561 lines
22 KiB
Text
561 lines
22 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.ComplexLorentz.PauliMatrices.Basis
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import HepLean.Tensors.ComplexLorentz.Lemmas
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/-!
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## Contractiong of indices of Pauli matrix.
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The main result of this file is `pauliMatrix_contract_pauliMatrix` which states that
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`η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`.
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The current way this result is proved is by using tensor tree manipulations.
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There is likely a more direct path to this result.
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace complexLorentzTensor
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open Fermion
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/-- The map to colors one gets when contracting the 4-vector indices pauli matrices. -/
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def pauliMatrixContrPauliMatrixMap := ((Sum.elim
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((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 1 ∘ Fin.succAbove 1) ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
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Fin.succAbove 0 ∘ Fin.succAbove 2)
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lemma pauliMatrix_contr_lower_0_0_0 :
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{(basisVector pauliCoMap (fun | 0 => 0 | 1 => 0 | 2 => 0)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
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+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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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_lower_0_1_1 :
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{(basisVector pauliCoMap (fun | 0 => 0 | 1 => 1 | 2 => 1)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
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+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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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_lower_1_0_1 :
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{(basisVector pauliCoMap (fun | 0 => 1 | 1 => 0 | 2 => 1)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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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_lower_1_1_0 :
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{(basisVector pauliCoMap (fun | 0 => 1 | 1 => 1 | 2 => 0)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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+ basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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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_lower_2_0_1 :
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{(basisVector pauliCoMap (fun | 0 => 2 | 1 => 0 | 2 => 1)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
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+ (I) •
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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congr 1
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· congr 2
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funext k
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fin_cases k <;> rfl
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· congr 2
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funext k
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fin_cases k <;> rfl
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lemma pauliMatrix_contr_lower_2_1_0 :
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{(basisVector pauliCoMap (fun | 0 => 2 | 1 => 1 | 2 => 0)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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(-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
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+ (I) •
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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congr 1
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· congr 2
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funext k
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fin_cases k <;> rfl
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· congr 2
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funext k
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fin_cases k <;> rfl
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lemma pauliMatrix_contr_lower_3_0_0 :
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{(basisVector pauliCoMap (fun | 0 => 3 | 1 => 0 | 2 => 0)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
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+ (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
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(fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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congr 1
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· congr 2
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funext k
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fin_cases k <;> rfl
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· congr 2
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funext k
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fin_cases k <;> rfl
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lemma pauliMatrix_contr_lower_3_1_1 :
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{(basisVector pauliCoMap (fun | 0 => 3 | 1 => 1 | 2 => 1)) | μ α β ⊗
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pauliContr | μ α' β'}ᵀ.tensor =
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
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+ (-1 : ℂ) •
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basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
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conv =>
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lhs
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rw [basis_contr_pauliMatrix_basis_tree_expand_tensor]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; lhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; lhs; rhs; rhs; lhs
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rw [contrBasisVectorMul_neg (by decide)]
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conv =>
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lhs; lhs; rhs; lhs;
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs; rhs; rhs; lhs;
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rw [contrBasisVectorMul_pos (by decide)]
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conv =>
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lhs
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simp only [_root_.zero_smul, one_smul, _root_.smul_zero, _root_.add_zero, _root_.zero_add]
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rw [basisVectorContrPauli, basisVectorContrPauli]
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/- Simplifying. -/
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congr 1
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· congr 2
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funext k
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fin_cases k <;> rfl
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· congr 2
