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feat: Add contraction of metric property for complex

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jstoobysmith 2024-10-24 16:35:15 +00:00
parent 8c584431c4
commit 833a570ce8
4 changed files with 185 additions and 2 deletions
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61
HepLean/SpaceTime/LorentzVector/Complex/Metric.lean
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@ -5,6 +5,8 @@ Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Complex.Two
import HepLean.SpaceTime.MinkowskiMetric
import HepLean.SpaceTime.LorentzVector.Complex.Contraction
import HepLean.SpaceTime.LorentzVector.Complex.Unit
/-!
# Metric for complex Lorentz vectors
@ -128,5 +130,64 @@ lemma coMetric_apply_one : coMetric.hom (1 : ℂ) = coMetricVal := by
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
coMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
/-!
## Contraction of metrics
-/
lemma contrCoContraction_apply_metric : (β_ complexContr complexCo).hom.hom
((complexContr.V ◁ (λ_ complexCo.V).hom)
((complexContr.V ◁ contrCoContraction.hom ▷ complexCo.V)
((complexContr.V ◁ (α_ complexContr.V complexCo.V complexCo.V).inv)
((α_ complexContr.V complexContr.V (complexCo.V ⊗ complexCo.V)).hom
(contrMetric.hom (1 : ℂ) ⊗ₜ[ℂ] coMetric.hom (1 : ℂ)))))) =
coContrUnit.hom (1 : ℂ) := by
rw [contrMetric_apply_one, coMetric_apply_one]
rw [contrMetricVal_expand_tmul, coMetricVal_expand_tmul]
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Fin.isValue, tmul_sub, add_tmul, neg_tmul, map_sub, map_add, map_neg, tmul_sub, sub_tmul]
have h1 (x1 x2 : complexContr) (y1 y2 :complexCo) :
(complexContr.V ◁ (λ_ complexCo.V).hom)
((complexContr.V ◁ contrCoContraction.hom ▷ complexCo.V) (((complexContr.V ◁
(α_ complexContr.V complexCo.V complexCo.V).inv)
((α_ complexContr.V complexContr.V (complexCo.V ⊗ complexCo.V)).hom
((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
= x1 ⊗ₜ[ℂ] ((λ_ complexCo.V).hom ((contrCoContraction.hom (x2 ⊗ₜ[ℂ] y1)) ⊗ₜ[ℂ] y2)) := rfl
repeat rw (config := { transparency := .instances }) [h1]
repeat rw [contrCoContraction_basis']
simp only [Fin.isValue, ↓reduceIte, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul,
reduceCtorEq, zero_tmul, map_zero, tmul_zero, sub_zero, zero_sub, Sum.inr.injEq, one_ne_zero,
Fin.reduceEq, sub_neg_eq_add, zero_ne_one, sub_self]
erw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]
rfl
lemma coContrContraction_apply_metric : (β_ complexCo complexContr).hom.hom
((complexCo.V ◁ (λ_ complexContr.V).hom)
((complexCo.V ◁ coContrContraction.hom ▷ complexContr.V)
((complexCo.V ◁ (α_ complexCo.V complexContr.V complexContr.V).inv)
((α_ complexCo.V complexCo.V (complexContr.V ⊗ complexContr.V)).hom
(coMetric.hom (1 : ℂ) ⊗ₜ[ℂ] contrMetric.hom (1 : ℂ)))))) =
contrCoUnit.hom (1 : ℂ) := by
rw [coMetric_apply_one, contrMetric_apply_one]
rw [coMetricVal_expand_tmul, contrMetricVal_expand_tmul]
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Fin.isValue, tmul_sub, add_tmul, neg_tmul, map_sub, map_add, map_neg, tmul_sub, sub_tmul]
have h1 (x1 x2 : complexCo) (y1 y2 :complexContr) :
(complexCo.V ◁ (λ_ complexContr.V).hom)
((complexCo.V ◁ coContrContraction.hom ▷ complexContr.V) (((complexCo.V ◁
(α_ complexCo.V complexContr.V complexContr.V).inv)
((α_ complexCo.V complexCo.V (complexContr.V ⊗ complexContr.V)).hom
((x1 ⊗ₜ[ℂ] x2) ⊗ₜ[ℂ] y1 ⊗ₜ[ℂ] y2)))))
= x1 ⊗ₜ[ℂ] ((λ_ complexContr.V).hom ((coContrContraction.hom (x2 ⊗ₜ[ℂ] y1)) ⊗ₜ[ℂ] y2)) := rfl
repeat rw (config := { transparency := .instances }) [h1]
repeat rw [coContrContraction_basis']
simp only [Fin.isValue, ↓reduceIte, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, one_smul,
reduceCtorEq, zero_tmul, map_zero, tmul_zero, sub_zero, zero_sub, Sum.inr.injEq, one_ne_zero,
Fin.reduceEq, sub_neg_eq_add, zero_ne_one, sub_self]
erw [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
rfl
end Lorentz
end