feat: Metric, unit, contract of complex Lorentz vec

HepLean/SpaceTime/LorentzVector/Complex/Contraction.lean

@@ -0,0 +1,101 @@
+/-
+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.Basic
+/-!
+
+# Contraction of Lorentz vectors
+
+
+-/
+
+noncomputable section
+
+open Matrix
+open MatrixGroups
+open Complex
+open TensorProduct
+open SpaceTime
+open CategoryTheory.MonoidalCategory
+namespace Lorentz
+
+/-- The bi-linear map corresponding to contraction of a contravariant Lorentz vector with a
+  covariant Lorentz vector. -/
+def contrCoContrBi : complexContr →ₗ[ℂ] complexCo →ₗ[ℂ] ℂ where
+  toFun ψ := {
+    toFun := fun φ => ψ.toFin13ℂ ⬝ᵥ φ.toFin13ℂ,
+    map_add' := by
+      intro φ φ'
+      simp only [map_add]
+      rw [dotProduct_add]
+    map_smul' := by
+      intro r φ
+      simp only [LinearEquiv.map_smul]
+      rw [dotProduct_smul]
+      rfl}
+  map_add' ψ ψ':= by
+    refine LinearMap.ext (fun φ => ?_)
+    simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
+    rw [add_dotProduct]
+  map_smul' r ψ := by
+    refine LinearMap.ext (fun φ => ?_)
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
+    rw [smul_dotProduct]
+    rfl
+
+/-- The bi-linear map corresponding to contraction of a covariant Lorentz vector with a
+  contravariant Lorentz vector. -/
+def contrContrCoBi : complexCo →ₗ[ℂ] complexContr →ₗ[ℂ] ℂ where
+  toFun φ := {
+    toFun := fun ψ => φ.toFin13ℂ ⬝ᵥ ψ.toFin13ℂ,
+    map_add' := by
+      intro ψ ψ'
+      simp only [map_add]
+      rw [dotProduct_add]
+    map_smul' := by
+      intro r ψ
+      simp only [LinearEquiv.map_smul]
+      rw [dotProduct_smul]
+      rfl}
+  map_add' φ φ' := by
+    refine LinearMap.ext (fun ψ => ?_)
+    simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
+    rw [add_dotProduct]
+  map_smul' r φ := by
+    refine LinearMap.ext (fun ψ => ?_)
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
+    rw [smul_dotProduct]
+    rfl
+
+/-- The linear map from complexContr ⊗ complexCo to ℂ given by
+    summing over components of contravariant Lorentz vector and
+    covariant Lorentz vector in the
+    standard basis (i.e. the dot product).
+    In terms of index notation this is the contraction is ψⁱ φᵢ. -/
+def contrCoContraction : complexContr ⊗ complexCo ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
+  hom := TensorProduct.lift contrCoContrBi
+  comm M := TensorProduct.ext' fun ψ φ => by
+    change ((LorentzGroup.toComplex (SL2C.toLorentzGroup M)) *ᵥ ψ.toFin13ℂ) ⬝ᵥ
+      ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ *ᵥ φ.toFin13ℂ) = ψ.toFin13ℂ ⬝ᵥ φ.toFin13ℂ
+    rw [dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
+    rw [inv_mul_of_invertible (LorentzGroup.toComplex (SL2C.toLorentzGroup M))]
+    simp
+
+/-- The linear map from complexCo ⊗ complexContr to ℂ given by
+    summing over components of covariant Lorentz vector and
+    contravariant Lorentz vector in the
+    standard basis (i.e. the dot product).
+    In terms of index notation this is the contraction is φᵢ ψⁱ. -/
+def coContrContraction : complexCo ⊗ complexContr ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
+  hom := TensorProduct.lift contrContrCoBi
+  comm M := TensorProduct.ext' fun φ ψ => by
+    change ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ  *ᵥ φ.toFin13ℂ) ⬝ᵥ
+      ((LorentzGroup.toComplex (SL2C.toLorentzGroup M)) *ᵥ ψ.toFin13ℂ) = φ.toFin13ℂ ⬝ᵥ ψ.toFin13ℂ
+    rw [dotProduct_mulVec, mulVec_transpose, vecMul_vecMul]
+    rw [inv_mul_of_invertible (LorentzGroup.toComplex (SL2C.toLorentzGroup M))]
+    simp
+
+end Lorentz
+end