feat: Add action node

HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean

@@ -185,4 +185,63 @@ lemma pauliContrDown_eq_metric_mul_pauliContr :
   apply perm_congr _ rfl
   decide
 
+/-!
+
+## Group actions
+
+-/
+
+/-- The tensor `pauliContr` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_pauliContr (g : SL(2,ℂ)) : {g •ₐ pauliContr | μ α β}ᵀ.tensor =
+    {pauliContr | μ α β}ᵀ.tensor := by
+  rw [tensorNode_pauliContr, constThreeNodeE]
+  rw [← action_constThreeNode _ g]
+  rfl
+
+/-- The tensor `pauliCo` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_pauliCo (g : SL(2,ℂ)) : {g •ₐ pauliCo | μ α β}ᵀ.tensor =
+    {pauliCo | μ α β}ᵀ.tensor := by
+  conv =>
+    lhs
+    rw [action_tensor_eq <| tensorNode_pauliCo]
+    rw [(contr_action _ _).symm]
+    rw [contr_tensor_eq <| (prod_action _ _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| action_constTwoNode _ _]
+    rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constThreeNode _ _]
+  rfl
+
+/-- The tensor `pauliCoDown` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_pauliCoDown (g : SL(2,ℂ)) : {g •ₐ pauliCoDown | μ α β}ᵀ.tensor =
+    {pauliCoDown | μ α β}ᵀ.tensor := by
+  conv =>
+    lhs
+    rw [action_tensor_eq <| tensorNode_pauliCoDown]
+    rw [(contr_action _ _).symm]
+    rw [contr_tensor_eq <| (prod_action _ _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| (contr_action _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| (prod_action _ _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_fst <|
+      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 _ _]
+  rfl
+
+/-- The tensor `pauliContrDown` is invariant under the action of `SL(2,ℂ)`. -/
+lemma action_pauliContrDown (g : SL(2,ℂ)) : {g •ₐ pauliContrDown | μ α β}ᵀ.tensor =
+    {pauliContrDown | μ α β}ᵀ.tensor := by
+  conv =>
+    lhs
+    rw [action_tensor_eq <| tensorNode_pauliContrDown]
+    rw [(contr_action _ _).symm]
+    rw [contr_tensor_eq <| (prod_action _ _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| (contr_action _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| (prod_action _ _ _).symm]
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_fst <|
+      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 _ _]
+  rfl
+
 end complexLorentzTensor