feat: Add Pauli Matrix defs

HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean

@@ -119,6 +119,7 @@ lemma contrBispinorDown_eq_metric_contr_contrBispinorUp  (p : complexContr) :
     | (1 : Fin 2) => rfl
   · rfl
 
+/- TODO: Remove maxHeartbeats from this result. -/
 set_option maxHeartbeats 400000 in
 set_option maxRecDepth 2000 in
 lemma contrBispinorDown_eq_contr_with_self (p : complexContr) :
@@ -154,8 +155,8 @@ lemma contrBispinorDown_eq_contr_with_self (p : complexContr) :
   apply OverColor.Hom.fin_ext
   intro i
   fin_cases i
-  exact rfl
-  exact rfl
+  · exact rfl
+  · exact rfl
 
 /-- Expansion of a `contrBispinorDown` into the original contravariant tensor nested
   between pauli matrices and metrics. -/

HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean

@@ -0,0 +1,78 @@
+/-
+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.ProdContr
+import HepLean.Tensors.Tree.NodeIdentities.Congr
+/-!
+
+## Pauli matrices 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 Pauli matrices as `σ^μ^α^{dot β}`. -/
+def pauliContr := {PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
+
+/-- The Pauli matrices as `σ_μ^α^{dot β}`. -/
+def pauliCo := {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
+
+/-- The Pauli matrices as `σ_μ_α_{dot β}`. -/
+def pauliCoDown := {Fermion.altLeftMetric | α α' ⊗
+  Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ.tensor
+
+/-- The Pauli matrices as `σ^μ_α_{dot β}`. -/
+def pauliContrDown := {Fermion.altLeftMetric | α α' ⊗
+  Fermion.altRightMetric | β β' ⊗ pauliContr | μ α β}ᵀ.tensor
+
+/-!
+
+## Tensor nodes.
+
+-/
+
+/-- The definitional tensor node relation for `pauliContr`. -/
+lemma tensorNode_pauliContr : {pauliContr | μ α β}ᵀ.tensor =
+    {PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
+  rfl
+
+/-- 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
+  rfl
+
+/-- The definitional tensor node relation for `pauliCoDown`. -/
+lemma tensorNode_pauliCoDown : {pauliCoDown | μ α β}ᵀ.tensor =
+    {Fermion.altLeftMetric | α α' ⊗
+    Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ.tensor := by
+  rfl
+
+/-- The definitional tensor node relation for `pauliContrDown`. -/
+lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
+    {Fermion.altLeftMetric | α α' ⊗
+    Fermion.altRightMetric | β β' ⊗ pauliContr | μ α β}ᵀ.tensor := by
+  rfl
+
+end complexLorentzTensor