refactor: Lint
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jstoobysmith
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commit
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9 changed files with 946 additions and 696 deletions
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@ -79,6 +79,15 @@ lemma perm_basisVector {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, LinearEquiv.ofLinear_apply]
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rw [basis_eq_FDiscrete]
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lemma perm_basisVector_tree {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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{c1 : Fin m → complexLorentzTensor.C} (σ : OverColor.mk c ⟶ OverColor.mk c1)
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(b : Π j, Fin (complexLorentzTensor.repDim (c j))) :
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(perm σ (tensorNode (basisVector c b))).tensor =
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(tensorNode (basisVector c1 (fun i => Fin.cast (perm_basisVector_cast σ i)
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(b ((OverColor.Hom.toEquiv σ).symm i))))).tensor := by
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exact perm_basisVector _ _
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/-- The scalar determining if contracting two basis vectors together gives zero or not. -/
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def contrBasisVectorMul {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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(i : Fin n.succ.succ) (j : Fin n.succ)
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) : ℂ :=
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@ -89,7 +98,7 @@ lemma contrBasisVectorMul_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzT
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(h : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
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contrBasisVectorMul i j b = 0 := by
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rw [contrBasisVectorMul]
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simp
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simp only [ite_eq_else, one_ne_zero, imp_false]
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exact h
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lemma contrBasisVectorMul_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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@ -97,7 +106,7 @@ lemma contrBasisVectorMul_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzT
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(h : (b i).val = (b (i.succAbove j)).val := by decide) :
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contrBasisVectorMul i j b = 1 := by
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rw [contrBasisVectorMul]
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simp
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simp only [ite_eq_then, zero_ne_one, imp_false, Decidable.not_not]
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exact h
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lemma contr_basisVector {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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@ -119,37 +128,41 @@ lemma contr_basisVector {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.
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erw [basis_contr]
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rfl
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lemma contr_basisVector_tree {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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lemma contr_basisVector_tree {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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(contr i j h (tensorNode (basisVector c b))).tensor =
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(smul (contrBasisVectorMul i j b) (tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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(fun k => b (i.succAbove (j.succAbove k)))) )).tensor := by
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(smul (contrBasisVectorMul i j b) (tensorNode
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(basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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(fun k => b (i.succAbove (j.succAbove k)))))).tensor := by
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exact contr_basisVector _
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lemma contr_basisVector_tree_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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lemma contr_basisVector_tree_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (hn : (b i).val = (b (i.succAbove j)).val := by decide) :
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(b : Π k, Fin (complexLorentzTensor.repDim (c k)))
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(hn : (b i).val = (b (i.succAbove j)).val := by decide) :
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(contr i j h (tensorNode (basisVector c b))).tensor =
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((tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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((tensorNode (basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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(fun k => b (i.succAbove (j.succAbove k)))))).tensor := by
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rw [contr_basisVector_tree, contrBasisVectorMul]
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rw [if_pos hn]
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simp [smul_tensor]
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lemma contr_basisVector_tree_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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lemma contr_basisVector_tree_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (hn : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
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(b : Π k, Fin (complexLorentzTensor.repDim (c k)))
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(hn : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
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(contr i j h (tensorNode (basisVector c b))).tensor =
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(tensorNode 0).tensor := by
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rw [contr_basisVector_tree, contrBasisVectorMul]
