162 lines
5.1 KiB
Text
162 lines
5.1 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.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 metric `ηᵢᵢ` as a complex Lorentz tensor. -/
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def coMetric := {Lorentz.coMetric | μ ν}ᵀ.tensor
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/-- The metric `ηⁱⁱ` as a complex Lorentz tensor. -/
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def contrMetric := {Lorentz.contrMetric | μ ν}ᵀ.tensor
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/-- The metric `εᵃᵃ` as a complex Lorentz tensor. -/
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def leftMetric := {Fermion.leftMetric | α α'}ᵀ.tensor
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/-- The metric `ε^{dot a}^{dot a}` as a complex Lorentz tensor. -/
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def rightMetric := {Fermion.rightMetric | β β'}ᵀ.tensor
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/-- The metric `εₐₐ` as a complex Lorentz tensor. -/
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def altLeftMetric := {Fermion.altLeftMetric | α α'}ᵀ.tensor
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/-- The metric `ε_{dot a}_{dot a}` as a complex Lorentz tensor. -/
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def altRightMetric := {Fermion.altRightMetric | β β'}ᵀ.tensor
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/-!
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## Notation
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-/
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/-- The metric `ηᵢᵢ` as a complex Lorentz tensors. -/
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scoped[complexLorentzTensor] notation "η'" => coMetric
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/-- The metric `ηⁱⁱ` as a complex Lorentz tensors. -/
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scoped[complexLorentzTensor] notation "η" => contrMetric
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/-- The metric `εᵃᵃ` as a complex Lorentz tensors. -/
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scoped[complexLorentzTensor] notation "εL" => leftMetric
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/-- The metric `ε^{dot a}^{dot a}` as a complex Lorentz tensors. -/
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scoped[complexLorentzTensor] notation "εR" => rightMetric
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/-- The metric `εₐₐ` as a complex Lorentz tensors. -/
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scoped[complexLorentzTensor] notation "εL'" => altLeftMetric
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/-- The metric `ε_{dot a}_{dot a}` as a complex Lorentz tensors. -/
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scoped[complexLorentzTensor] notation "εR'" => altRightMetric
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/-!
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## Tensor nodes.
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-/
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/-- The definitional tensor node relation for `coMetric`. -/
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lemma tensorNode_coMetric : {η' | μ ν}ᵀ.tensor = {Lorentz.coMetric | μ ν}ᵀ.tensor := by
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rfl
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/-- The definitional tensor node relation for `contrMetric`. -/
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lemma tensorNode_contrMetric : {η | μ ν}ᵀ.tensor = {Lorentz.contrMetric | μ ν}ᵀ.tensor := by
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rfl
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/-- The definitional tensor node relation for `leftMetric`. -/
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lemma tensorNode_leftMetric : {εL | α α'}ᵀ.tensor = {Fermion.leftMetric | α α'}ᵀ.tensor := by
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rfl
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/-- The definitional tensor node relation for `rightMetric`. -/
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lemma tensorNode_rightMetric : {εR | β β'}ᵀ.tensor = {Fermion.rightMetric | β β'}ᵀ.tensor := by
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rfl
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/-- The definitional tensor node relation for `altLeftMetric`. -/
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lemma tensorNode_altLeftMetric :
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{εL' | α α'}ᵀ.tensor = {Fermion.altLeftMetric | α α'}ᵀ.tensor := by
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rfl
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/-- The definitional tensor node relation for `altRightMetric`. -/
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lemma tensorNode_altRightMetric :
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{εR' | β β'}ᵀ.tensor = {Fermion.altRightMetric | β β'}ᵀ.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 `coMetric` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_coMetric (g : SL(2,ℂ)) : {g •ₐ η' | μ ν}ᵀ.tensor =
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{η' | μ ν}ᵀ.tensor := by
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rw [tensorNode_coMetric, constTwoNodeE]
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rw [← action_constTwoNode _ g]
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rfl
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/-- The tensor `contrMetric` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_contrMetric (g : SL(2,ℂ)) : {g •ₐ η | μ ν}ᵀ.tensor =
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{η | μ ν}ᵀ.tensor := by
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rw [tensorNode_contrMetric, constTwoNodeE]
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rw [← action_constTwoNode _ g]
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rfl
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/-- The tensor `leftMetric` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_leftMetric (g : SL(2,ℂ)) : {g •ₐ εL | α α'}ᵀ.tensor =
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{εL | α α'}ᵀ.tensor := by
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rw [tensorNode_leftMetric, constTwoNodeE]
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rw [← action_constTwoNode _ g]
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rfl
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/-- The tensor `rightMetric` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_rightMetric (g : SL(2,ℂ)) : {g •ₐ εR | β β'}ᵀ.tensor =
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{εR | β β'}ᵀ.tensor := by
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rw [tensorNode_rightMetric, constTwoNodeE]
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rw [← action_constTwoNode _ g]
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rfl
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/-- The tensor `altLeftMetric` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_altLeftMetric (g : SL(2,ℂ)) : {g •ₐ εL' | α α'}ᵀ.tensor =
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{εL' | α α'}ᵀ.tensor := by
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rw [tensorNode_altLeftMetric, constTwoNodeE]
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rw [← action_constTwoNode _ g]
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rfl
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/-- The tensor `altRightMetric` is invariant under the action of `SL(2,ℂ)`. -/
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lemma action_altRightMetric (g : SL(2,ℂ)) : {g •ₐ εR' | β β'}ᵀ.tensor =
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{εR' | β β'}ᵀ.tensor := by
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rw [tensorNode_altRightMetric, constTwoNodeE]
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rw [← action_constTwoNode _ g]
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rfl
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end complexLorentzTensor
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