refactor: Pauli matrices
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jstoobysmith
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7 changed files with 650 additions and 700 deletions
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@ -41,7 +41,7 @@ open Fermion
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def pauliContr := {PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
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/-- The Pauli matrices as the complex Lorentz tensor `σ_μ^α^{dot β}`. -/
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def pauliCo := {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
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def pauliCo := {Lorentz.coMetric | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor
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/-- The Pauli matrices as the complex Lorentz tensor `σ_μ_α_{dot β}`. -/
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def pauliCoDown := {pauliCo | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
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636
HepLean/Tensors/ComplexLorentz/PauliMatrices/Basis.lean
Normal file
636
HepLean/Tensors/ComplexLorentz/PauliMatrices/Basis.lean
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@ -0,0 +1,636 @@
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.ComplexLorentz.PauliMatrices.Basic
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import HepLean.Tensors.ComplexLorentz.Basis
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/-!
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## Pauli matrices and the basis of complex Lorentz tensors
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace complexLorentzTensor
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open Fermion
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/-!
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## Expanding pauliContr in a basis.
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-/
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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 pauliContr_in_basis : {pauliContr | 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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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 1 | 2 => 1)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 0 | 2 => 1)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 1 | 2 => 0)
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- I • basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 0 | 2 => 1)
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+ I • basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 1 | 2 => 0)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 0 | 2 => 0)
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- basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
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rw [tensorNode_pauliContr]
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constThreeNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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erw [PauliMatrix.asConsTensor_apply_one, PauliMatrix.asTensor_expand]
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simp only [Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, Action.instMonoidalCategory_tensorObj_V,
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Fin.isValue, map_sub, map_add, _root_.map_smul]
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congr 1
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congr 1
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congr 1
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congr 1
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congr 1
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congr 1
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congr 1
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all_goals
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erw [tripleIsoSep_tmul, basisVector]
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apply congrArg
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try apply congrArg
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funext i
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match i with
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| (0 : Fin 3) =>
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
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cons_val_zero, Fin.cases_zero]
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change _ = Lorentz.complexContrBasisFin4 _
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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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| (1 : Fin 3) => rfl
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| (2 : Fin 3) => rfl
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lemma pauliContr_basis_expand_tree : {pauliContr | 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]
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(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
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rw [pauliContr_in_basis]
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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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/-- The map to colors one gets when contracting with Pauli matrices on the right. -/
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abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) :=
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(Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
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lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) :
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(TensorTree.prod t (tensorNode pauliContr)).tensor = (((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
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((TensorTree.smul (-I) ((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
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((TensorTree.smul I ((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
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((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0))).add
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(TensorTree.smul (-1) (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR]
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fun | 0 => 3 | 1 => 1 | 2 => 1))))))))))).tensor := by
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rw [prod_tensor_eq_snd <| pauliContr_basis_expand_tree]
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rw [prod_add _ _ _]
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rw [add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
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prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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/- Moving smuls. -/
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd<| add_tensor_eq_snd
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<| add_tensor_eq_snd <| prod_smul _ _ _]
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lemma contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) =
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complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
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(contr i j h (TensorTree.prod t (tensorNode pauliContr))).tensor =
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
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((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
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((TensorTree.smul I (contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0)))).add
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(TensorTree.smul (-1) (contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR]
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fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
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rw [contr_tensor_eq <| prod_pauliMatrix_basis_tree_expand _]
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/- Moving contr over add. -/
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rw [contr_add]
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rw [add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
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/- Moving contr over smul. -/
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
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add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
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contr_smul _ _]
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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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((pauliMatrixContrMap c) i))
