2024-10-15 13:19:46 +00:00
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/-
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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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/-!
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# Unit 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 contra-co unit for complex lorentz vectors. Usually denoted `δⁱᵢ`. -/
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def contrCoUnitVal : (complexContr ⊗ complexCo).V :=
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contrCoToMatrix.symm 1
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/-- The contra-co unit for complex lorentz vectors as a morphism
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`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo`, manifesting the invaraince under
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the `SL(2, ℂ)` action. -/
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2024-10-15 13:36:48 +00:00
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def contrCoUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo where
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2024-10-15 13:19:46 +00:00
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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' • contrCoUnitVal,
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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' • contrCoUnitVal =
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(TensorProduct.map (complexContr.ρ M) (complexCo.ρ M)) (x' • contrCoUnitVal)
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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, contrCoUnitVal]
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erw [contrCoToMatrix_ρ_symm]
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apply congrArg
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simp
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/-- The co-contra unit for complex lorentz vectors. Usually denoted `δᵢⁱ`. -/
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def coContrUnitVal : (complexCo ⊗ complexContr).V :=
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coContrToMatrix.symm 1
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/-- The co-contra unit for complex lorentz vectors as a morphism
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`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr`, manifesting the invaraince under
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the `SL(2, ℂ)` action. -/
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2024-10-15 13:36:48 +00:00
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def coContrUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr where
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2024-10-15 13:19:46 +00:00
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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' • coContrUnitVal,
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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' • coContrUnitVal =
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(TensorProduct.map (complexCo.ρ M) (complexContr.ρ M)) (x' • coContrUnitVal)
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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, coContrUnitVal]
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erw [coContrToMatrix_ρ_symm]
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apply congrArg
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symm
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refine transpose_eq_one.mp ?h.h.h.a
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simp
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end Lorentz
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end
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