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feat: Important fix to ContrContr

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jstoobysmith 2024-10-29 10:37:18 +00:00
parent 83ff8f5358
commit 4756466001
7 changed files with 184 additions and 2 deletions
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12
HepLean/Tensors/ComplexLorentz/Basic.lean
View file

@ -209,5 +209,17 @@ lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.re
| Color.up => Lorentz.contrCoContraction_basis _ _
| Color.down => Lorentz.coContrContraction_basis _ _
instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
DecidableEq (OverColor.mk c).left := instDecidableEqFin n
instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
Fintype (OverColor.mk c).left := Fin.fintype n
instance {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
{c1 : Fin m → complexLorentzTensor.C} (σ σ' : OverColor.mk c ⟶ OverColor.mk c1) :
Decidable (σ = σ') :=
decidable_of_iff _ (OverColor.Hom.ext_iff σ σ')
end complexLorentzTensor
end

52
HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean
View file

@ -87,5 +87,57 @@ lemma tensorNode_coBispinorDown (p : complexCo) :
⊗ coBispinorUp p | α β}ᵀ.tensor := by
rw [coBispinorDown, tensorNode_tensor]
/-!
## Basic equalities.
-/
lemma contrBispinorDown_expand (p : complexContr) :
{contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
(pauliCo | μ α β ⊗ p | μ)}ᵀ.tensor := by
rw [tensorNode_contrBispinorDown p]
rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_contrBispinorUp p]
set_option maxRecDepth 5000 in
lemma contrBispinorDown_eq_pauliCoDown_contr (p : complexContr) :
{contrBispinorDown p | α β = pauliCoDown | μ α β ⊗ p | μ}ᵀ := by
conv =>
rhs
rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <| pauliCoDown_eq_metric_mul_pauliCo]
rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
apply (perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _).trans
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_contr_congr 0 4]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
conv =>
lhs
rw [contrBispinorDown_expand p]
rw [contr_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _]
rw [contr_tensor_eq <| perm_contr_congr 0 3]
rw [perm_contr_congr 1 2]
apply (perm_tensor_eq <| contr_tensor_eq <| contr_contr _ _ _).trans
rw [perm_tensor_eq <| perm_contr _ _]
rw [perm_perm]
rw [perm_tensor_eq <| contr_contr _ _ _]
rw [perm_perm]
apply perm_congr _ rfl
decide
end complexLorentzTensor
end

61
HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean
View file

@ -4,7 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.Tensors.Tree.NodeIdentities.ProdAssoc
import HepLean.Tensors.Tree.NodeIdentities.ProdComm
import HepLean.Tensors.Tree.NodeIdentities.ProdContr
import HepLean.Tensors.Tree.NodeIdentities.ContrContr
import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
import HepLean.Tensors.Tree.NodeIdentities.PermContr
import HepLean.Tensors.Tree.NodeIdentities.Congr
/-!
@ -75,4 +79,61 @@ lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
Fermion.altRightMetric | β β'}ᵀ.tensor := by
rfl
/-!
## Basic equalities
-/
set_option maxRecDepth 5000 in
/-- A rearanging of `pauliCoDown` to place the pauli matrices on the right. -/
lemma pauliCoDown_eq_metric_mul_pauliCo :
{pauliCoDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ := by
conv =>
lhs
rw [tensorNode_pauliCoDown]
rw [contr_tensor_eq <| contr_prod _ _ _]
rw [perm_contr]
erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
rw [perm_tensor_eq <| contr_congr 1 2]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_congr 1 2]
rw [perm_tensor_eq <| perm_contr _ _]
rw [perm_perm]
rw [perm_tensor_eq <| contr_congr 1 2]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 1 2]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
rw [perm_tensor_eq <| perm_contr _ _]
rw [perm_perm]
rw [perm_tensor_eq <| contr_congr 1 2]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_comm _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
rw [perm_tensor_eq <| perm_contr _ _]
rw [perm_perm]
rw [perm_tensor_eq <| contr_congr 4 1]
rw [perm_perm]
conv =>
rhs
rw [perm_tensor_eq <| contr_swap _ _]
rw [perm_perm]
erw [perm_tensor_eq <| contr_congr 4 1]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_swap _ _]
erw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
rw [perm_tensor_eq <| perm_contr _ _]
rw [perm_perm]
rw [perm_tensor_eq <| contr_congr 4 1]
rw [perm_perm]
apply perm_congr _ rfl
decide
end complexLorentzTensor

7
HepLean/Tensors/OverColor/Basic.lean
View file

@ -45,6 +45,13 @@ variable {C : Type} {f g h : OverColor C}
lemma ext (m n : f ⟶ g) (h : m.hom = n.hom) : m = n := by
apply CategoryTheory.Iso.ext h
lemma ext_iff (m n : f ⟶ g) : (∀ x, m.hom.left x = n.hom.left x) ↔ m = n := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· apply ext
ext x
exact h x
· rw [h]
exact fun x => rfl
/-- Given a hom in `OverColor C` the underlying equivalence between types. -/
def toEquiv (m : f ⟶ g) : f.left ≃ g.left where

