feat: Fix pauli matrices as tensors. Speedup
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jstoobysmith
parent
c565e7ea1c
commit
e6ef68d7e6
9 changed files with 692 additions and 322 deletions
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@ -18,9 +18,6 @@ import HepLean.SpaceTime.PauliMatrices.AsTensor
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## Complex Lorentz tensors
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-/
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namespace Fermion
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open Matrix
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open MatrixGroups
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open Complex
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@ -29,6 +26,8 @@ open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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namespace complexLorentzTensor
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/-- The colors associated with complex representations of SL(2, ℂ) of intrest to physics. -/
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inductive Color
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| upL : Color
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@ -78,11 +77,13 @@ instance : DecidableEq Color := fun x y =>
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| Color.down, Color.downR => isFalse fun h => Color.noConfusion h
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| Color.down, Color.up => isFalse fun h => Color.noConfusion h
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noncomputable section
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end complexLorentzTensor
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noncomputable section
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open complexLorentzTensor in
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/-- The tensor structure for complex Lorentz tensors. -/
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def complexLorentzTensor : TensorSpecies where
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C := Fermion.Color
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C := complexLorentzTensor.Color
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G := SL(2, ℂ)
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G_group := inferInstance
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k := ℂ
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@ -191,8 +192,8 @@ def complexLorentzTensor : TensorSpecies where
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| Color.downR => by simpa using Fermion.altRightContraction_apply_metric
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| Color.up => by simpa using Lorentz.contrCoContraction_apply_metric
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| Color.down => by simpa using Lorentz.coContrContraction_apply_metric
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instance : DecidableEq complexLorentzTensor.C := Fermion.instDecidableEqColor
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namespace complexLorentzTensor
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instance : DecidableEq complexLorentzTensor.C := complexLorentzTensor.instDecidableEqColor
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lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.repDim c))
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(j : Fin (complexLorentzTensor.repDim (complexLorentzTensor.τ c))) :
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@ -208,5 +209,6 @@ 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 complexLorentzTensor
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end
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end Fermion
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@ -32,8 +32,8 @@ open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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open Fermion
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noncomputable section
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namespace Fermion
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namespace complexLorentzTensor
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/-- Basis vectors for complex Lorentz tensors. -/
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@ -95,7 +95,7 @@ def contrBasisVectorMul {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.
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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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(h : ¬ (b i).val = (b (i.succAbove j)).val) :
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contrBasisVectorMul i j b = 0 := by
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rw [contrBasisVectorMul]
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simp only [ite_eq_else, one_ne_zero, imp_false]
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@ -103,7 +103,7 @@ lemma contrBasisVectorMul_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzT
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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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(h : (b i).val = (b (i.succAbove j)).val) :
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contrBasisVectorMul i j b = 1 := by
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rw [contrBasisVectorMul]
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simp only [ite_eq_then, zero_ne_one, imp_false, Decidable.not_not]
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@ -156,7 +156,7 @@ lemma contr_basisVector_tree_neg {n : ℕ} {c : Fin n.succ.succ → complexLoren
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(tensorNode 0).tensor := by
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rw [contr_basisVector_tree, contrBasisVectorMul]
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rw [if_neg hn]
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simp only [Nat.succ_eq_add_one, smul_tensor, tensorNode_tensor, zero_smul]
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simp only [Nat.succ_eq_add_one, smul_tensor, tensorNode_tensor, _root_.zero_smul]
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/-- Equivalence of Fin types appearing in the product of two basis vectors. -/
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def prodBasisVecEquiv {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
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@ -214,7 +214,7 @@ lemma eval_basisVector {n : ℕ} {c : Fin n.succ → complexLorentzTensor.C}
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basisVector (c ∘ Fin.succAbove i) (fun k => b (i.succAbove k)) := by
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rw [eval_tensor, basisVector, basisVector]
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simp only [Nat.succ_eq_add_one, Functor.id_obj, OverColor.mk_hom, tensorNode_tensor,
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Function.comp_apply, one_smul, zero_smul]
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Function.comp_apply, one_smul, _root_.zero_smul]
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erw [TensorSpecies.evalMap_tprod]
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congr 1
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have h1 : Fin.ofNat' ↑j (@Fin.size_pos' (complexLorentzTensor.repDim (c i)) _) = j := by
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@ -448,5 +448,4 @@ lemma pauliMatrix_basis_expand_tree : {PauliMatrix.asConsTensor | μ α β}ᵀ.t
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rfl
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end complexLorentzTensor
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end Fermion
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end
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@ -36,8 +36,8 @@ open OverColor.Discrete
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noncomputable section
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namespace Fermion
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open complexLorentzTensor
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namespace complexLorentzTensor
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open Fermion
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/-!
