Merge branch 'master' into Update-versions
This commit is contained in:
Joseph Tooby-Smith
GitHub
commit
f476f21039
3 changed files with 547 additions and 44 deletions
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@ -15,7 +15,6 @@ marked index. This is equivalent to contracting two Lorentz tensors.
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We prove various results about this multiplication.
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-/
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/-! TODO: Generalize to contracting any marked index of a marked tensor. -/
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/-! TODO: Set up a good notation for the multiplication. -/
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namespace RealLorentzTensor
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@ -38,6 +37,31 @@ def mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
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S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2,
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oneMarkedIndexValue $ colorsIndexDualCast h x))
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/-- The index value appearing in the multiplication of Marked tensors on the left. -/
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def mulFstArg {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
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(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
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(x : ColorsIndex d (T.color (markedPoint X 0))) : IndexValue d T.color :=
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splitIndexValue.symm ((indexValueSumEquiv i).1, oneMarkedIndexValue x)
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lemma mulFstArg_inr {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
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(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
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(x : ColorsIndex d (T.color (markedPoint X 0))) :
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mulFstArg i x (Sum.inr 0) = x := by
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rfl
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lemma mulFstArg_inl {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
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(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
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(x : ColorsIndex d (T.color (markedPoint X 0))) (c : X):
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mulFstArg i x (Sum.inl c) = i (Sum.inl c) := by
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rfl
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/-- The index value appearing in the multiplication of Marked tensors on the right. -/
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def mulSndArg {X Y : Type} {T : Marked d X 1} {S : Marked d Y 1}
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(i : IndexValue d (Sum.elim T.unmarkedColor S.unmarkedColor))
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(x : ColorsIndex d (T.color (markedPoint X 0))) (h : T.markedColor 0 = τ (S.markedColor 0)) :
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IndexValue d S.color :=
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splitIndexValue.symm ((indexValueSumEquiv i).2, oneMarkedIndexValue $ colorsIndexDualCast h x)
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/-!
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## Expressions for multiplication
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@ -117,7 +141,6 @@ lemma mul_mapIso {X Y Z : Type} (T : Marked d X 1) (S : Marked d Y 1) (f : X ≃
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rw [mapIso_apply_coord, mapIso_apply_coord]
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refine Mathlib.Tactic.Ring.mul_congr ?_ ?_ rfl
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· apply congrArg
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simp only [IndexValue]
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exact (Equiv.symm_apply_eq splitIndexValue).mpr rfl
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· apply congrArg
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exact (Equiv.symm_apply_eq splitIndexValue).mpr rfl
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@ -188,40 +211,38 @@ lemma mul_unmarkedLorentzAction (T : Marked d X 1) (S : Marked d Y 1)
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T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
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toTensorRepMat Λ (indexValueSumEquiv i).2 k *
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S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
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apply Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul_sum]
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apply Finset.sum_congr rfl (fun j _ => ?_)
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apply Finset.sum_congr rfl (fun k _ => ?_)
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ring
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· apply Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul_sum ]
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apply Finset.sum_congr rfl (fun j _ => ?_)
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apply Finset.sum_congr rfl (fun k _ => ?_)
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ring
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rw [Finset.sum_comm]
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trans ∑ j, ∑ k, ∑ x, (toTensorRepMat Λ (indexValueSumEquiv i).1 j *
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T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))*
