Refactor: Metrics as complex lorentz tensors

HepLean/Tensors/ComplexLorentz/Metrics/Basic.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.Tensors.Tree.NodeIdentities.ProdAssoc
+import HepLean.Tensors.Tree.NodeIdentities.ProdComm
+import HepLean.Tensors.Tree.NodeIdentities.ProdContr
+import HepLean.Tensors.Tree.NodeIdentities.ContrContr
+import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
+import HepLean.Tensors.Tree.NodeIdentities.PermContr
+import HepLean.Tensors.Tree.NodeIdentities.Congr
+/-!
+
+## Metrics as complex Lorentz tensors
+
+-/
+open IndexNotation
+open CategoryTheory
+open MonoidalCategory
+open Matrix
+open MatrixGroups
+open Complex
+open TensorProduct
+open IndexNotation
+open CategoryTheory
+open TensorTree
+open OverColor.Discrete
+noncomputable section
+
+namespace complexLorentzTensor
+open Fermion
+
+/-!
+
+## Definitions.
+
+-/
+
+/-- The metric `ηᵢᵢ` as a complex Lorentz tensor. -/
+def coMetric := {Lorentz.coMetric | μ ν}ᵀ.tensor
+
+/-- The metric `ηⁱⁱ` as a complex Lorentz tensor. -/
+def contrMetric := {Lorentz.contrMetric | μ ν}ᵀ.tensor
+
+/-- The metric `εᵃᵃ` as a complex Lorentz tensor. -/
+def leftMetric := {Fermion.leftMetric | α α'}ᵀ.tensor
+
+/-- The metric `ε^{dot a}^{dot a}` as a complex Lorentz tensor. -/
+def rightMetric := {Fermion.rightMetric | β β'}ᵀ.tensor
+
+/-- The metric `εₐₐ` as a complex Lorentz tensor. -/
+def altLeftMetric := {Fermion.altLeftMetric | α α'}ᵀ.tensor
+
+/-- The metric `ε_{dot a}_{dot a}` as a complex Lorentz tensor. -/
+def altRightMetric := {Fermion.altRightMetric | β β'}ᵀ.tensor
+
+/-!
+
+## Notation
+
+-/
+
+/-- The metric `ηᵢᵢ` as a complex Lorentz tensors. -/
+scoped[complexLorentzTensor] notation "η'" => coMetric
+
+/-- The metric `ηⁱⁱ` as a complex Lorentz tensors. -/
+scoped[complexLorentzTensor] notation "η" => contrMetric
+
+/-- The metric `εᵃᵃ` as a complex Lorentz tensors. -/
+scoped[complexLorentzTensor] notation "εL" => leftMetric
+
+/-- The metric `ε^{dot a}^{dot a}` as a complex Lorentz tensors. -/
+scoped[complexLorentzTensor] notation "εR" => rightMetric
+
+/-- The metric `εₐₐ` as a complex Lorentz tensors. -/
+scoped[complexLorentzTensor] notation "εL'" => altLeftMetric
+
+/-- The metric `ε_{dot a}_{dot a}` as a complex Lorentz tensors. -/
+scoped[complexLorentzTensor] notation "εR'" => altRightMetric
+
+/-!
+
+## Tensor nodes.
+
+-/
+
+/-- The definitional tensor node relation for `coMetric`. -/
+lemma tensorNode_coMetric : {η' | μ ν}ᵀ.tensor = {Lorentz.coMetric | μ ν}ᵀ.tensor := by
+  rfl
+
+/-- The definitional tensor node relation for `contrMetric`. -/
+lemma tensorNode_contrMetric : {η | μ ν}ᵀ.tensor = {Lorentz.contrMetric | μ ν}ᵀ.tensor := by
+  rfl
+
+/-- The definitional tensor node relation for `leftMetric`. -/
+lemma tensorNode_leftMetric : {εL | α α'}ᵀ.tensor = {Fermion.leftMetric | α α'}ᵀ.tensor := by
+  rfl
+
+/-- The definitional tensor node relation for `rightMetric`. -/
+lemma tensorNode_rightMetric : {εR | β β'}ᵀ.tensor = {Fermion.rightMetric | β β'}ᵀ.tensor := by
+  rfl
+
+/-- The definitional tensor node relation for `altLeftMetric`. -/
+lemma tensorNode_altLeftMetric :
+    {εL' | α α'}ᵀ.tensor = {Fermion.altLeftMetric | α α'}ᵀ.tensor := by
+  rfl
+
+/-- The definitional tensor node relation for `altRightMetric`. -/
+lemma tensorNode_altRightMetric :
+    {εR' | β β'}ᵀ.tensor = {Fermion.altRightMetric | β β'}ᵀ.tensor := by
+  rfl
+
+/-!
+
+## Group actions
+
+-/
+
+/-- The tensor `coMetric` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_coMetric (g : SL(2,ℂ)) : {g •ₐ η' | μ ν}ᵀ.tensor =
+    {η' | μ ν}ᵀ.tensor := by
+  rw [tensorNode_coMetric, constTwoNodeE]
+  rw [← action_constTwoNode _ g]
+  rfl
+
+/-- The tensor `contrMetric` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_contrMetric (g : SL(2,ℂ)) : {g •ₐ η | μ ν}ᵀ.tensor =
+    {η | μ ν}ᵀ.tensor := by
+  rw [tensorNode_contrMetric, constTwoNodeE]
+  rw [← action_constTwoNode _ g]
+  rfl
+
+/-- The tensor `leftMetric` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_leftMetric (g : SL(2,ℂ)) : {g •ₐ εL | α α'}ᵀ.tensor =
+    {εL | α α'}ᵀ.tensor := by
+  rw [tensorNode_leftMetric, constTwoNodeE]
+  rw [← action_constTwoNode _ g]
+  rfl
+
+/-- The tensor `rightMetric` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_rightMetric (g : SL(2,ℂ)) : {g •ₐ εR | β β'}ᵀ.tensor =
+    {εR | β β'}ᵀ.tensor := by
+  rw [tensorNode_rightMetric, constTwoNodeE]
+  rw [← action_constTwoNode _ g]
+  rfl
+
+/-- The tensor `altLeftMetric` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_altLeftMetric (g : SL(2,ℂ)) : {g •ₐ εL' | α α'}ᵀ.tensor =
+    {εL' | α α'}ᵀ.tensor := by
+  rw [tensorNode_altLeftMetric, constTwoNodeE]
+  rw [← action_constTwoNode _ g]
+  rfl
+
+/-- The tensor `altRightMetric` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_altRightMetric (g : SL(2,ℂ)) : {g •ₐ εR' | β β'}ᵀ.tensor =
+    {εR' | β β'}ᵀ.tensor := by
+  rw [tensorNode_altRightMetric, constTwoNodeE]
+  rw [← action_constTwoNode _ g]
+  rfl
+
+end complexLorentzTensor