2
HepLean/Tensors/Tree/NodeIdentities/Basic.lean
View file

@ -92,7 +92,7 @@ lemma add_neg (t : TensorTree S c) : (add t (neg t)).tensor = 0 := by
-/
/-- Applying two permutations is the same as applying the transitive permutation. -/
lemma perm_perm {n : ℕ} {c : Fin n → S.C} {c1 : Fin n → S.C} {c2 : Fin n → S.C}
lemma perm_perm {n n1 n2 : ℕ} {c : Fin n → S.C} {c1 : Fin n1 → S.C} {c2 : Fin n2 → S.C}
(σ : (OverColor.mk c) ⟶ (OverColor.mk c1)) (σ2 : (OverColor.mk c1) ⟶ (OverColor.mk c2))
(t : TensorTree S c) : (perm σ2 (perm σ t)).tensor = (perm (σ ≫ σ2) t).tensor := by
simp [perm_tensor]

16
HepLean/Tensors/Tree/NodeIdentities/ContrContr.lean
View file

@ -281,6 +281,20 @@ lemma contr_contr (t : TensorTree S c) :
end
end ContrQuartet
def contrContrPerm {n : ℕ} {c : Fin n.succ.succ.succ.succ → S.C} {i : Fin n.succ.succ.succ.succ}
{j : Fin n.succ.succ.succ} {k : Fin n.succ.succ} {l : Fin n.succ}
(hij : c (i.succAbove j) = S.τ (c i)) (hkl : (c ∘ i.succAbove ∘ j.succAbove) (k.succAbove l) =
S.τ ((c ∘ i.succAbove ∘ j.succAbove) k)) :
OverColor.mk
((c ∘
(ContrQuartet.mk i j k l hij hkl).swapI.succAbove ∘
(ContrQuartet.mk i j k l hij hkl).swapJ.succAbove) ∘
(ContrQuartet.mk i j k l hij hkl).swapK.succAbove ∘
(ContrQuartet.mk i j k l hij hkl).swapL.succAbove) ⟶
OverColor.mk
((c ∘ i.succAbove ∘ j.succAbove) ∘ k.succAbove ∘ l.succAbove)
:= (ContrQuartet.mk i j k l hij hkl).contrSwapHom
/-- Contraction nodes commute on adjusting indices. -/
theorem contr_contr {n : ℕ} {c : Fin n.succ.succ.succ.succ → S.C} {i : Fin n.succ.succ.succ.succ}
{j : Fin n.succ.succ.succ} {k : Fin n.succ.succ} {l : Fin n.succ}
@ -288,7 +302,7 @@ theorem contr_contr {n : ℕ} {c : Fin n.succ.succ.succ.succ → S.C} {i : Fin n
S.τ ((c ∘ i.succAbove ∘ j.succAbove) k))
(t : TensorTree S c) :
(contr k l hkl (contr i j hij t)).tensor =
(perm (ContrQuartet.mk i j k l hij hkl).contrSwapHom
(perm (contrContrPerm hij hkl)
(contr (ContrQuartet.mk i j k l hij hkl).swapK (ContrQuartet.mk i j k l hij hkl).swapL
(ContrQuartet.mk i j k l hij hkl).swap.hkl (contr (ContrQuartet.mk i j k l hij hkl).swapI
(ContrQuartet.mk i j k l hij hkl).swapJ

36
HepLean/Tensors/Tree/NodeIdentities/PermContr.lean
View file

@ -4,6 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.Tensors.Tree.Basic
import HepLean.Tensors.Tree.NodeIdentities.Congr
import HepLean.Tensors.Tree.NodeIdentities.Basic
/-!
# The commutativity of Permutations and contractions.
@ -256,4 +258,38 @@ lemma perm_contr {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ
rw [S.contrMap_naturality σ]
rfl
lemma perm_contr_congr_mkIso_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
{i : Fin n.succ.succ} {j : Fin n.succ}
{σ : (OverColor.mk c) ⟶ (OverColor.mk c1)}
{i' : Fin n.succ.succ} {j' : Fin n.succ}
(hi : i' = ((Hom.toEquiv σ).symm i))
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j)) :
c ∘ i'.succAbove ∘ j'.succAbove =
c ∘ Fin.succAbove ((Hom.toEquiv σ).symm i) ∘ Fin.succAbove ((Hom.toEquiv (extractOne i σ)).symm j) := by
rw [hi, hj]
lemma perm_contr_congr_contr_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
{i : Fin n.succ.succ} {j : Fin n.succ}
(h : c1 (i.succAbove j) = S.τ (c1 i))
{σ : (OverColor.mk c) ⟶ (OverColor.mk c1)}
{i' : Fin n.succ.succ} {j' : Fin n.succ}
(hi : i' = ((Hom.toEquiv σ).symm i))
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j)) :
c (i'.succAbove j') = S.τ (c i') := by
rw [hi, hj]
exact S.perm_contr_cond h σ
lemma perm_contr_congr {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
{i : Fin n.succ.succ} {j : Fin n.succ}
{h : c1 (i.succAbove j) = S.τ (c1 i)} {t : TensorTree S c}
{σ : (OverColor.mk c) ⟶ (OverColor.mk c1)}
(i' : Fin n.succ.succ) (j' : Fin n.succ)
(hi : i' = ((Hom.toEquiv σ).symm i) := by decide)
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j) := by decide) :
(contr i j h (perm σ t)).tensor = (perm ((mkIso (perm_contr_congr_mkIso_cond hi hj)).hom ≫
extractTwo i j σ) (contr i' j' (perm_contr_congr_contr_cond h hi hj) t)).tensor := by
rw [perm_contr]
rw [perm_tensor_eq <| contr_congr i' j' hi.symm hj.symm]
rw [perm_perm]
end TensorTree