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toTensorRepMat Λ (indexValueSumEquiv i).2 k *
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S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
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apply Finset.sum_congr rfl (fun j _ => ?_)
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rw [Finset.sum_comm]
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· exact Finset.sum_congr rfl (fun j _ => Finset.sum_comm)
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trans ∑ j, ∑ k,
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((toTensorRepMat Λ (indexValueSumEquiv i).1 j) * toTensorRepMat Λ (indexValueSumEquiv i).2 k)
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* ∑ x, (T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))
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* S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
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apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
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rw [Finset.mul_sum]
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apply Finset.sum_congr rfl (fun x _ => ?_)
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ring
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· apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
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rw [Finset.mul_sum]
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apply Finset.sum_congr rfl (fun x _ => ?_)
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ring
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trans ∑ j, ∑ k, toTensorRepMat Λ i (indexValueSumEquiv.symm (j, k)) *
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∑ x, (T.coord (splitIndexValue.symm (j, oneMarkedIndexValue x)))
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* S.coord (splitIndexValue.symm (k, oneMarkedIndexValue $ colorsIndexDualCast h x))
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apply Finset.sum_congr rfl (fun j _ => Finset.sum_congr rfl (fun k _ => ?_))
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rw [toTensorRepMat_of_indexValueSumEquiv']
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congr
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simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply, mul_color]
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· rw [toTensorRepMat_of_indexValueSumEquiv']
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congr
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simp only [IndexValue, Finset.mem_univ, Prod.mk.eta, Equiv.symm_apply_apply, mul_color]
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trans ∑ p, toTensorRepMat Λ i p *
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∑ x, (T.coord (splitIndexValue.symm ((indexValueSumEquiv p).1, oneMarkedIndexValue x)))
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* S.coord (splitIndexValue.symm ((indexValueSumEquiv p).2,
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oneMarkedIndexValue $ colorsIndexDualCast h x))
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erw [← Equiv.sum_comp indexValueSumEquiv.symm]
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rw [Fintype.sum_prod_type]
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rfl
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· erw [← Equiv.sum_comp indexValueSumEquiv.symm]
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exact Eq.symm Fintype.sum_prod_type
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rfl
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/-- The Lorentz action commutes with multiplication. -/
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@ -231,4 +252,234 @@ lemma mul_lorentzAction (T : Marked d X 1) (S : Marked d Y 1)
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simp only [← marked_unmarked_action_eq_action]
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rw [mul_markedLorentzAction, mul_unmarkedLorentzAction]
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/-!
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## Multiplication on selected indices
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-/
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variable {n m : ℕ} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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{X' Y' Z : Type} [Fintype X'] [DecidableEq X'] [Fintype Y'] [DecidableEq Y']
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[Fintype Z] [DecidableEq Z]
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/-- The multiplication of two real Lorentz Tensors along specified indices. -/
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@[simps!]
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def mulS (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y) (x : X) (y : Y)
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(h : T.color x = τ (S.color y)) : RealLorentzTensor d ({x' // x' ≠ x} ⊕ {y' // y' ≠ y}) :=
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mul (markSingle x T) (markSingle y S) h
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/-- The first index value appearing in the multiplication of two Lorentz tensors. -/
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def mulSFstArg {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
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(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))) :
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IndexValue d T.color := (markSingleIndexValue T x).symm (mulFstArg i a)
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lemma mulSFstArg_ext {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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{i j : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor)}
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{a b : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))}
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(hij : i = j) (hab : a = b) : mulSFstArg i a = mulSFstArg j b := by
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subst hij; subst hab
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rfl
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lemma mulSFstArg_on_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
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(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))) :
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mulSFstArg i a x = a := by
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rw [mulSFstArg, markSingleIndexValue]
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simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
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Sum.map_inr, id_eq]
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erw [markEmbeddingIndexValue_apply_symm_on_mem]
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swap
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Finset.univ_unique, Fin.default_eq_zero,
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Fin.isValue, Finset.image_singleton, embedSingleton_apply, Finset.mem_singleton]
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rw [indexValueIso_symm_apply']
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erw [Equiv.symm_apply_eq, Equiv.symm_apply_eq]
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simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_symm, Equiv.cast_apply, cast_cast]
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symm
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apply cast_eq_iff_heq.mpr
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rw [embedSingleton_toEquivRange_symm]
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rfl
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lemma mulSFstArg_on_not_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
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(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
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(c : X) (hc : c ≠ x) : mulSFstArg i a c = i (Sum.inl ⟨c, hc⟩) := by
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rw [mulSFstArg, markSingleIndexValue]
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simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
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Sum.map_inr, id_eq]
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erw [markEmbeddingIndexValue_apply_symm_on_not_mem]
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swap
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simpa using hc
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rfl
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/-- The second index value appearing in the multiplication of two Lorentz tensors. -/
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def mulSSndArg {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
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(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
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(h : T.color x = τ (S.color y)) : IndexValue d S.color :=
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(markSingleIndexValue S y).symm (mulSndArg i a h)
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lemma mulSSndArg_on_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
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(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
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(h : T.color x = τ (S.color y)) : mulSSndArg i a h y = colorsIndexDualCast h a := by
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rw [mulSSndArg, markSingleIndexValue]
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simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
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Sum.map_inr, id_eq]
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erw [markEmbeddingIndexValue_apply_symm_on_mem]
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swap
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Finset.univ_unique, Fin.default_eq_zero,
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Fin.isValue, Finset.image_singleton, embedSingleton_apply, Finset.mem_singleton]
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rw [indexValueIso_symm_apply']
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erw [Equiv.symm_apply_eq, Equiv.symm_apply_eq]
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simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_symm, Equiv.cast_apply, cast_cast]
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symm
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apply cast_eq_iff_heq.mpr
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rw [embedSingleton_toEquivRange_symm]
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rfl
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lemma mulSSndArg_on_not_mem {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor))
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(a : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0)))
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(h : T.color x = τ (S.color y)) (c : Y) (hc : c ≠ y) :
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mulSSndArg i a h c = i (Sum.inr ⟨c, hc⟩) := by
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rw [mulSSndArg, markSingleIndexValue]
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simp only [ne_eq, Fintype.univ_ofSubsingleton, Fin.zero_eta, Fin.isValue, Equiv.symm_trans_apply,
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Sum.map_inr, id_eq]
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erw [markEmbeddingIndexValue_apply_symm_on_not_mem]
