refactor: Simplify proofs
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jstoobysmith
parent
33a42c7e06
commit
95857993b5
6 changed files with 366 additions and 496 deletions
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@ -88,7 +88,6 @@ lemma complexCoBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
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def complexCoBasisFin4 : Basis (Fin 4) ℂ complexCo :=
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Basis.reindex complexCoBasis finSumFinEquiv
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/-!
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## Relation to real
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@ -184,6 +184,5 @@ lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.re
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| Color.up => Lorentz.contrCoContraction_basis _ _
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| Color.down => Lorentz.coContrContraction_basis _ _
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end
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end Fermion
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@ -79,11 +79,32 @@ 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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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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(if (b i).val = (b (i.succAbove j)).val then (1 : ℂ) else 0)
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lemma contrBasisVectorMul_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} {b : Π k, Fin (complexLorentzTensor.repDim (c k))}
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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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exact h
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lemma contrBasisVectorMul_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} {b : Π k, Fin (complexLorentzTensor.repDim (c k))}
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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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exact h
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lemma contr_basisVector {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 = (if (b i).val = (b (i.succAbove j)).val
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then (1 : ℂ) else 0) • basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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(contr i j h (tensorNode (basisVector c b))).tensor = (contrBasisVectorMul i j b) •
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basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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(fun k => b (i.succAbove (j.succAbove k))) := by
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rw [contr_tensor]
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simp only [Nat.succ_eq_add_one, tensorNode_tensor]
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@ -102,8 +123,7 @@ lemma contr_basisVector_tree {n : ℕ} {c : Fin n.succ.succ → complexLorentzT
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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 ((if (b i).val = (b (i.succAbove j)).val
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then (1 : ℂ) else 0)) (tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
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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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exact contr_basisVector _
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@ -113,7 +133,7 @@ lemma contr_basisVector_tree_pos {n : ℕ} {c : Fin n.succ.succ → complexLore
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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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(fun k => b (i.succAbove (j.succAbove k)))))).tensor := by
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rw [contr_basisVector_tree]
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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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@ -122,10 +142,11 @@ lemma contr_basisVector_tree_neg {n : ℕ} {c : Fin n.succ.succ → complexLore
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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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(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]
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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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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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@ -153,18 +153,59 @@ lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.righ
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funext x
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fin_cases x <;> rfl
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def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 0 ∘ Fin.succAbove 1)
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abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
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lemma pauliMatrix_contr_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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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) = complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
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(contr i j h (TensorTree.prod t (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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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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@ -180,26 +221,7 @@ lemma pauliMatrix_contr_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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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) (contr i j h (t.prod (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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rw [contr_tensor_eq <| prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
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rw [contr_tensor_eq <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| 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 [contr_tensor_eq <| 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 [contr_tensor_eq <| 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 [contr_tensor_eq <| 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 [contr_tensor_eq <| 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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rw [contr_tensor_eq <| prod_pauliMatrix_basis_tree_expand _]
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/- Moving contr over add. -/
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rw [contr_add]
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rw [add_tensor_eq_snd <| contr_add _ _]
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@ -219,15 +241,26 @@ lemma pauliMatrix_contr_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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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 <| add_tensor_eq_snd <|
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contr_smul _ _]
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rfl
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lemma pauliMatrix_contr_down_0 :
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(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor)))).tensor
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= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
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+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
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rw [pauliMatrix_contr_expand]
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lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ ((pauliMatrixContrMap c) i))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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let c' := Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm
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let b' (i1 i2 i3 : Fin 4) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
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((HepLean.PiTensorProduct.elimPureTensor b (fun | 0 => i1 | 1 => i2 | 2 => i3)) (finSumFinEquiv.symm i))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b)) (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor = ((contr i j h ((tensorNode
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(basisVector c' (b' 0 0 0))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 0 1 1))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 1 0 1))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 1 1 0))))).add
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((TensorTree.smul (-I) (contr i j h ((tensorNode (basisVector c' (b' 2 0 1)))))).add
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((TensorTree.smul I (contr i j h ((tensorNode (basisVector c' (b' 2 1 0)))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 3 0 0))))).add
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(TensorTree.smul (-1) (contr i j h ((tensorNode
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(basisVector c' (b' 3 1 1))))))))))))).tensor := by
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rw [contr_pauliMatrix_basis_tree_expand]
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/- Product of basis vectors . -/
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rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
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@ -246,26 +279,62 @@ lemma pauliMatrix_contr_down_0 :
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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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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
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<| contr_tensor_eq <| prod_basisVector_tree _ _]
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rfl
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lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ ((pauliMatrixContrMap c) i))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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let c' := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm)
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∘ Fin.succAbove i ∘ Fin.succAbove j
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let b'' (i1 i2 i3 : Fin 4) : (i : Fin (n + (Nat.succ 0).succ.succ)) →
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Fin (complexLorentzTensor.repDim (Sum.elim c ![Color.up, Color.upL, Color.upR] (finSumFinEquiv.symm i))) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
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((HepLean.PiTensorProduct.elimPureTensor b (fun | (0 : Fin 3) => i1 | 1 => i2 | 2 => i3)) (finSumFinEquiv.symm i))
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let b' (i1 i2 i3 : Fin 4) := fun k => (b'' i1 i2 i3) (i.succAbove (j.succAbove k))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor = (((
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TensorTree.smul (contrBasisVectorMul i j (b'' 0 0 0)) (tensorNode (basisVector c' (b' 0 0 0))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1)) (tensorNode (basisVector c' (b' 0 1 1))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1)) (tensorNode (basisVector c' (b' 1 0 1))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0)) (tensorNode (basisVector c' (b' 1 1 0))))).add
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((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1)) (tensorNode (basisVector c' (b' 2 0 1)))))).add
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((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0)) (tensorNode (basisVector c' (b' 2 1 0)))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 3 0 0)) (tensorNode (basisVector c' (b' 3 0 0))))).add
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(TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (b'' 3 1 1)) (tensorNode
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(basisVector c' (b' 3 1 1))))))))))))).tensor := by
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rw [basis_contr_pauliMatrix_basis_tree_expand']
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/- Contracting basis vectors. -/
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rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
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rw [add_tensor_eq_fst <| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
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<| contr_basisVector_tree_neg _ ]
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<| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
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<| add_tensor_eq_fst <| contr_basisVector_tree _]
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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 <| smul_tensor_eq <| contr_basisVector_tree_neg _]
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree _]
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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 <| smul_tensor_eq
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<| contr_basisVector_tree_neg _]
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<| contr_basisVector_tree _]
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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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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
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<| 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_neg _]
|
||||
/- Simplifying. -/
|
||||
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 [smul_zero, add_zero]
|
||||
simp only [one_smul, zero_smul, smul_zero, add_zero]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -274,51 +343,28 @@ lemma pauliMatrix_contr_down_0 :
|
|||
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 : (contr 0 1 rfl
|
||||
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 1)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
|
||||
rw [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_pos _ rfl]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
/- Simplifying. -/
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -341,45 +387,14 @@ lemma pauliMatrix_contr_down_2 : (contr 0 1 rfl
|
|||
PauliMatrix.asConsTensor)))).tensor
|
||||
= (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
+ (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||||
rw [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_pos _]
|
||||
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_pos _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -388,51 +403,30 @@ lemma pauliMatrix_contr_down_2 : (contr 0 1 rfl
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_2_tree : (contr 0 1 rfl
|
||||
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 2)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 : (contr 0 1 rfl
|
||||
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 3)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_pos _]
|
||||
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_pos _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -441,6 +435,16 @@ lemma pauliMatrix_contr_down_3 : (contr 0 1 rfl
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_3_tree : (contr 0 1 rfl
|
||||
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 3)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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)
|
||||
|
|
@ -453,45 +457,14 @@ lemma pauliMatrix_contr_lower_0_0_0 : (contr 0 2 rfl
|
|||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -505,45 +478,14 @@ lemma pauliMatrix_contr_lower_0_1_1 : (contr 0 2 rfl
|
|||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -552,51 +494,19 @@ lemma pauliMatrix_contr_lower_0_1_1 : (contr 0 2 rfl
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_0_1 : (contr 0 2 rfl
|
||||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1))).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_pos _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -610,45 +520,14 @@ lemma pauliMatrix_contr_lower_1_1_0 : (contr 0 2 rfl
|
|||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_pos _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -663,45 +542,14 @@ lemma pauliMatrix_contr_lower_2_0_1 : (contr 0 2 rfl
|
|||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_pos _]
|
||||
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_pos _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -716,45 +564,14 @@ lemma pauliMatrix_contr_lower_2_1_0 : (contr 0 2 rfl
|
|||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_pos _]
|
||||
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_pos _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -763,52 +580,21 @@ lemma pauliMatrix_contr_lower_2_1_0 : (contr 0 2 rfl
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_0_0 : (contr 0 2 rfl
|
||||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_pos _]
|
||||
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_pos _]
|
||||
+ (-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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -817,52 +603,20 @@ lemma pauliMatrix_contr_lower_3_0_0 : (contr 0 2 rfl
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_1_1 : (contr 0 2 rfl
|
||||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1))).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).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 [pauliMatrix_contr_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 _ _]
|
||||
/- Contracting basis vectors. -/
|
||||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree_neg _ ]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||||
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_neg _]
|
||||
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_neg _]
|
||||
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_pos _]
|
||||
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_pos _]
|
||||
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 [smul_zero, add_zero, zero_add]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -871,18 +625,15 @@ lemma pauliMatrix_contr_lower_3_1_1 : (contr 0 2 rfl
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
|
||||
|
||||
lemma pauliMatrix_lower :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
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
|
||||
- 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 _ _ _]
|
||||
|
|
@ -903,16 +654,117 @@ lemma pauliMatrix_lower :
|
|||
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. -/
|
||||
sorry
|
||||
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
|
||||
(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 _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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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 _ _]
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/- Moving contraction through smul. -/
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
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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 _ _]
|
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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 _ _]
|
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/- Replacing the contractions. -/
|
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rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_1_1]
|
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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]
|
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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
|
||||
= basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
- basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||||
- basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
sorry
|
||||
{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
|
||||
|
||||
end Fermion
|
||||
|
||||
end
|
||||
|
|
|
|||
|
|
@ -286,7 +286,6 @@ lemma μ_tmul_tprod_mk {X Y : Type} {cX : X → C} {cY : Y → C}
|
|||
| Sum.inl i => rfl
|
||||
| Sum.inr i => rfl
|
||||
|
||||
|
||||
lemma μ_natural_left {X Y : OverColor C} (f : X ⟶ Y) (Z : OverColor C) :
|
||||
MonoidalCategory.whiskerRight (objMap' F f) (objObj' F Z) ≫ (μ F Y Z).hom =
|
||||
(μ F X Z).hom ≫ objMap' F (MonoidalCategory.whiskerRight f Z) := by
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue