Refactor: Metrics as complex lorentz tensors

HepLean.lean

@@ -111,6 +111,9 @@ import HepLean.Tensors.ComplexLorentz.Basic
 import HepLean.Tensors.ComplexLorentz.Basis
 import HepLean.Tensors.ComplexLorentz.Bispinors.Basic
 import HepLean.Tensors.ComplexLorentz.Lemmas
+import HepLean.Tensors.ComplexLorentz.Metrics.Basic
+import HepLean.Tensors.ComplexLorentz.Metrics.Basis
+import HepLean.Tensors.ComplexLorentz.Metrics.Lemmas
 import HepLean.Tensors.ComplexLorentz.PauliMatrices.Basic
 import HepLean.Tensors.ComplexLorentz.PauliMatrices.Basis
 import HepLean.Tensors.ComplexLorentz.PauliMatrices.CoContractContr

HepLean/Tensors/ComplexLorentz/Basis.lean

@@ -223,168 +223,5 @@ lemma eval_basisVector {n : ℕ} {c : Fin n.succ → complexLorentzTensor.C}
   exact ite_congr (Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a))
     (congrFun rfl) (congrFun rfl)
 
-/-!
-
-## Useful expansions.
-
--/
-
-/-- The expansion of the Lorentz covariant metric in terms of basis vectors. -/
-lemma coMetric_basis_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
-    basisVector ![Color.down, Color.down] (fun _ => 0)
-    - basisVector ![Color.down, Color.down] (fun _ => 1)
-    - basisVector ![Color.down, Color.down] (fun _ => 2)
-    - basisVector ![Color.down, Color.down] (fun _ => 3) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
-    Functor.id_obj, Fin.isValue]
-  erw [Lorentz.coMetric_apply_one, Lorentz.coMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_sub]
-  congr 1
-  congr 1
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, basisVector]
-    apply congrArg
-    funext i
-    fin_cases i
-    all_goals
-      simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
-        cons_val_zero, Fin.cases_zero]
-      change _ = Lorentz.complexCoBasisFin4 _
-      simp only [Fin.isValue, Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
-      rfl
-
-lemma coMetric_basis_expand_tree : {Lorentz.coMetric | μ ν}ᵀ.tensor =
-    (TensorTree.add (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 0))) <|
-    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 1)))) <|
-    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 2)))) <|
-    (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 3))))).tensor :=
-  coMetric_basis_expand
-
-/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors. -/
-lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
-    basisVector ![Color.up, Color.up] (fun _ => 0)
-    - basisVector ![Color.up, Color.up] (fun _ => 1)
-    - basisVector ![Color.up, Color.up] (fun _ => 2)
-    - basisVector ![Color.up, Color.up] (fun _ => 3) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V]
-  erw [Lorentz.contrMetric_apply_one, Lorentz.contrMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_sub]
-  congr 1
-  congr 1
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, basisVector]
-    apply congrArg
-    funext i
-    fin_cases i
-    all_goals
-      simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
-        cons_val_zero, Fin.cases_zero]
-      change _ = Lorentz.complexContrBasisFin4 _
-      simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
-      rfl
-
-lemma contrMatrix_basis_expand_tree : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
-    (TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
-    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 1)))) <|
-    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 2)))) <|
-    (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 3))))).tensor :=
-  contrMatrix_basis_expand
-
-lemma leftMetric_expand : {Fermion.leftMetric | α β}ᵀ.tensor =
-    - basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
-    + basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
-  erw [Fermion.leftMetric_apply_one, Fermion.leftMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_add, map_neg]
-  congr 1
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, basisVector]
-    apply congrArg
-    funext i
-    fin_cases i
-    · rfl
-    · rfl
-
-lemma leftMetric_expand_tree : {Fermion.leftMetric | α β}ᵀ.tensor =
-    (TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL]
-    (fun | 0 => 0 | 1 => 1)))) <|
-    (tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
-  leftMetric_expand
-
-lemma altLeftMetric_expand : {Fermion.altLeftMetric | α β}ᵀ.tensor =
-    basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1)
-    - basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
-  erw [Fermion.altLeftMetric_apply_one, Fermion.altLeftMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_sub]
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, basisVector]
-    apply congrArg
-    funext i
-    fin_cases i
-    · rfl
-    · rfl
-
-lemma altLeftMetric_expand_tree : {Fermion.altLeftMetric | α β}ᵀ.tensor =
-    (TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL]
-    (fun | 0 => 0 | 1 => 1))) <|
-    (smul (-1) (tensorNode (basisVector ![Color.downL, Color.downL]
-    (fun | 0 => 1 | 1 => 0))))).tensor :=
-  altLeftMetric_expand
-
-lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
-    - basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)
-    + basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
-  erw [Fermion.rightMetric_apply_one, Fermion.rightMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_add, map_neg]
-  congr 1
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, basisVector]
-    apply congrArg
-    funext i
-    fin_cases i
-    · rfl
-    · rfl
-
-lemma rightMetric_expand_tree : {Fermion.rightMetric | α β}ᵀ.tensor =
-    (TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR]
-    (fun | 0 => 0 | 1 => 1)))) <|
-    (tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
-  rightMetric_expand
-
-lemma altRightMetric_expand : {Fermion.altRightMetric | α β}ᵀ.tensor =
-    basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1)
-    - basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
-  erw [Fermion.altRightMetric_apply_one, Fermion.altRightMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_sub]
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, basisVector]
-    apply congrArg
-    funext i
-    fin_cases i
-    · rfl
-    · rfl
-
-lemma altRightMetric_expand_tree : {Fermion.altRightMetric | α β}ᵀ.tensor =
-    (TensorTree.add (tensorNode (basisVector
-    ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|
-    (smul (-1) (tensorNode (basisVector ![Color.downR, Color.downR]
-    (fun | 0 => 1 | 1 => 0))))).tensor :=
-  altRightMetric_expand
-
 end complexLorentzTensor
 end

HepLean/Tensors/ComplexLorentz/Lemmas.lean

@@ -100,65 +100,6 @@ lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
   apply TensorTree.add_tensor_eq_snd
   rw [neg_tensor_eq hA, neg_tensor_eq (neg_perm _ _), neg_neg]
 
-/-!
-
-## The contraction of Pauli matrices with Pauli matrices
-
-And related results.
-
--/
-
-/-- The map to color one gets when multiplying left and right metrics. -/
-def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
-  finSumFinEquiv.symm
-
-lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
-    = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
-    - basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
-    - basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
-    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
-  rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
-  rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
-  rw [prod_add_both]
-  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
-  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
-  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
-  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
-  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
-  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
-  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
-  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
-  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
-  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
-  rw [← add_assoc]
-  simp only [add_tensor, smul_tensor, tensorNode_tensor]
-  change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
-    +- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
-    +- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
-    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
-  congr 1
-  congr 1
-  congr 1
-  all_goals
-    congr
-    funext x
-    fin_cases x <;> rfl
-
-lemma leftMetric_mul_rightMetric_tree :
-    {Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
-    = (TensorTree.add (tensorNode
-        (basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|
-      TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
-        (basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
-      TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
-        (basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
-      (tensorNode
-        (basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
-  rw [leftMetric_mul_rightMetric]
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
-    smul_tensor, neg_smul, one_smul]
-  rfl
-
 end complexLorentzTensor
 
 end

HepLean/Tensors/ComplexLorentz/Metrics/Basic.lean

@@ -0,0 +1,162 @@
+/-
+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

HepLean/Tensors/ComplexLorentz/Metrics/Basis.lean

@@ -0,0 +1,191 @@
+/-
+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.ComplexLorentz.Metrics.Basic
+import HepLean.Tensors.ComplexLorentz.Basis
+/-!
+
+## Metrics and basis expansions
+
+-/
+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
+
+/-- The expansion of the Lorentz covariant metric in terms of basis vectors. -/
+lemma coMetric_basis_expand : {η' | μ ν}ᵀ.tensor =
+    basisVector ![Color.down, Color.down] (fun _ => 0)
+    - basisVector ![Color.down, Color.down] (fun _ => 1)
+    - basisVector ![Color.down, Color.down] (fun _ => 2)
+    - basisVector ![Color.down, Color.down] (fun _ => 3) := by
+  rw [tensorNode_coMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+    Functor.id_obj, Fin.isValue]
+  erw [Lorentz.coMetric_apply_one, Lorentz.coMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_sub]
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul, basisVector]
+    apply congrArg
+    funext i
+    fin_cases i
+    all_goals
+      simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
+        cons_val_zero, Fin.cases_zero]
+      change _ = Lorentz.complexCoBasisFin4 _
+      simp only [Fin.isValue, Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
+      rfl
+
+lemma coMetric_basis_expand_tree : {η' | μ ν}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 0))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 1)))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 2)))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 3))))).tensor :=
+  coMetric_basis_expand
+
+/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors. -/
+lemma contrMatrix_basis_expand : {η | μ ν}ᵀ.tensor =
+    basisVector ![Color.up, Color.up] (fun _ => 0)
+    - basisVector ![Color.up, Color.up] (fun _ => 1)
+    - basisVector ![Color.up, Color.up] (fun _ => 2)
+    - basisVector ![Color.up, Color.up] (fun _ => 3) := by
+  rw [tensorNode_contrMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V]
+  erw [Lorentz.contrMetric_apply_one, Lorentz.contrMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_sub]
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul, basisVector]
+    apply congrArg
+    funext i
+    fin_cases i
+    all_goals
+      simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
+        cons_val_zero, Fin.cases_zero]
+      change _ = Lorentz.complexContrBasisFin4 _
+      simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
+      rfl
+
+lemma contrMatrix_basis_expand_tree : {η | μ ν}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 1)))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 2)))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 3))))).tensor :=
+  contrMatrix_basis_expand
+
+lemma leftMetric_expand : {εL | α β}ᵀ.tensor =
+    - basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
+    + basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0) := by
+  rw [tensorNode_leftMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
+  erw [Fermion.leftMetric_apply_one, Fermion.leftMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_add, map_neg]
+  congr 1
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul, basisVector]
+    apply congrArg
+    funext i
+    fin_cases i
+    · rfl
+    · rfl
+
+lemma leftMetric_expand_tree : {εL | α β}ᵀ.tensor =
+    (TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL]
+    (fun | 0 => 0 | 1 => 1)))) <|
+    (tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
+  leftMetric_expand
+
+lemma altLeftMetric_expand : {εL' | α β}ᵀ.tensor =
+    basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1)
+    - basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0) := by
+  rw [tensorNode_altLeftMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
+  erw [Fermion.altLeftMetric_apply_one, Fermion.altLeftMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_sub]
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul, basisVector]
+    apply congrArg
+    funext i
+    fin_cases i
+    · rfl
+    · rfl
+
+lemma altLeftMetric_expand_tree : {εL' | α β}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL]
+    (fun | 0 => 0 | 1 => 1))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.downL, Color.downL]
+    (fun | 0 => 1 | 1 => 0))))).tensor :=
+  altLeftMetric_expand
+
+lemma rightMetric_expand : {εR | α β}ᵀ.tensor =
+    - basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)
+    + basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0) := by
+  rw [tensorNode_rightMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
+  erw [Fermion.rightMetric_apply_one, Fermion.rightMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_add, map_neg]
+  congr 1
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul, basisVector]
+    apply congrArg
+    funext i
+    fin_cases i
+    · rfl
+    · rfl
+
+lemma rightMetric_expand_tree : {εR | α β}ᵀ.tensor =
+    (TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR]
+    (fun | 0 => 0 | 1 => 1)))) <|
+    (tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
+  rightMetric_expand
+
+lemma altRightMetric_expand : {εR' | α β}ᵀ.tensor =
+    basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1)
+    - basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0) := by
+  rw [tensorNode_altRightMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
+  erw [Fermion.altRightMetric_apply_one, Fermion.altRightMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_sub]
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul, basisVector]
+    apply congrArg
+    funext i
+    fin_cases i
+    · rfl
+    · rfl
+
+lemma altRightMetric_expand_tree : {εR' | α β}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector
+    ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.downR, Color.downR]
+    (fun | 0 => 1 | 1 => 0))))).tensor :=
+  altRightMetric_expand
+
+end complexLorentzTensor

HepLean/Tensors/ComplexLorentz/Metrics/Lemmas.lean

@@ -0,0 +1,79 @@
+/-
+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.ComplexLorentz.Metrics.Basis
+import HepLean.Tensors.ComplexLorentz.Basis
+/-!
+
+## Metrics and basis expansions
+
+-/
+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
+
+/-- The map to color one gets when multiplying left and right metrics. -/
+def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
+  finSumFinEquiv.symm
+
+lemma leftMetric_mul_rightMetric : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
+    = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    - basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    - basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
+  rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
+  rw [prod_add_both]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
+  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
+  rw [← add_assoc]
+  simp only [add_tensor, smul_tensor, tensorNode_tensor]
+  change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    +- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    +- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    congr
+    funext x
+    fin_cases x <;> rfl
+
+lemma leftMetric_mul_rightMetric_tree :
+    {εL | α α' ⊗ εR | β β'}ᵀ.tensor
+    = (TensorTree.add (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|
+      TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
+      TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
+      (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
+  rw [leftMetric_mul_rightMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
+    smul_tensor, neg_smul, one_smul]
+  rfl
+
+end complexLorentzTensor

HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean

@@ -10,6 +10,7 @@ import HepLean.Tensors.Tree.NodeIdentities.ContrContr
 import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
 import HepLean.Tensors.Tree.NodeIdentities.PermContr
 import HepLean.Tensors.Tree.NodeIdentities.Congr
+import HepLean.Tensors.ComplexLorentz.Metrics.Lemmas
 /-!
 
 ## Pauli matrices as complex Lorentz tensors
@@ -41,15 +42,13 @@ open Fermion
 def pauliContr := {PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
 
 /-- The Pauli matrices as the complex Lorentz tensor `σ_μ^α^{dot β}`. -/
-def pauliCo := {Lorentz.coMetric | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor
+def pauliCo := {η' | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor
 
 /-- The Pauli matrices as the complex Lorentz tensor `σ_μ_α_{dot β}`. -/
-def pauliCoDown := {pauliCo | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
+def pauliCoDown := {pauliCo | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor
-  Fermion.altRightMetric | β β'}ᵀ.tensor
 
 /-- The Pauli matrices as the complex Lorentz tensor `σ^μ_α_{dot β}`. -/
-def pauliContrDown := {pauliContr | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
+def pauliContrDown := {pauliContr | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor
-  Fermion.altRightMetric | β β'}ᵀ.tensor
 
 /-!
 
@@ -64,19 +63,17 @@ lemma tensorNode_pauliContr : {pauliContr | μ α β}ᵀ.tensor =
 
 /-- The definitional tensor node relation for `pauliCo`. -/
 lemma tensorNode_pauliCo : {pauliCo | μ α β}ᵀ.tensor =
-    {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
+    {η' | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
   rfl
 
 /-- The definitional tensor node relation for `pauliCoDown`. -/
 lemma tensorNode_pauliCoDown : {pauliCoDown | μ α β}ᵀ.tensor =
-    {pauliCo | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
+    {pauliCo | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor := by
-    Fermion.altRightMetric | β β'}ᵀ.tensor := by
   rfl
 
 /-- The definitional tensor node relation for `pauliContrDown`. -/
 lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
-    {pauliContr | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
+    {pauliContr | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor := by
-    Fermion.altRightMetric | β β'}ᵀ.tensor := by
   rfl
 
 /-!
@@ -88,8 +85,7 @@ lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
 set_option maxRecDepth 5000 in
 /-- A rearanging of `pauliCoDown` to place the pauli matrices on the right. -/
 lemma pauliCoDown_eq_metric_mul_pauliCo :
-    {pauliCoDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
+    {pauliCoDown | μ α' β' = εL' | α α' ⊗ εR' | β β' ⊗ pauliCo | μ α β}ᵀ := by
-    Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ := by
   conv =>
     lhs
     rw [tensorNode_pauliCoDown]
@@ -138,8 +134,8 @@ lemma pauliCoDown_eq_metric_mul_pauliCo :
 set_option maxRecDepth 5000 in
 /-- A rearanging of `pauliContrDown` to place the pauli matrices on the right. -/
 lemma pauliContrDown_eq_metric_mul_pauliContr :
-    {pauliContrDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
+    {pauliContrDown | μ α' β' = εL' | α α' ⊗
-    Fermion.altRightMetric | β β' ⊗ pauliContr | μ α β}ᵀ := by
+    εR' | β β' ⊗ pauliContr | μ α β}ᵀ := by
   conv =>
     lhs
     rw [tensorNode_pauliContrDown]
@@ -224,7 +220,7 @@ lemma action_pauliCoDown (g : SL(2,ℂ)) : {g •ₐ pauliCoDown | μ α β}ᵀ.
       action_pauliCo _]
     rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_snd <|
       action_constTwoNode _ _]
-    rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
+    erw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
   rfl
 
 /-- The tensor `pauliContrDown` is invariant under the action of `SL(2,ℂ)`. -/
@@ -241,7 +237,7 @@ lemma action_pauliContrDown (g : SL(2,ℂ)) : {g •ₐ pauliContrDown | μ α
       action_pauliContr _]
     rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_snd <|
       action_constTwoNode _ _]
-    rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
+    erw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
   rfl
 
 end complexLorentzTensor

HepLean/Tensors/ComplexLorentz/PauliMatrices/CoContractContr.lean

@@ -517,8 +517,7 @@ lemma pauliCo_contr_pauliContr_expand :
 
 /-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
 theorem pauliCo_contr_pauliContr :
-    {pauliCo | ν α β ⊗ pauliContr | ν α' β' =
+    {pauliCo | ν α β ⊗ pauliContr | ν α' β' = 2 •ₜ εL | α α' ⊗ εR | β β'}ᵀ := by
-    2 •ₜ Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ := by
   rw [pauliCo_contr_pauliContr_expand]
   rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
   rw [perm_smul]