feat: Metric, unit, contract of complex Lorentz vec

HepLean/SpaceTime/LorentzVector/Complex/Metric.lean

@@ -0,0 +1,88 @@
+/-
+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
+import HepLean.SpaceTime.MinkowskiMetric
+/-!
+
+# Metric for complex Lorentz vectors
+
+-/
+noncomputable section
+
+open Matrix
+open MatrixGroups
+open Complex
+open TensorProduct
+open SpaceTime
+open CategoryTheory.MonoidalCategory
+namespace Lorentz
+
+/-- The metric `ηᵃᵃ` as an element of `(complexContr ⊗ complexContr).V`. -/
+def contrMetricVal : (complexContr ⊗ complexContr).V :=
+  contrContrToMatrix.symm ((@minkowskiMatrix 3).map ofReal)
+
+/-- The metric `ηᵃᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexContr`,
+  making its invariance under the action of `SL(2,ℂ)`. -/
+def contrMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexContr where
+  hom := {
+    toFun := fun a =>
+      let a' : ℂ := a
+      a' • contrMetricVal,
+    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' • contrMetricVal =
+      (TensorProduct.map (complexContr.ρ M) (complexContr.ρ M)) (x' • contrMetricVal)
+    simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
+    apply congrArg
+    simp only [Action.instMonoidalCategory_tensorObj_V, contrMetricVal]
+    erw [contrContrToMatrix_ρ_symm]
+    apply congrArg
+    simp only [LorentzGroup.toComplex_mul_minkowskiMatrix_mul_transpose]
+
+/-- The metric `ηᵢᵢ` as an element of `(complexCo ⊗ complexCo).V`. -/
+def coMetricVal : (complexCo ⊗ complexCo).V :=
+  coCoToMatrix.symm ((@minkowskiMatrix 3).map ofReal)
+
+/-- The metric `ηᵢᵢ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo`,
+  making its invariance under the action of `SL(2,ℂ)`. -/
+def coMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo where
+  hom := {
+    toFun := fun a =>
+      let a' : ℂ := a
+      a' • coMetricVal,
+    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' • coMetricVal =
+      (TensorProduct.map (complexCo.ρ M) (complexCo.ρ M)) (x' • coMetricVal)
+    simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
+    apply congrArg
+    simp only [Action.instMonoidalCategory_tensorObj_V, coMetricVal]
+    erw [coCoToMatrix_ρ_symm]
+    apply congrArg
+    rw [LorentzGroup.toComplex_inv]
+    simp only [lorentzGroupIsGroup_inv, SL2C.toLorentzGroup_apply_coe,
+      LorentzGroup.toComplex_transpose_mul_minkowskiMatrix_mul_self]
+
+end Lorentz
+end