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rw [if_neg hn]
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simp [smul_tensor]
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simp only [Nat.succ_eq_add_one, smul_tensor, tensorNode_tensor, zero_smul]
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/-- Equivalence of Fin types appearing in the product of two basis vectors. -/
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def prodBasisVecEquiv {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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{c1 : Fin m → complexLorentzTensor.C} (i : Fin n ⊕ Fin m) :
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Sum.elim (fun i => Fin (complexLorentzTensor.repDim (c i))) (fun i => Fin (complexLorentzTensor.repDim (c1 i)))
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Sum.elim (fun i => Fin (complexLorentzTensor.repDim (c i))) (fun i =>
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Fin (complexLorentzTensor.repDim (c1 i)))
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i ≃ Fin (complexLorentzTensor.repDim ((Sum.elim c c1 i))) :=
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match i with
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| Sum.inl _ => Equiv.refl _
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@ -184,17 +197,17 @@ lemma prod_basisVector {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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| Sum.inl k => rfl
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| Sum.inr k => rfl
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lemma prod_basisVector_tree {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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lemma prod_basisVector_tree {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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{c1 : Fin m → complexLorentzTensor.C}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k)))
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(b1 : Π k, Fin (complexLorentzTensor.repDim (c1 k))) :
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(prod (tensorNode (basisVector c b)) (tensorNode (basisVector c1 b1))).tensor =
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(prod (tensorNode (basisVector c b)) (tensorNode (basisVector c1 b1))).tensor =
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(tensorNode (basisVector (Sum.elim c c1 ∘ finSumFinEquiv.symm) (fun i =>
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prodBasisVecEquiv (finSumFinEquiv.symm i)
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((HepLean.PiTensorProduct.elimPureTensor b b1) (finSumFinEquiv.symm i))))).tensor := by
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exact prod_basisVector _ _
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lemma eval_basisVector {n : ℕ} {c : Fin n.succ → complexLorentzTensor.C}
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lemma eval_basisVector {n : ℕ} {c : Fin n.succ → complexLorentzTensor.C}
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{i : Fin n.succ} (j : Fin (complexLorentzTensor.repDim (c i)))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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(eval i j (tensorNode (basisVector c b))).tensor = (if j = b i then (1 : ℂ) else 0) •
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@ -282,7 +295,7 @@ lemma contrMatrix_basis_expand_tree : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
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contrMatrix_basis_expand
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lemma leftMetric_expand : {Fermion.leftMetric | α β}ᵀ.tensor =
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- basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
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- basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
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+ basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0) := by
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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@ -299,7 +312,8 @@ lemma leftMetric_expand : {Fermion.leftMetric | α β}ᵀ.tensor =
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· rfl
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lemma leftMetric_expand_tree : {Fermion.leftMetric | α β}ᵀ.tensor =
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(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)))) <|
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(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL]
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(fun | 0 => 0 | 1 => 1)))) <|
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(tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
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leftMetric_expand
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@ -320,8 +334,10 @@ lemma altLeftMetric_expand : {Fermion.altLeftMetric | α β}ᵀ.tensor =
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· rfl
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lemma altLeftMetric_expand_tree : {Fermion.altLeftMetric | α β}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1))) <|
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(smul (-1) (tensorNode (basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0))))).tensor :=
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(TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL]
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(fun | 0 => 0 | 1 => 1))) <|
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(smul (-1) (tensorNode (basisVector ![Color.downL, Color.downL]
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(fun | 0 => 1 | 1 => 0))))).tensor :=
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altLeftMetric_expand
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lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
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@ -342,7 +358,8 @@ lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
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· rfl
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lemma rightMetric_expand_tree : {Fermion.rightMetric | α β}ᵀ.tensor =
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(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)))) <|
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(TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR]
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(fun | 0 => 0 | 1 => 1)))) <|
