feat: Modules for real Lorentz tensors

HepLean/SpaceTime/LorentzVector/Real/Modules.lean

@@ -0,0 +1,127 @@
+/-
+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.Meta.Informal
+import HepLean.SpaceTime.SL2C.Basic
+import Mathlib.RepresentationTheory.Rep
+import Mathlib.Logic.Equiv.TransferInstance
+/-!
+
+## Modules associated with Real Lorentz vectors
+
+We define the modules underlying real Lorentz vectors.
+
+-/
+
+namespace Lorentz
+
+noncomputable section
+open Matrix
+open MatrixGroups
+open Complex
+
+/-- The module for contravariant (up-index) real Lorentz vectors. -/
+structure ContrℝModule (d : ℕ) where
+  /-- The underlying value as a vector `Fin 1 ⊕ Fin d → ℝ`. -/
+  val : Fin 1 ⊕ Fin d → ℝ
+
+namespace ContrℝModule
+
+variable {d : ℕ}
+
+/-- The equivalence between `ContrℝModule` and `Fin 1 ⊕ Fin d → ℂ`. -/
+def toFin1dℝFun : ContrℝModule d  ≃ (Fin 1 ⊕ Fin d → ℝ) where
+  toFun v := v.val
+  invFun f := ⟨f⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+/-- The instance of `AddCommMonoid` on `ContrℝModule` defined via its equivalence
+  with `Fin 1 ⊕ Fin d → ℝ`. -/
+instance : AddCommMonoid (ContrℝModule d) := Equiv.addCommMonoid toFin1dℝFun
+
+/-- The instance of `AddCommGroup` on `ContrℝModule` defined via its equivalence
+  with `Fin 1 ⊕ Fin d → ℝ`. -/
+instance : AddCommGroup (ContrℝModule d) := Equiv.addCommGroup toFin1dℝFun
+
+/-- The instance of `Module` on `ContrℝModule` defined via its equivalence
+  with `Fin 1 ⊕ Fin d → ℝ`. -/
+instance : Module ℝ (ContrℝModule d) := Equiv.module ℝ toFin1dℝFun
+
+@[ext]
+lemma ext (ψ ψ' : ContrℝModule d) (h : ψ.val = ψ'.val) : ψ = ψ' := by
+  cases ψ
+  cases ψ'
+  subst h
+  rfl
+
+@[simp]
+lemma val_add (ψ ψ' : ContrℝModule d) : (ψ + ψ').val = ψ.val + ψ'.val := rfl
+
+@[simp]
+lemma val_smul (r : ℝ) (ψ : ContrℝModule d) : (r • ψ).val = r • ψ.val := rfl
+
+/-- The linear equivalence between `ContrℝModule` and `(Fin 1 ⊕ Fin d → ℝ)`. -/
+@[simps!]
+def toFin1dℝEquiv : ContrℝModule d ≃ₗ[ℝ] (Fin 1 ⊕ Fin d → ℝ) where
+  toFun := toFin1dℝFun
+  map_add' := fun _ _ => rfl
+  map_smul' := fun _ _ => rfl
+  invFun := toFin1dℝFun.symm
+  left_inv := fun _ => rfl
+  right_inv := fun _ => rfl
+
+/-- The underlying element of `Fin 1 ⊕ Fin d → ℝ` of a element in `ContrℝModule` defined
+  through the linear equivalence `toFin1dℝEquiv`. -/
+abbrev toFin1dℝ (ψ : ContrℝModule d) := toFin1dℝEquiv ψ
+
+end ContrℝModule
+
+/-- The module for covariant (up-index) complex Lorentz vectors. -/
+structure CoℝModule (d : ℕ) where
+  /-- The underlying value as a vector `Fin 1 ⊕ Fin d → ℝ`. -/
+  val : Fin 1 ⊕ Fin d → ℝ
+
+namespace CoℝModule
+
+variable {d : ℕ}
+
+/-- The equivalence between `CoℝModule` and `Fin 1 ⊕ Fin d → ℝ`. -/
+def toFin1dℝFun : CoℝModule d ≃ (Fin 1 ⊕ Fin d → ℝ) where
+  toFun v := v.val
+  invFun f := ⟨f⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+/-- The instance of `AddCommMonoid` on `CoℂModule` defined via its equivalence
+  with `Fin 1 ⊕ Fin d → ℝ`. -/
+instance : AddCommMonoid (CoℝModule d) := Equiv.addCommMonoid toFin1dℝFun
+
+/-- The instance of `AddCommGroup` on `CoℝModule` defined via its equivalence
+  with `Fin 1 ⊕ Fin d → ℝ`. -/
+instance : AddCommGroup (CoℝModule d) := Equiv.addCommGroup toFin1dℝFun
+
+/-- The instance of `Module` on `CoℝModule` defined via its equivalence
+  with `Fin 1 ⊕ Fin d → ℝ`. -/
+instance : Module ℝ (CoℝModule d) := Equiv.module ℝ toFin1dℝFun
+
+/-- The linear equivalence between `CoℝModule` and `(Fin 1 ⊕ Fin d → ℝ)`. -/
+@[simps!]
+def toFin1dℝEquiv : CoℝModule d ≃ₗ[ℝ] (Fin 1 ⊕ Fin d → ℝ) where
+  toFun := toFin1dℝFun
+  map_add' := fun _ _ => rfl
+  map_smul' := fun _ _ => rfl
+  invFun := toFin1dℝFun.symm
+  left_inv := fun _ => rfl
+  right_inv := fun _ => rfl
+
+/-- The underlying element of `Fin 1 ⊕ Fin d → ℝ` of a element in `CoℝModule` defined
+  through the linear equivalence `toFin1dℝEquiv`. -/
+abbrev toFin13ℂ (ψ : CoℝModule d) := toFin1dℝEquiv ψ
+
+end CoℝModule
+
+end
+end Lorentz