feat: Prove multiplication commute Lorentz action
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5 changed files with 649 additions and 348 deletions
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@ -6,6 +6,8 @@ Authors: Joseph Tooby-Smith
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import Mathlib.Logic.Function.CompTypeclasses
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import Mathlib.Data.Real.Basic
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import Mathlib.Analysis.Normed.Field.Basic
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import Mathlib.CategoryTheory.Core
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import Mathlib.CategoryTheory.Types
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/-!
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# Real Lorentz Tensors
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@ -59,35 +61,11 @@ structure RealLorentzTensor (d : ℕ) (X : Type) where
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coord : RealLorentzTensor.IndexValue d color → ℝ
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namespace RealLorentzTensor
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open Matrix
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open Matrix CategoryTheory
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universe u1
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variable {d : ℕ} {X Y Z : Type}
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variable (c : X → Colors)
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/-!
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## Some equivalences of types
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These come in use casting Lorentz tensors.
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There is likely a better way to deal with these castings.
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-/
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/-- An equivalence from `Empty ⊕ PUnit.{1}` to `Empty ⊕ Σ _ : Fin 1, PUnit`. -/
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def equivPUnitToSigma :
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(Empty ⊕ PUnit.{1}) ≃ (Empty ⊕ Σ _ : Fin 1, PUnit) where
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toFun x := match x with
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| Sum.inr x => Sum.inr ⟨0, x⟩
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invFun x := match x with
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| Sum.inr ⟨0, x⟩ => Sum.inr x
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left_inv x := match x with
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| Sum.inr _ => rfl
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right_inv x := match x with
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| Sum.inr ⟨0, _⟩ => rfl
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/-!
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## Colors
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@ -105,11 +83,19 @@ lemma τ_involutive : Function.Involutive τ := by
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intro x
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cases x <;> rfl
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lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
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(Function.Involutive.eq_iff τ_involutive).mp h.symm
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/-- The color associated with an element of `x ∈ X` for a tensor `T`. -/
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def ch {X : Type} (x : X) (T : RealLorentzTensor d X) : Colors := T.color x
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/-- An equivalence of `ColorsIndex` types given an equality of a colors. -/
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def colorsIndexCast {d : ℕ} {μ₁ μ₂ : RealLorentzTensor.Colors} (h : μ₁ = μ₂) :
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ColorsIndex d μ₁ ≃ ColorsIndex d μ₂ :=
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Equiv.cast (by rw [h])
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/-- An equivalence of `ColorsIndex` between that of a color and that of its dual. -/
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def dualColorsIndex {d : ℕ} {μ : RealLorentzTensor.Colors}:
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def colorsIndexDualCastSelf {d : ℕ} {μ : RealLorentzTensor.Colors}:
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ColorsIndex d μ ≃ ColorsIndex d (τ μ) where
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toFun x :=
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match μ with
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@ -122,26 +108,18 @@ def dualColorsIndex {d : ℕ} {μ : RealLorentzTensor.Colors}:
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left_inv x := by cases μ <;> rfl
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right_inv x := by cases μ <;> rfl
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/-- An equivalence of `ColorsIndex` types given an equality of a colors. -/
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def castColorsIndex {d : ℕ} {μ₁ μ₂ : RealLorentzTensor.Colors} (h : μ₁ = μ₂) :
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ColorsIndex d μ₁ ≃ ColorsIndex d μ₂ :=
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Equiv.cast (by rw [h])
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/-- An equivalence of `ColorsIndex` types given an equality of a color and the dual of a color. -/
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def congrColorsDual {μ ν : Colors} (h : μ = τ ν) :
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def colorsIndexDualCast {μ ν : Colors} (h : μ = τ ν) :
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ColorsIndex d μ ≃ ColorsIndex d ν :=
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(castColorsIndex h).trans dualColorsIndex.symm
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(colorsIndexCast h).trans colorsIndexDualCastSelf.symm
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lemma congrColorsDual_symm {μ ν : Colors} (h : μ = τ ν) :
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(congrColorsDual h).symm =
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@congrColorsDual d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
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lemma colorsIndexDualCast_symm {μ ν : Colors} (h : μ = τ ν) :
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(colorsIndexDualCast h).symm =
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@colorsIndexDualCast d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
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match μ, ν with
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| Colors.up, Colors.down => rfl
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| Colors.down, Colors.up => rfl
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lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
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(Function.Involutive.eq_iff τ_involutive).mp h.symm
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/-!
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## Index values
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@ -153,19 +131,54 @@ instance [Fintype X] [DecidableEq X] : Fintype (IndexValue d c) := Pi.fintype
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instance [Fintype X] [DecidableEq X] : DecidableEq (IndexValue d c) :=
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Fintype.decidablePiFintype
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/-- An equivalence of Index values from an equality of color maps. -/
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def castIndexValue {X : Type} {T S : X → Colors} (h : T = S) :
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IndexValue d T ≃ IndexValue d S where
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toFun i := (fun μ => castColorsIndex (congrFun h μ) (i μ))
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invFun i := (fun μ => (castColorsIndex (congrFun h μ)).symm (i μ))
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left_inv i := by
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simp
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right_inv i := by
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simp
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/-!
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lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h : T₁ = T₂) :
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IndexValue d T₁ = IndexValue d T₂ :=
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pi_congr fun a => congrArg (ColorsIndex d) (congrFun h a)
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## Induced isomorphisms between IndexValue sets
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-/
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@[simps!]
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def indexValueIso (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors} (h : i = j ∘ f) :
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IndexValue d i ≃ IndexValue d j :=
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(Equiv.piCongrRight (fun μ => colorsIndexCast (congrFun h μ))).trans $
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Equiv.piCongrLeft (fun y => RealLorentzTensor.ColorsIndex d (j y)) f
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lemma indexValueIso_symm_apply' (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors}
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(h : i = j ∘ f) (y : IndexValue d j) (x : X) :
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(indexValueIso d f h).symm y x = (colorsIndexCast (congrFun h x)).symm (y (f x)) := by
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rfl
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@[simp]
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lemma indexValueIso_trans (d : ℕ) (f : X ≃ Y) (g : Y ≃ Z) {i : X → Colors}
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{j : Y → Colors} {k : Z → Colors} (h : i = j ∘ f) (h' : j = k ∘ g) :
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(indexValueIso d f h).trans (indexValueIso d g h') =
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indexValueIso d (f.trans g) (by rw [h, h', Function.comp.assoc]; rfl) := by
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have h1 : ((indexValueIso d f h).trans (indexValueIso d g h')).symm =
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(indexValueIso d (f.trans g) (by rw [h, h', Function.comp.assoc]; rfl)).symm := by
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subst h' h
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ext x : 2
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rfl
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simpa only [Equiv.symm_symm] using congrArg (fun e => e.symm) h1
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lemma indexValueIso_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
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(indexValueIso d f h).symm = indexValueIso d f.symm (by rw [h, Function.comp.assoc]; simp) := by
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ext i : 1
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rw [← Equiv.symm_apply_eq]
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funext y
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rw [indexValueIso_symm_apply', indexValueIso_symm_apply']
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simp [colorsIndexCast]
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apply cast_eq_iff_heq.mpr
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rw [Equiv.apply_symm_apply]
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lemma indexValueIso_eq_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
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indexValueIso d f h = (indexValueIso d f.symm (by rw [h, Function.comp.assoc]; simp)).symm := by
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rw [indexValueIso_symm]
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congr
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@[simp]
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lemma indexValueIso_refl (d : ℕ) (i : X → Colors) :
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indexValueIso d (Equiv.refl X) (rfl : i = i) = Equiv.refl _ := by
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rfl
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/-!
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@ -174,19 +187,7 @@ lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h :
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-/
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lemma ext {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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(h' : T₁.coord = T₂.coord ∘ Equiv.cast (indexValue_eq d h)) : T₁ = T₂ := by
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cases T₁
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cases T₂
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simp_all only [IndexValue, mk.injEq]
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apply And.intro h
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simp only at h
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subst h
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simp only [Equiv.cast_refl, Equiv.coe_refl, CompTriple.comp_eq] at h'
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subst h'
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rfl
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lemma ext' {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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(h' : T₁.coord = fun i => T₂.coord (castIndexValue h i)) :
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(h' : T₁.coord = fun i => T₂.coord (indexValueIso d (Equiv.refl X) h i)) :
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T₁ = T₂ := by
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cases T₁
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cases T₂
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@ -199,60 +200,69 @@ lemma ext' {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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/-!
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## Congruence
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## Mapping isomorphisms.
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-/
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/-- An equivalence between `X → Fin 1 ⊕ Fin d` and `Y → Fin 1 ⊕ Fin d` given an isomorphism
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between `X` and `Y`. -/
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@[simps!]
