refactor: Move ComplexTensor
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14 changed files with 29 additions and 29 deletions
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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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/-!
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## Metrics 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 unit `δᵢⁱ` as a complex Lorentz tensor. -/
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def coContrUnit := (TensorTree.constTwoNodeE complexLorentzTensor Color.down Color.up
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Lorentz.coContrUnit).tensor
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/-- The unit `δⁱᵢ` as a complex Lorentz tensor. -/
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def contrCoUnit := (TensorTree.constTwoNodeE complexLorentzTensor Color.up Color.down
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Lorentz.contrCoUnit).tensor
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/-- The unit `δₐᵃ` as a complex Lorentz tensor. -/
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def altLeftLeftUnit := (TensorTree.constTwoNodeE complexLorentzTensor Color.downL Color.upL
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Fermion.altLeftLeftUnit).tensor
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/-- The unit `δᵃₐ` as a complex Lorentz tensor. -/
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def leftAltLeftUnit := (TensorTree.constTwoNodeE complexLorentzTensor Color.upL Color.downL
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Fermion.leftAltLeftUnit).tensor
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/-- The unit `δ_{dot a}^{dot a}` as a complex Lorentz tensor. -/
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def altRightRightUnit := (TensorTree.constTwoNodeE complexLorentzTensor Color.downR Color.upR
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Fermion.altRightRightUnit).tensor
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/-- The unit `δ^{dot a}_{dot a}` as a complex Lorentz tensor. -/
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def rightAltRightUnit := (TensorTree.constTwoNodeE complexLorentzTensor Color.upR Color.downR
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Fermion.rightAltRightUnit).tensor
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/-!
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## Notation
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-/
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/-- The unit `δᵢⁱ` as a complex Lorentz tensor. -/
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scoped[complexLorentzTensor] notation "δ'" => coContrUnit
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/-- The unit `δⁱᵢ` as a complex Lorentz tensor. -/
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scoped[complexLorentzTensor] notation "δ" => contrCoUnit
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/-- The unit `δₐᵃ` as a complex Lorentz tensor. -/
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scoped[complexLorentzTensor] notation "δL'" => altLeftLeftUnit
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/-- The unit `δᵃₐ` as a complex Lorentz tensor. -/
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scoped[complexLorentzTensor] notation "δL" => leftAltLeftUnit
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/-- The unit `δ_{dot a}^{dot a}` as a complex Lorentz tensor. -/
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scoped[complexLorentzTensor] notation "δR'" => altRightRightUnit
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/-- The unit `δ^{dot a}_{dot a}` as a complex Lorentz tensor. -/
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scoped[complexLorentzTensor] notation "δR" => rightAltRightUnit
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/-!
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## Tensor nodes.
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-/
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/-- The definitional tensor node relation for `coContrUnit`. -/
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lemma tensorNode_coContrUnit : {δ' | μ ν}ᵀ.tensor = (TensorTree.constTwoNodeE complexLorentzTensor
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Color.down Color.up Lorentz.coContrUnit).tensor:= by
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rfl
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/-- The definitional tensor node relation for `contrCoUnit`. -/
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lemma tensorNode_contrCoUnit: {δ | μ ν}ᵀ.tensor = (TensorTree.constTwoNodeE complexLorentzTensor
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Color.up Color.down Lorentz.contrCoUnit).tensor := by
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rfl
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/-- The definitional tensor node relation for `altLeftLeftUnit`. -/
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lemma tensorNode_altLeftLeftUnit : {δL' | μ ν}ᵀ.tensor = (TensorTree.constTwoNodeE
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complexLorentzTensor Color.downL Color.upL Fermion.altLeftLeftUnit).tensor := by
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rfl
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/-- The definitional tensor node relation for `leftAltLeftUnit`. -/
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lemma tensorNode_leftAltLeftUnit : {δL | μ ν}ᵀ.tensor = (TensorTree.constTwoNodeE
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complexLorentzTensor Color.upL Color.downL Fermion.leftAltLeftUnit).tensor := by
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rfl
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/-- The definitional tensor node relation for `altRightRightUnit`. -/
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lemma tensorNode_altRightRightUnit: {δR' | μ ν}ᵀ.tensor = (TensorTree.constTwoNodeE
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complexLorentzTensor Color.downR Color.upR Fermion.altRightRightUnit).tensor := by
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rfl
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/-- The definitional tensor node relation for `rightAltRightUnit`. -/
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lemma tensorNode_rightAltRightUnit: {δR | μ ν}ᵀ.tensor = (TensorTree.constTwoNodeE
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complexLorentzTensor Color.upR Color.downR Fermion.rightAltRightUnit).tensor := by
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rfl
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/-!
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## Group actions
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-/
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/-- The tensor `coContrUnit` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_coContrUnit (g : SL(2,ℂ)) : {g •ₐ δ' | μ ν}ᵀ.tensor = {δ' | μ ν}ᵀ.tensor := by
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rw [tensorNode_coContrUnit, constTwoNodeE, ← action_constTwoNode _ g]
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rfl
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/-- The tensor `contrCoUnit` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_contrCoUnit (g : SL(2,ℂ)) : {g •ₐ δ | μ ν}ᵀ.tensor = {δ | μ ν}ᵀ.tensor := by
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rw [tensorNode_contrCoUnit, constTwoNodeE, ← action_constTwoNode _ g]
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rfl
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/-- The tensor `altLeftLeftUnit` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_altLeftLeftUnit (g : SL(2,ℂ)) : {g •ₐ δL' | μ ν}ᵀ.tensor = {δL' | μ ν}ᵀ.tensor := by
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rw [tensorNode_altLeftLeftUnit, constTwoNodeE, ← action_constTwoNode _ g]
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rfl
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/-- The tensor `leftAltLeftUnit` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_leftAltLeftUnit (g : SL(2,ℂ)) : {g •ₐ δL | μ ν}ᵀ.tensor = {δL | μ ν}ᵀ.tensor := by
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rw [tensorNode_leftAltLeftUnit, constTwoNodeE, ← action_constTwoNode _ g]
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rfl
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/-- The tensor `altRightRightUnit` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_altRightRightUnit (g : SL(2,ℂ)) : {g •ₐ δR' | μ ν}ᵀ.tensor = {δR' | μ ν}ᵀ.tensor := by
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rw [tensorNode_altRightRightUnit, constTwoNodeE, ← action_constTwoNode _ g]
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rfl
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/-- The tensor `rightAltRightUnit` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_rightAltRightUnit (g : SL(2,ℂ)) : {g •ₐ δR | μ ν}ᵀ.tensor = {δR | μ ν}ᵀ.tensor := by
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rw [tensorNode_rightAltRightUnit, constTwoNodeE, ← action_constTwoNode _ g]
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rfl
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end complexLorentzTensor
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