161 lines
5.4 KiB
Text
161 lines
5.4 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.Lorentz.SL2C.Basic
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/-!
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## Modules associated with complex Lorentz vectors
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We define the modules underlying complex Lorentz vectors.
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These definitions are preludes to the definitions of
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`Lorentz.complexContr` and `Lorentz.complexCo`.
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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) complex Lorentz vectors. -/
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structure ContrℂModule where
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/-- The underlying value as a vector `Fin 1 ⊕ Fin 3 → ℂ`. -/
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val : Fin 1 ⊕ Fin 3 → ℂ
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namespace ContrℂModule
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/-- The equivalence between `ContrℂModule` and `Fin 1 ⊕ Fin 3 → ℂ`. -/
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def toFin13ℂFun : ContrℂModule ≃ (Fin 1 ⊕ Fin 3 → ℂ) 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 `ContrℂModule` defined via its equivalence
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with `Fin 1 ⊕ Fin 3 → ℂ`. -/
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instance : AddCommMonoid ContrℂModule := Equiv.addCommMonoid toFin13ℂFun
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/-- The instance of `AddCommGroup` on `ContrℂModule` defined via its equivalence
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with `Fin 1 ⊕ Fin 3 → ℂ`. -/
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instance : AddCommGroup ContrℂModule := Equiv.addCommGroup toFin13ℂFun
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/-- The instance of `Module` on `ContrℂModule` defined via its equivalence
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with `Fin 1 ⊕ Fin 3 → ℂ`. -/
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instance : Module ℂ ContrℂModule := Equiv.module ℂ toFin13ℂFun
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@[ext]
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lemma ext (ψ ψ' : ContrℂModule) (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' : ContrℂModule) : (ψ + ψ').val = ψ.val + ψ'.val := rfl
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@[simp]
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lemma val_smul (r : ℂ) (ψ : ContrℂModule) : (r • ψ).val = r • ψ.val := rfl
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/-- The linear equivalence between `ContrℂModule` and `(Fin 1 ⊕ Fin 3 → ℂ)`. -/
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@[simps!]
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def toFin13ℂEquiv : ContrℂModule ≃ₗ[ℂ] (Fin 1 ⊕ Fin 3 → ℂ) :=
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Equiv.linearEquiv ℂ toFin13ℂFun
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/-- The underlying element of `Fin 1 ⊕ Fin 3 → ℂ` of a element in `ContrℂModule` defined
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through the linear equivalence `toFin13ℂEquiv`. -/
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abbrev toFin13ℂ (ψ : ContrℂModule) := toFin13ℂEquiv ψ
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/-- The representation of the Lorentz group on `ContrℂModule`. -/
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def lorentzGroupRep : Representation ℂ (LorentzGroup 3) ContrℂModule where
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toFun M := {
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toFun := fun v => toFin13ℂEquiv.symm (LorentzGroup.toComplex M *ᵥ v.toFin13ℂ),
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map_add' := by
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intro ψ ψ'
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simp [mulVec_add]
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map_smul' := by
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intro r ψ
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simp [mulVec_smul]}
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map_one' := by
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ext i
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simp
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map_mul' M N := by
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ext i
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simp
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/-- The representation of the SL(2, ℂ) on `ContrℂModule` induced by the representation of the
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Lorentz group. -/
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def SL2CRep : Representation ℂ SL(2, ℂ) ContrℂModule :=
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MonoidHom.comp lorentzGroupRep Lorentz.SL2C.toLorentzGroup
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end ContrℂModule
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/-- The module for covariant (up-index) complex Lorentz vectors. -/
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structure CoℂModule where
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/-- The underlying value as a vector `Fin 1 ⊕ Fin 3 → ℂ`. -/
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val : Fin 1 ⊕ Fin 3 → ℂ
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namespace CoℂModule
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/-- The equivalence between `CoℂModule` and `Fin 1 ⊕ Fin 3 → ℂ`. -/
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def toFin13ℂFun : CoℂModule ≃ (Fin 1 ⊕ Fin 3 → ℂ) 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 3 → ℂ`. -/
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instance : AddCommMonoid CoℂModule := Equiv.addCommMonoid toFin13ℂFun
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/-- The instance of `AddCommGroup` on `CoℂModule` defined via its equivalence
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with `Fin 1 ⊕ Fin 3 → ℂ`. -/
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instance : AddCommGroup CoℂModule := Equiv.addCommGroup toFin13ℂFun
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/-- The instance of `Module` on `CoℂModule` defined via its equivalence
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with `Fin 1 ⊕ Fin 3 → ℂ`. -/
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instance : Module ℂ CoℂModule := Equiv.module ℂ toFin13ℂFun
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/-- The linear equivalence between `CoℂModule` and `(Fin 1 ⊕ Fin 3 → ℂ)`. -/
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@[simps!]
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def toFin13ℂEquiv : CoℂModule ≃ₗ[ℂ] (Fin 1 ⊕ Fin 3 → ℂ) :=
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Equiv.linearEquiv ℂ toFin13ℂFun
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/-- The underlying element of `Fin 1 ⊕ Fin 3 → ℂ` of a element in `CoℂModule` defined
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through the linear equivalence `toFin13ℂEquiv`. -/
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abbrev toFin13ℂ (ψ : CoℂModule) := toFin13ℂEquiv ψ
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/-- The representation of the Lorentz group on `CoℂModule`. -/
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def lorentzGroupRep : Representation ℂ (LorentzGroup 3) CoℂModule where
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toFun M := {
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toFun := fun v => toFin13ℂEquiv.symm ((LorentzGroup.toComplex M)⁻¹ᵀ *ᵥ v.toFin13ℂ),
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map_add' := by
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intro ψ ψ'
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simp [mulVec_add]
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map_smul' := by
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intro r ψ
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simp [mulVec_smul]}
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map_one' := by
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ext i
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simp
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map_mul' M N := by
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ext1 x
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simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply,
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LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
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refine (congrFun (congrArg _ ?_) _)
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simp only [_root_.map_mul]
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rw [Matrix.mul_inv_rev]
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exact transpose_mul _ _
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/-- The representation of the SL(2, ℂ) on `ContrℂModule` induced by the representation of the
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Lorentz group. -/
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def SL2CRep : Representation ℂ SL(2, ℂ) CoℂModule :=
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MonoidHom.comp lorentzGroupRep Lorentz.SL2C.toLorentzGroup
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end CoℂModule
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end
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end Lorentz
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