79 lines
2.2 KiB
Text
79 lines
2.2 KiB
Text
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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.ProdContr
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import HepLean.Tensors.Tree.NodeIdentities.Congr
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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 `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 `σ_μ^α^{dot β}`. -/
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def pauliCo := {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
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/-- The Pauli matrices as `σ_μ_α_{dot β}`. -/
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def pauliCoDown := {Fermion.altLeftMetric | α α' ⊗
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Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ.tensor
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/-- The Pauli matrices as `σ^μ_α_{dot β}`. -/
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def pauliContrDown := {Fermion.altLeftMetric | α α' ⊗
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Fermion.altRightMetric | β β' ⊗ pauliContr | μ α 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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{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
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rfl
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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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{Fermion.altLeftMetric | α α' ⊗
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Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ.tensor := by
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rfl
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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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{Fermion.altLeftMetric | α α' ⊗
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Fermion.altRightMetric | β β' ⊗ pauliContr | μ α β}ᵀ.tensor := by
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rfl
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end complexLorentzTensor
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