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
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let c' := Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ finSumFinEquiv.symm
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let b' (i1 i2 i3 : Fin 4) := fun i => prodBasisVecEquiv (finSumFinEquiv.symm i)
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((HepLean.PiTensorProduct.elimPureTensor b (fun | 0 => i1 | 1 => i2 | 2 => i3))
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(finSumFinEquiv.symm i))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
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(tensorNode pauliContr))).tensor = ((contr i j h ((tensorNode
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(basisVector c' (b' 0 0 0))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 0 1 1))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 1 0 1))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 1 1 0))))).add
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((TensorTree.smul (-I) (contr i j h ((tensorNode (basisVector c' (b' 2 0 1)))))).add
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((TensorTree.smul I (contr i j h ((tensorNode (basisVector c' (b' 2 1 0)))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 3 0 0))))).add
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(TensorTree.smul (-1) (contr i j h ((tensorNode
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(basisVector c' (b' 3 1 1))))))))))))).tensor := by
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rw [contr_pauliMatrix_basis_tree_expand]
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/- Product of basis vectors . -/
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rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
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<| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
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<| contr_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
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<| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
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<| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
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<| contr_tensor_eq <| prod_basisVector_tree _ _]
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rfl
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/-- The map to color which appears when contracting a basis vector with
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puali matrices. -/
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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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/-- The new basis vectors which appear when contracting pauli matrices with
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basis vectors. -/
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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)
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(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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((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)
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(i.succAbove (j.succAbove k))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
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(tensorNode pauliContr))).tensor =
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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 (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 (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 (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
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(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
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(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 (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
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(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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rw [add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||||
<| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
smul_tensor_eq <| contr_basisVector_tree _]
|
||||
rfl
|
||||
|
||||
lemma basis_contr_pauliMatrix_basis_tree_expand_tensor {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(i : Fin (n + 3)) (j : Fin (n +2))
|
||||
(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ
|
||||
((pauliMatrixContrMap c) i))
|
||||
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
|
||||
(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
|
||||
(tensorNode pauliContr))).tensor =
|
||||
(contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0)) •
|
||||
(basisVectorContrPauli i j b 0 0 0)
|
||||
+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1)) •
|
||||
(basisVectorContrPauli i j b 0 1 1)
|
||||
+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 0 1)) •
|
||||
(basisVectorContrPauli i j b 1 0 1)
|
||||
+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 1 1 0)) •
|
||||
(basisVectorContrPauli i j b 1 1 0)
|
||||
+ (-I) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 0 1)) •
|
||||
(basisVectorContrPauli i j b 2 0 1)
|
||||
+ I • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 2 1 0)) •
|
||||
(basisVectorContrPauli i j b 2 1 0)
|
||||
+ (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 0 0)) •
|
||||
(basisVectorContrPauli i j b 3 0 0)
|
||||
+ (-1 : ℂ) • (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 3 1 1)) •
|
||||
(basisVectorContrPauli i j b 3 1 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val_one, head_cons, Fin.val_zero,
|
||||
Nat.cast_zero, cons_val_two, Fin.val_one, Nat.cast_one, add_tensor, smul_tensor,
|
||||
tensorNode_tensor, neg_smul, one_smul, Int.reduceNeg]
|
||||
simp_all only [Function.comp_apply, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
|
||||
rfl
|
||||
|
||||
|
||||
/-!
|
||||
|
||||
## Expanding pauliCo in a basis.
|
||||
|
||||
-/
|
||||
|
||||
|
||||
/-- The map to color one gets when lowering the indices of pauli matrices. -/
|
||||
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 1 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
|
||||
(tensorNode pauliContr)))).tensor
|
||||
= 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
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_1 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 1) | ν μ ⊗
|
||||
pauliContr | μ α β}ᵀ.tensor
|
||||
= 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
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_2 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
pauliContr | 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
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_3 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
pauliContr | 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
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· rw [basisVectorContrPauli]
|
||||
congr 1
|
||||
congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
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 _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| smul_prod _ _ _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_smul _ _]
|
||||
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 pauliCo_basis_expand_tree : {pauliCo | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
|
||||
TensorTree.add (tensorNode
|
||||
(basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul I (tensorNode
|
||||
(basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-I) (tensorNode
|
||||
(basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(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 pauliCo_prod_basis_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(prod (tensorNode pauliCo) t).tensor =
|
||||
(((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul I ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-I) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
t)))))))).tensor := by
|
||||
rw [prod_tensor_eq_fst <| pauliCo_basis_expand_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
|
||||
end complexLorentzTensor
|
||||
|
|
@ -0,0 +1,562 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.ComplexLorentz.PauliMatrices.Basis
|
||||
import HepLean.Tensors.ComplexLorentz.Lemmas
|
||||
/-!
|
||||
|
||||
## Contractiong of indices of Pauli matrix.
|
||||
|
||||
The main result of this file is `pauliMatrix_contract_pauliMatrix` which states that
|
||||
`η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`.
|
||||
|
||||
The current way this result is proved is by using tensor tree manipulations.
|
||||
There is likely a more direct path to this result.
|
||||
|
||||
-/
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open MonoidalCategory
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
open TensorProduct
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open TensorTree
|
||||
open OverColor.Discrete
|
||||
noncomputable section
|
||||
|
||||
namespace complexLorentzTensor
|
||||
open Fermion
|
||||
|
||||
/-- The map to colors one gets when contracting the 4-vector indices pauli matrices. -/
|
||||
def pauliMatrixContrPauliMatrixMap := ((Sum.elim
|
||||
((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 1 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 2)
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_0_0 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
pauliContr | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_1_1 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_0_1 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_1_0 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_0_1 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_1_0 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_0_0 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_1_1 :
|
||||
{(basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
pauliContr | 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
|
||||
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. -/
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
/-! 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 pauliCo_prod_basis_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
|
||||
(t : TensorTree complexLorentzTensor c) :
|
||||
(TensorTree.prod (tensorNode pauliCo) t).tensor =
|
||||
((((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
|
||||
(((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul I ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||||
((TensorTree.smul (-I) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||||
((TensorTree.smul (-1) ((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||||
((tensorNode
|
||||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||||
t))))))))).tensor := by
|
||||
exact pauliCo_prod_basis_expand _
|
||||
|
||||
lemma pauliCo_contr_pauliContr_expand_aux :
|
||||
{pauliCo | μ α β ⊗ pauliContr | 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
|
||||
((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1)).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul I (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
I •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-I) (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1))).add
|
||||
(tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| pauliCo_prod_basis_expand' _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_0_1_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_tensor_eq <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_2_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor
|
||||
<| pauliMatrix_contr_lower_2_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_3_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_3_1_1]
|
||||
|
||||
lemma pauliCo_contr_pauliContr_expand :
|
||||
{pauliCo | ν α β ⊗ pauliContr | ν α' β'}ᵀ.tensor =
|
||||
2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)
|
||||
+ 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) := by
|
||||
rw [pauliCo_contr_pauliContr_expand_aux]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, neg_smul,
|
||||
one_smul, add_tensor, tensorNode_tensor, smul_tensor, smul_add, smul_neg, _root_.smul_smul,
|
||||
neg_mul, _root_.neg_neg]
|
||||
ring_nf
|
||||
rw [Complex.I_sq]
|
||||
simp only [neg_smul, one_smul, _root_.neg_neg]
|
||||
abel
|
||||
|
||||
/-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
|
||||
theorem pauliCo_contr_pauliContr :
|
||||
{pauliCo | ν α β ⊗ pauliContr | ν α' β' =
|
||||
2 •ₜ Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ := by
|
||||
rw [pauliCo_contr_pauliContr_expand]
|
||||
rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
|
||||
rw [perm_smul]
|
||||
/- Moving perm through adds. -/
|
||||
rw [smul_tensor_eq <| perm_add _ _ _]
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| perm_add _ _ _]
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| perm_add _ _ _]
|
||||
/- Moving perm through smul. -/
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| perm_smul _ _ _]
|
||||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| add_tensor_eq_fst <| perm_smul _ _ _]
|
||||
/- Perm acting on basis. -/
|
||||
erw [smul_tensor_eq <| add_tensor_eq_fst <| perm_basisVector_tree _ _]
|
||||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
perm_basisVector_tree _ _]
|
||||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_tensor_eq <| perm_basisVector_tree _ _]
|
||||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
perm_basisVector_tree _ _]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
have h1 (b0011 b1100 b0110 b1001 : CoeSort.coe (complexLorentzTensor.F.obj
|
||||
(OverColor.mk pauliMatrixContrPauliMatrixMap))) :
|
||||
((2 • b0011 + 2 • b1100) - 2 • b0110 - 2 • b1001) = (2 : ℂ) • ((b0011) +
|
||||
(((-1 : ℂ)• b0110) + (((-1 : ℂ) •b1001) + b1100))) := by
|
||||
trans (2 : ℂ) • b0011 + (2 : ℂ) • b1100 - ((2 : ℂ) • b0110) - ((2 : ℂ) • b1001)
|
||||
· repeat rw [two_smul]
|
||||
· simp only [neg_smul, one_smul, smul_add, smul_neg]
|
||||
abel
|
||||
rw [h1]
|
||||
congr
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
· funext i
|
||||
fin_cases i <;> rfl
|
||||
|
||||
end complexLorentzTensor
|
||||
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