894 lines
48 KiB
Text
894 lines
48 KiB
Text
/-
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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.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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/-!
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## Lemmas related to 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 Fermion
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set_option maxRecDepth 20000 in
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lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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{T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
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{T1 | μ ν ⊗ T2 | μ ν = T2 | μ ν ⊗ T1 | μ ν}ᵀ := by
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rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_comm _ _ _ _)))]
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rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
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rw [perm_tensor_eq (perm_contr _ _)]
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rw [perm_perm]
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rw [perm_eq_id]
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· rw [(contr_tensor_eq (contr_swap _ _))]
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rw [perm_contr]
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rw [perm_tensor_eq (contr_swap _ _)]
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rw [perm_perm]
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rw [perm_eq_id]
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· rfl
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· rfl
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· apply OverColor.Hom.ext
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ext x
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exact Fin.elim0 x
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lemma contr_rank_2_symm' {T1 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
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{T2 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V} :
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{T1 | μ ν ⊗ T2 | μ ν = T2 | μ ν ⊗ T1 | μ ν}ᵀ := by
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rw [perm_tensor_eq contr_rank_2_symm]
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rw [perm_perm]
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rw [perm_eq_id]
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apply OverColor.Hom.ext
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ext x
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exact Fin.elim0 x
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set_option maxRecDepth 20000 in
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/-- Contracting a rank-2 anti-symmetric tensor with a rank-2 symmetric tensor gives zero. -/
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lemma antiSymm_contr_symm {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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{S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
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(hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
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{A | μ ν ⊗ S | μ ν}ᵀ.tensor = 0 := by
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have h1 {M : Type} [AddCommGroup M] [Module ℂ M] {x : M} (h : x = - x) : x = 0 := by
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rw [eq_neg_iff_add_eq_zero, ← two_smul ℂ x] at h
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simpa using h
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refine h1 ?_
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rw [← neg_tensor]
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rw [neg_perm] at hA
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nth_rewrite 1 [contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_fst hA))]
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nth_rewrite 1 [(contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_snd hs)))]
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rw [contr_tensor_eq (contr_tensor_eq (neg_fst_prod _ _))]
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rw [contr_tensor_eq (neg_contr _)]
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rw [neg_contr]
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rw [neg_tensor]
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apply congrArg
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rw [contr_tensor_eq (contr_tensor_eq (prod_perm_left _ _ _ _))]
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rw [contr_tensor_eq (perm_contr _ _)]
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rw [perm_contr]
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rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_perm_right _ _ _ _)))]
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rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
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rw [perm_tensor_eq (perm_contr _ _)]
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rw [perm_perm]
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nth_rewrite 1 [perm_tensor_eq (contr_contr _ _ _)]
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rw [perm_perm]
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rw [perm_eq_id]
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· rfl
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· rfl
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lemma symm_contr_antiSymm {S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
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{A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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(hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
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{S | μ ν ⊗ A | μ ν}ᵀ.tensor = 0 := by
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rw [contr_rank_2_symm', perm_tensor, antiSymm_contr_symm hA hs]
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rfl
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lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
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(hA : {A | μ ν = - (A | ν μ)}ᵀ) :
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{A | μ ν + A | ν μ}ᵀ.tensor = 0 := by
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rw [← TensorTree.add_neg (twoNodeE complexLorentzTensor Color.up Color.up A)]
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apply TensorTree.add_tensor_eq_snd
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rw [neg_tensor_eq hA, neg_tensor_eq (neg_perm _ _), neg_neg]
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/-!
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## The contraction of Pauli matrices with Pauli matrices
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And related results.
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-/
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open complexLorentzTensor
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def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘ finSumFinEquiv.symm
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lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
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= basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
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- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
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rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
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rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
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rw [prod_add_both]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
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rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
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rw [← add_assoc]
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simp only [add_tensor, smul_tensor, tensorNode_tensor]
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change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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+- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
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+- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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+ basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
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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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congr
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funext x
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fin_cases x <;> rfl
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def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 0 ∘ Fin.succAbove 1)
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abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
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lemma pauliMatrix_contr_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
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(h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
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(contr i j h (TensorTree.prod t (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor = (
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(contr i j h (t.prod (tensorNode
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(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] fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
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rw [contr_tensor_eq <| prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
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rw [contr_tensor_eq <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
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/- Moving smuls. -/
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
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rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd<| add_tensor_eq_snd
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<| add_tensor_eq_snd <| prod_smul _ _ _]
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/- 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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rfl
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lemma pauliMatrix_contr_down_0 :
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(contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor)))).tensor
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= basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
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+ basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
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rw [pauliMatrix_contr_expand]
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/- 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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/- Contracting basis vectors. -/
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rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
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<| contr_basisVector_tree_neg _ ]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
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<| contr_basisVector_tree_neg _]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
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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_neg _]
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/- Simplifying. -/
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simp only [smul_tensor, add_tensor, tensorNode_tensor]
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simp only [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_1 : (contr 0 1 rfl
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(((tensorNode (basisVector ![Color.down, Color.down] fun x => 1)).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 => 1 | 1 => 0 | 2 => 1)
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+ basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
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rw [pauliMatrix_contr_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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/- Contracting basis vectors. -/
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rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
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rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
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<| contr_basisVector_tree_pos _ rfl]
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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_pos _ rfl]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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_down_2 : (contr 0 1 rfl
|
||
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 2)).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor
|
||
= (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
|
||
+ (I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 1 rfl
|
||
(((tensorNode (basisVector ![Color.down, Color.down] fun x => 3)).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor
|
||
= basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
|
||
+ (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_pos _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [smul_zero, add_zero, zero_add]
|
||
congr 1
|
||
· congr 2
|
||
funext k
|
||
fin_cases k <;> rfl
|
||
· congr 2
|
||
funext k
|
||
fin_cases k <;> rfl
|
||
|
||
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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_pos _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_pos _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor =
|
||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||
+ (I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor =
|
||
(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||
+ (I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor =
|
||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
|
||
+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_pos _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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 : (contr 0 2 rfl
|
||
(((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1))).prod
|
||
(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
|
||
PauliMatrix.asConsTensor)))).tensor =
|
||
basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||
+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
|
||
rw [pauliMatrix_contr_expand]
|
||
/- Product of basis vectors . -/
|
||
rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
|
||
<| prod_basisVector_tree _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
|
||
<| contr_tensor_eq <| prod_basisVector_tree _ _]
|
||
/- Contracting basis vectors. -/
|
||
rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
|
||
<| contr_basisVector_tree_neg _ ]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
|
||
<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
|
||
<| contr_basisVector_tree_neg _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
|
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<| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
|
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rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
smul_tensor_eq <| contr_basisVector_tree_pos _]
|
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/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
simp only [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]
|
||
rw [contr_tensor_eq <| prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
|
||
|
||
sorry -/
|
||
|
||
|
||
lemma pauliMatrix_contract_pauliMatrix :
|
||
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗ PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
|
||
= basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
|
||
- basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||
- basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
|
||
+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
|
||
|
||
end Fermion
|
||
|
||
end
|