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(tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
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rightMetric_expand
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@ -363,8 +380,10 @@ lemma altRightMetric_expand : {Fermion.altRightMetric | α β}ᵀ.tensor =
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· rfl
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lemma altRightMetric_expand_tree : {Fermion.altRightMetric | α β}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|
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(smul (-1) (tensorNode (basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0))))).tensor :=
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(TensorTree.add (tensorNode (basisVector
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![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|
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(smul (-1) (tensorNode (basisVector ![Color.downR, Color.downR]
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(fun | 0 => 1 | 1 => 0))))).tensor :=
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altRightMetric_expand
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/-- The expansion of the Pauli matrices `σ^μ^a^{dot a}` in terms of basis vectors. -/
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@ -406,13 +425,13 @@ lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
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| (2 : Fin 3) => rfl
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lemma pauliMatrix_basis_expand_tree : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
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(TensorTree.add (tensorNode
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(TensorTree.add (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
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TensorTree.add (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
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TensorTree.add (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 0 | 2 => 1))) <|
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TensorTree.add (tensorNode
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TensorTree.add (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 1 | 2 => 0))) <|
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TensorTree.add (smul (-I) (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
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@ -421,7 +440,8 @@ lemma pauliMatrix_basis_expand_tree : {PauliMatrix.asConsTensor | μ α β}ᵀ.t
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TensorTree.add (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 0 | 2 => 0))) <|
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(smul (-1) (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
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(basisVector ![Color.up, Color.upL, Color.upR]
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(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
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rw [pauliMatrix_basis_expand]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
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smul_tensor, neg_smul, one_smul]
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233
HepLean/Tensors/ComplexLorentz/BasisTrees.lean
Normal file
233
HepLean/Tensors/ComplexLorentz/BasisTrees.lean
Normal file
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@ -0,0 +1,233 @@
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.Tree.Elab
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import HepLean.Tensors.ComplexLorentz.Basic
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import Mathlib.LinearAlgebra.TensorProduct.Basis
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import HepLean.Tensors.Tree.NodeIdentities.Basic
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import HepLean.Tensors.Tree.NodeIdentities.PermProd
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import HepLean.Tensors.Tree.NodeIdentities.PermContr
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import HepLean.Tensors.Tree.NodeIdentities.ProdComm
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import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
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import HepLean.Tensors.Tree.NodeIdentities.ContrContr
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import HepLean.Tensors.ComplexLorentz.Basis
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/-!
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## Basis trees
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When contracting e.g. Pauli matrices with other tensors, it is sometimes convienent
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to rewrite the contraction in terms of a basis.
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The lemmas in this file allow us to do this.
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace Fermion
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open complexLorentzTensor
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/-!
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## Tree expansions for Pauli matrices.
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-/
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/-- The map to colors one gets when contracting with Pauli matrices on the right. -/
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abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) :=
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(Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
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lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) :
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(TensorTree.prod t (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor)).tensor = (((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
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((TensorTree.smul (-I) ((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
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((TensorTree.smul I ((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
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((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0))).add
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(TensorTree.smul (-1) (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR]
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fun | 0 => 3 | 1 => 1 | 2 => 1))))))))))).tensor := by
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rw [prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
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rw [prod_add _ _ _]
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rw [add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
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prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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/- Moving smuls. -/
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd<| add_tensor_eq_snd
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<| add_tensor_eq_snd <| prod_smul _ _ _]
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rfl
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lemma contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) =
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complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
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(contr i j h (TensorTree.prod t
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor =
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
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((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
|
||||
((TensorTree.smul I (contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
|
||||
((contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0)))).add
|
||||
(TensorTree.smul (-1) (contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR]
|
||||
fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| prod_pauliMatrix_basis_tree_expand _]
|
||||
/- Moving contr over add. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contr over smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
contr_smul _ _]
|
||||
|
||||
lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(i : Fin (n + 3)) (j : Fin (n +2))
|
||||
(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
|
||||
((pauliMatrixContrMap c) i))
|
||||
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
|
||||
let c' := Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm
|
||||
let b' (i1 i2 i3 : Fin 4) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
|
||||
((HepLean.PiTensorProduct.elimPureTensor b (fun | 0 => i1 | 1 => i2 | 2 => i3))
|
||||
(finSumFinEquiv.symm i))
|
||||
(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor))).tensor = ((contr i j h ((tensorNode
|
||||
(basisVector c' (b' 0 0 0))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 0 1 1))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 1 0 1))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 1 1 0))))).add
|
||||
((TensorTree.smul (-I) (contr i j h ((tensorNode (basisVector c' (b' 2 0 1)))))).add
|
||||
((TensorTree.smul I (contr i j h ((tensorNode (basisVector c' (b' 2 1 0)))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 3 0 0))))).add
|
||||
(TensorTree.smul (-1) (contr i j h ((tensorNode
|
||||
(basisVector c' (b' 3 1 1))))))))))))).tensor := by
|
||||
rw [contr_pauliMatrix_basis_tree_expand]
|
||||
/- Product of basis vectors . -/
|
||||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||||
<| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||||
<| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||||
<| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rfl
|
||||
|
||||
lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(i : Fin (n + 3)) (j : Fin (n +2))
|
||||
(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
|
||||
((pauliMatrixContrMap c) i))
|
||||
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
|
||||
let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
|
||||
∘ Fin.succAbove i ∘ Fin.succAbove j
|
||||
let b'' (i1 i2 i3 : Fin 4) : (i : Fin (n + (Nat.succ 0).succ.succ)) →
|
||||
Fin (complexLorentzTensor.repDim (Sum.elim c ![Color.up, Color.upL, Color.upR]
|
||||
(finSumFinEquiv.symm i))) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
|
||||
((HepLean.PiTensorProduct.elimPureTensor b (fun | (0 : Fin 3) => i1 | 1 => i2 | 2 => i3))
|
||||
(finSumFinEquiv.symm i))
|
||||
let b' (i1 i2 i3 : Fin 4) := fun k => (b'' i1 i2 i3) (i.succAbove (j.succAbove k))
|
||||
(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor))).tensor = (((
|
||||
TensorTree.smul (contrBasisVectorMul i j (b'' 0 0 0))
|
||||
(tensorNode (basisVector c' (b' 0 0 0))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1))
|
||||
(tensorNode (basisVector c' (b' 0 1 1))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1))
|
||||
(tensorNode (basisVector c' (b' 1 0 1))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0))
|
||||
(tensorNode (basisVector c' (b' 1 1 0))))).add
|
||||
((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1))
|
||||
(tensorNode (basisVector c' (b' 2 0 1)))))).add
|
||||
((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0))
|
||||
(tensorNode (basisVector c' (b' 2 1 0)))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 3 0 0))
|
||||
(tensorNode (basisVector c' (b' 3 0 0))))).add
|
||||
(TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (b'' 3 1 1)) (tensorNode
|
||||
(basisVector c' (b' 3 1 1))))))))))))).tensor := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand']
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||||
<| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
smul_tensor_eq <| contr_basisVector_tree _]
|
||||
|
||||
end Fermion
|
||||
|
||||
end
|
||||
|
|
@ -3,17 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.Tree.Elab
|
||||
import HepLean.Tensors.ComplexLorentz.Basic
|
||||
import Mathlib.LinearAlgebra.TensorProduct.Basis
|
||||
import HepLean.Tensors.Tree.NodeIdentities.Basic
|
||||
import HepLean.Tensors.Tree.NodeIdentities.PermProd
|
||||
import HepLean.Tensors.Tree.NodeIdentities.PermContr
|
||||
import HepLean.Tensors.Tree.NodeIdentities.ProdComm
|
||||
import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
|
||||
import HepLean.Tensors.Tree.NodeIdentities.ContrContr
|
||||
import HepLean.Tensors.ComplexLorentz.Basis
|
||||
import LLMLean
|
||||
import HepLean.Tensors.ComplexLorentz.BasisTrees
|
||||
/-!
|
||||
|
||||
## Lemmas related to complex Lorentz tensors.
|
||||
|
|
@ -33,7 +23,7 @@ open OverColor.Discrete
|
|||
noncomputable section
|
||||
|
||||
namespace Fermion
|
||||
|
||||
open complexLorentzTensor
|
||||
set_option maxRecDepth 20000 in
|
||||
lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
|
||||
{T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
|
||||
|
|
@ -117,8 +107,8 @@ lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
|
|||
And related results.