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@ -177,6 +177,25 @@ lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → comple
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<| contr_tensor_eq <| prod_basisVector_tree _ _]
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rfl
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def pauliMatrixBasisProdMap
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{n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4) :
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(i : Fin (n + (Nat.succ 0).succ.succ)) →
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Fin (complexLorentzTensor.repDim (Sum.elim c ![Color.up, Color.upL, Color.upR]
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(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))
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(finSumFinEquiv.symm i))
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def basisVectorContrPauli {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k)))
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(i1 i2 i3 : Fin 4) :=
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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) := fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
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basisVector c' (b' i1 i2 i3)
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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
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@ -184,30 +203,25 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
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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]
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(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))
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(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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let b' (i1 i2 i3 : Fin 4) := fun k => (pauliMatrixBasisProdMap 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))
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TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0))
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(tensorNode (basisVector c' (b' 0 0 0))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 0 1 1))
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1))
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(tensorNode (basisVector c' (b' 0 1 1))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 0 1))
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1))
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(tensorNode (basisVector c' (b' 1 0 1))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 1 1 0))
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0))
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(tensorNode (basisVector c' (b' 1 1 0))))).add
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((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 0 1))
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((TensorTree.smul (-I) ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1))
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(tensorNode (basisVector c' (b' 2 0 1)))))).add
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((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (b'' 2 1 0))
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((TensorTree.smul I ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0))
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(tensorNode (basisVector c' (b' 2 1 0)))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (b'' 3 0 0))
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0))
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(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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(TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap 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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@ -227,7 +241,46 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
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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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smul_tensor_eq <| contr_basisVector_tree _]
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rfl
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end Fermion
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def pauilMatrixBasisContrMap {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(i : Fin (n + 3)) (j : Fin (n +2))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4) :
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(k : Fin (n + 1)) →
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Fin
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(complexLorentzTensor.repDim
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(Sum.elim c ![Color.up, Color.upL, Color.upR] (finSumFinEquiv.symm (i.succAbove (j.succAbove k))))) :=
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fun k => (pauliMatrixBasisProdMap b i1 i2 i3) (i.succAbove (j.succAbove k))
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lemma basis_contr_pauliMatrix_basis_tree_expand_tensor {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
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((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) := fun k => (pauliMatrixBasisProdMap 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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(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0)) • (basisVectorContrPauli i j b 0 0 0)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1)) • (basisVectorContrPauli i j b 0 1 1)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1)) • (basisVectorContrPauli i j b 1 0 1)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0)) • (basisVectorContrPauli i j b 1 1 0)
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+ (-I) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1)) • (basisVectorContrPauli i j b 2 0 1)
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+ I • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0)) • (basisVectorContrPauli i j b 2 1 0)
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+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0)) • (basisVectorContrPauli i j b 3 0 0)
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+ (-1 : ℂ) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) • (basisVectorContrPauli i j b 3 1 1) := by
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rw [basis_contr_pauliMatrix_basis_tree_expand]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val_one, head_cons, Fin.val_zero,
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Nat.cast_zero, cons_val_two, Fin.val_one, Nat.cast_one, add_tensor, smul_tensor,
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tensorNode_tensor, neg_smul, one_smul, Int.reduceNeg]
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simp_all only [Function.comp_apply, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
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rfl
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end complexLorentzTensor
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end
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@ -3,15 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.Tree.Elab
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import HepLean.Tensors.ComplexLorentz.Basic
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import Mathlib.LinearAlgebra.TensorProduct.Basis
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import HepLean.Tensors.Tree.NodeIdentities.Basic
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import HepLean.Tensors.Tree.NodeIdentities.PermProd
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import HepLean.Tensors.Tree.NodeIdentities.PermContr
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import HepLean.Tensors.Tree.NodeIdentities.ProdComm
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import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
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import HepLean.Tensors.Tree.NodeIdentities.ContrContr
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import HepLean.Tensors.ComplexLorentz.PauliLower
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/-!
