feat: Contractions with pauli matrices
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jstoobysmith
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ca55da6a34
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685c1b293c
4 changed files with 940 additions and 81 deletions
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@ -98,6 +98,34 @@ 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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{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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(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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{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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(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 [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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{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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(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 [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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@ -135,6 +163,16 @@ 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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{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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(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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{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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@ -183,6 +221,13 @@ lemma coMetric_basis_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
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simp only [Fin.isValue, Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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lemma coMetric_basis_expand_tree : {Lorentz.coMetric | μ ν}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 0))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 1)))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 2)))) <|
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(smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 3))))).tensor :=
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coMetric_basis_expand
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/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors. -/
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lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
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basisVector ![Color.up, Color.up] (fun _ => 0)
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@ -208,6 +253,13 @@ lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
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simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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lemma contrMatrix_basis_expand_tree : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 1)))) <|
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TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 2)))) <|
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(smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 3))))).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 => 1 | 1 => 0) := by
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@ -225,6 +277,11 @@ lemma leftMetric_expand : {Fermion.leftMetric | α β}ᵀ.tensor =
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· rfl
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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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(tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
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leftMetric_expand
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lemma altLeftMetric_expand : {Fermion.altLeftMetric | α β}ᵀ.tensor =
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basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1)
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- basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0) := by
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@ -241,6 +298,11 @@ lemma altLeftMetric_expand : {Fermion.altLeftMetric | α β}ᵀ.tensor =
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· rfl
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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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altLeftMetric_expand
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lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
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- basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)
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+ basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0) := by
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@ -258,6 +320,11 @@ lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
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· rfl
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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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(tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
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rightMetric_expand
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lemma altRightMetric_expand : {Fermion.altRightMetric | α β}ᵀ.tensor =
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basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1)
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- basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0) := by
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@ -274,6 +341,11 @@ lemma altRightMetric_expand : {Fermion.altRightMetric | α β}ᵀ.tensor =
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· rfl
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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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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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lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
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basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 0 | 2 => 0)
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@ -312,6 +384,27 @@ lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
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| (1 : Fin 3) => rfl
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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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(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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(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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TensorTree.add (smul I (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
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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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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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rfl
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end complexLorentzTensor
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end Fermion
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