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funext k
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fin_cases k <;> rfl
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/-! TODO: Work out why `pauliCo_prod_basis_expand'` is needed. -/
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/-- This lemma is exactly the same as `pauliCo_prod_basis_expand`.
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It is needed here for `pauliMatrix_contract_pauliMatrix_aux`. It is unclear why
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`pauliMatrix_lower_basis_expand_prod` does not work. -/
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private lemma pauliCo_prod_basis_expand' {n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(t : TensorTree complexLorentzTensor c) :
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(TensorTree.prod (tensorNode pauliCo) t).tensor =
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((((tensorNode
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(basisVector pauliCoMap fun | 0 => 0 | 1 => 0 | 2 => 0)).prod t).add
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(((tensorNode
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(basisVector pauliCoMap fun | 0 => 0 | 1 => 1 | 2 => 1)).prod t).add
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((TensorTree.smul (-1) ((tensorNode
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(basisVector pauliCoMap fun | 0 => 1 | 1 => 0 | 2 => 1)).prod t)).add
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((TensorTree.smul (-1) ((tensorNode
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(basisVector pauliCoMap fun | 0 => 1 | 1 => 1 | 2 => 0)).prod t)).add
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((TensorTree.smul I ((tensorNode
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(basisVector pauliCoMap fun | 0 => 2 | 1 => 0 | 2 => 1)).prod t)).add
|
||
((TensorTree.smul (-I) ((tensorNode
|
||
(basisVector pauliCoMap fun | 0 => 2 | 1 => 1 | 2 => 0)).prod t)).add
|
||
((TensorTree.smul (-1) ((tensorNode
|
||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 0 | 2 => 0)).prod t)).add
|
||
((tensorNode
|
||
(basisVector pauliCoMap fun | 0 => 3 | 1 => 1 | 2 => 1)).prod
|
||
t))))))))).tensor := by
|
||
exact pauliCo_prod_basis_expand _
|
||
|
||
lemma pauliCo_contr_pauliContr_expand_aux :
|
||
{pauliCo | μ α β ⊗ pauliContr | μ α' β'}ᵀ.tensor
|
||
= ((tensorNode
|
||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)).add
|
||
((tensorNode
|
||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1)).add
|
||
((TensorTree.smul (-1) (tensorNode
|
||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||
((TensorTree.smul (-1) (tensorNode
|
||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||
((TensorTree.smul I (tensorNode
|
||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1) +
|
||
I •
|
||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0))).add
|
||
((TensorTree.smul (-I) (tensorNode
|
||
((-I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) +
|
||
I • basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0))).add
|
||
((TensorTree.smul (-1) (tensorNode
|
||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0) +
|
||
(-1 : ℂ) •
|
||
basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1))).add
|
||
(tensorNode
|
||
((basisVector pauliMatrixContrPauliMatrixMap fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0) +
|
||
(-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap
|
||
fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1))))))))).tensor := by
|
||
rw [contr_tensor_eq <| pauliCo_prod_basis_expand' _]
|
||
/- Moving contraction through addition. -/
|
||
rw [contr_add]
|
||
rw [add_tensor_eq_snd <| contr_add _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
contr_add _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| contr_add _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
|
||
/- Moving contraction through smul. -/
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||
contr_smul _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_fst <| contr_smul _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
|
||
/- Replacing the contractions. -/
|
||
rw [add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_0_0_0]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_fst <| eq_tensorNode_of_eq_tensor <|
|
||
pauliMatrix_contr_lower_0_1_1]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_0_1]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||
smul_tensor_eq <| eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_1_1_0]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor <|
|
||
pauliMatrix_contr_lower_2_0_1]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| eq_tensorNode_of_eq_tensor
|
||
<| pauliMatrix_contr_lower_2_1_0]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||
eq_tensorNode_of_eq_tensor <| pauliMatrix_contr_lower_3_0_0]
|
||
rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| eq_tensorNode_of_eq_tensor <|
|
||
pauliMatrix_contr_lower_3_1_1]
|
||
|
||
lemma pauliCo_contr_pauliContr_expand :
|
||
{pauliCo | ν α β ⊗ pauliContr | ν α' β'}ᵀ.tensor =
|
||
2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1)
|
||
+ 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
|
||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
|
||
- 2 • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1) := by
|
||
rw [pauliCo_contr_pauliContr_expand_aux]
|
||
simp only [Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, neg_smul,
|
||
one_smul, add_tensor, tensorNode_tensor, smul_tensor, smul_add, smul_neg, _root_.smul_smul,
|
||
neg_mul, _root_.neg_neg]
|
||
ring_nf
|
||
rw [Complex.I_sq]
|
||
simp only [neg_smul, one_smul, _root_.neg_neg]
|
||
abel
|
||
|
||
/-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
|
||
theorem pauliCo_contr_pauliContr :
|
||
{pauliCo | ν α β ⊗ pauliContr | ν α' β' = 2 •ₜ εL | α α' ⊗ εR | β β'}ᵀ := by
|
||
rw [pauliCo_contr_pauliContr_expand]
|
||
rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
|
||
rw [perm_smul]
|
||
/- Moving perm through adds. -/
|
||
rw [smul_tensor_eq <| perm_add _ _ _]
|
||
rw [smul_tensor_eq <| add_tensor_eq_snd <| perm_add _ _ _]
|
||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| perm_add _ _ _]
|
||
/- Moving perm through smul. -/
|
||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| perm_smul _ _ _]
|
||
rw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd
|
||
<| add_tensor_eq_fst <| perm_smul _ _ _]
|
||
/- Perm acting on basis. -/
|
||
erw [smul_tensor_eq <| add_tensor_eq_fst <| perm_basisVector_tree _ _]
|
||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <|
|
||
perm_basisVector_tree _ _]
|
||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <|
|
||
smul_tensor_eq <| perm_basisVector_tree _ _]
|
||
erw [smul_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
|
||
perm_basisVector_tree _ _]
|
||
/- Simplifying. -/
|
||
simp only [smul_tensor, add_tensor, tensorNode_tensor]
|
||
have h1 (b0011 b1100 b0110 b1001 : CoeSort.coe (complexLorentzTensor.F.obj
|
||
(OverColor.mk pauliMatrixContrPauliMatrixMap))) :
|
||
((2 • b0011 + 2 • b1100) - 2 • b0110 - 2 • b1001) = (2 : ℂ) • ((b0011) +
|
||
(((-1 : ℂ)• b0110) + (((-1 : ℂ) •b1001) + b1100))) := by
|
||
trans (2 : ℂ) • b0011 + (2 : ℂ) • b1100 - ((2 : ℂ) • b0110) - ((2 : ℂ) • b1001)
|
||
· repeat rw [two_smul]
|
||
· simp only [neg_smul, one_smul, smul_add, smul_neg]
|
||
abel
|
||
rw [h1]
|
||
congr
|
||
· funext i
|
||
fin_cases i <;> rfl
|
||
· funext i
|
||
fin_cases i <;> rfl
|
||
· funext i
|
||
fin_cases i <;> rfl
|
||
· funext i
|
||
fin_cases i <;> rfl
|
||
|
||
end complexLorentzTensor
|