refactor: Lorentz action on tensors
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jstoobysmith
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3 changed files with 298 additions and 119 deletions
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@ -17,12 +17,13 @@ The Lorentz action is currently only defined for finite and decidable types `X`.
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namespace RealLorentzTensor
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variable {d : ℕ} {X : Type} [Fintype X] [DecidableEq X] (T : RealLorentzTensor d X) (c : X → Colors)
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variable {d : ℕ} {X Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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variable (T : RealLorentzTensor d X) (c : X → Colors)
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variable (Λ Λ' : LorentzGroup d)
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open LorentzGroup
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open BigOperators
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instance : Fintype (IndexValue d c) := Pi.fintype
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variable {μ : Colors}
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/-- Monoid homomorphism from the Lorentz group to matrices indexed by `ColorsIndex d μ` for a
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@ -56,88 +57,159 @@ def colorMatrix (μ : Colors) : LorentzGroup d →* Matrix (ColorsIndex d μ) (C
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Matrix.transpose_mul, Matrix.transpose_apply]
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rfl
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/-- A real number occuring in the action of the Lorentz group on Lorentz tensors. -/
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@[simp]
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def prodColorMatrixOnIndexValue (i j : IndexValue d c) : ℝ :=
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∏ x, colorMatrix (c x) Λ (i x) (j x)
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/-- `prodColorMatrixOnIndexValue` evaluated at `1` on the diagonal returns `1`. -/
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lemma one_prodColorMatrixOnIndexValue_on_diag (i : IndexValue d c) :
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prodColorMatrixOnIndexValue c 1 i i = 1 := by
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simp only [prodColorMatrixOnIndexValue]
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rw [Finset.prod_eq_one]
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intro x _
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simp only [colorMatrix, MonoidHom.map_one, Matrix.one_apply]
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lemma colorMatrix_cast {μ ν : Colors} (h : μ = ν) (Λ : LorentzGroup d) :
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colorMatrix μ Λ =
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Matrix.reindex (castColorsIndex h).symm (castColorsIndex h).symm (colorMatrix ν Λ) := by
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subst h
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rfl
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/-- `prodColorMatrixOnIndexValue` evaluated at `1` off the diagonal returns `0`. -/
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lemma one_prodColorMatrixOnIndexValue_off_diag {i j : IndexValue d c} (hij : j ≠ i) :
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prodColorMatrixOnIndexValue c 1 i j = 0 := by
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simp only [prodColorMatrixOnIndexValue]
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obtain ⟨x, hijx⟩ := Function.ne_iff.mp hij
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rw [@Finset.prod_eq_zero _ _ _ _ _ x]
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exact Finset.mem_univ x
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simp only [map_one]
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exact Matrix.one_apply_ne' hijx
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/-- A real number occuring in the action of the Lorentz group on Lorentz tensors. -/
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@[simps!]
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def toTensorRepMat {c : X → Colors} :
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LorentzGroup d →* Matrix (IndexValue d c) (IndexValue d c) ℝ where
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toFun Λ := fun i j => ∏ x, colorMatrix (c x) Λ (i x) (j x)
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map_one' := by
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ext i j
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by_cases hij : i = j
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· subst hij
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simp only [map_one, Matrix.one_apply_eq, Finset.prod_const_one]
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· obtain ⟨x, hijx⟩ := Function.ne_iff.mp hij
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simp only [map_one]
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rw [@Finset.prod_eq_zero _ _ _ _ _ x]
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exact Eq.symm (Matrix.one_apply_ne' fun a => hij (id (Eq.symm a)))
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exact Finset.mem_univ x
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exact Matrix.one_apply_ne' (id (Ne.symm hijx))
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map_mul' Λ Λ' := by
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ext i j
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rw [Matrix.mul_apply]
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trans ∑ (k : IndexValue d c), ∏ x,
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(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x))
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have h1 : ∑ (k : IndexValue d c), ∏ x,
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(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) =
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∏ x, ∑ y, (colorMatrix (c x) Λ (i x) y) * (colorMatrix (c x) Λ' y (j x)) := by
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rw [Finset.prod_sum]
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simp only [Finset.prod_attach_univ, Finset.sum_univ_pi]
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apply Finset.sum_congr
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simp only [IndexValue, Fintype.piFinset_univ]
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intro x _
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rfl
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rw [h1]
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simp only [map_mul]
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exact Finset.prod_congr rfl (fun x _ => rfl)
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refine Finset.sum_congr rfl (fun k _ => ?_)
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rw [Finset.prod_mul_distrib]
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lemma mul_prodColorMatrixOnIndexValue (i j : IndexValue d c) :
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prodColorMatrixOnIndexValue c (Λ * Λ') i j =
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∑ (k : IndexValue d c),
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∏ x, (colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) := by
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have h1 : ∑ (k : IndexValue d c), ∏ x,
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(colorMatrix (c x) Λ (i x) (k x)) * (colorMatrix (c x) Λ' (k x) (j x)) =
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∏ x, ∑ y, (colorMatrix (c x) Λ (i x) y) * (colorMatrix (c x) Λ' y (j x)) := by
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rw [Finset.prod_sum]
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simp only [Finset.prod_attach_univ, Finset.sum_univ_pi]
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apply Finset.sum_congr
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simp only [IndexValue, Fintype.piFinset_univ]
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intro x _
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rfl
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rw [h1]
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simp only [prodColorMatrixOnIndexValue, map_mul]
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exact Finset.prod_congr rfl (fun x _ => rfl)
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lemma toTensorRepMat_mul' (i j : IndexValue d c) :
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toTensorRepMat (Λ * Λ') i j = ∑ (k : IndexValue d c),
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∏ x, colorMatrix (c x) Λ (i x) (k x) * colorMatrix (c x) Λ' (k x) (j x) := by
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simp [Matrix.mul_apply]
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refine Finset.sum_congr rfl (fun k _ => ?_)
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rw [Finset.prod_mul_distrib]
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rfl
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@[simp]
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lemma toTensorRepMat_on_sum {cX : X → Colors} {cY : Y → Colors}
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(i j : IndexValue d (sumElimIndexColor cX cY)) :
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toTensorRepMat Λ i j = toTensorRepMat Λ (inlIndexValue i) (inlIndexValue j) *
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toTensorRepMat Λ (inrIndexValue i) (inrIndexValue j) := by
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simp only [toTensorRepMat_apply]
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rw [Fintype.prod_sum_type]
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rfl
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open Marked
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lemma toTensorRepMap_on_splitIndexValue (T : Marked d X n)
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(i : T.UnmarkedIndexValue) (k : T.MarkedIndexValue) (j : IndexValue d T.color) :
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toTensorRepMat Λ (splitIndexValue.symm (i, k)) j =
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toTensorRepMat Λ i (toUnmarkedIndexValue j) *
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toTensorRepMat Λ k (toMarkedIndexValue j) := by
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simp only [toTensorRepMat_apply]
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rw [Fintype.prod_sum_type]
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rfl
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/-!
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## Definition of the Lorentz group action on Real Lorentz Tensors.
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-/
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/-- Action of the Lorentz group on `X`-indexed Real Lorentz Tensors. -/
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@[simps!]
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instance lorentzAction : MulAction (LorentzGroup d) (RealLorentzTensor d X) where
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smul Λ T := {color := T.color,
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coord := fun i => ∑ j, prodColorMatrixOnIndexValue T.color Λ i j * T.coord j}
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coord := fun i => ∑ j, toTensorRepMat Λ i j * T.coord j}
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one_smul T := by
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refine ext' rfl ?_
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funext i
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simp only [HSMul.hSMul, map_one]
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erw [Finset.sum_eq_single_of_mem i]
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rw [one_prodColorMatrixOnIndexValue_on_diag]
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simp only [one_mul, IndexValue]
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simp only [Matrix.one_apply_eq, one_mul, IndexValue]
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rfl
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exact Finset.mem_univ i
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intro j _ hij
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rw [one_prodColorMatrixOnIndexValue_off_diag]
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simp only [zero_mul]
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exact hij
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exact fun j _ hij => mul_eq_zero.mpr (Or.inl (Matrix.one_apply_ne' hij))
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mul_smul Λ Λ' T := by
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refine ext' rfl ?_
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simp only [HSMul.hSMul]
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funext i
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have h1 : ∑ j : IndexValue d T.color, prodColorMatrixOnIndexValue T.color (Λ * Λ') i j
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have h1 : ∑ j : IndexValue d T.color, toTensorRepMat (Λ * Λ') i j
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* T.coord j = ∑ j : IndexValue d T.color, ∑ (k : IndexValue d T.color),
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(∏ x, ((colorMatrix (T.color x) Λ (i x) (k x)) *
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(colorMatrix (T.color x) Λ' (k x) (j x)))) * T.coord j := by
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [mul_prodColorMatrixOnIndexValue, Finset.sum_mul]
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rw [toTensorRepMat_mul', Finset.sum_mul]
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rw [h1]
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rw [Finset.sum_comm]
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refine Finset.sum_congr rfl (fun j _ => ?_)
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rw [Finset.mul_sum]
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refine Finset.sum_congr rfl (fun k _ => ?_)
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simp only [prodColorMatrixOnIndexValue, IndexValue]
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simp only [toTensorRepMat, IndexValue]
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rw [← mul_assoc]
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congr
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rw [Finset.prod_mul_distrib]
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rfl
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/-!
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## The Lorentz action on marked tensors.
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-/
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@[simps!]
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instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
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instance : MulAction (LorentzGroup d) (Marked d X n) := lorentzAction
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lemma lorentzAction_on_splitIndexValue' (T : Marked d X n)
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(i : T.UnmarkedIndexValue) (k : T.MarkedIndexValue) :
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(Λ • T).coord (splitIndexValue.symm (i, k)) =
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∑ (x : T.UnmarkedIndexValue), ∑ (y : T.MarkedIndexValue),
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(toTensorRepMat Λ i x * toTensorRepMat Λ k y) * T.coord (splitIndexValue.symm (x, y)) := by
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erw [lorentzAction_smul_coord]
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erw [← Equiv.sum_comp splitIndexValue.symm]
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rw [Fintype.sum_prod_type]
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refine Finset.sum_congr rfl (fun x _ => ?_)
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [toTensorRepMap_on_splitIndexValue]
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rfl
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@[simp]
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lemma lorentzAction_on_splitIndexValue (T : Marked d X n)
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(i : T.UnmarkedIndexValue) (k : T.MarkedIndexValue) :
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(Λ • T).coord (splitIndexValue.symm (i, k)) =
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∑ (x : T.UnmarkedIndexValue), toTensorRepMat Λ i x *
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∑ (y : T.MarkedIndexValue), toTensorRepMat Λ k y *
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T.coord (splitIndexValue.symm (x, y)) := by
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rw [lorentzAction_on_splitIndexValue']
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.mul_sum]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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rw [NonUnitalRing.mul_assoc]
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/-!
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## Properties of the Lorentz action.
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-/
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/-- The action on an empty Lorentz tensor is trivial. -/
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lemma lorentzAction_on_isEmpty [IsEmpty X] (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
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@ -146,8 +218,62 @@ lemma lorentzAction_on_isEmpty [IsEmpty X] (Λ : LorentzGroup d) (T : RealLorent
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funext i
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erw [lorentzAction_smul_coord]
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simp only [Finset.univ_unique, Finset.univ_eq_empty, Finset.prod_empty, one_mul,
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Finset.sum_singleton]
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simp only [IndexValue, Unique.eq_default]
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Finset.sum_singleton, toTensorRepMat_apply]
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erw [toTensorRepMat_apply]
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simp only [IndexValue, toTensorRepMat, Unique.eq_default]
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rw [@mul_left_eq_self₀]
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exact Or.inl rfl
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/-- The Lorentz action commutes with `congrSet`. -/
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lemma lorentzAction_comm_congrSet (f : X ≃ Y) (Λ : LorentzGroup d) (T : RealLorentzTensor d X) :
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congrSet f (Λ • T) = Λ • (congrSet f T) := by
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refine ext' rfl ?_
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funext i
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erw [lorentzAction_smul_coord, lorentzAction_smul_coord]
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erw [← Equiv.sum_comp (congrSetIndexValue d f T.color)]
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refine Finset.sum_congr rfl (fun j _ => ?_)
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simp [toTensorRepMat]
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erw [← Equiv.prod_comp f]
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apply Or.inl
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congr
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funext x
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have h1 : (T.color (f.symm (f x))) = T.color x := by
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simp only [Equiv.symm_apply_apply]
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rw [colorMatrix_cast h1]
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simp only [Matrix.reindex_apply, Equiv.symm_symm, Matrix.submatrix_apply]
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erw [castColorsIndex_comp_congrSetIndexValue]
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apply congrFun
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apply congrArg
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symm
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refine cast_eq_iff_heq.mpr ?_
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simp only [congrSetIndexValue, Equiv.piCongrLeft'_symm_apply, heq_eqRec_iff_heq, heq_eq_eq]
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rfl
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open Marked
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lemma lorentzAction_comm_mul (T : Marked d X 1) (S : Marked d Y 1)
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(h : T.markedColor 0 = τ (S.markedColor 1)) :
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mul (Λ • T) (Λ • S) h = Λ • mul T S h := by
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refine ext' rfl ?_
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funext i
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trans ∑ j, toTensorRepMat Λ (inlIndexValue i) (inlIndexValue j) *
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toTensorRepMat Λ (inrIndexValue i) (inrIndexValue j)
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* (mul T S h).coord j
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swap
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refine Finset.sum_congr rfl (fun j _ => ?_)
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erw [toTensorRepMat_on_sum]
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rfl
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change ∑ x, (∑ j, toTensorRepMat Λ (splitIndexValue.symm
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(inlIndexValue i, T.oneMarkedIndexValue x)) j * T.coord j) *
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(∑ k, toTensorRepMat Λ _ k * S.coord k) = _
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trans ∑ x, (∑ j,
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toTensorRepMat Λ (inlIndexValue i) (toUnmarkedIndexValue j)
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* toTensorRepMat Λ (T.oneMarkedIndexValue x) (toMarkedIndexValue j)
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* T.coord j) *
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sorry
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/-! TODO: Show that the Lorentz action commutes with multiplication. -/
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/-! TODO: Show that the Lorentz action commutes with contraction. -/
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