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def congrSetIndexValue (d : ℕ) (f : X ≃ Y) (i : X → Colors) :
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IndexValue d i ≃ IndexValue d (i ∘ f.symm) :=
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Equiv.piCongrLeft' _ f
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@[simp]
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lemma castColorsIndex_comp_congrSetIndexValue (c : X → Colors) (j : IndexValue d c) (f : X ≃ Y)
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(h1 : (c <| f.symm <| f <| x) = c x) : (castColorsIndex h1 <| congrSetIndexValue d f c j <| f x)
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= j x := by
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rw [congrSetIndexValue_apply]
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refine cast_eq_iff_heq.mpr ?_
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rw [Equiv.symm_apply_apply]
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/-- Given an equivalence of indexing sets, a map on Lorentz tensors. -/
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@[simps!]
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def congrSetMap (f : X ≃ Y) (T : RealLorentzTensor d X) : RealLorentzTensor d Y where
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color := T.color ∘ f.symm
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coord := T.coord ∘ (congrSetIndexValue d f T.color).symm
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lemma congrSetMap_trans (f : X ≃ Y) (g : Y ≃ Z) (T : RealLorentzTensor d X) :
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congrSetMap g (congrSetMap f T) = congrSetMap (f.trans g) T := by
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apply ext (by rfl)
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have h1 : congrSetIndexValue d (f.trans g) T.color = (congrSetIndexValue d f T.color).trans
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(congrSetIndexValue d g $ Equiv.piCongrLeft' (fun _ => Colors) f T.color) := by
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exact Equiv.coe_inj.mp rfl
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simp only [congrSetMap, Equiv.piCongrLeft'_apply, IndexValue, Equiv.symm_trans_apply, h1,
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Equiv.cast_refl, Equiv.coe_refl, CompTriple.comp_eq]
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rfl
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/-- An equivalence of Tensors given an equivalence of underlying sets. -/
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@[simps!]
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def congrSet (f : X ≃ Y) : RealLorentzTensor d X ≃ RealLorentzTensor d Y where
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toFun := congrSetMap f
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invFun := congrSetMap f.symm
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def mapIso (d : ℕ) (f : X ≃ Y) : RealLorentzTensor d X ≃ RealLorentzTensor d Y where
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toFun T := {
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color := T.color ∘ f.symm,
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coord := T.coord ∘ (indexValueIso d f (by simp : T.color = T.color ∘ f.symm ∘ f)).symm}
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invFun T := {
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color := T.color ∘ f,
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coord := T.coord ∘ (indexValueIso d f.symm (by simp : T.color = T.color ∘ f ∘ f.symm)).symm}
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left_inv T := by
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rw [congrSetMap_trans, Equiv.self_trans_symm]
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rfl
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refine ext ?_ ?_
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· simp [Function.comp.assoc]
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· funext i
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simp only [IndexValue, Function.comp_apply, Function.comp_id]
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apply congrArg
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funext x
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erw [indexValueIso_symm_apply', indexValueIso_symm_apply', indexValueIso_eq_symm,
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indexValueIso_symm_apply']
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rw [← Equiv.apply_eq_iff_eq_symm_apply]
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simp only [Equiv.refl_symm, Equiv.coe_refl, Function.comp_apply, id_eq, colorsIndexCast,
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Equiv.cast_symm, Equiv.cast_apply, cast_cast, Equiv.refl_apply]
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apply cast_eq_iff_heq.mpr
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congr
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exact Equiv.symm_apply_apply f x
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right_inv T := by
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rw [congrSetMap_trans, Equiv.symm_trans_self]
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rfl
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refine ext ?_ ?_
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· simp [Function.comp.assoc]
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· funext i
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simp only [IndexValue, Function.comp_apply, Function.comp_id]
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apply congrArg
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funext x
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erw [indexValueIso_symm_apply', indexValueIso_symm_apply', indexValueIso_eq_symm,
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indexValueIso_symm_apply']
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rw [← Equiv.apply_eq_iff_eq_symm_apply]
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simp only [Equiv.refl_symm, Equiv.coe_refl, Function.comp_apply, id_eq, colorsIndexCast,
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Equiv.cast_symm, Equiv.cast_apply, cast_cast, Equiv.refl_apply]
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apply cast_eq_iff_heq.mpr
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congr
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exact Equiv.apply_symm_apply f x
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lemma congrSet_trans (f : X ≃ Y) (g : Y ≃ Z) :
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(@congrSet d _ _ f).trans (congrSet g) = congrSet (f.trans g) := by
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@[simp]
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lemma mapIso_trans (f : X ≃ Y) (g : Y ≃ Z) :
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(mapIso d f).trans (mapIso d g) = mapIso d (f.trans g) := by
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refine Equiv.coe_inj.mp ?_
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funext T
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exact congrSetMap_trans f g T
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refine ext rfl ?_
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simp only [Equiv.trans_apply, IndexValue, mapIso_apply_color, Equiv.symm_trans_apply,
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indexValueIso_refl, Equiv.refl_apply, mapIso_apply_coord]
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funext i
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rw [mapIso_apply_coord, mapIso_apply_coord]
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apply congrArg
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rw [← indexValueIso_trans]
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rfl
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simp only [Function.comp.assoc, Equiv.symm_comp_self, CompTriple.comp_eq]
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lemma congrSet_refl : @congrSet d _ _ (Equiv.refl X) = Equiv.refl _ := rfl
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lemma mapIso_symm (f : X ≃ Y) : (mapIso d f).symm = mapIso d f.symm := by
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rfl
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lemma mapIso_refl : mapIso d (Equiv.refl X) = Equiv.refl _ := rfl
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/-!
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@ -260,54 +270,24 @@ lemma congrSet_refl : @congrSet d _ _ (Equiv.refl X) = Equiv.refl _ := rfl
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-/
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/-- The sum of two color maps. -/
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def sumElimIndexColor (Tc : X → Colors) (Sc : Y → Colors) :
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(X ⊕ Y) → Colors :=
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Sum.elim Tc Sc
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/-- The symmetry property on `sumElimIndexColor`. -/
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lemma sumElimIndexColor_symm (Tc : X → Colors) (Sc : Y → Colors) : sumElimIndexColor Tc Sc =
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Equiv.piCongrLeft' _ (Equiv.sumComm X Y).symm (sumElimIndexColor Sc Tc) := by
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ext1 x
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simp_all only [Equiv.piCongrLeft'_apply, Equiv.sumComm_symm, Equiv.sumComm_apply]
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cases x <;> rfl
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/-- The sum of two index values for different color maps. -/
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@[simp]
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def sumElimIndexValue {X Y : Type} {TX : X → Colors} {TY : Y → Colors}
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(i : IndexValue d TX) (j : IndexValue d TY) :
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IndexValue d (sumElimIndexColor TX TY) :=
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fun c => match c with
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| Sum.inl x => i x
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| Sum.inr x => j x
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/-- The projection of an index value on a sum of color maps to its left component. -/
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def inlIndexValue {Tc : X → Colors} {Sc : Y → Colors} (i : IndexValue d (sumElimIndexColor Tc Sc)) :
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IndexValue d Tc := fun x => i (Sum.inl x)
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/-- The projection of an index value on a sum of color maps to its right component. -/
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def inrIndexValue {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d (sumElimIndexColor Tc Sc)) :
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IndexValue d Sc := fun y => i (Sum.inr y)
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def indexValueSumEquiv {X Y : Type} {TX : X → Colors} {TY : Y → Colors} :
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IndexValue d (Sum.elim TX TY) ≃ IndexValue d TX × IndexValue d TY where
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toFun i := (fun x => i (Sum.inl x), fun x => i (Sum.inr x))
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invFun p := fun c => match c with
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| Sum.inl x => (p.1 x)
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| Sum.inr x => (p.2 x)
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left_inv i := by
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simp only [IndexValue]
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ext1 x
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cases x with
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| inl val => rfl
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| inr val_1 => rfl
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right_inv p := rfl
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/-- An equivalence between index values formed by commuting sums. -/
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def sumCommIndexValue {X Y : Type} (Tc : X → Colors) (Sc : Y → Colors) :
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IndexValue d (sumElimIndexColor Tc Sc) ≃ IndexValue d (sumElimIndexColor Sc Tc) :=
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(congrSetIndexValue d (Equiv.sumComm X Y) (sumElimIndexColor Tc Sc)).trans
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(castIndexValue (sumElimIndexColor_symm Sc Tc).symm)
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lemma sumCommIndexValue_inlIndexValue {X Y : Type} {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d <| sumElimIndexColor Tc Sc) :
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inlIndexValue (sumCommIndexValue Tc Sc i) = inrIndexValue i := rfl
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lemma sumCommIndexValue_inrIndexValue {X Y : Type} {Tc : X → Colors} {Sc : Y → Colors}
|
||||
(i : IndexValue d <| sumElimIndexColor Tc Sc) :
|
||||
inrIndexValue (sumCommIndexValue Tc Sc i) = inlIndexValue i := rfl
|
||||
|
||||
/-- Equivalence between sets of `RealLorentzTensor` formed by commuting sums. -/
|
||||
@[simps!]
