refactor: Lint
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jstoobysmith
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9 changed files with 946 additions and 696 deletions
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@ -3,17 +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.Basis
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import LLMLean
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import HepLean.Tensors.ComplexLorentz.BasisTrees
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/-!
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## Lemmas related to complex Lorentz tensors.
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@ -33,7 +23,7 @@ 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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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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@ -117,8 +107,8 @@ lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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And related results.
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-/
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open complexLorentzTensor
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/-- The map to color one gets when multiplying left and right metrics. -/
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def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
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finSumFinEquiv.symm
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@ -154,659 +144,20 @@ lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.righ
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funext x
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fin_cases x <;> rfl
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def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
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⇑finSumFinEquiv.symm) ∘ Fin.succAbove 0 ∘ Fin.succAbove 1)
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abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) :=
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(Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
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lemma prod_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) :
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(TensorTree.prod t (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor)).tensor = (((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
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(((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
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((TensorTree.smul (-I) ((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
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((TensorTree.smul I ((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
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((t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0))).add
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(TensorTree.smul (-1) (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR]
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fun | 0 => 3 | 1 => 1 | 2 => 1))))))))))).tensor := by
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rw [prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
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rw [prod_add _ _ _]
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rw [add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
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prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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/- Moving smuls. -/
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd<| add_tensor_eq_snd
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<| add_tensor_eq_snd <| prod_smul _ _ _]
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rfl
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lemma contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) =
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complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
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(contr i j h (TensorTree.prod t
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor =
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 0 | 2 => 1)))).add
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((contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
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((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
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(basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
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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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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor = ((contr i j h ((tensorNode
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(basisVector c' (b' 0 0 0))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 0 1 1))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 1 0 1))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 1 1 0))))).add
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((TensorTree.smul (-I) (contr i j h ((tensorNode (basisVector c' (b' 2 0 1)))))).add
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((TensorTree.smul I (contr i j h ((tensorNode (basisVector c' (b' 2 1 0)))))).add
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((contr i j h ((tensorNode (basisVector c' (b' 3 0 0))))).add
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(TensorTree.smul (-1) (contr i j h ((tensorNode
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(basisVector c' (b' 3 1 1))))))))))))).tensor := by
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rw [contr_pauliMatrix_basis_tree_expand]
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/- Product of basis vectors . -/
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rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
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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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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) : (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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(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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(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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(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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(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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(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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(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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(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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(tensorNode (basisVector c' (b' 3 0 0))))).add
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(TensorTree.smul (-1) ((TensorTree.smul (contrBasisVectorMul i j (b'' 3 1 1)) (tensorNode
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(basisVector c' (b' 3 1 1))))))))))))).tensor := by
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rw [basis_contr_pauliMatrix_basis_tree_expand']
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/- Contracting basis vectors. -/
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rw [add_tensor_eq_fst <| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
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<| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_fst <| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
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<| contr_basisVector_tree _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree _]
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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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lemma pauliMatrix_contr_down_0 :
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(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor)))).tensor
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= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
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+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
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rw [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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simp only [smul_tensor, add_tensor, tensorNode_tensor]
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simp only [one_smul, zero_smul, smul_zero, add_zero]
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congr 1
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· congr 1
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funext k
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fin_cases k <;> rfl
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· congr 1
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funext k
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fin_cases k <;> rfl
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lemma pauliMatrix_contr_down_0_tree :
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(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor)))).tensor
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lemma leftMetric_mul_rightMetric_tree :
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{Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
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= (TensorTree.add (tensorNode
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(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)))
