refactor: Move ComplexTensor
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253
HepLean/Lorentz/ComplexTensor/PauliMatrices/Basic.lean
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253
HepLean/Lorentz/ComplexTensor/PauliMatrices/Basic.lean
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.Tree.NodeIdentities.ProdAssoc
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import HepLean.Tensors.Tree.NodeIdentities.ProdComm
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import HepLean.Tensors.Tree.NodeIdentities.ProdContr
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import HepLean.Tensors.Tree.NodeIdentities.ContrContr
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import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
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import HepLean.Tensors.Tree.NodeIdentities.PermContr
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import HepLean.Tensors.Tree.NodeIdentities.Congr
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import HepLean.Lorentz.ComplexTensor.Metrics.Lemmas
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/-!
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## Pauli matrices as complex Lorentz tensors
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open IndexNotation
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open CategoryTheory
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open TensorTree
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open OverColor.Discrete
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noncomputable section
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namespace complexLorentzTensor
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open Fermion
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/-!
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## Definitions.
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-/
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/-- The Pauli matrices as the complex Lorentz tensor `σ^μ^α^{dot β}`. -/
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def pauliContr := {PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
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/-- The Pauli matrices as the complex Lorentz tensor `σ_μ^α^{dot β}`. -/
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def pauliCo := {η' | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor
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/-- The Pauli matrices as the complex Lorentz tensor `σ_μ_α_{dot β}`. -/
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def pauliCoDown := {pauliCo | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor
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/-- The Pauli matrices as the complex Lorentz tensor `σ^μ_α_{dot β}`. -/
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def pauliContrDown := {pauliContr | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor
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/-!
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## Tensor nodes.
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-/
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/-- The definitional tensor node relation for `pauliContr`. -/
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lemma tensorNode_pauliContr : {pauliContr | μ α β}ᵀ.tensor =
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{PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
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rfl
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/-- The definitional tensor node relation for `pauliCo`. -/
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lemma tensorNode_pauliCo : {pauliCo | μ α β}ᵀ.tensor =
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{η' | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor := by
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rw [pauliCo, tensorNode_tensor]
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/-- The definitional tensor node relation for `pauliCoDown`. -/
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lemma tensorNode_pauliCoDown : {pauliCoDown | μ α β}ᵀ.tensor =
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{pauliCo | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor := by
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rw [pauliCoDown, tensorNode_tensor]
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/-- The definitional tensor node relation for `pauliContrDown`. -/
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lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
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{pauliContr | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor := by
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rw [pauliContr, tensorNode_tensor]
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rfl
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/-!
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## Basic equalities
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-/
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set_option maxRecDepth 5000 in
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/-- A rearanging of `pauliCoDown` to place the pauli matrices on the right. -/
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lemma pauliCoDown_eq_metric_mul_pauliCo :
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{pauliCoDown | μ α' β' = εL' | α α' ⊗ εR' | β β' ⊗ pauliCo | μ α β}ᵀ := by
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conv =>
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lhs
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rw [tensorNode_pauliCoDown]
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rw [contr_tensor_eq <| contr_prod _ _ _]
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rw [perm_contr]
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erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
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rw [perm_tensor_eq <| contr_congr 1 2]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_congr 1 2]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 1 2]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 1 2]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 1 2]
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rw [perm_perm]
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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 <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 4 1]
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rw [perm_perm]
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conv =>
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rhs
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rw [perm_tensor_eq <| contr_swap _ _]
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rw [perm_perm]
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erw [perm_tensor_eq <| contr_congr 4 1]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_swap _ _]
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erw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 4 1]
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rw [perm_perm]
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apply perm_congr _ rfl
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decide
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set_option maxRecDepth 5000 in
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/-- A rearanging of `pauliContrDown` to place the pauli matrices on the right. -/
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lemma pauliContrDown_eq_metric_mul_pauliContr :
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{pauliContrDown | μ α' β' = εL' | α α' ⊗
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εR' | β β' ⊗ pauliContr | μ α β}ᵀ := by
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conv =>
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lhs
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rw [tensorNode_pauliContrDown]
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rw [contr_tensor_eq <| contr_prod _ _ _]
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rw [perm_contr]
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erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
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rw [perm_tensor_eq <| contr_congr 1 2]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_congr 1 2]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 1 2]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 1 2]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 1 2]
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rw [perm_perm]
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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 <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 4 1]
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rw [perm_perm]
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conv =>
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rhs
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rw [perm_tensor_eq <| contr_swap _ _]
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rw [perm_perm]
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erw [perm_tensor_eq <| contr_congr 4 1]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_tensor_eq <| contr_swap _ _]
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erw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
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rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
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rw [perm_tensor_eq <| perm_contr _ _]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_congr 4 1]
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rw [perm_perm]
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apply perm_congr _ rfl
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decide
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/-!
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## Group actions
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-/
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/-- The tensor `pauliContr` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_pauliContr (g : SL(2,ℂ)) : {g •ₐ pauliContr | μ α β}ᵀ.tensor =
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{pauliContr | μ α β}ᵀ.tensor := by
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rw [tensorNode_pauliContr, constThreeNodeE]
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rw [← action_constThreeNode _ g]
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rfl
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/-- The tensor `pauliCo` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_pauliCo (g : SL(2,ℂ)) : {g •ₐ pauliCo | μ α β}ᵀ.tensor =
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{pauliCo | μ α β}ᵀ.tensor := by
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conv =>
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lhs
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rw [action_tensor_eq <| tensorNode_pauliCo]
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rw [action_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_pauliContr]
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rw [(contr_action _ _).symm]
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rw [contr_tensor_eq <| (prod_action _ _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| action_constTwoNode _ _]
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rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constThreeNode _ _]
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conv =>
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rhs
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rw [tensorNode_pauliCo]
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rw [contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_pauliContr]
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rfl
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/-- The tensor `pauliCoDown` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_pauliCoDown (g : SL(2,ℂ)) : {g •ₐ pauliCoDown | μ α β}ᵀ.tensor =
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{pauliCoDown | μ α β}ᵀ.tensor := by
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conv =>
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lhs
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rw [action_tensor_eq <| tensorNode_pauliCoDown]
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rw [(contr_action _ _).symm]
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rw [contr_tensor_eq <| (prod_action _ _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| (contr_action _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| (prod_action _ _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_fst <|
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action_pauliCo _]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_snd <|
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action_altLeftMetric _]
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rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_altRightMetric _]
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conv =>
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rhs
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rw [tensorNode_pauliCoDown]
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/-- The tensor `pauliContrDown` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_pauliContrDown (g : SL(2,ℂ)) : {g •ₐ pauliContrDown | μ α β}ᵀ.tensor =
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{pauliContrDown | μ α β}ᵀ.tensor := by
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conv =>
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lhs
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rw [action_tensor_eq <| tensorNode_pauliContrDown]
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rw [(contr_action _ _).symm]
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rw [contr_tensor_eq <| (prod_action _ _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| (contr_action _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| (prod_action _ _ _).symm]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_fst <|
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action_pauliContr _]
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_snd <|
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action_altLeftMetric _]
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erw [contr_tensor_eq <| prod_tensor_eq_snd <| action_altRightMetric _]
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conv =>
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rhs
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rw [tensorNode_pauliContrDown]
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end complexLorentzTensor
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