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swap
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simpa using hc
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rfl
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lemma mulSSndArg_ext {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y} {x : X} {y : Y}
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{i j : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor)}
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{a b : ColorsIndex d ((markSingle x T).color (markedPoint {x' // x' ≠ x} 0))}
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(h : T.color x = τ (S.color y)) (hij : i = j) (hab : a = b) :
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mulSSndArg i a h = mulSSndArg j b h := by
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subst hij
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subst hab
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rfl
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lemma mulS_coord_arg (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y) (x : X) (y : Y)
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(h : T.color x = τ (S.color y))
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(i : IndexValue d (Sum.elim (markSingle x T).unmarkedColor (markSingle y S).unmarkedColor)) :
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(mulS T S x y h).coord i = ∑ a, T.coord (mulSFstArg i a) * S.coord (mulSSndArg i a h) := by
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rfl
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lemma mulS_mapIso (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y)
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(eX : X ≃ X') (eY : Y ≃ Y') (x : X) (y : Y) (x' : X') (y' : Y') (hx : eX x = x')
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(hy : eY y = y') (h : T.color x = τ (S.color y)) :
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mulS (mapIso d eX T) (mapIso d eY S) x' y' (by subst hx hy; simpa using h) =
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mapIso d (Equiv.sumCongr (equivSingleCompl eX hx) (equivSingleCompl eY hy))
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(mulS T S x y h) := by
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rw [mulS, mulS, mul_mapIso]
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congr 1 <;> rw [markSingle_mapIso]
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lemma mulS_lorentzAction (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y)
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(x : X) (y : Y) (h : T.color x = τ (S.color y)) (Λ : LorentzGroup d) :
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mulS (Λ • T) (Λ • S) x y h = Λ • mulS T S x y h := by
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rw [mulS, mulS, ← mul_lorentzAction]
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congr 1
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all_goals
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rw [markSingle, markEmbedding, Equiv.trans_apply]
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erw [lorentzAction_mapIso, lorentzAction_mapIso]
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rfl
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lemma mulS_symm (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y)
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(x : X) (y : Y) (h : T.color x = τ (S.color y)) :
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mapIso d (Equiv.sumComm _ _) (mulS T S x y h) = mulS S T y x (color_eq_dual_symm h) := by
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rw [mulS, mulS, mul_symm]
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/-- An equivalence of types associated with multiplying two consecutive indices,
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with the second index appearing on the left. -/
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def mulSSplitLeft {y y' : Y} (hy : y ≠ y') (z : Z) :
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{yz // yz ≠ (Sum.inl ⟨y, hy⟩ : {y'' // y'' ≠ y'} ⊕ {z' // z' ≠ z})} ≃
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{y'' // y'' ≠ y' ∧ y'' ≠ y} ⊕ {z' // z' ≠ z} :=
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Equiv.subtypeSum.trans <|
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Equiv.sumCongr (
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(Equiv.subtypeEquivRight (fun a => by
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obtain ⟨a, p⟩ := a; simp only [ne_eq, Sum.inl.injEq, Subtype.mk.injEq])).trans
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(Equiv.subtypeSubtypeEquivSubtypeInter _ _)) <|
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Equiv.subtypeUnivEquiv (fun a => Sum.inr_ne_inl)
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/-- An equivalence of types associated with multiplying two consecutive indices with the
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second index appearing on the right. -/
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def mulSSplitRight {y y' : Y} (hy : y ≠ y') (z : Z) :
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{yz // yz ≠ (Sum.inr ⟨y', hy.symm⟩ : {z' // z' ≠ z} ⊕ {y'' // y'' ≠ y})} ≃
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{z' // z' ≠ z} ⊕ {y'' // y'' ≠ y' ∧ y'' ≠ y} :=
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Equiv.subtypeSum.trans <|
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Equiv.sumCongr (Equiv.subtypeUnivEquiv (fun a => Sum.inl_ne_inr)) <|
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(Equiv.subtypeEquivRight (fun a => by
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obtain ⟨a, p⟩ := a; simp only [ne_eq, Sum.inr.injEq, Subtype.mk.injEq])).trans <|
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((Equiv.subtypeSubtypeEquivSubtypeInter _ _).trans
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(Equiv.subtypeEquivRight (fun y'' => And.comm)))
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/-- An equivalence of types associated with the associativity property of multiplication. -/
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def mulSAssocIso (x : X) {y y' : Y} (hy : y ≠ y') (z : Z) :
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{x' // x' ≠ x} ⊕ {yz // yz ≠ (Sum.inl ⟨y, hy⟩ : {y'' // y'' ≠ y'} ⊕ {z' // z' ≠ z})}
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≃ {xy // xy ≠ (Sum.inr ⟨y', hy.symm⟩ : {x' // x' ≠ x} ⊕ {y'' // y'' ≠ y})} ⊕ {z' // z' ≠ z} :=
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(Equiv.sumCongr (Equiv.refl _) (mulSSplitLeft hy z)).trans <|
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(Equiv.sumAssoc _ _ _).symm.trans <|
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(Equiv.sumCongr (mulSSplitRight hy x).symm (Equiv.refl _))
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lemma mulS_assoc_color {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y}
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{U : RealLorentzTensor d Z} {x : X} {y y' : Y} (hy : y ≠ y') {z : Z}
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(h : T.color x = τ (S.color y))
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(h' : S.color y' = τ (U.color z)) : (mulS (mulS T S x y h) U (Sum.inr ⟨y', hy.symm⟩) z h').color
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= (mapIso d (mulSAssocIso x hy z) (mulS T (mulS S U y' z h') x (Sum.inl ⟨y, hy⟩) h)).color := by
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funext a
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match a with
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| .inl ⟨.inl _, _⟩ => rfl
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| .inl ⟨.inr _, _⟩ => rfl
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| .inr _ => rfl
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/-- An equivalence of index values associated with the associativity property of multiplication. -/
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def mulSAssocIndexValue {T : RealLorentzTensor d X} {S : RealLorentzTensor d Y}
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{U : RealLorentzTensor d Z} {x : X} {y y' : Y} (hy : y ≠ y') {z : Z}
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(h : T.color x = τ (S.color y)) (h' : S.color y' = τ (U.color z)) :
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IndexValue d ((T.mulS S x y h).mulS U (Sum.inr ⟨y', hy.symm⟩) z h').color ≃
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IndexValue d (T.mulS (S.mulS U y' z h') x (Sum.inl ⟨y, hy⟩) h).color :=
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indexValueIso d (mulSAssocIso x hy z).symm (mulS_assoc_color hy h h')
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/-- Multiplication of indices is associative, up to a `mapIso` equivalence. -/
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lemma mulS_assoc (T : RealLorentzTensor d X) (S : RealLorentzTensor d Y) (U : RealLorentzTensor d Z)
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(x : X) (y y' : Y) (hy : y ≠ y') (z : Z) (h : T.color x = τ (S.color y))
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(h' : S.color y' = τ (U.color z)) : mulS (mulS T S x y h) U (Sum.inr ⟨y', hy.symm⟩) z h' =
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mapIso d (mulSAssocIso x hy z) (mulS T (mulS S U y' z h') x (Sum.inl ⟨y, hy⟩) h) := by
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apply ext (mulS_assoc_color _ _ _) ?_
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funext i
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trans ∑ a, (∑ b, T.coord (mulSFstArg (mulSFstArg i a) b) *
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S.coord (mulSSndArg (mulSFstArg i a) b h)) * U.coord (mulSSndArg i a h')
|
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rfl
|
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trans ∑ a, T.coord (mulSFstArg (mulSAssocIndexValue hy h h' i) a) *
|
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(∑ b, S.coord (mulSFstArg (mulSSndArg (mulSAssocIndexValue hy h h' i) a h) b) *
|
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U.coord (mulSSndArg (mulSSndArg (mulSAssocIndexValue hy h h' i) a h) b h'))
|
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swap
|
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rw [mapIso_apply_coord, mulS_coord_arg, indexValueIso_symm]
|
||||
rfl
|
||||
rw [Finset.sum_congr rfl (fun x _ => Finset.sum_mul _ _ _)]
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rw [Finset.sum_congr rfl (fun x _ => Finset.mul_sum _ _ _)]
|
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rw [Finset.sum_comm]
|
||||
refine Finset.sum_congr rfl (fun a _ => Finset.sum_congr rfl (fun b _ => ?_))
|
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rw [mul_assoc]
|
||||
refine Mathlib.Tactic.Ring.mul_congr rfl (Mathlib.Tactic.Ring.mul_congr ?_ rfl rfl) rfl
|
||||
apply congrArg
|
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funext c
|
||||
by_cases hcy : c = y
|
||||
· subst hcy
|
||||
rw [mulSSndArg_on_mem, mulSFstArg_on_not_mem, mulSSndArg_on_mem]
|
||||
rfl
|
||||
· by_cases hcy' : c = y'
|
||||
· subst hcy'
|
||||
rw [mulSFstArg_on_mem, mulSSndArg_on_not_mem, mulSFstArg_on_mem]
|
||||
· rw [mulSFstArg_on_not_mem, mulSSndArg_on_not_mem, mulSSndArg_on_not_mem,
|
||||
mulSFstArg_on_not_mem]
|
||||
rw [mulSAssocIndexValue, indexValueIso_eq_symm, indexValueIso_symm_apply']
|
||||
simp only [ne_eq, Function.comp_apply, Equiv.symm_symm_apply, mulS_color, Sum.elim_inr,
|
||||
colorsIndexCast, Equiv.cast_refl, Equiv.refl_symm]
|
||||
erw [Equiv.refl_apply]
|
||||
rfl
|
||||
exact hcy'
|
||||
simpa using hcy
|
||||
|
||||
end RealLorentzTensor
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue