feat: defined mult and contract of Lorentz tensors
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1 changed files with 273 additions and 35 deletions
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@ -6,6 +6,7 @@ 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.CategoryTheory.FintypeCat
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import Mathlib.Analysis.Normed.Field.Basic
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/-!
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# Lorentz Tensors
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@ -40,40 +41,33 @@ def RealLorentzTensor.ColorsIndex (d : ℕ) (μ : RealLorentzTensor.Colors) : Ty
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| RealLorentzTensor.Colors.up => Fin 1 ⊕ Fin d
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| RealLorentzTensor.Colors.down => Fin 1 ⊕ Fin d
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instance (d : ℕ) (μ : RealLorentzTensor.Colors) : Fintype (RealLorentzTensor.ColorsIndex d μ) :=
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match μ with
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| RealLorentzTensor.Colors.up => instFintypeSum (Fin 1) (Fin d)
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| RealLorentzTensor.Colors.down => instFintypeSum (Fin 1) (Fin d)
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/-- An `IndexValue` is a set of actual values an index can take. e.g. for a
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3-tensor (0, 1, 2). -/
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@[simp]
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def RealLorentzTensor.IndexValue {X : FintypeCat} (d : ℕ ) (c : X → RealLorentzTensor.Colors) :
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Type 0 := (x : X) → RealLorentzTensor.ColorsIndex d (c x)
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/-- A Lorentz Tensor defined by its coordinate map. -/
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structure RealLorentzTensor (d : ℕ) (X : FintypeCat) where
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/-- The color associated to each index of the tensor. -/
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color : X → RealLorentzTensor.Colors
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/-- The coordinate map for the tensor. -/
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coord : ((x : X) → RealLorentzTensor.ColorsIndex d (color x)) → ℝ
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coord : RealLorentzTensor.IndexValue d color → ℝ
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namespace RealLorentzTensor
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open CategoryTheory
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universe u1
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variable {d : ℕ} {X Y Z : FintypeCat.{u1}}
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variable {d : ℕ} {X Y Z : FintypeCat.{0}}
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/-!
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/-- An `IndexType` for a tensor is an element of
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`(x : X) → RealLorentzTensor.ColorsIndex d (T.color x)`. -/
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@[simp]
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def IndexType (T : RealLorentzTensor d X) : Type u1 :=
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(x : X) → RealLorentzTensor.ColorsIndex d (T.color x)
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## Colors
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lemma indexType_eq {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color) :
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T₁.IndexType = T₂.IndexType := by
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simp only [IndexType]
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rw [h]
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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 (indexType_eq h)) : T₁ = T₂ := by
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cases T₁
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cases T₂
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simp_all only [IndexType, 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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-/
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/-- The involution acting on colors. -/
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def τ : Colors → Colors
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@ -81,6 +75,7 @@ def τ : Colors → Colors
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| Colors.down => Colors.up
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/-- The map τ is an involution. -/
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@[simp]
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lemma τ_involutive : Function.Involutive τ := by
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intro x
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cases x <;> rfl
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@ -88,6 +83,90 @@ lemma τ_involutive : Function.Involutive τ := by
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/-- The color associated with an element of `x ∈ X` for a tensor `T`. -/
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def ch {X : FintypeCat} (x : X) (T : RealLorentzTensor d X) : Colors := T.color x
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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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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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| RealLorentzTensor.Colors.up => x
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| RealLorentzTensor.Colors.down => x
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invFun x :=
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match μ with
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| RealLorentzTensor.Colors.up => x
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| RealLorentzTensor.Colors.down => x
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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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ColorsIndex d μ ≃ ColorsIndex d ν :=
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(castColorsIndex h).trans dualColorsIndex.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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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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-/
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/-- An equivalence of Index values from an equality of color maps. -/
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def castIndexValue {X : FintypeCat} {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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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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/-!
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## Extensionality
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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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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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rfl
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/-!