|
||||
def sumComm : RealLorentzTensor d (X ⊕ Y) ≃ RealLorentzTensor d (Y ⊕ X) :=
|
||||
congrSet (Equiv.sumComm X Y)
|
||||
def indexValueSumComm {X Y : Type} (Tc : X → Colors) (Sc : Y → Colors) :
|
||||
IndexValue d (Sum.elim Tc Sc) ≃ IndexValue d (Sum.elim Sc Tc) :=
|
||||
indexValueIso d (Equiv.sumComm X Y) (by aesop)
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -325,7 +305,6 @@ namespace Marked
|
|||
|
||||
variable {n m : ℕ}
|
||||
|
||||
|
||||
/-- The marked point. -/
|
||||
def markedPoint (X : Type) (i : Fin n) : (X ⊕ Fin n) :=
|
||||
Sum.inr i
|
||||
|
|
@ -342,43 +321,31 @@ def markedColor (T : Marked d X n) : Fin n → Colors :=
|
|||
def UnmarkedIndexValue (T : Marked d X n) : Type :=
|
||||
IndexValue d T.unmarkedColor
|
||||
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.UnmarkedIndexValue :=
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.UnmarkedIndexValue :=
|
||||
Pi.fintype
|
||||
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : DecidableEq T.UnmarkedIndexValue :=
|
||||
Fintype.decidablePiFintype
|
||||
|
||||
/-- The index values restricted to marked indices. -/
|
||||
def MarkedIndexValue (T : Marked d X n) : Type :=
|
||||
IndexValue d T.markedColor
|
||||
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.MarkedIndexValue :=
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : Fintype T.MarkedIndexValue :=
|
||||
Pi.fintype
|
||||
|
||||
lemma sumElimIndexColor_of_marked (T : Marked d X n) :
|
||||
sumElimIndexColor T.unmarkedColor T.markedColor = T.color := by
|
||||
instance [Fintype X] [DecidableEq X] (T : Marked d X n) : DecidableEq T.MarkedIndexValue :=
|
||||
Fintype.decidablePiFintype
|
||||
|
||||
lemma color_eq_elim (T : Marked d X n) :
|
||||
T.color = Sum.elim T.unmarkedColor T.markedColor := by
|
||||
ext1 x
|
||||
cases' x <;> rfl
|
||||
|
||||
def toUnmarkedIndexValue {T : Marked d X n} (i : IndexValue d T.color) : UnmarkedIndexValue T :=
|
||||
inlIndexValue <| castIndexValue T.sumElimIndexColor_of_marked.symm <| i
|
||||
|
||||
def toMarkedIndexValue {T : Marked d X n} (i : IndexValue d T.color) : MarkedIndexValue T :=
|
||||
inrIndexValue <| castIndexValue T.sumElimIndexColor_of_marked.symm <| i
|
||||
|
||||
def splitIndexValue {T : Marked d X n} :
|
||||
IndexValue d T.color ≃ UnmarkedIndexValue T × MarkedIndexValue T where
|
||||
toFun i := ⟨toUnmarkedIndexValue i, toMarkedIndexValue i⟩
|
||||
invFun p := castIndexValue T.sumElimIndexColor_of_marked $
|
||||
sumElimIndexValue p.1 p.2
|
||||
left_inv i := by
|
||||
simp_all only [IndexValue]
|
||||
ext1 x
|
||||
cases x with
|
||||
| inl _ => rfl
|
||||
| inr _ => rfl
|
||||
right_inv p := by
|
||||
simp_all only [IndexValue]
|
||||
obtain ⟨fst, snd⟩ := p
|
||||
simp_all only [Prod.mk.injEq]
|
||||
apply And.intro rfl rfl
|
||||
IndexValue d T.color ≃ T.UnmarkedIndexValue × T.MarkedIndexValue :=
|
||||
(indexValueIso d (Equiv.refl _) T.color_eq_elim).trans
|
||||
indexValueSumEquiv
|
||||
|
||||
@[simp]
|
||||
lemma splitIndexValue_sum {T : Marked d X n} [Fintype X] [DecidableEq X]
|
||||
|
|
@ -406,7 +373,6 @@ def twoMarkedIndexValue (T : Marked d X 2) (x : ColorsIndex d (T.color (markedPo
|
|||
| 0 => x
|
||||
| 1 => y
|
||||
|
||||
|
||||
/-- An equivalence of types used to turn the first marked index into an unmarked index. -/
|
||||
def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃
|
||||
(X ⊕ Fin 1) ⊕ Fin n :=
|
||||
|
|
@ -416,62 +382,24 @@ def unmarkFirstSet (X : Type) (n : ℕ) : (X ⊕ Fin n.succ) ≃
|
|||
|
||||
/-- Unmark the first marked index of a marked thensor. -/
|
||||
def unmarkFirst {X : Type} : Marked d X n.succ ≃ Marked d (X ⊕ Fin 1) n :=
|
||||
congrSet (unmarkFirstSet X n)
|
||||
mapIso d (unmarkFirstSet X n)
|
||||
|
||||
end Marked
|
||||
|
||||
/-!
|
||||
|
||||
## Multiplication
|
||||
|
||||
-/
|
||||
open Marked
|
||||
|
||||
/-- The contraction of the marked indices of two tensors each with one marked index, which
|
||||
is dual to the others. The contraction is done via
|
||||
`φ^μ ψ_μ = φ^0 ψ_0 + φ^1 ψ_1 + ...`. -/
|
||||
@[simps!]
|
||||
def mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
RealLorentzTensor d (X ⊕ Y) where
|
||||
color := sumElimIndexColor T.unmarkedColor S.unmarkedColor
|
||||
coord := fun i => ∑ x,
|
||||
T.coord (splitIndexValue.symm (inlIndexValue i, oneMarkedIndexValue x)) *
|
||||
S.coord (splitIndexValue.symm (inrIndexValue i, oneMarkedIndexValue $ congrColorsDual h x))
|
||||
|
||||
/-- Multiplication is well behaved with regard to swapping tensors. -/
|
||||
lemma sumComm_mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
sumComm (mul T S h) = mul S T (color_eq_dual_symm h) := by
|
||||
refine ext' (sumElimIndexColor_symm S.unmarkedColor T.unmarkedColor).symm ?_
|
||||
change (mul T S h).coord ∘
|
||||
(congrSetIndexValue d (Equiv.sumComm X Y) (mul T S h).color).symm = _
|
||||
rw [Equiv.comp_symm_eq]
|
||||
funext i
|
||||
simp only [mul_coord, IndexValue, mul_color, Function.comp_apply, sumComm_apply_color]
|
||||
erw [sumCommIndexValue_inlIndexValue, sumCommIndexValue_inrIndexValue,
|
||||
← Equiv.sum_comp (congrColorsDual h)]
|
||||
refine Fintype.sum_congr _ _ (fun a => ?_)
|
||||
rw [mul_comm]
|
||||
repeat apply congrArg
|
||||
rw [← congrColorsDual_symm h]
|
||||
exact (Equiv.apply_eq_iff_eq_symm_apply <| congrColorsDual h).mp rfl
|
||||
|
||||
/-! TODO: Following the ethos of modular operads, prove properties of multiplication. -/
|
||||
|
||||
/-! TODO: Use `mul` to generalize to any pair of marked index. -/
|
||||
/-!
|
||||
|
||||
## Contraction of indices
|
||||
|
||||
-/
|
||||
|
||||
open Marked
|
||||
|
||||
/-- The contraction of the marked indices in a tensor with two marked indices. -/
|
||||
def contr {X : Type} (T : Marked d X 2) (h : T.markedColor 0 = τ (T.markedColor 1)) :
|
||||
RealLorentzTensor d X where
|
||||
color := T.unmarkedColor
|
||||
coord := fun i =>
|
||||
∑ x, T.coord (splitIndexValue.symm (i, T.twoMarkedIndexValue x $ congrColorsDual h x))
|
||||
∑ x, T.coord (splitIndexValue.symm (i, T.twoMarkedIndexValue x $ colorsIndexDualCast h x))
|
||||
|
||||
/-! TODO: Following the ethos of modular operads, prove properties of contraction. -/
|
||||
|
||||
|
|
|
|||
|
|
@ -5,6 +5,7 @@ Authors: Joseph Tooby-Smith
|
|||
-/
|
||||
import HepLean.SpaceTime.LorentzTensor.Real.Basic
|
||||
import HepLean.SpaceTime.LorentzTensor.Real.LorentzAction
|
||||
import HepLean.SpaceTime.LorentzTensor.Real.Multiplication
|
||||
/-!