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## Bispinors
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@ -30,18 +22,17 @@ open TensorTree
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open OverColor.Discrete
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open Fermion
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noncomputable section
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namespace Lorentz
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namespace complexContr
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namespace complexLorentzTensor
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open Lorentz
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/-- A bispinor `pᵃᵃ` created from a lorentz vector `p^μ`. -/
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def bispinorUp (p : complexContr) :=
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{p | μ ⊗ (Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β)}ᵀ.tensor
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def contrBispinorUp (p : complexContr) :=
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{p | μ ⊗ pauliCo | μ α β}ᵀ.tensor
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/-- A bispinor `pₐₐ` created from a lorentz vector `p^μ`. -/
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def bispinorDown (p : complexContr) :=
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{Fermion.altRightMetric | β β' ⊗ Fermion.altLeftMetric | α α' ⊗
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(complexContr.bispinorUp p) | α β}ᵀ.tensor
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end complexContr
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end Lorentz
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end complexLorentzTensor
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end
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@ -22,8 +22,8 @@ open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace Fermion
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open complexLorentzTensor
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namespace complexLorentzTensor
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open Fermion
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set_option maxRecDepth 20000 in
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lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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{T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
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@ -159,6 +159,6 @@ lemma leftMetric_mul_rightMetric_tree :
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smul_tensor, neg_smul, one_smul]
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rfl
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end Fermion
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end complexLorentzTensor
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end
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@ -29,28 +29,52 @@ open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace Fermion
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open complexLorentzTensor
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namespace complexLorentzTensor
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open Fermion
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/-- The map to colors one gets when contracting the 4-vector indices pauli matrices. -/
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def pauliMatrixContrPauliMatrixMap := ((Sum.elim
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((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 0 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 1 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 0 ∘ Fin.succAbove 2)
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lemma pauliMatrix_contr_lower_0_0_0 :
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{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
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{(basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
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PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
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+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
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rw [basis_contr_pauliMatrix_basis_tree_expand]
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rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
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contrBasisVectorMul_neg, contrBasisVectorMul_neg,
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contrBasisVectorMul_neg, contrBasisVectorMul_neg,
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contrBasisVectorMul_neg, contrBasisVectorMul_neg]
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -60,18 +84,42 @@ lemma pauliMatrix_contr_lower_0_0_0 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -81,18 +129,42 @@ lemma pauliMatrix_contr_lower_0_1_1 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -102,18 +174,42 @@ lemma pauliMatrix_contr_lower_1_0_1 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
|
|
@ -123,19 +219,43 @@ lemma pauliMatrix_contr_lower_1_1_0 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -145,19 +265,43 @@ lemma pauliMatrix_contr_lower_2_0_1 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -167,19 +311,43 @@ lemma pauliMatrix_contr_lower_2_1_0 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
(fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -189,19 +357,43 @@ lemma pauliMatrix_contr_lower_3_0_0 :
|
|||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
{(basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
rw [basisVectorContrPauli, basisVectorContrPauli]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
|
|
@ -210,35 +402,34 @@ lemma pauliMatrix_contr_lower_3_1_1 :
|
|||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
/-! TODO: Work out why `pauliMatrix_lower_basis_expand_prod'` is needed. -/
|
||||
/-- This lemma is exactly the same as `pauliMatrix_lower_basis_expand_prod'`.
|
||||
/-! TODO: Work out why `pauliCo_prod_basis_expand'` is needed. -/
|
||||
/-- This lemma is exactly the same as `pauliCo_prod_basis_expand`.
