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