|
||||
|
||||
# Constructors for real Lorentz tensors
|
||||
|
|
@ -186,7 +187,7 @@ open Marked
|
|||
/-- Contracting the indices of `ofMatUpDown` returns the trace of the matrix. -/
|
||||
lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
contr (ofMatUpDown M) (by rfl) = ofReal d M.trace := by
|
||||
refine ext' ?_ ?_
|
||||
refine ext ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
|
|
@ -199,7 +200,7 @@ lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1
|
|||
/-- Contracting the indices of `ofMatDownUp` returns the trace of the matrix. -/
|
||||
lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
contr (ofMatDownUp M) (by rfl) = ofReal d M.trace := by
|
||||
refine ext' ?_ ?_
|
||||
refine ext ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
|
|
@ -214,9 +215,9 @@ lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1
|
|||
/-- Multiplying `ofVecUp` with `ofVecDown` gives the dot product. -/
|
||||
@[simp]
|
||||
lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
|
||||
mapIso d (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
|
||||
(mul (ofVecUp v₁) (ofVecDown v₂) (by rfl)) = ofReal d (v₁ ⬝ᵥ v₂) := by
|
||||
refine ext' ?_ ?_
|
||||
refine ext ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
|
|
@ -225,9 +226,9 @@ lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d
|
|||
/-- Multiplying `ofVecDown` with `ofVecUp` gives the dot product. -/
|
||||
@[simp]
|
||||
lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet (Equiv.equivEmpty (Empty ⊕ Empty))
|
||||
mapIso d (Equiv.equivEmpty (Empty ⊕ Empty))
|
||||
(mul (ofVecDown v₁) (ofVecUp v₂) rfl) = ofReal d (v₁ ⬝ᵥ v₂) := by
|
||||
refine ext' ?_ ?_
|
||||
refine ext ?_ ?_
|
||||
· funext i
|
||||
exact Empty.elim i
|
||||
· funext i
|
||||
|
|
@ -235,11 +236,11 @@ lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d
|
|||
|
||||
lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
|
||||
(v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet ((Equiv.sumEmpty (Empty ⊕ Fin 1) Empty))
|
||||
mapIso d ((Equiv.sumEmpty (Empty ⊕ Fin 1) Empty))
|
||||
(mul (unmarkFirst $ ofMatUpDown M) (ofVecUp v) rfl) = ofVecUp (M *ᵥ v) := by
|
||||
refine ext' ?_ ?_
|
||||
refine ext ?_ ?_
|
||||
· funext i
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, mapIso_apply_color, mul_color, Equiv.symm_symm]
|
||||
fin_cases i
|
||||
rfl
|
||||
· funext i
|
||||
|
|
@ -247,11 +248,11 @@ lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d)
|
|||
|
||||
lemma mul_ofMatDownUp_ofVecDown_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
|
||||
(v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
congrSet (Equiv.sumEmpty (Empty ⊕ Fin 1) Empty)
|
||||
mapIso d (Equiv.sumEmpty (Empty ⊕ Fin 1) Empty)
|
||||
(mul (unmarkFirst $ ofMatDownUp M) (ofVecDown v) rfl) = ofVecDown (M *ᵥ v) := by
|
||||
refine ext' ?_ ?_
|
||||
refine ext ?_ ?_
|
||||
· funext i
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, mapIso_apply_color, mul_color, Equiv.symm_symm]
|
||||
fin_cases i
|
||||
rfl
|
||||
· funext i
|
||||
|
|
@ -277,7 +278,7 @@ lemma lorentzAction_ofReal (r : ℝ) : Λ • ofReal d r = ofReal d r :=
|
|||
@[simp]
|
||||
lemma lorentzAction_ofVecUp (v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
Λ • ofVecUp v = ofVecUp (Λ *ᵥ v) := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
simp only [ofVecUp, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
|
|
@ -289,7 +290,7 @@ lemma lorentzAction_ofVecUp (v : Fin 1 ⊕ Fin d → ℝ) :
|
|||
intro i
|
||||
simp_all only [Finset.mem_univ, Fin.isValue, Equiv.coe_fn_mk]
|
||||
intro j _
|
||||
erw [Finset.prod_singleton]
|
||||
erw [toTensorRepMat_apply, Finset.prod_singleton]
|
||||
rfl
|
||||
|
||||
/-- The action of the Lorentz group on `ofVecDown v` is
|
||||
|
|
@ -297,7 +298,7 @@ lemma lorentzAction_ofVecUp (v : Fin 1 ⊕ Fin d → ℝ) :
|
|||
@[simp]
|
||||
lemma lorentzAction_ofVecDown (v : Fin 1 ⊕ Fin d → ℝ) :
|
||||
Λ • ofVecDown v = ofVecDown ((LorentzGroup.transpose Λ⁻¹) *ᵥ v) := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
simp only [ofVecDown, IndexValue, Fin.isValue, Fintype.prod_sum_type, Finset.univ_eq_empty,
|
||||
|
|
@ -309,13 +310,13 @@ lemma lorentzAction_ofVecDown (v : Fin 1 ⊕ Fin d → ℝ) :
|
|||
intro i
|
||||
simp_all only [Finset.mem_univ, Fin.isValue, Equiv.coe_fn_mk]
|
||||
intro j _
|
||||
erw [Finset.prod_singleton]
|
||||
erw [toTensorRepMat_apply, Finset.prod_singleton]
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma lorentzAction_ofMatUpUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatUpUp M = ofMatUpUp (Λ.1 * M * (LorentzGroup.transpose Λ).1) := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatUpUpIndexValue M).symm]
|
||||
|
|
@ -326,7 +327,9 @@ lemma lorentzAction_ofMatUpUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
|
|||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
rw [Fin.prod_univ_two]
|
||||
erw [toTensorRepMat_apply]
|
||||
simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
|
||||
Fin.isValue, one_mul, indexValueIso_refl, IndexValue]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
|
@ -334,7 +337,7 @@ lemma lorentzAction_ofMatUpUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
|
|||
@[simp]
|
||||
lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatDownDown M = ofMatDownDown ((LorentzGroup.transpose Λ⁻¹).1 * M * (Λ⁻¹).1) := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatDownDownIndexValue M).symm]
|
||||
|
|
@ -345,7 +348,9 @@ lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d
|
|||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
rw [Fin.prod_univ_two]
|
||||
erw [toTensorRepMat_apply]
|
||||
simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
|
||||
Fin.isValue, one_mul, lorentzGroupIsGroup_inv, indexValueIso_refl, IndexValue]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
|
@ -353,7 +358,7 @@ lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d
|
|||
@[simp]
|
||||
lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatUpDown M = ofMatUpDown (Λ.1 * M * (Λ⁻¹).1) := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatUpDownIndexValue M).symm]
|
||||
|
|
@ -364,7 +369,9 @@ lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
|
|||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
rw [Fin.prod_univ_two]
|
||||
erw [toTensorRepMat_apply]
|
||||
simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
|
||||
Fin.isValue, one_mul, indexValueIso_refl, IndexValue, lorentzGroupIsGroup_inv]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
|
@ -373,7 +380,7 @@ lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
|
|||
lemma lorentzAction_ofMatDownUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
|
||||
Λ • ofMatDownUp M =
|
||||
ofMatDownUp ((LorentzGroup.transpose Λ⁻¹).1 * M * (LorentzGroup.transpose Λ).1) := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (ofMatDownUpIndexValue M).symm]
|
||||
|
|
@ -384,7 +391,9 @@ lemma lorentzAction_ofMatDownUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
|
|||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
rw [Fin.prod_univ_two]
|
||||
erw [toTensorRepMat_apply]
|
||||
simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
|
||||
Fin.isValue, one_mul, lorentzGroupIsGroup_inv, indexValueIso_refl, IndexValue]
|
||||
simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
|
||||
rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
|
||||
congr
|
||||
|
|
|
|||
|
|
@ -18,13 +18,9 @@ The Lorentz action is currently only defined for finite and decidable types `X`.
|
|||
namespace RealLorentzTensor
|
||||
|
||||
variable {d : ℕ} {X Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
||||
variable (T : RealLorentzTensor d X) (c : X → Colors)
|
||||
variable (Λ Λ' : LorentzGroup d)
|
||||
open LorentzGroup
|
||||
open BigOperators
|
||||
(T : RealLorentzTensor d X) (c : X → Colors) (Λ Λ' : LorentzGroup d) {μ : Colors}
|
||||
|
||||
|
||||
variable {μ : Colors}
|
||||
open LorentzGroup BigOperators Marked
|
||||
|
||||
/-- Monoid homomorphism from the Lorentz group to matrices indexed by `ColorsIndex d μ` for a
|
||||
color `μ`.
|
||||
|
|
@ -59,11 +55,35 @@ def colorMatrix (μ : Colors) : LorentzGroup d →* Matrix (ColorsIndex d μ) (C
|
|||
|
||||
lemma colorMatrix_cast {μ ν : Colors} (h : μ = ν) (Λ : LorentzGroup d) :
|
||||
colorMatrix μ Λ =
|
||||
Matrix.reindex (castColorsIndex h).symm (castColorsIndex h).symm (colorMatrix ν Λ) := by
|
||||
Matrix.reindex (colorsIndexCast h).symm (colorsIndexCast h).symm (colorMatrix ν Λ) := by
|
||||
subst h
|
||||
rfl
|
||||
|
||||
/-- A real number occuring in the action of the Lorentz group on Lorentz tensors. -/
|
||||
lemma colorMatrix_dual_cast {μ : Colors} (Λ : LorentzGroup d) :
|
||||
colorMatrix (τ μ) Λ = Matrix.reindex (colorsIndexDualCastSelf) (colorsIndexDualCastSelf)
|
||||
(colorMatrix μ (LorentzGroup.transpose Λ⁻¹)) := by
|
||||
match μ with
|
||||
| .up => rfl
|
||||
| .down =>
|
||||
ext i j
|
||||
simp only [τ, colorMatrix, MonoidHom.coe_mk, OneHom.coe_mk, colorsIndexDualCastSelf, transpose,
|
||||
lorentzGroupIsGroup_inv, Matrix.transpose_apply, minkowskiMetric.dual_transpose,
|
||||
minkowskiMetric.dual_dual, Matrix.reindex_apply, Equiv.coe_fn_symm_mk, Matrix.submatrix_apply]
|
||||
lemma colorMatrix_transpose {μ : Colors} (Λ : LorentzGroup d) :
|
||||
colorMatrix μ (LorentzGroup.transpose Λ) = (colorMatrix μ Λ).transpose := by
|
||||
match μ with
|
||||
| .up => rfl
|
||||
| .down =>
|
||||
ext i j
|
||||
simp only [colorMatrix, transpose, lorentzGroupIsGroup_inv, Matrix.transpose_apply,
|
||||
MonoidHom.coe_mk, OneHom.coe_mk, minkowskiMetric.dual_transpose]
|
||||
|
||||
/-!
|
||||
|
||||
## Lorentz group to tensor representation matrices.
|
||||
|
||||
-/
|
||||
|
||||
@[simps!]
|
||||
def toTensorRepMat {c : X → Colors} :
|
||||
LorentzGroup d →* Matrix (IndexValue d c) (IndexValue d c) ℝ where
|
||||
|
|
@ -108,27 +128,83 @@ lemma toTensorRepMat_mul' (i j : IndexValue d c) :
|
|||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma toTensorRepMat_on_sum {cX : X → Colors} {cY : Y → Colors}
|
||||
(i j : IndexValue d (sumElimIndexColor cX cY)) :
|
||||
toTensorRepMat Λ i j = toTensorRepMat Λ (inlIndexValue i) (inlIndexValue j) *
|
||||
toTensorRepMat Λ (inrIndexValue i) (inrIndexValue j) := by
|
||||
lemma toTensorRepMat_of_indexValueSumEquiv {cX : X → Colors} {cY : Y → Colors}
|
||||
(i j : IndexValue d (Sum.elim cX cY)) :
|
||||
toTensorRepMat Λ i j = toTensorRepMat Λ (indexValueSumEquiv i).1 (indexValueSumEquiv j).1 *
|
||||
toTensorRepMat Λ (indexValueSumEquiv i).2 (indexValueSumEquiv j).2 := by
|
||||
simp only [toTensorRepMat_apply]
|
||||
rw [Fintype.prod_sum_type]
|
||||
rfl
|
||||
|
||||
open Marked
|
||||
|
||||
lemma toTensorRepMap_on_splitIndexValue (T : Marked d X n)
|
||||
(i : T.UnmarkedIndexValue) (k : T.MarkedIndexValue) (j : IndexValue d T.color) :
|
||||
toTensorRepMat Λ (splitIndexValue.symm (i, k)) j =
|
||||
toTensorRepMat Λ i (toUnmarkedIndexValue j) *
|
||||
toTensorRepMat Λ k (toMarkedIndexValue j) := by
|
||||
lemma toTensorRepMat_of_indexValueSumEquiv' {cX : X → Colors} {cY : Y → Colors}
|
||||
(i j : IndexValue d cX) (k l : IndexValue d cY) :
|
||||
toTensorRepMat Λ i j * toTensorRepMat Λ k l =
|
||||
toTensorRepMat Λ (indexValueSumEquiv.symm (i, k)) (indexValueSumEquiv.symm (j, l)) := by
|
||||
simp only [toTensorRepMat_apply]
|
||||
rw [Fintype.prod_sum_type]
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Tensor representation matrices and marked tensors.
|
||||
|
||||
-/
|
||||
|
||||
lemma toTensorRepMat_of_splitIndexValue' (T : Marked d X n)
|
||||
(i j : T.UnmarkedIndexValue) (k l : T.MarkedIndexValue) :
|
||||
toTensorRepMat Λ i j * toTensorRepMat Λ k l =
|
||||
toTensorRepMat Λ (splitIndexValue.symm (i, k)) (splitIndexValue.symm (j, l)) := by
|
||||
simp only [toTensorRepMat_apply]
|
||||
rw [Fintype.prod_sum_type]
|
||||
rfl
|
||||
|
||||
lemma toTensorRepMat_oneMarkedIndexValue_dual (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) (x : ColorsIndex d (T.markedColor 0))
|
||||
(k : S.MarkedIndexValue) :
|
||||
toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k =
|
||||
toTensorRepMat Λ⁻¹ (oneMarkedIndexValue
|
||||
$ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k)
|
||||
(oneMarkedIndexValue x) := by
|
||||
rw [toTensorRepMat_apply, toTensorRepMat_apply]
|
||||
erw [Finset.prod_singleton, Finset.prod_singleton]
|
||||
simp
|
||||
rw [colorMatrix_cast h, colorMatrix_dual_cast]
|
||||
rw [Matrix.reindex_apply, Matrix.reindex_apply]
|
||||
simp
|
||||
rw [colorMatrix_transpose]
|
||||
simp
|
||||
apply congrArg
|
||||
simp only [Fin.isValue, oneMarkedIndexValue, colorsIndexDualCast, Equiv.coe_fn_symm_mk,
|
||||
Equiv.symm_trans_apply, Equiv.symm_symm, Equiv.coe_fn_mk, Equiv.apply_symm_apply,
|
||||
Equiv.symm_apply_apply]
|
||||
|
||||
lemma toTensorRepMap_sum_dual (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) (j : T.MarkedIndexValue) (k : S.MarkedIndexValue) :
|
||||
∑ x, toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k
|
||||
* toTensorRepMat Λ (oneMarkedIndexValue x) j =
|
||||
toTensorRepMat 1
|
||||
(oneMarkedIndexValue $ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k) j := by
|
||||
trans ∑ x, toTensorRepMat Λ⁻¹ (oneMarkedIndexValue$ (colorsIndexDualCast h).symm $
|
||||
oneMarkedIndexValue.symm k) (oneMarkedIndexValue x) * toTensorRepMat Λ (oneMarkedIndexValue x) j
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [toTensorRepMat_oneMarkedIndexValue_dual]
|
||||
rw [← Equiv.sum_comp oneMarkedIndexValue.symm]
|
||||
change ∑ i, toTensorRepMat Λ⁻¹ (oneMarkedIndexValue $ (colorsIndexDualCast h).symm $
|
||||
oneMarkedIndexValue.symm k) i * toTensorRepMat Λ i j = _
|
||||
rw [← Matrix.mul_apply, ← toTensorRepMat.map_mul, inv_mul_self Λ]
|
||||
|
||||
lemma toTensorRepMat_one_coord_sum (T : Marked d X n) (i : T.UnmarkedIndexValue)
|
||||
(k : T.MarkedIndexValue) : T.coord (splitIndexValue.symm (i, k)) = ∑ j, toTensorRepMat 1 k j *
|
||||
T.coord (splitIndexValue.symm (i, j)) := by
|
||||
erw [Finset.sum_eq_single_of_mem k]
|
||||
simp only [IndexValue, map_one, Matrix.one_apply_eq, one_mul]
|
||||
exact Finset.mem_univ k
|
||||
intro j _ hjk
|
||||
simp [hjk]
|
||||
exact Or.inl (Matrix.one_apply_ne' hjk)
|
||||
|
||||
/-!
|
||||
|
||||
## Definition of the Lorentz group action on Real Lorentz Tensors.
|
||||
|
||||
-/
|
||||
|
|
@ -139,16 +215,16 @@ instance lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) wher
|
|||
smul Λ T := {color := T.color,
|
||||
coord := fun i => ∑ j, toTensorRepMat Λ i j * T.coord j}
|
||||
one_smul T := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
simp only [HSMul.hSMul, map_one]
|
||||
erw [Finset.sum_eq_single_of_mem i]
|
||||
simp only [Matrix.one_apply_eq, one_mul, IndexValue]
|
||||
rfl
|
||||
exact Finset.mem_univ i
|
||||
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
||||
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
||||
mul_smul Λ Λ' T := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
simp only [HSMul.hSMul]
|
||||
funext i
|
||||
have h1 : ∑ j : IndexValue d T.color, toTensorRepMat (Λ * Λ') i j
|
||||
|
|
@ -168,53 +244,21 @@ instance lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) wher
|
|||
rw [Finset.prod_mul_distrib]
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## The Lorentz action on marked tensors.
|
||||
|
||||
-/
|
||||
|
||||
@[simps!]
|
||||
instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
|
||||
|
||||
lemma lorentzAction_on_splitIndexValue' (T : Marked d X n)
|
||||
(i : T.UnmarkedIndexValue) (k : T.MarkedIndexValue) :
|
||||
(Λ • T).coord (splitIndexValue.symm (i, k)) =
|
||||
∑ (x : T.UnmarkedIndexValue), ∑ (y : T.MarkedIndexValue),
|
||||
(toTensorRepMat Λ i x * toTensorRepMat Λ k y) * T.coord (splitIndexValue.symm (x, y)) := by
|
||||
erw [lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp splitIndexValue.symm]
|
||||
rw [Fintype.sum_prod_type]
|
||||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
erw [toTensorRepMap_on_splitIndexValue]
|
||||
lemma lorentzAction_smul_coord' {d : ℕ} {X : Type} [Fintype X] [DecidableEq X] (Λ : ↑(𝓛 d))
|
||||
(T : RealLorentzTensor d X) (i : IndexValue d T.color) :
|
||||
(Λ • T).coord i = ∑ j : IndexValue d T.color, toTensorRepMat Λ i j * T.coord j := by
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma lorentzAction_on_splitIndexValue (T : Marked d X n)
|
||||
(i : T.UnmarkedIndexValue) (k : T.MarkedIndexValue) :
|
||||
(Λ • T).coord (splitIndexValue.symm (i, k)) =
|
||||
∑ (x : T.UnmarkedIndexValue), toTensorRepMat Λ i x *
|
||||
∑ (y : T.MarkedIndexValue), toTensorRepMat Λ k y *
|
||||
T.coord (splitIndexValue.symm (x, y)) := by
|
||||
rw [lorentzAction_on_splitIndexValue']
|
||||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.mul_sum]
|
||||
refine Finset.sum_congr rfl (fun y _ => ?_)
|
||||
rw [NonUnitalRing.mul_assoc]
|
||||
|
||||
|
||||
/-!
|
||||
|
||||
## Properties of the Lorentz action.
|
||||
|
||||
-/
|
||||
|
||||
|
||||
/-- The action on an empty Lorentz tensor is trivial. -/
|
||||
lemma lorentzAction_on_isEmpty [IsEmpty X] (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
|
||||
Λ • T = T := by
|
||||
refine ext' rfl ?_
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord]
|
||||
simp only [Finset.univ_unique, Finset.univ_eq_empty, Finset.prod_empty, one_mul,
|
||||
|
|
@ -224,58 +268,183 @@ lemma lorentzAction_on_isEmpty [IsEmpty X] (Λ : LorentzGroup d) (T : RealLorent
|
|||
rw [@mul_left_eq_self₀]
|
||||
exact Or.inl rfl
|
||||
|
||||
/-- The Lorentz action commutes with `congrSet`. -/
|
||||
lemma lorentzAction_comm_congrSet (f : X ≃ Y) (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
|
||||
congrSet f (Λ • T) = Λ • (congrSet f T) := by
|
||||
refine ext' rfl ?_
|
||||
/-- The Lorentz action commutes with `mapIso`. -/
|
||||
lemma lorentzAction_mapIso (f : X ≃ Y) (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
|
||||
mapIso d f (Λ • T) = Λ • (mapIso d f T) := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
erw [lorentzAction_smul_coord, lorentzAction_smul_coord]
|
||||
erw [← Equiv.sum_comp (congrSetIndexValue d f T.color)]
|
||||
rw [mapIso_apply_coord]
|
||||
rw [lorentzAction_smul_coord', lorentzAction_smul_coord']
|
||||
let is : IndexValue d T.color ≃ IndexValue d ((mapIso d f) T).color :=
|
||||
indexValueIso d f (by funext x ; simp)
|
||||
rw [← Equiv.sum_comp is]
|
||||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||||
simp [toTensorRepMat]
|
||||
erw [← Equiv.prod_comp f]
|
||||
apply Or.inl
|
||||
congr
|
||||
funext x
|
||||
have h1 : (T.color (f.symm (f x))) = T.color x := by
|
||||
simp only [Equiv.symm_apply_apply]
|
||||
rw [colorMatrix_cast h1]
|
||||
simp only [Matrix.reindex_apply, Equiv.symm_symm, Matrix.submatrix_apply]
|
||||
erw [castColorsIndex_comp_congrSetIndexValue]
|
||||
apply congrFun
|
||||
apply congrArg
|
||||
symm
|
||||
refine cast_eq_iff_heq.mpr ?_
|
||||
simp only [congrSetIndexValue, Equiv.piCongrLeft'_symm_apply, heq_eqRec_iff_heq, heq_eq_eq]
|
||||
rw [mapIso_apply_coord]
|
||||
refine Mathlib.Tactic.Ring.mul_congr ?_ ?_ rfl
|
||||
· simp only [IndexValue, toTensorRepMat, MonoidHom.coe_mk, OneHom.coe_mk, mapIso_apply_color,
|
||||
indexValueIso_refl]
|
||||
rw [← Equiv.prod_comp f]
|
||||
apply Finset.prod_congr rfl (fun x _ => ?_)
|
||||
have h1 : (T.color (f.symm (f x))) = T.color x := by
|
||||
simp only [Equiv.symm_apply_apply]
|
||||
rw [colorMatrix_cast h1]
|
||||
apply congrArg
|
||||
simp only [is]
|
||||
erw [indexValueIso_eq_symm, indexValueIso_symm_apply']
|
||||
simp only [colorsIndexCast, Function.comp_apply, mapIso_apply_color, Equiv.cast_refl,
|
||||
Equiv.refl_symm, Equiv.refl_apply, Equiv.cast_apply]
|
||||
symm
|
||||
refine cast_eq_iff_heq.mpr ?_
|
||||
congr
|
||||
exact Equiv.symm_apply_apply f x
|
||||
· apply congrArg
|
||||
funext a
|
||||
simp only [IndexValue, mapIso_apply_color, Equiv.symm_apply_apply, is]
|
||||
|
||||
/-!
|
||||
|
||||
## The Lorentz action on marked tensors.
|
||||
|
||||
-/
|
||||
|
||||
@[simps!]
|
||||
instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
|
||||
|
||||
/-- Action of the Lorentz group on just marked indices. -/
|
||||
@[simps!]
|
||||
def markedLorentzAction : MulAction (LorentzGroup d) (Marked d X n) where
|
||||
smul Λ T := {
|
||||
color := T.color,
|
||||
coord := fun i => ∑ j, toTensorRepMat Λ (splitIndexValue i).2 j *
|
||||
T.coord (splitIndexValue.symm ((splitIndexValue i).1, j))}
|
||||
one_smul T := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
simp only [HSMul.hSMul, map_one]
|
||||
erw [Finset.sum_eq_single_of_mem (splitIndexValue i).2]
|
||||
erw [Matrix.one_apply_eq (splitIndexValue i).2]
|
||||
simp only [IndexValue, one_mul, indexValueIso_refl, Equiv.refl_apply]
|
||||
apply congrArg
|
||||
exact Equiv.symm_apply_apply splitIndexValue i
|
||||
exact Finset.mem_univ (splitIndexValue i).2
|
||||
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
||||
mul_smul Λ Λ' T := by
|
||||
refine ext rfl ?_
|
||||
simp only [HSMul.hSMul]
|
||||
funext i
|
||||
have h1 : ∑ (j : T.MarkedIndexValue), toTensorRepMat (Λ * Λ') (splitIndexValue i).2 j
|
||||
* T.coord (splitIndexValue.symm ((splitIndexValue i).1, j)) =
|
||||
∑ (j : T.MarkedIndexValue), ∑ (k : T.MarkedIndexValue),
|
||||
(∏ x, ((colorMatrix (T.markedColor x) Λ ((splitIndexValue i).2 x) (k x)) *
|
||||
(colorMatrix (T.markedColor x) Λ' (k x) (j x)))) *
|
||||
T.coord (splitIndexValue.symm ((splitIndexValue i).1, j)) := by
|
||||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [toTensorRepMat_mul', Finset.sum_mul]
|
||||
rfl
|
||||
erw [h1]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.mul_sum]
|
||||
refine Finset.sum_congr rfl (fun k _ => ?_)
|
||||
simp only [toTensorRepMat, IndexValue]
|
||||
rw [← mul_assoc]
|
||||
congr
|
||||
rw [Finset.prod_mul_distrib]
|
||||
rfl
|
||||
|
||||
/-- Action of the Lorentz group on just unmarked indices. -/
|
||||
@[simps!]