|
||||
It is needed here for `pauliMatrix_contract_pauliMatrix_aux`. It is unclear why
|
||||
`pauliMatrix_lower_basis_expand_prod` does not work. -/
|
||||
private lemma pauliMatrix_lower_basis_expand_prod' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
private lemma pauliCo_prod_basis_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
|
||||
(TensorTree.prod (tensorNode pauliCo) t).tensor =
|
||||
((((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul I ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-I) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
t))))))))).tensor := by
|
||||
exact pauliMatrix_lower_basis_expand_prod _
|
||||
exact pauliCo_prod_basis_expand _
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix_aux :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
|
||||
lemma pauliCo_contr_pauliContr_expand_aux :
|
||||
{pauliCo | μ α β ⊗ PauliMatrix.asConsTensor | μ 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
|
||||
|
|
@ -266,7 +457,7 @@ lemma pauliMatrix_contract_pauliMatrix_aux :
|
|||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| pauliMatrix_lower_basis_expand_prod' _]
|
||||
rw [contr_tensor_eq <| pauliCo_prod_basis_expand' _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
|
|
@ -309,9 +500,8 @@ lemma pauliMatrix_contract_pauliMatrix_aux :
|
|||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_3_1_1]
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix_expand :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor =
|
||||
lemma pauliCo_contr_pauliContr_expand :
|
||||
{pauliCo | ν 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)
|
||||
|
|
@ -326,9 +516,8 @@ lemma pauliMatrix_contract_pauliMatrix_expand :
|
|||
abel
|
||||
|
||||
/-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
|
||||
theorem pauliMatrix_contract_pauliMatrix :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β' =
|
||||
theorem pauliCo_contr_pauliContr :
|
||||
{pauliCo | ν α β ⊗ PauliMatrix.asConsTensor | ν α' β' =
|
||||
2 •ₜ Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ := by
|
||||
rw [pauliMatrix_contract_pauliMatrix_expand]
|
||||
rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
|
||||
|
|
@ -370,4 +559,4 @@ theorem pauliMatrix_contract_pauliMatrix :
|
|||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
|
||||
end Fermion
|
||||
end complexLorentzTensor
|
||||
|
|
|
|||
|
|
@ -26,135 +26,212 @@ namespace Fermion
|
|||
open complexLorentzTensor
|
||||
|
||||
/-- The map to color one gets when lowering the indices of pauli matrices. -/
|
||||
def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
|
||||
⇑finSumFinEquiv.symm) ∘ Fin.succAbove 0 ∘ Fin.succAbove 1)
|
||||
def pauliCoMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
|
||||
⇑finSumFinEquiv.symm) ∘ Fin.succAbove 1 ∘ Fin.succAbove 1)
|
||||
|
||||
lemma pauliMatrix_contr_down_0 :
|
||||
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(contr 1 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [one_smul, zero_smul, smul_zero, add_zero]
|
||||
= basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
congr 1
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_0_tree :
|
||||
(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||||
PauliMatrix.asConsTensor)))).tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)))
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
exact pauliMatrix_contr_down_0
|
||||
|
||||
lemma pauliMatrix_contr_down_1 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | ν μ ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
= basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
+ basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
congr 1
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_1_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))).tensor := by
|
||||
exact pauliMatrix_contr_down_1
|
||||
|
||||
lemma pauliMatrix_contr_down_2 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
+ (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
|
||||
= (- I) • basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
+ (I) • basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_2_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
(smul (- I) (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))))
|
||||
(smul I (tensorNode (basisVector
|
||||
pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))))).tensor := by
|
||||
exact pauliMatrix_contr_down_2
|
||||
|
||||
lemma pauliMatrix_contr_down_3 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
|
||||
= basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ (- 1 : ℂ) • basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
conv =>
|
||||
lhs
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; lhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; lhs; rhs; rhs; lhs
|
||||
rw [contrBasisVectorMul_neg (by decide)]
|
||||
conv =>
|
||||
lhs; lhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs;
|