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## Congruence
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@ -96,24 +175,23 @@ def ch {X : FintypeCat} (x : X) (T : RealLorentzTensor d X) : Colors := T.color
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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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def congrSetIndexType (d : ℕ) (f : X ≃ Y) (i : X → Colors) :
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((x : X) → ColorsIndex d (i x)) ≃ ((y : Y) → ColorsIndex d ((Equiv.piCongrLeft' _ f) i y)) :=
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Equiv.piCongrLeft' _ (f)
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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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/-- 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 := (Equiv.piCongrLeft' _ f) T.color
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coord := (Equiv.piCongrLeft' _ (congrSetIndexType d f T.color)) T.coord
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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 : (congrSetIndexType d (f.trans g) T.color) = (congrSetIndexType d f T.color).trans
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(congrSetIndexType d g ((Equiv.piCongrLeft' (fun _ => Colors) f) T.color)) := by
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simp only [Equiv.piCongrLeft'_apply, Equiv.symm_trans_apply, congrSetIndexType]
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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, IndexType, Equiv.symm_trans_apply, h1,
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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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@ -140,19 +218,179 @@ lemma congrSet_refl : @congrSet d _ _ (Equiv.refl X) = Equiv.refl _ := by
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/-!
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## Multiplication
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## Sums
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-/
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/-! TODO: Following the ethos of modular operads, define multiplication of Lorentz tensors. -/
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/-- An equivalence through commuting sums between types casted from `FintypeCat.of`.-/
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def sumCommFintypeCat (X Y : FintypeCat) : FintypeCat.of (X ⊕ Y) ≃ FintypeCat.of (Y ⊕ X) :=
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Equiv.sumComm X Y
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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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FintypeCat.of (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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def sumElimIndexValue {X Y : FintypeCat} {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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/-- An equivalence between index values formed by commuting sums.-/
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def sumCommIndexValue {X Y : FintypeCat} (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 (sumCommFintypeCat 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 : FintypeCat} {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 : FintypeCat} {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d (sumElimIndexColor Tc Sc)) :
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inrIndexValue (sumCommIndexValue Tc Sc i) = inlIndexValue i := rfl
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/-- Equivalence between sets of `RealLorentzTensor` formed by commuting sums. -/
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@[simps!]
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def sumComm :
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RealLorentzTensor d (FintypeCat.of (X ⊕ Y)) ≃ RealLorentzTensor d (FintypeCat.of (Y ⊕ X)) :=
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congrSet (Equiv.sumComm X Y)
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/-!
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## Marked Lorentz tensors
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To define contraction and multiplication of Lorentz tensors we need to mark indices.
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-/
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/-- A `RealLorentzTensor` with `n` marked indices. -/
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def Marked (d : ℕ) (X : FintypeCat) (n : ℕ) : Type :=
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RealLorentzTensor d (FintypeCat.of (X ⊕ Σ _ : Fin n, PUnit))
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namespace Marked
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variable {n m : ℕ}
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/-- The marked point. -/
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def markedPoint (X : FintypeCat) (i : Fin n) : FintypeCat.of (X ⊕ Σ _ : Fin n, PUnit) :=
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Sum.inr ⟨i, PUnit.unit⟩
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/-- The colors of unmarked indices. -/
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def unmarkedColor (T : Marked d X n) : X → Colors :=
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T.color ∘ Sum.inl
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/-- The colors of marked indices. -/
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def markedColor (T : Marked d X n) : FintypeCat.of (Σ _ : Fin n, PUnit) → Colors :=
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T.color ∘ Sum.inr
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/-- The index values restricted to unmarked indices. -/
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def UnmarkedIndexValue (T : Marked d X n) : Type :=
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IndexValue d T.unmarkedColor
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/-- The index values restricted to marked indices. -/
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def MarkedIndexValue (T : Marked d X n) : Type :=
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IndexValue d T.markedColor
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lemma sumElimIndexColor_of_marked (T : Marked d X n) :
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sumElimIndexColor T.unmarkedColor T.markedColor = T.color := by
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ext1 x
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cases' x <;> rfl
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/-- Contruction of marked index values for the case of 1 marked index. -/
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def oneMarkedIndexValue (T : Marked d X 1) (x : ColorsIndex d (T.color (markedPoint X 0))) :
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T.MarkedIndexValue := fun i => match i with
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| ⟨0, PUnit.unit⟩ => x
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/-- Contruction of marked index values for the case of 2 marked index. -/
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def twoMarkedIndexValue (T : Marked d X 2) (x : ColorsIndex d (T.color (markedPoint X 0)))
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(y : ColorsIndex d (T.color (markedPoint X 1))) :
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T.MarkedIndexValue := fun i =>
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match i with
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| ⟨0, PUnit.unit⟩ => x
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| ⟨1, PUnit.unit⟩ => y
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end Marked
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/-!