|
||||
|
||||
-/
|
||||
open complexLorentzTensor
|
||||
|
||||
/-- The map to color one gets when multiplying left and right metrics. -/
|
||||
def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
|
||||
finSumFinEquiv.symm
|
||||
|
||||
|
|
@ -154,659 +144,20 @@ lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.righ
|
|||
funext x
|
||||
fin_cases x <;> rfl
|
||||
|
||||
def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
|
||||
⇑finSumFinEquiv.symm) ∘ Fin.succAbove 0 ∘ Fin.succAbove 1)
|
||||
|
||||
abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) :=
|
||||
(Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
|
||||
|
||||
lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(TensorTree.prod t (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)).tensor = (((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
|
||||
(((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
|
||||
(((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
|
||||
(((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
|
||||
((TensorTree.smul (-I) ((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
|
||||
((TensorTree.smul I ((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
|
||||
((t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0))).add
|
||||
(TensorTree.smul (-1) (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR]
|
||||
fun | 0 => 3 | 1 => 1 | 2 => 1))))))))))).tensor := by
|
||||
rw [prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
|
||||
rw [prod_add _ _ _]
|
||||
rw [add_tensor_eq_snd <| prod_add _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
prod_add _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
|
||||
/- Moving smuls. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd<| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| prod_smul _ _ _]
|
||||
rfl
|
||||
|
||||
lemma contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
|
||||
(h : (pauliMatrixContrMap c) (i.succAbove j) =
|
||||
complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
|
||||
(contr i j h (TensorTree.prod t
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor))).tensor =
|
||||
((contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
|
||||
((contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
|
||||
((contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
|
||||
((contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
|
||||
((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
|
||||
((TensorTree.smul I (contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
|
||||
((contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0)))).add
|
||||
(TensorTree.smul (-1) (contr i j h (t.prod (tensorNode
|
||||
(basisVector ![Color.up, Color.upL, Color.upR]
|
||||
fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| prod_pauliMatrix_basis_tree_expand _]
|
||||
/- Moving contr over add. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contr over smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
contr_smul _ _]
|
||||
|
||||
lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(i : Fin (n + 3)) (j : Fin (n +2))
|
||||
(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
|
||||
((pauliMatrixContrMap c) i))
|
||||
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
|
||||
let c' := Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm
|
||||
let b' (i1 i2 i3 : Fin 4) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
|
||||
((HepLean.PiTensorProduct.elimPureTensor b (fun | 0 => i1 | 1 => i2 | 2 => i3))
|
||||
(finSumFinEquiv.symm i))
|
||||
(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor))).tensor = ((contr i j h ((tensorNode
|
||||
(basisVector c' (b' 0 0 0))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 0 1 1))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 1 0 1))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 1 1 0))))).add
|
||||
((TensorTree.smul (-I) (contr i j h ((tensorNode (basisVector c' (b' 2 0 1)))))).add
|
||||
((TensorTree.smul I (contr i j h ((tensorNode (basisVector c' (b' 2 1 0)))))).add
|
||||
((contr i j h ((tensorNode (basisVector c' (b' 3 0 0))))).add
|
||||
(TensorTree.smul (-1) (contr i j h ((tensorNode
|
||||
(basisVector c' (b' 3 1 1))))))))))))).tensor := by
|
||||
rw [contr_pauliMatrix_basis_tree_expand]
|
||||
/- Product of basis vectors . -/
|
||||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||||
<| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||||
<| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||||
<| prod_basisVector_tree _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||||
rfl
|
||||
|
||||
lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(i : Fin (n + 3)) (j : Fin (n +2))
|
||||
(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
|
||||
((pauliMatrixContrMap c) i))
|
||||
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
|
||||
let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
|
||||
∘ Fin.succAbove i ∘ Fin.succAbove j
|
||||
let b'' (i1 i2 i3 : Fin 4) : (i : Fin (n + (Nat.succ 0).succ.succ)) →
|
||||
Fin (complexLorentzTensor.repDim (Sum.elim c ![Color.up, Color.upL, Color.upR]
|
||||
(finSumFinEquiv.symm i))) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
|
||||
((HepLean.PiTensorProduct.elimPureTensor b (fun | (0 : Fin 3) => i1 | 1 => i2 | 2 => i3))
|
||||
(finSumFinEquiv.symm i))
|
||||
let b' (i1 i2 i3 : Fin 4) := fun k => (b'' i1 i2 i3) (i.succAbove (j.succAbove k))
|
||||
(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor))).tensor = (((
|
||||
TensorTree.smul (contrBasisVectorMul i j (b'' 0 0 0))
|
||||
(tensorNode (basisVector c' (b' 0 0 0))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1))
|
||||
(tensorNode (basisVector c' (b' 0 1 1))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1))
|
||||
(tensorNode (basisVector c' (b' 1 0 1))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0))
|
||||
(tensorNode (basisVector c' (b' 1 1 0))))).add
|
||||
((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1))
|
||||
(tensorNode (basisVector c' (b' 2 0 1)))))).add
|
||||
((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0))
|
||||
(tensorNode (basisVector c' (b' 2 1 0)))))).add
|
||||
(((TensorTree.smul (contrBasisVectorMul i j (b'' 3 0 0))
|
||||
(tensorNode (basisVector c' (b' 3 0 0))))).add
|
||||
(TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (b'' 3 1 1)) (tensorNode
|
||||
(basisVector c' (b' 3 1 1))))))))))))).tensor := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand']
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||||
<| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
smul_tensor_eq <| contr_basisVector_tree _]
|
||||
|
||||
lemma pauliMatrix_contr_down_0 :
|
||||
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [one_smul, zero_smul, smul_zero, add_zero]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_0_tree :
|
||||
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
lemma leftMetric_mul_rightMetric_tree :
|
||||
{Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)))
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
exact pauliMatrix_contr_down_0
|
||||
|
||||
lemma pauliMatrix_contr_down_1 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_1_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))).tensor := by
|
||||
exact pauliMatrix_contr_down_1
|
||||
|
||||
lemma pauliMatrix_contr_down_2 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
+ (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_2_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
(smul (- I) (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))))
|
||||
(smul I (tensorNode (basisVector
|
||||
pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))))).tensor := by
|
||||
exact pauliMatrix_contr_down_2
|
||||
|
||||
lemma pauliMatrix_contr_down_3 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_3_tree : {(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))))
|
||||
(smul (-1) (tensorNode (basisVector pauliMatrixLowerMap
|
||||
(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
|
||||
exact pauliMatrix_contr_down_3
|
||||
|
||||
def pauliMatrixContrPauliMatrixMap := ((Sum.elim
|
||||
((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 2)
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
(fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_lower : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
|
||||
+ I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
- I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [contr_tensor_eq <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| smul_prod _ _ _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| pauliMatrix_contr_down_0_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| pauliMatrix_contr_down_1_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_2_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_3_tree]
|
||||
/- Simplifying -/
|
||||
simp only [add_tensor, smul_tensor, tensorNode_tensor, smul_add,_root_.smul_smul]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, neg_smul, one_smul, mul_neg, neg_mul, one_mul,
|
||||
_root_.neg_neg, mul_one]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_lower_tree : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
|
||||
TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul I (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-I) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)))) <|
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
rw [pauliMatrix_lower]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, add_tensor,
|
||||
tensorNode_tensor, smul_tensor, neg_smul, one_smul]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix_aux :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
|
||||
= ((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)).add
|
||||
((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1)).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul I (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
I •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-I) (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1))).add
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
|
||||
(tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| pauliMatrix_lower_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [contr_tensor_eq <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_0_1_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_tensor_eq <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_2_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor
|
||||
<| pauliMatrix_contr_lower_2_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_3_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_3_1_1]
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor =
|
||||
2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)
|
||||
+ 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) := by
|
||||
rw [pauliMatrix_contract_pauliMatrix_aux]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, neg_smul,
|
||||
one_smul, add_tensor, tensorNode_tensor, smul_tensor, smul_add, smul_neg, _root_.smul_smul,
|
||||
neg_mul, _root_.neg_neg]
|
||||
ring_nf
|
||||
rw [Complex.I_sq]
|
||||
simp only [neg_smul, one_smul, _root_.neg_neg]
|
||||
abel
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
|
||||
rw [leftMetric_mul_rightMetric]
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
|
||||
smul_tensor, neg_smul, one_smul]
|
||||
rfl
|
||||
|
||||
end Fermion
|
||||
|
||||
|
|
|
|||
373
HepLean/Tensors/ComplexLorentz/PauliContr.lean
Normal file
373
HepLean/Tensors/ComplexLorentz/PauliContr.lean
Normal file
|
|
@ -0,0 +1,373 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.ComplexLorentz.PauliLower
|
||||
import HepLean.Tensors.ComplexLorentz.Lemmas
|
||||
/-!