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(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)))).tensor := by
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exact pauliMatrix_contr_down_0
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lemma pauliMatrix_contr_down_1 :
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{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
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PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
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= basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
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+ basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
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rw [basis_contr_pauliMatrix_basis_tree_expand]
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rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
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contrBasisVectorMul_pos, contrBasisVectorMul_pos,
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contrBasisVectorMul_neg, contrBasisVectorMul_neg,
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contrBasisVectorMul_neg, contrBasisVectorMul_neg]
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simp only [smul_tensor, add_tensor, tensorNode_tensor]
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simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
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congr 1
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· congr 1
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funext k
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fin_cases k <;> rfl
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· congr 1
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funext k
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fin_cases k <;> rfl
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lemma pauliMatrix_contr_down_1_tree :
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{(basisVector ![Color.down, Color.down] fun x => 1) | μ ν ⊗
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PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
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= (TensorTree.add (tensorNode
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(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))
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(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))).tensor := by
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exact pauliMatrix_contr_down_1
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|
||||
lemma pauliMatrix_contr_down_2 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
+ (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_2_tree :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 2) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
(smul (- I) (tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))))
|
||||
(smul I (tensorNode (basisVector
|
||||
pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))))).tensor := by
|
||||
exact pauliMatrix_contr_down_2
|
||||
|
||||
lemma pauliMatrix_contr_down_3 :
|
||||
{(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_down_3_tree : {(basisVector ![Color.down, Color.down] fun x => 3) | μ ν ⊗
|
||||
PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
|
||||
(TensorTree.add
|
||||
((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))))
|
||||
(smul (-1) (tensorNode (basisVector pauliMatrixLowerMap
|
||||
(fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
|
||||
exact pauliMatrix_contr_down_3
|
||||
|
||||
def pauliMatrixContrPauliMatrixMap := ((Sum.elim
|
||||
((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
|
||||
Fin.succAbove 0 ∘ Fin.succAbove 2)
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_0_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_1_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 1
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_0_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_2_1_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||||
+ (I) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_0_0 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
(fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_contr_lower_3_1_1 :
|
||||
{(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | μ α' β'}ᵀ.tensor =
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
+ (-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||||
rw [basis_contr_pauliMatrix_basis_tree_expand]
|
||||
rw [contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_neg, contrBasisVectorMul_neg,
|
||||
contrBasisVectorMul_pos, contrBasisVectorMul_pos]
|
||||
/- Simplifying. -/
|
||||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||||
simp only [zero_smul, one_smul, smul_zero, add_zero, zero_add]
|
||||
congr 1
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
· congr 2
|
||||
funext k
|
||||
fin_cases k <;> rfl
|
||||
|
||||
lemma pauliMatrix_lower : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
|
||||
+ I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||||
- I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
|
||||
- basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||||
+ basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [contr_tensor_eq <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||||
<| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||||
<| smul_prod _ _ _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| pauliMatrix_contr_down_0_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| pauliMatrix_contr_down_1_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_2_tree]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq <|
|
||||
pauliMatrix_contr_down_3_tree]
|
||||
/- Simplifying -/
|
||||
simp only [add_tensor, smul_tensor, tensorNode_tensor, smul_add,_root_.smul_smul]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, neg_smul, one_smul, mul_neg, neg_mul, one_mul,
|
||||
_root_.neg_neg, mul_one]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_lower_tree : {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
|
||||
= (TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
|
||||
TensorTree.add (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul I (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
|
||||
TensorTree.add (TensorTree.smul (-I) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1) (tensorNode
|
||||
(basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)))) <|
|
||||
(tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1)))).tensor := by
|
||||
rw [pauliMatrix_lower]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, add_tensor,
|
||||
tensorNode_tensor, smul_tensor, neg_smul, one_smul]
|
||||
rfl
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix_aux :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
|
||||
= ((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)).add
|
||||
((tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1)).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul I (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||||
I •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-I) (tensorNode
|
||||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||||
I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||||
((TensorTree.smul (-1) (tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) •
|
||||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1))).add
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|
|
||||
TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
|
||||
TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
|
||||
(tensorNode
|
||||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||||
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||||
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
|
||||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| pauliMatrix_lower_tree]
|
||||
/- Moving the prod through additions. -/
|
||||
rw [contr_tensor_eq <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_prod _ _ _]
|
||||
/- Moving the prod through smuls. -/
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_prod _ _ _]
|
||||
rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_prod _ _ _]
|
||||
/- Moving contraction through addition. -/
|
||||
rw [contr_add]
|
||||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| contr_add _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||||
/- Moving contraction through smul. -/
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||||
/- Replacing the contractions. -/
|
||||
rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_0_1_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||||
smul_tensor_eq <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_2_0_1]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor
|
||||
<| pauliMatrix_contr_lower_2_1_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_3_0_0]
|
||||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
|
||||
pauliMatrix_contr_lower_3_1_1]
|
||||
|
||||
lemma pauliMatrix_contract_pauliMatrix :
|
||||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗
|
||||
PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor =
|
||||
2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)
|
||||
+ 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) := by
|
||||
rw [pauliMatrix_contract_pauliMatrix_aux]
|
||||
simp only [Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, neg_smul,
|
||||
one_smul, add_tensor, tensorNode_tensor, smul_tensor, smul_add, smul_neg, _root_.smul_smul,
|
||||
neg_mul, _root_.neg_neg]
|
||||
ring_nf
|
||||
rw [Complex.I_sq]
|
||||
simp only [neg_smul, one_smul, _root_.neg_neg]
|
||||
abel
|
||||
(basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
|
||||
rw [leftMetric_mul_rightMetric]
|
||||
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
|
||||
smul_tensor, neg_smul, one_smul]
|
||||
rfl
|
||||
|
||||
end Fermion
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue