feat: Add contraction of metric property for complex
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jstoobysmith
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4 changed files with 185 additions and 2 deletions
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@ -7,6 +7,7 @@ import HepLean.SpaceTime.WeylFermion.Basic
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import HepLean.SpaceTime.WeylFermion.Contraction
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import Mathlib.LinearAlgebra.TensorProduct.Matrix
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import HepLean.SpaceTime.WeylFermion.Two
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import HepLean.SpaceTime.WeylFermion.Unit
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/-!
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# Metrics of Weyl fermions
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@ -274,5 +275,119 @@ lemma altRightMetric_apply_one : altRightMetric.hom (1 : ℂ) = altRightMetricVa
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change altRightMetric.hom.toFun (1 : ℂ) = altRightMetricVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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altRightMetric, 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 leftAltContraction_apply_metric : (β_ leftHanded altLeftHanded).hom.hom
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((leftHanded.V ◁ (λ_ altLeftHanded.V).hom)
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((leftHanded.V ◁ leftAltContraction.hom ▷ altLeftHanded.V)
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((leftHanded.V ◁ (α_ leftHanded.V altLeftHanded.V altLeftHanded.V).inv)
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((α_ leftHanded.V leftHanded.V (altLeftHanded.V ⊗ altLeftHanded.V)).hom
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(leftMetric.hom (1 : ℂ) ⊗ₜ[ℂ] altLeftMetric.hom (1 : ℂ)))))) =
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altLeftLeftUnit.hom (1 : ℂ) := by
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rw [leftMetric_apply_one, altLeftMetric_apply_one]
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rw [leftMetricVal_expand_tmul, altLeftMetricVal_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]
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have h1 (x1 x2 : leftHanded) (y1 y2 :altLeftHanded) :
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(leftHanded.V ◁ (λ_ altLeftHanded.V).hom)
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((leftHanded.V ◁ leftAltContraction.hom ▷ altLeftHanded.V) (((leftHanded.V ◁
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(α_ leftHanded.V altLeftHanded.V altLeftHanded.V).inv)
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((α_ leftHanded.V leftHanded.V (altLeftHanded.V ⊗ altLeftHanded.V)).hom
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((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
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= x1 ⊗ₜ[ℂ] ((λ_ altLeftHanded.V).hom ((leftAltContraction.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 [leftAltContraction_basis]
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simp only [Fin.isValue, Fin.val_one, Fin.val_zero, one_ne_zero, ↓reduceIte, zero_tmul, map_zero,
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tmul_zero, neg_zero, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul, zero_add,
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zero_ne_one, add_zero, sub_neg_eq_add]
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erw [altLeftLeftUnit_apply_one, altLeftLeftUnitVal_expand_tmul]
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rw [add_comm]
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rfl
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lemma altLeftContraction_apply_metric : (β_ altLeftHanded leftHanded).hom.hom
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((altLeftHanded.V ◁ (λ_ leftHanded.V).hom)
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((altLeftHanded.V ◁ altLeftContraction.hom ▷ leftHanded.V)
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((altLeftHanded.V ◁ (α_ altLeftHanded.V leftHanded.V leftHanded.V).inv)
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((α_ altLeftHanded.V altLeftHanded.V (leftHanded.V ⊗ leftHanded.V)).hom
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(altLeftMetric.hom (1 : ℂ) ⊗ₜ[ℂ] leftMetric.hom (1 : ℂ)))))) =
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leftAltLeftUnit.hom (1 : ℂ) := by
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rw [leftMetric_apply_one, altLeftMetric_apply_one]
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rw [leftMetricVal_expand_tmul, altLeftMetricVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Fin.isValue, tmul_add, tmul_neg, sub_tmul, map_add, map_neg, map_sub]
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have h1 (x1 x2 : altLeftHanded) (y1 y2 : leftHanded) :
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(altLeftHanded.V ◁ (λ_ leftHanded.V).hom)
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((altLeftHanded.V ◁ altLeftContraction.hom ▷ leftHanded.V) (((altLeftHanded.V ◁
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(α_ altLeftHanded.V leftHanded.V leftHanded.V).inv)
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((α_ altLeftHanded.V altLeftHanded.V (leftHanded.V ⊗ leftHanded.V)).hom
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((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
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= x1 ⊗ₜ[ℂ] ((λ_ leftHanded.V).hom ((altLeftContraction.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 [altLeftContraction_basis]
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simp only [Fin.isValue, Fin.val_one, Fin.val_zero, one_ne_zero, ↓reduceIte, zero_tmul, map_zero,
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tmul_zero, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul, zero_sub, neg_neg,
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zero_ne_one, sub_zero]
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erw [leftAltLeftUnit_apply_one, leftAltLeftUnitVal_expand_tmul]
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rw [add_comm]
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rfl
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lemma rightAltContraction_apply_metric : (β_ rightHanded altRightHanded).hom.hom
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((rightHanded.V ◁ (λ_ altRightHanded.V).hom)
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((rightHanded.V ◁ rightAltContraction.hom ▷ altRightHanded.V)
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((rightHanded.V ◁ (α_ rightHanded.V altRightHanded.V altRightHanded.V).inv)
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((α_ rightHanded.V rightHanded.V (altRightHanded.V ⊗ altRightHanded.V)).hom
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(rightMetric.hom (1 : ℂ) ⊗ₜ[ℂ] altRightMetric.hom (1 : ℂ)))))) =
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altRightRightUnit.hom (1 : ℂ) := by
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rw [rightMetric_apply_one, altRightMetric_apply_one]
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rw [rightMetricVal_expand_tmul, altRightMetricVal_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]
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have h1 (x1 x2 : rightHanded) (y1 y2 : altRightHanded) :
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(rightHanded.V ◁ (λ_ altRightHanded.V).hom)
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((rightHanded.V ◁ rightAltContraction.hom ▷ altRightHanded.V) (((rightHanded.V ◁
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(α_ rightHanded.V altRightHanded.V altRightHanded.V).inv)
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((α_ rightHanded.V rightHanded.V (altRightHanded.V ⊗ altRightHanded.V)).hom
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((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
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= x1 ⊗ₜ[ℂ] ((λ_ altRightHanded.V).hom ((rightAltContraction.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 [rightAltContraction_basis]
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simp only [Fin.isValue, Fin.val_one, Fin.val_zero, one_ne_zero, ↓reduceIte, zero_tmul, map_zero,
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tmul_zero, neg_zero, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul, zero_add,
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zero_ne_one, add_zero, sub_neg_eq_add]
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erw [altRightRightUnit_apply_one, altRightRightUnitVal_expand_tmul]
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rw [add_comm]
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rfl
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lemma altRightContraction_apply_metric : (β_ altRightHanded rightHanded).hom.hom
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((altRightHanded.V ◁ (λ_ rightHanded.V).hom)
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((altRightHanded.V ◁ altRightContraction.hom ▷ rightHanded.V)
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((altRightHanded.V ◁ (α_ altRightHanded.V rightHanded.V rightHanded.V).inv)
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((α_ altRightHanded.V altRightHanded.V (rightHanded.V ⊗ rightHanded.V)).hom
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(altRightMetric.hom (1 : ℂ) ⊗ₜ[ℂ] rightMetric.hom (1 : ℂ)))))) =
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rightAltRightUnit.hom (1 : ℂ) := by
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rw [rightMetric_apply_one, altRightMetric_apply_one]
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rw [rightMetricVal_expand_tmul, altRightMetricVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Fin.isValue, tmul_add, tmul_neg, sub_tmul, map_add, map_neg, map_sub]
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have h1 (x1 x2 : altRightHanded) (y1 y2 : rightHanded) : (altRightHanded.V ◁ (λ_ rightHanded.V).hom)
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((altRightHanded.V ◁ altRightContraction.hom ▷ rightHanded.V) (((altRightHanded.V ◁
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(α_ altRightHanded.V rightHanded.V rightHanded.V).inv)
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((α_ altRightHanded.V altRightHanded.V (rightHanded.V ⊗ rightHanded.V)).hom
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((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
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= x1 ⊗ₜ[ℂ] ((λ_ rightHanded.V).hom ((altRightContraction.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 [altRightContraction_basis]
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simp only [Fin.isValue, Fin.val_one, Fin.val_zero, one_ne_zero, ↓reduceIte, zero_tmul, map_zero,
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tmul_zero, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul, zero_sub, neg_neg,
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zero_ne_one, sub_zero]
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erw [rightAltRightUnit_apply_one, rightAltRightUnitVal_expand_tmul]
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rw [add_comm]
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rfl
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end
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end Fermion
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