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## Multiplication
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-/
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open Marked
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/-- The contraction of the marked indices of two tensors each with one marked index, which
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is dual to the others. The contraction is done via
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`φ^μ ψ_μ = φ^0 ψ_0 + φ^1 ψ_1 + ...`. -/
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@[simps!]
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def mul {X Y : FintypeCat} (T : Marked d X 1) (S : Marked d Y 1)
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(h : T.markedColor ⟨0, PUnit.unit⟩ = τ (S.markedColor ⟨0, PUnit.unit⟩)) :
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RealLorentzTensor d (FintypeCat.of (X ⊕ Y)) where
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color := sumElimIndexColor T.unmarkedColor S.unmarkedColor
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coord := fun i => ∑ x,
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T.coord (Equiv.cast (indexValue_eq d T.sumElimIndexColor_of_marked)
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(sumElimIndexValue (inlIndexValue i) (T.oneMarkedIndexValue x))) *
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S.coord (Equiv.cast (indexValue_eq d S.sumElimIndexColor_of_marked) $
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sumElimIndexValue (inrIndexValue i) (S.oneMarkedIndexValue $ congrColorsDual h x))
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/-- Multiplication is well behaved with regard to swapping tensors. -/
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lemma sumComm_mul {X Y : FintypeCat} (T : Marked d X 1) (S : Marked d Y 1)
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(h : T.markedColor ⟨0, PUnit.unit⟩ = τ (S.markedColor ⟨0, PUnit.unit⟩)) :
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sumComm (mul T S h) = mul S T (color_eq_dual_symm h) := by
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refine ext' (sumElimIndexColor_symm S.unmarkedColor T.unmarkedColor).symm ?_
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change (mul T S h).coord ∘
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(congrSetIndexValue d (sumCommFintypeCat X Y) (mul T S h).color).symm = _
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rw [Equiv.comp_symm_eq]
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funext i
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simp only [mul_coord, IndexValue, mul_color, Function.comp_apply, sumComm_apply_color]
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erw [sumCommIndexValue_inlIndexValue, sumCommIndexValue_inrIndexValue,
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← Equiv.sum_comp (congrColorsDual h)]
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refine Fintype.sum_congr _ _ (fun a => ?_)
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rw [mul_comm]
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repeat apply congrArg
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rw [← congrColorsDual_symm h]
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exact (Equiv.apply_eq_iff_eq_symm_apply (congrColorsDual h)).mp rfl
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/-! TODO: Following the ethos of modular operads, prove properties of multiplication. -/
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/-! TODO: Use `mul` to generalize to any pair of marked index. -/
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/-!
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## Contraction of indices
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-/
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/-! TODO: Following the ethos of modular operads, define contraction of Lorentz tensors. -/
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/-- The contraction of the marked indices in a tensor with two marked indices. -/
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def contr {X : FintypeCat} (T : Marked d X 2)
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(h : T.markedColor ⟨0, PUnit.unit⟩ = τ (T.markedColor ⟨1, PUnit.unit⟩)) :
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RealLorentzTensor d X where
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color := T.unmarkedColor
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coord := fun i =>
|
||||
∑ x, T.coord (Equiv.cast (indexValue_eq d T.sumElimIndexColor_of_marked)
|
||||
(sumElimIndexValue i (T.twoMarkedIndexValue x ((congrColorsDual h) x))))
|
||||
|
||||
/-! TODO: Following the ethos of modular operads, prove properties of contraction. -/
|
||||
|
||||
/-! TODO: Use `contr` to generalize to any pair of marked index. -/
|
||||
|
||||
/-!
|
||||
|
||||
|
|
|
|||
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Add table
Add a link
Reference in a new issue