||||
rw [contrBasisVectorMul_pos (by decide)]
|
||||
conv =>
|
||||
lhs
|
||||
simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
|
||||
congr 1
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_3_tree : {(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))))
|
||||
(smul (-1) (tensorNode (basisVector pauliMatrixLowerMap
|
||||
(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
|
||||
exact pauliMatrix_contr_down_3
|
||||
def pauliCo := {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν 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
|
||||
lemma tensoreNode_pauliCo : (tensorNode pauliCo).tensor =
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
|
||||
rw [pauliCo]
|
||||
rfl
|
||||
|
||||
set_option profiler true
|
||||
set_option profiler.threshold 10
|
||||
lemma pauliCo_basis_expand : pauliCo
|
||||
= basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
|
||||
- basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
- basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
|
||||
+ I • basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
- I • basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
|
||||
- basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
conv =>
|
||||
lhs
|
||||
rw [pauliCo]
|
||||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [contr_tensor_eq <| add_prod _ _ _]
|
||||
|
|
@ -174,61 +251,67 @@ lemma pauliMatrix_lower : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor
|
|||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| pauliMatrix_contr_down_0_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| pauliMatrix_contr_down_1_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_2_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_3_tree]
|
||||
/- Simplifying -/
|
||||
simp only [add_tensor, smul_tensor, tensorNode_tensor, smul_add,_root_.smul_smul]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, neg_smul, one_smul, mul_neg, neg_mul, one_mul,
|
||||
_root_.neg_neg, mul_one]
|
||||
rfl
|
||||
simp only [tensorNode_tensor, add_tensor, smul_tensor]
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, neg_smul, one_smul]
|
||||
conv =>
|
||||
lhs; lhs;
|
||||
rw [pauliMatrix_contr_down_0]
|
||||
conv =>
|
||||
lhs; rhs; lhs; rhs;
|
||||
rw [pauliMatrix_contr_down_1]
|
||||
conv =>
|
||||
lhs; rhs; rhs; lhs; rhs;
|
||||
rw [pauliMatrix_contr_down_2]
|
||||
conv =>
|
||||
lhs; rhs; rhs; rhs; rhs;
|
||||
rw [pauliMatrix_contr_down_3]
|
||||
simp only [neg_smul, one_smul]
|
||||
abel
|
||||
|
||||
lemma pauliMatrix_lower_tree : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
|
||||
lemma pauliCo_basis_expand_tree : pauliCo
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
|
||||
(basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
|
||||
TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
|
||||
(basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
|
||||
(basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
|
||||
(basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul I (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
|
||||
(basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-I) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
|
||||
(basisVector pauliCoMap (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]
|
||||
(basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)))) <|
|
||||
(tensorNode (basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
rw [pauliCo_basis_expand]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor, smul_tensor, neg_smul,
|
||||
one_smul]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_lower_basis_expand_prod {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
lemma pauliCo_prod_basis_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(prod {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ t).tensor =
|
||||
(prod (tensorNode pauliCo) t).tensor =
|
||||
(((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul I ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-I) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
((tensorNode
|
||||
(basisVector pauliMatrixLowerMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
t)))))))).tensor := by
|
||||
rw [prod_tensor_eq_fst <| pauliMatrix_lower_tree]
|
||||
rw [prod_tensor_eq_fst <| tensoreNode_pauliCo]
|
||||
rw [prod_tensor_eq_fst <| pauliCo_basis_expand_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_prod _ _ _]
|
||||
|
|
|
|||
|
|
@ -770,10 +770,63 @@ lemma smul_tensor_eq {T1 T2 : TensorTree S c} {a : S.k} (h : T1.tensor = T2.tens
|
|||
simp only [smul_tensor]
|
||||
rw [h]
|
||||
|
||||
lemma smul_mul_eq {T1 : TensorTree S c} {a b : S.k} (h : a = b) :
|
||||
(smul a T1).tensor = (smul b T1).tensor := by
|
||||
rw [h]
|
||||
|
||||
lemma eq_tensorNode_of_eq_tensor {T1 : TensorTree S c} {t : S.F.obj (OverColor.mk c)}
|
||||
(h : T1.tensor = t) : T1.tensor = (tensorNode t).tensor := by
|
||||
simpa using h
|
||||
|
||||
/-!
