feat: Metric, unit, contract of complex Lorentz vec

HepLean/SpaceTime/LorentzVector/Complex/Unit.lean

@@ -0,0 +1,89 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzVector.Complex.Two
+/-!
+
+# Unit for complex Lorentz vectors
+
+-/
+noncomputable section
+
+open Matrix
+open MatrixGroups
+open Complex
+open TensorProduct
+open SpaceTime
+open CategoryTheory.MonoidalCategory
+namespace Lorentz
+
+/-- The contra-co unit for complex lorentz vectors. Usually denoted `δⁱᵢ`. -/
+def contrCoUnitVal : (complexContr ⊗ complexCo).V :=
+  contrCoToMatrix.symm 1
+
+/-- The contra-co unit for complex lorentz vectors as a morphism
+  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo`, manifesting the invaraince under
+  the `SL(2, ℂ)` action. -/
+def contrCoUnit :  𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo where
+  hom := {
+    toFun := fun a =>
+      let a' : ℂ := a
+      a' • contrCoUnitVal,
+    map_add' := fun x y => by
+      simp only [add_smul],
+    map_smul' := fun m x => by
+      simp only [smul_smul]
+      rfl}
+  comm M := by
+    ext x : 2
+    simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+      Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
+      Function.comp_apply]
+    let x' : ℂ := x
+    change x' • contrCoUnitVal =
+      (TensorProduct.map (complexContr.ρ M) (complexCo.ρ M)) (x' • contrCoUnitVal)
+    simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
+    apply congrArg
+    simp only [Action.instMonoidalCategory_tensorObj_V, contrCoUnitVal]
+    erw [contrCoToMatrix_ρ_symm]
+    apply congrArg
+    simp
+
+/-- The co-contra unit for complex lorentz vectors. Usually denoted `δᵢⁱ`. -/
+def coContrUnitVal : (complexCo ⊗ complexContr).V :=
+  coContrToMatrix.symm 1
+
+/-- The co-contra unit for complex lorentz vectors as a morphism
+  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr`, manifesting the invaraince under
+  the `SL(2, ℂ)` action. -/
+def coContrUnit :  𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr where
+  hom := {
+    toFun := fun a =>
+      let a' : ℂ := a
+      a' • coContrUnitVal,
+    map_add' := fun x y => by
+      simp only [add_smul],
+    map_smul' := fun m x => by
+      simp only [smul_smul]
+      rfl}
+  comm M := by
+    ext x : 2
+    simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+      Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
+      Function.comp_apply]
+    let x' : ℂ := x
+    change x' • coContrUnitVal =
+      (TensorProduct.map (complexCo.ρ M) (complexContr.ρ M)) (x' • coContrUnitVal)
+    simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
+    apply congrArg
+    simp only [Action.instMonoidalCategory_tensorObj_V, coContrUnitVal]
+    erw [coContrToMatrix_ρ_symm]
+    apply congrArg
+    symm
+    refine transpose_eq_one.mp ?h.h.h.a
+    simp
+
+end Lorentz
+end