|
||||
def unmarkedLorentzAction : MulAction (LorentzGroup d) (Marked d X n) where
|
||||
smul Λ T := {
|
||||
color := T.color,
|
||||
coord := fun i => ∑ j, toTensorRepMat Λ (splitIndexValue i).1 j *
|
||||
T.coord (splitIndexValue.symm (j, (splitIndexValue i).2))}
|
||||
one_smul T := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
simp only [HSMul.hSMul, map_one]
|
||||
erw [Finset.sum_eq_single_of_mem (splitIndexValue i).1]
|
||||
erw [Matrix.one_apply_eq (splitIndexValue i).1]
|
||||
simp only [IndexValue, one_mul, indexValueIso_refl, Equiv.refl_apply]
|
||||
apply congrArg
|
||||
exact Equiv.symm_apply_apply splitIndexValue i
|
||||
exact Finset.mem_univ (splitIndexValue i).1
|
||||
exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
|
||||
mul_smul Λ Λ' T := by
|
||||
refine ext rfl ?_
|
||||
simp only [HSMul.hSMul]
|
||||
funext i
|
||||
have h1 : ∑ (j : T.UnmarkedIndexValue), toTensorRepMat (Λ * Λ') (splitIndexValue i).1 j
|
||||
* T.coord (splitIndexValue.symm (j, (splitIndexValue i).2)) =
|
||||
∑ (j : T.UnmarkedIndexValue), ∑ (k : T.UnmarkedIndexValue),
|
||||
(∏ x, ((colorMatrix (T.unmarkedColor x) Λ ((splitIndexValue i).1 x) (k x)) *
|
||||
(colorMatrix (T.unmarkedColor x) Λ' (k x) (j x)))) *
|
||||
T.coord (splitIndexValue.symm (j, (splitIndexValue i).2)) := by
|
||||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [toTensorRepMat_mul', Finset.sum_mul]
|
||||
rfl
|
||||
erw [h1]
|
||||
rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.mul_sum]
|
||||
refine Finset.sum_congr rfl (fun k _ => ?_)
|
||||
simp only [toTensorRepMat, IndexValue]
|
||||
rw [← mul_assoc]
|
||||
congr
|
||||
rw [Finset.prod_mul_distrib]
|
||||
rfl
|
||||
|
||||
scoped[RealLorentzTensor] infixr:73 " •ₘ " => markedLorentzAction.smul
|
||||
scoped[RealLorentzTensor] infixr:73 " •ᵤₘ " => unmarkedLorentzAction.smul
|
||||
|
||||
/-- Acting on unmarked and then marked indices is equivalent to acting on all indices. -/
|
||||
lemma marked_unmarked_action_eq_action (T : Marked d X n) : Λ •ₘ (Λ •ᵤₘ T) = Λ • T := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
change ∑ j, toTensorRepMat Λ (splitIndexValue i).2 j *
|
||||
(∑ k, toTensorRepMat Λ (splitIndexValue i).1 k * T.coord (splitIndexValue.symm (k, j))) = _
|
||||
trans ∑ j, ∑ k, (toTensorRepMat Λ (splitIndexValue i).2 j *
|
||||
toTensorRepMat Λ (splitIndexValue i).1 k) * T.coord (splitIndexValue.symm (k, j))
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.mul_sum]
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
exact Eq.symm (mul_assoc _ _ _)
|
||||
trans ∑ j, ∑ k, (toTensorRepMat Λ i (splitIndexValue.symm (k, j))
|
||||
* T.coord (splitIndexValue.symm (k, j)))
|
||||
apply Finset.sum_congr rfl (fun j _ => (Finset.sum_congr rfl (fun k _ => ?_)))
|
||||
rw [mul_comm (toTensorRepMat _ _ _), toTensorRepMat_of_splitIndexValue']
|
||||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply]
|
||||
trans ∑ p, (toTensorRepMat Λ i p * T.coord p)
|
||||
rw [← Equiv.sum_comp splitIndexValue.symm, Fintype.sum_prod_type, Finset.sum_comm]
|
||||
rfl
|
||||
rfl
|
||||
|
||||
open Marked
|
||||
|
||||
lemma lorentzAction_comm_mul (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 1)) :
|
||||
mul (Λ • T) (Λ • S) h = Λ • mul T S h := by
|
||||
refine ext' rfl ?_
|
||||
/-- Acting on marked and then unmarked indices is equivalent to acting on all indices. -/
|
||||
lemma unmarked_marked_action_eq_action (T : Marked d X n) : Λ •ᵤₘ (Λ •ₘ T) = Λ • T := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
trans ∑ j, toTensorRepMat Λ (inlIndexValue i) (inlIndexValue j) *
|
||||
toTensorRepMat Λ (inrIndexValue i) (inrIndexValue j)
|
||||
* (mul T S h).coord j
|
||||
swap
|
||||
refine Finset.sum_congr rfl (fun j _ => ?_)
|
||||
erw [toTensorRepMat_on_sum]
|
||||
change ∑ j, toTensorRepMat Λ (splitIndexValue i).1 j *
|
||||
(∑ k, toTensorRepMat Λ (splitIndexValue i).2 k * T.coord (splitIndexValue.symm (j, k))) = _
|
||||
trans ∑ j, ∑ k, (toTensorRepMat Λ (splitIndexValue i).1 j *
|
||||
toTensorRepMat Λ (splitIndexValue i).2 k) * T.coord (splitIndexValue.symm (j, k))
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.mul_sum]
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
exact Eq.symm (mul_assoc _ _ _)
|
||||
trans ∑ j, ∑ k, (toTensorRepMat Λ i (splitIndexValue.symm (j, k))
|
||||
* T.coord (splitIndexValue.symm (j, k)))
|
||||
apply Finset.sum_congr rfl (fun j _ => (Finset.sum_congr rfl (fun k _ => ?_)))
|
||||
rw [toTensorRepMat_of_splitIndexValue']
|
||||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply]
|
||||
trans ∑ p, (toTensorRepMat Λ i p * T.coord p)
|
||||
rw [← Equiv.sum_comp splitIndexValue.symm, Fintype.sum_prod_type]
|
||||
rfl
|
||||
rfl
|
||||
change ∑ x, (∑ j, toTensorRepMat Λ (splitIndexValue.symm
|
||||
(inlIndexValue i, T.oneMarkedIndexValue x)) j * T.coord j) *
|
||||
(∑ k, toTensorRepMat Λ _ k * S.coord k) = _
|
||||
trans ∑ x, (∑ j,
|
||||
toTensorRepMat Λ (inlIndexValue i) (toUnmarkedIndexValue j)
|
||||
* toTensorRepMat Λ (T.oneMarkedIndexValue x) (toMarkedIndexValue j)
|
||||
* T.coord j) *
|
||||
|
||||
sorry
|
||||
/-- The marked and unmarked actions commute. -/
|
||||
lemma marked_unmarked_action_comm (T : Marked d X n) : Λ •ᵤₘ (Λ •ₘ T) = Λ •ₘ (Λ •ᵤₘ T) := by
|
||||
rw [unmarked_marked_action_eq_action, marked_unmarked_action_eq_action]
|
||||
|
||||
|
||||
|
||||
/-! TODO: Show that the Lorentz action commutes with multiplication. -/
|
||||
/-! TODO: Show that the Lorentz action commutes with contraction. -/
|
||||
/-! TODO: Show that the Lorentz action commutes with rising and lowering indices. -/
|
||||
end RealLorentzTensor
|
||||
|
|
|
|||
194
HepLean/SpaceTime/LorentzTensor/Real/Multiplication.lean
Normal file
194
HepLean/SpaceTime/LorentzTensor/Real/Multiplication.lean
Normal file
|
|
@ -0,0 +1,194 @@
|
|||
/-
|
||||
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.LorentzTensor.Real.Basic
|
||||
import HepLean.SpaceTime.LorentzTensor.Real.LorentzAction
|
||||
/-!
|
||||
|
||||
# Multiplication of Real Lorentz Tensors along an index
|
||||
|
||||
We define the multiplication of two singularly marked Lorentz tensors along the
|
||||
marked index. This is equivalent to contracting two Lorentz tensors.
|
||||
|
||||
We prove various results about this multiplication.
|
||||
|
||||
-/
|
||||
/-! TODO: Add unit to the multiplication. -/
|
||||
/-! TODO: Generalize to contracting any marked index of a marked tensor. -/
|
||||
/-! TODO: Set up a good notation for the multiplication. -/
|
||||
|
||||
namespace RealLorentzTensor
|
||||
|
||||
variable {d : ℕ} {X Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
||||
(T : RealLorentzTensor d X) (c : X → Colors) (Λ Λ' : LorentzGroup d) {μ : Colors}
|
||||
|
||||
open Marked
|
||||
|
||||
/-- The contraction of the marked indices of two tensors each with one marked index, which
|
||||
is dual to the others. The contraction is done via
|
||||
`φ^μ ψ_μ = φ^0 ψ_0 + φ^1 ψ_1 + ...`. -/
|
||||
@[simps!]
|
||||
def mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
RealLorentzTensor d (X ⊕ Y) where
|
||||
color := Sum.elim T.unmarkedColor S.unmarkedColor
|
||||
coord := fun i => ∑ x,
|
||||
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1, oneMarkedIndexValue x)) *
|
||||
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2,
|
||||
oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
|
||||
/-- Multiplication is well behaved with regard to swapping tensors. -/
|
||||
lemma mul_symm {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
mapIso d (Equiv.sumComm X Y) (mul T S h) = mul S T (color_eq_dual_symm h) := by
|
||||
refine ext ?_ ?_
|
||||
· funext a
|
||||
cases a with
|
||||
| inl _ => rfl
|
||||
| inr _ => rfl
|
||||
· funext i
|
||||
rw [mapIso_apply_coord, mul_coord, mul_coord]
|
||||
erw [← Equiv.sum_comp (colorsIndexDualCast h).symm]
|
||||
refine Fintype.sum_congr _ _ (fun x => ?_)
|
||||
rw [mul_comm]
|
||||
congr
|
||||
· exact Equiv.apply_symm_apply (colorsIndexDualCast h) x
|
||||
· exact colorsIndexDualCast_symm h
|
||||
|
||||
lemma mul_mapIso {X Y Z : Type} (T : Marked d X 1) (S : Marked d Y 1) (f : X ≃ W)
|
||||
(g : Y ≃ Z) (h : T.markedColor 0 = τ (S.markedColor 0)) :
|
||||
mapIso d (Equiv.sumCongr f g) (mul T S h) = mul (mapIso d (Equiv.sumCongr f (Equiv.refl _)) T)
|
||||
(mapIso d (Equiv.sumCongr g (Equiv.refl _)) S) h := by
|
||||
refine ext ?_ ?_
|
||||
· funext a
|
||||
cases a with
|
||||
| inl _ => rfl
|
||||
| inr _ => rfl
|
||||
· funext i
|
||||
rw [mapIso_apply_coord, mul_coord, mul_coord]
|
||||
refine Fintype.sum_congr _ _ (fun x => ?_)
|
||||
rw [mapIso_apply_coord, mapIso_apply_coord]
|
||||
refine Mathlib.Tactic.Ring.mul_congr ?_ ?_ rfl
|
||||
· apply congrArg
|
||||
ext1 r
|
||||
cases r with
|
||||
| inl val => rfl
|
||||
| inr val_1 => rfl
|
||||
· apply congrArg
|
||||
ext1 r
|
||||
cases r with
|
||||
| inl val => rfl
|
||||
| inr val_1 => rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Lorentz action and multiplication.
|
||||
|
||||
-/
|
||||
|
||||
/-- The marked Lorentz Action leaves multiplication invariant. -/
|
||||
lemma mul_markedLorentzAction (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 1)) :
|
||||
mul (Λ •ₘ T) (Λ •ₘ S) h = mul T S h := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
change ∑ x, (∑ j, toTensorRepMat Λ (oneMarkedIndexValue x) j *
|
||||
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1, j))) *
|
||||
(∑ k, toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k *
|
||||
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, k))) = _
|
||||
trans ∑ x, ∑ j, ∑ k, (toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k
|
||||
* toTensorRepMat Λ (oneMarkedIndexValue x) j) *
|
||||
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1, j))
|
||||
* S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, k))
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul_sum ]
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
ring
|
||||
rw [Finset.sum_comm]
|
||||
trans ∑ j, ∑ k, ∑ x, (toTensorRepMat Λ (oneMarkedIndexValue $ colorsIndexDualCast h x) k
|
||||
* toTensorRepMat Λ (oneMarkedIndexValue x) j) *
|
||||
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1, j))
|
||||
* S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, k))
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.sum_comm]
|
||||
trans ∑ j, ∑ k, (toTensorRepMat 1
|
||||
(oneMarkedIndexValue $ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k) j) *
|
||||
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1, j))
|
||||
* S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, k))
|
||||
apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
|
||||
rw [← Finset.sum_mul, ← Finset.sum_mul]
|
||||
erw [toTensorRepMap_sum_dual]
|
||||
rfl
|
||||
rw [Finset.sum_comm]
|
||||
trans ∑ k,
|
||||
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1,
|
||||
(oneMarkedIndexValue $ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm k)))*
|
||||
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, k))
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
rw [← Finset.sum_mul, ← toTensorRepMat_one_coord_sum T]
|
||||
rw [← Equiv.sum_comp (oneMarkedIndexValue)]
|
||||
erw [← Equiv.sum_comp (colorsIndexDualCast h)]
|
||||
simp
|
||||
rfl
|
||||
|
||||
/-- The unmarked Lorentz Action commutes with multiplication. -/
|
||||
lemma mul_unmarkedLorentzAction (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 1)) :
|
||||
mul (Λ •ᵤₘ T) (Λ •ᵤₘ S) h = Λ • mul T S h := by
|
||||
refine ext rfl ?_
|
||||
funext i
|
||||
change ∑ x, (∑ j, toTensorRepMat Λ (indexValueSumEquiv i).1 j *
|
||||
T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
|
||||
∑ k, toTensorRepMat Λ (indexValueSumEquiv i).2 k *
|
||||
S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x)) = _
|
||||
trans ∑ x, ∑ j, ∑ k, (toTensorRepMat Λ (indexValueSumEquiv i).1 j *
|
||||
T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
|
||||
toTensorRepMat Λ (indexValueSumEquiv i).2 k *
|
||||
S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
rw [Finset.sum_mul_sum ]
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
apply Finset.sum_congr rfl (fun k _ => ?_)
|
||||
ring
|
||||
rw [Finset.sum_comm]
|
||||
trans ∑ j, ∑ k, ∑ x, (toTensorRepMat Λ (indexValueSumEquiv i).1 j *
|
||||
T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
|
||||
toTensorRepMat Λ (indexValueSumEquiv i).2 k *
|
||||
S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun j _ => ?_)
|
||||
rw [Finset.sum_comm]
|
||||
trans ∑ j, ∑ k,
|
||||
((toTensorRepMat Λ (indexValueSumEquiv i).1 j) * toTensorRepMat Λ (indexValueSumEquiv i).2 k)
|
||||
* ∑ x, (T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))
|
||||
* S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
|
||||
rw [Finset.mul_sum]
|
||||
apply Finset.sum_congr rfl (fun x _ => ?_)
|
||||
ring
|
||||
trans ∑ j, ∑ k, toTensorRepMat Λ i (indexValueSumEquiv.symm (j, k)) *
|
||||
∑ x, (T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))
|
||||
* S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
|
||||
rw [toTensorRepMat_of_indexValueSumEquiv']
|
||||
congr
|
||||
simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply, mul_color]
|
||||
trans ∑ p, toTensorRepMat Λ i p *
|
||||
∑ x, (T.coord (splitIndexValue.symm ((indexValueSumEquiv p).1, oneMarkedIndexValue x)))
|
||||
* S.coord (splitIndexValue.symm ((indexValueSumEquiv p).2,
|
||||
oneMarkedIndexValue $ colorsIndexDualCast h x))
|
||||
erw [← Equiv.sum_comp indexValueSumEquiv.symm]
|
||||
rw [Fintype.sum_prod_type]
|
||||
rfl
|
||||
rfl
|
||||
|
||||
/-- The Lorentz action commutes with multiplication. -/
|
||||
lemma mul_lorentzAction (T : Marked d X 1) (S : Marked d Y 1)
|
||||
(h : T.markedColor 0 = τ (S.markedColor 1)) :
|
||||
mul (Λ • T) (Λ • S) h = Λ • mul T S h := by
|
||||
simp only [← marked_unmarked_action_eq_action]
|
||||
rw [mul_markedLorentzAction, mul_unmarkedLorentzAction]
|
||||
|
||||
end RealLorentzTensor
|
||||
Loading…
Add table
Add a link
Reference in a new issue