|
||||
|
||||
## The zero tensor tree
|
||||
|
||||
-/
|
||||
|
||||
/-- The zero tensor tree. -/
|
||||
def zeroTree {n : ℕ} {c : Fin n → S.C} : TensorTree S c := tensorNode 0
|
||||
|
||||
@[simp]
|
||||
lemma zeroTree_tensor {n : ℕ} {c : Fin n → S.C} : (zeroTree (c := c)).tensor = 0 := by
|
||||
rfl
|
||||
|
||||
lemma zero_smul {T1 : TensorTree S c} :
|
||||
(smul 0 T1).tensor = zeroTree.tensor := by
|
||||
simp only [smul_tensor, _root_.zero_smul, zeroTree_tensor]
|
||||
|
||||
lemma smul_zero {a : S.k} : (smul a (zeroTree (c :=c ))).tensor = zeroTree.tensor := by
|
||||
simp only [smul_tensor, zeroTree_tensor, _root_.smul_zero]
|
||||
|
||||
lemma zero_add {T1 : TensorTree S c} : (add zeroTree T1).tensor = T1.tensor := by
|
||||
simp only [add_tensor, zeroTree_tensor, _root_.zero_add]
|
||||
|
||||
lemma add_zero {T1 : TensorTree S c} : (add T1 zeroTree).tensor = T1.tensor := by
|
||||
simp only [add_tensor, zeroTree_tensor, _root_.add_zero]
|
||||
|
||||
lemma perm_zero {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C} (σ : (OverColor.mk c) ⟶
|
||||
(OverColor.mk c1)) : (perm σ zeroTree).tensor = zeroTree.tensor := by
|
||||
simp only [perm_tensor, zeroTree_tensor, map_zero]
|
||||
|
||||
lemma neg_zero : (neg (zeroTree (c := c))).tensor = zeroTree.tensor := by
|
||||
simp only [neg_tensor, zeroTree_tensor, _root_.neg_zero]
|
||||
|
||||
lemma contr_zero {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ}
|
||||
{h : c (i.succAbove j) = S.τ (c i)} : (contr i j h zeroTree).tensor = zeroTree.tensor := by
|
||||
simp only [contr_tensor, zeroTree_tensor, map_zero]
|
||||
|
||||
lemma zero_prod {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C} (t : TensorTree S c1) :
|
||||
(prod (zeroTree (c := c)) t).tensor = zeroTree.tensor := by
|
||||
simp only [prod_tensor, Functor.id_obj, OverColor.mk_hom, Action.instMonoidalCategory_tensorObj_V,
|
||||
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
|
||||
Action.FunctorCategoryEquivalence.functor_obj_obj, zeroTree_tensor, zero_tmul, map_zero]
|
||||
|
||||
lemma prod_zero {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C} (t : TensorTree S c) :
|
||||
(prod t (zeroTree (c := c1))).tensor = zeroTree.tensor := by
|
||||
simp only [prod_tensor, Functor.id_obj, OverColor.mk_hom, Action.instMonoidalCategory_tensorObj_V,
|
||||
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
|
||||
Action.FunctorCategoryEquivalence.functor_obj_obj, zeroTree_tensor, tmul_zero, map_zero]
|
||||
|
||||
/-- A structure containing a pair of indices (i, j) to be contracted in a tensor.
|
||||
This is used in some proofs of node identities for tensor trees. -/
|
||||
structure ContrPair {n : ℕ} (c : Fin n.succ.succ → S.C) where
|
||||
|
|
|
|||
|
|
@ -23,7 +23,7 @@ open Lean.Elab
|
|||
open Lean.Elab.Term
|
||||
open Lean Meta Elab Tactic
|
||||
open IndexNotation
|
||||
|
||||
open complexLorentzTensor
|
||||
namespace TensorTree
|
||||
|
||||
/-!
|
||||
|
|
@ -182,60 +182,60 @@ def stringToTerm (str : String) : TermElabM Term := do
|
|||
/-- Specific types of tensors which appear which we want to elaborate in specific ways. -/
|
||||
def specialTypes : List (String × (Term → Term)) := [
|
||||
("CoeSort.coe Lorentz.complexCo", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.down, T]),
|
||||
Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.down, T]),
|
||||
("CoeSort.coe Lorentz.complexContr", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.up, T]),
|
||||
Syntax.mkApp (mkIdent ``TensorTree.vecNodeE) #[mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.up, T]),
|
||||
("ModuleCat.carrier (Lorentz.complexContr ⊗ Lorentz.complexCo).V", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.up, mkIdent ``Fermion.Color.down, T]),
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.up, mkIdent ``complexLorentzTensor.Color.down, T]),
|
||||
("ModuleCat.carrier (Lorentz.complexContr ⊗ Lorentz.complexContr).V", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.up, mkIdent ``Fermion.Color.up, T]),
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.up, mkIdent ``complexLorentzTensor.Color.up, T]),
|
||||
("ModuleCat.carrier (Lorentz.complexCo ⊗ Lorentz.complexCo).V", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.down, mkIdent ``Fermion.Color.down, T]),
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.down, mkIdent ``complexLorentzTensor.Color.down, T]),
|
||||
("ModuleCat.carrier (Lorentz.complexCo ⊗ Lorentz.complexContr).V", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.twoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.down,
|
||||
mkIdent ``Fermion.Color.up, T]),
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.down,
|
||||
mkIdent ``complexLorentzTensor.Color.up, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Lorentz.complexCo ⊗ Lorentz.complexCo", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.down,
|
||||
mkIdent ``Fermion.Color.down, T]),
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.down,
|
||||
mkIdent ``complexLorentzTensor.Color.down, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Lorentz.complexContr ⊗ Lorentz.complexContr", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.up,
|
||||
mkIdent ``Fermion.Color.up, T]),
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.up,
|
||||
mkIdent ``complexLorentzTensor.Color.up, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Lorentz.complexContr ⊗ Fermion.leftHanded ⊗ Fermion.rightHanded", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constThreeNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor, mkIdent ``Fermion.Color.up,
|
||||
mkIdent ``Fermion.Color.upL,
|
||||
mkIdent ``Fermion.Color.upR, T]),
|
||||
mkIdent ``complexLorentzTensor, mkIdent ``complexLorentzTensor.Color.up,
|
||||
mkIdent ``complexLorentzTensor.Color.upL,
|
||||
mkIdent ``complexLorentzTensor.Color.upR, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.leftHanded ⊗ Fermion.leftHanded", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.upL,
|
||||
mkIdent ``Fermion.Color.upL, T]),
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.upL,
|
||||
mkIdent ``complexLorentzTensor.Color.upL, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.altLeftHanded ⊗ Fermion.altLeftHanded", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.downL,
|
||||
mkIdent ``Fermion.Color.downL, T]),
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.downL,
|
||||
mkIdent ``complexLorentzTensor.Color.downL, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.altRightHanded ⊗ Fermion.altRightHanded", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.downR,
|
||||
mkIdent ``Fermion.Color.downR, T]),
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.downR,
|
||||
mkIdent ``complexLorentzTensor.Color.downR, T]),
|
||||
("𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ Fermion.rightHanded ⊗ Fermion.rightHanded", fun T =>
|
||||
Syntax.mkApp (mkIdent ``TensorTree.constTwoNodeE) #[
|
||||
mkIdent ``Fermion.complexLorentzTensor,
|
||||
mkIdent ``Fermion.Color.upR,
|
||||
mkIdent ``Fermion.Color.upR, T])]
|
||||
mkIdent ``complexLorentzTensor,
|
||||
mkIdent ``complexLorentzTensor.Color.upR,
|
||||
mkIdent ``complexLorentzTensor.Color.upR, T])]
|
||||
|
||||
/-- The syntax associated with a terminal node of a tensor tree. -/
|
||||
def termNodeSyntax (T : Term) : TermElabM Term := do
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue