 :complexContr) :
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(complexCo.V ◁ (λ_ complexContr.V).hom)
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((complexCo.V ◁ coContrContraction.hom ▷ complexContr.V) (((complexCo.V ◁
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(α_ complexCo.V complexContr.V complexContr.V).inv)
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((α_ complexCo.V complexCo.V (complexContr.V ⊗ complexContr.V)).hom
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((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
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= x1 ⊗ₜ[ℂ] ((λ_ complexContr.V).hom ((coContrContraction.hom (x2 ⊗ₜ[ℂ] y1)) ⊗ₜ[ℂ] y2)) := rfl
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repeat rw (config := { transparency := .instances }) [h1]
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repeat rw [coContrContraction_basis']
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simp only [Fin.isValue, ↓reduceIte, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul,
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reduceCtorEq, zero_tmul, map_zero, tmul_zero, sub_zero, zero_sub, Sum.inr.injEq, one_ne_zero,
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Fin.reduceEq, sub_neg_eq_add, zero_ne_one, sub_self]
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erw [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
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rfl
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end Lorentz
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end
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