|
||||
|
||||
## Contractiong of indices of Pauli matrix.
|
||||
|
||||
The main result of this file is `pauliMatrix_contract_pauliMatrix` which states that
|
||||
`η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`.
|
||||
|
||||
The current way this result is proved is by using tensor tree manipulations.
|
||||
There is likely a more direct path to this result.
|
||||
|
||||
-/
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open MonoidalCategory
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
open TensorProduct
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open TensorTree
|
||||
open OverColor.Discrete
|
||||
noncomputable section
|
||||
|
||||
namespace Fermion
|
||||
open complexLorentzTensor
|
||||
|
||||
/-- The map to colors one gets when contracting the 4-vector indices pauli matrices. -/
|
||||
def pauliMatrixContrPauliMatrixMap := ((Sum.elim
|
||||
((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 2)
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
(fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
/-! TODO: Work out why `pauliMatrix_lower_basis_expand_prod'` is needed. -/
|
||||
/-- This lemma is exactly the same as `pauliMatrix_lower_basis_expand_prod'`.
|
||||
It is needed here for `pauliMatrix_contract_pauliMatrix_aux`. It is unclear why
|
||||
`pauliMatrix_lower_basis_expand_prod` does not work. -/
|
||||
private lemma pauliMatrix_lower_basis_expand_prod' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
|
||||
((((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul I ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-I) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
t))))))))).tensor := by
|
||||
exact pauliMatrix_lower_basis_expand_prod _
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix_aux :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
|
||||
= ((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)).add
|
||||
((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1)).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul I (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
I •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-I) (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1))).add
|
||||
(tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| pauliMatrix_lower_basis_expand_prod' _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_0_1_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_tensor_eq <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_2_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor
|
||||
<| pauliMatrix_contr_lower_2_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_3_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_3_1_1]
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix_expand :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor =
|
||||
2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)
|
||||
+ 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) := by
|
||||
rw [pauliMatrix_contract_pauliMatrix_aux]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, neg_smul,
|
||||
one_smul, add_tensor, tensorNode_tensor, smul_tensor, smul_add, smul_neg, _root_.smul_smul,
|
||||
neg_mul, _root_.neg_neg]
|
||||
ring_nf
|
||||
rw [Complex.I_sq]
|
||||
simp only [neg_smul, one_smul, _root_.neg_neg]
|
||||
abel
|
||||
|
||||
/-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
|
||||
theorem pauliMatrix_contract_pauliMatrix :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β' =
|
||||
2 •ₜ Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ := by
|
||||
rw [pauliMatrix_contract_pauliMatrix_expand]
|
||||
rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
|
||||
rw [perm_smul]
|
||||
/- Moving perm through adds. -/
|
||||
rw [smul_tensor_eq <| perm_add _ _ _]
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| perm_add _ _ _]
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| perm_add _ _ _]
|
||||
/- Moving perm through smul. -/
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| perm_smul _ _ _]
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| perm_smul _ _ _]
|
||||
/- Perm acting on basis. -/
|
||||
erw [smul_tensor_eq <| add_tensor_eq_fst <| perm_basisVector_tree _ _]
|
||||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
perm_basisVector_tree _ _]
|
||||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_tensor_eq <| perm_basisVector_tree _ _]
|
||||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
perm_basisVector_tree _ _]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
have h1 (b0011 b1100 b0110 b1001 : CoeSort.coe (complexLorentzTensor.F.obj
|
||||
(OverColor.mk pauliMatrixContrPauliMatrixMap))) :
|
||||
((2 • b0011 + 2 • b1100) - 2 • b0110 - 2 • b1001) = (2 : ℂ) • ((b0011) +
|
||||
(((-1 : ℂ)• b0110) + (((-1 : ℂ) •b1001) + b1100))) := by
|
||||
trans (2 : ℂ) • b0011 + (2 : ℂ) • b1100 - ((2 : ℂ) • b0110) - ((2 : ℂ) • b1001)
|
||||
· repeat rw [two_smul]
|
||||
· simp only [neg_smul, one_smul, smul_add, smul_neg]
|
||||
abel
|
||||
rw [h1]
|
||||
congr
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
|
||||
end Fermion
|
||||
257
HepLean/Tensors/ComplexLorentz/PauliLower.lean
Normal file
257
HepLean/Tensors/ComplexLorentz/PauliLower.lean
Normal file
|
|
@ -0,0 +1,257 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.ComplexLorentz.BasisTrees
|
||||
/-!
|
||||
|
||||
## Lowering indices of Pauli matrices.
|
||||
|
||||
-/
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open MonoidalCategory
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
open TensorProduct
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open TensorTree
|
||||
open OverColor.Discrete
|
||||
noncomputable section
|
||||
|
||||
namespace Fermion
|
||||
open complexLorentzTensor
|
||||
|
||||
/-- The map to color one gets when lowering the indices of pauli matrices. -/
|
||||
def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
|
||||
⇑finSumFinEquiv.symm) ∘ Fin.succAbove 0 ∘ Fin.succAbove 1)
|
||||
|
||||
lemma pauliMatrix_contr_down_0 :
|
||||
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [one_smul, zero_smul, smul_zero, add_zero]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_0_tree :
|
||||
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)))
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
exact pauliMatrix_contr_down_0
|
||||
|
||||
lemma pauliMatrix_contr_down_1 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_1_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))).tensor := by
|
||||
exact pauliMatrix_contr_down_1
|
||||
|
||||
lemma pauliMatrix_contr_down_2 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
+ (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_2_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
(smul (- I) (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))))
|
||||
(smul I (tensorNode (basisVector
|
||||
pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))))).tensor := by
|
||||
exact pauliMatrix_contr_down_2
|
||||
|
||||
lemma pauliMatrix_contr_down_3 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_3_tree : {(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))))
|
||||
(smul (-1) (tensorNode (basisVector pauliMatrixLowerMap
|
||||
(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
|
||||
exact pauliMatrix_contr_down_3
|
||||
|
||||
lemma pauliMatrix_lower : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
|
||||
+ I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
- I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [contr_tensor_eq <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| smul_prod _ _ _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| pauliMatrix_contr_down_0_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| pauliMatrix_contr_down_1_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_2_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_3_tree]
|
||||
/- Simplifying -/
|
||||
simp only [add_tensor, smul_tensor, tensorNode_tensor, smul_add,_root_.smul_smul]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, neg_smul, one_smul, mul_neg, neg_mul, one_mul,
|
||||
_root_.neg_neg, mul_one]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_lower_tree : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
|
||||
TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul I (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-I) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)))) <|
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
rw [pauliMatrix_lower]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, add_tensor,
|
||||
tensorNode_tensor, smul_tensor, neg_smul, one_smul]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_lower_basis_expand_prod {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
|
||||
(((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul I ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-I) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
t)))))))).tensor := by
|
||||
rw [prod_tensor_eq_fst <| pauliMatrix_lower_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
|
||||
end Fermion
|
||||
Loading…
Add table
Add a link
Reference in a new issue