Merge pull request #196 from HEPLean/IndexNotation

refactor: Index notation

HepLean.lean

@@ -111,10 +111,7 @@ import HepLean.StandardModel.HiggsBoson.Potential
 import HepLean.StandardModel.Representations
 import HepLean.Tensors.Basic
 import HepLean.Tensors.ComplexLorentz.Basic
-import HepLean.Tensors.ComplexLorentz.ColorFun
-import HepLean.Tensors.ComplexLorentz.ContrNatTransform
 import HepLean.Tensors.ComplexLorentz.Examples
-import HepLean.Tensors.ComplexLorentz.TensorStruct
 import HepLean.Tensors.Contraction
 import HepLean.Tensors.EinsteinNotation.Basic
 import HepLean.Tensors.EinsteinNotation.IndexNotation
@@ -138,6 +135,7 @@ import HepLean.Tensors.IndexNotation.IndexString
 import HepLean.Tensors.IndexNotation.TensorIndex
 import HepLean.Tensors.MulActionTensor
 import HepLean.Tensors.OverColor.Basic
+import HepLean.Tensors.OverColor.Discrete
 import HepLean.Tensors.OverColor.Functors
 import HepLean.Tensors.OverColor.Iso
 import HepLean.Tensors.OverColor.Lift

HepLean/Tensors/ComplexLorentz/Basic.lean

@@ -4,7 +4,15 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.Tensors.OverColor.Basic
+import HepLean.Tensors.Tree.Dot
+import HepLean.SpaceTime.WeylFermion.Contraction
+import HepLean.SpaceTime.WeylFermion.Metric
+import HepLean.SpaceTime.WeylFermion.Unit
+import HepLean.SpaceTime.LorentzVector.Complex.Contraction
+import HepLean.SpaceTime.LorentzVector.Complex.Metric
+import HepLean.SpaceTime.LorentzVector.Complex.Unit
 import HepLean.Mathematics.PiTensorProduct
+import HepLean.SpaceTime.PauliMatrices.AsTensor
 /-!
 
 ## Complex Lorentz tensors
@@ -30,34 +38,71 @@ inductive Color
   | up : Color
   | down : Color
 
-/-- The involution taking a colour to its dual. -/
-def τ : Color → Color
-  | Color.upL => Color.downL
-  | Color.downL => Color.upL
-  | Color.upR => Color.downR
-  | Color.downR => Color.upR
-  | Color.up => Color.down
-  | Color.down => Color.up
-
-/-- The function taking a color to the dimension of the basis of vectors. -/
-def evalNo : Color → ℕ
-  | Color.upL => 2
-  | Color.downL => 2
-  | Color.upR => 2
-  | Color.downR => 2
-  | Color.up => 4
-  | Color.down => 4
-
 noncomputable section
-/-- The corresponding representations associated with a color. -/
-def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
-  match c with
-  | Color.upL => Fermion.altLeftHanded
-  | Color.downL => Fermion.leftHanded
-  | Color.upR => Fermion.altRightHanded
-  | Color.downR => Fermion.rightHanded
-  | Color.up => Lorentz.complexContr
-  | Color.down => Lorentz.complexCo
+
+/-- The tensor structure for complex Lorentz tensors. -/
+def complexLorentzTensor : TensorStruct where
+  C := Fermion.Color
+  G := SL(2, ℂ)
+  G_group := inferInstance
+  k := ℂ
+  k_commRing := inferInstance
+  FDiscrete := Discrete.functor fun c =>
+    match c with
+    | Color.upL => Fermion.leftHanded
+    | Color.downL => Fermion.altLeftHanded
+    | Color.upR => Fermion.rightHanded
+    | Color.downR => Fermion.altRightHanded
+    | Color.up => Lorentz.complexContr
+    | Color.down => Lorentz.complexCo
+  τ := fun c =>
+    match c with
+    | Color.upL => Color.downL
+    | Color.downL => Color.upL
+    | Color.upR => Color.downR
+    | Color.downR => Color.upR
+    | Color.up => Color.down
+    | Color.down => Color.up
+  τ_involution c := by
+    match c with
+    | Color.upL => rfl
+    | Color.downL => rfl
+    | Color.upR => rfl
+    | Color.downR => rfl
+    | Color.up => rfl
+    | Color.down => rfl
+  contr := Discrete.natTrans fun c =>
+    match c with
+    | Discrete.mk Color.upL => Fermion.leftAltContraction
+    | Discrete.mk Color.downL => Fermion.altLeftContraction
+    | Discrete.mk Color.upR => Fermion.rightAltContraction
+    | Discrete.mk Color.downR => Fermion.altRightContraction
+    | Discrete.mk Color.up => Lorentz.contrCoContraction
+    | Discrete.mk Color.down => Lorentz.coContrContraction
+  metric := Discrete.natTrans fun c =>
+    match c with
+    | Discrete.mk Color.upL => Fermion.leftMetric
+    | Discrete.mk Color.downL => Fermion.altLeftMetric
+    | Discrete.mk Color.upR => Fermion.rightMetric
+    | Discrete.mk Color.downR => Fermion.altRightMetric
+    | Discrete.mk Color.up => Lorentz.contrMetric
+    | Discrete.mk Color.down => Lorentz.coMetric
+  unit := Discrete.natTrans fun c =>
+    match c with
+    | Discrete.mk Color.upL => Fermion.altLeftLeftUnit
+    | Discrete.mk Color.downL => Fermion.leftAltLeftUnit
+    | Discrete.mk Color.upR => Fermion.altRightRightUnit
+    | Discrete.mk Color.downR => Fermion.rightAltRightUnit
+    | Discrete.mk Color.up => Lorentz.coContrUnit
+    | Discrete.mk Color.down => Lorentz.contrCoUnit
+  evalNo := fun c =>
+    match c with
+    | Color.upL => 2
+    | Color.downL => 2
+    | Color.upR => 2
+    | Color.downR => 2
+    | Color.up => 4
+    | Color.down => 4
 
 end
 end Fermion