213 lines
7.3 KiB
Text
213 lines
7.3 KiB
Text
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Meta.Informal
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import HepLean.SpaceTime.SL2C.Basic
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import Mathlib.RepresentationTheory.Rep
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import HepLean.Tensors.Basic
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import Mathlib.Logic.Equiv.TransferInstance
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/-!
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## Modules associated with Fermions
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Weyl fermions live in the vector space `ℂ^2`, defined here as `Fin 2 → ℂ`.
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However if we simply define the Module of Weyl fermions as `Fin 2 → ℂ` we get casting problems,
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where e.g. left-handed fermions can be cast to right-handed fermions etc.
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To overcome this, for each type of Weyl fermion we define a structure that wraps `Fin 2 → ℂ`,
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and these structures we define the instance of a module. This prevents casting between different
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types of fermions.
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-/
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namespace Fermion
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noncomputable section
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section LeftHanded
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/-- The module in which left handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
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structure LeftHandedModule where
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/-- The underlying value in `Fin 2 → ℂ`. -/
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val : Fin 2 → ℂ
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namespace LeftHandedModule
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/-- The equivalence between `LeftHandedModule` and `Fin 2 → ℂ`. -/
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def toFin2ℂFun : LeftHandedModule ≃ (Fin 2 → ℂ) where
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toFun v := v.val
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invFun f := ⟨f⟩
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left_inv _ := rfl
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right_inv _ := rfl
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/-- The instance of `AddCommMonoid` on `LeftHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommMonoid LeftHandedModule := Equiv.addCommMonoid toFin2ℂFun
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/-- The instance of `AddCommGroup` on `LeftHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommGroup LeftHandedModule := Equiv.addCommGroup toFin2ℂFun
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/-- The instance of `Module` on `LeftHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : Module ℂ LeftHandedModule := Equiv.module ℂ toFin2ℂFun
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/-- The linear equivalence between `LeftHandedModule` and `(Fin 2 → ℂ)`. -/
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@[simps!]
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def toFin2ℂEquiv : LeftHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
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toFun := toFin2ℂFun
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map_add' := fun _ _ => rfl
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map_smul' := fun _ _ => rfl
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invFun := toFin2ℂFun.symm
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left_inv := fun _ => rfl
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right_inv := fun _ => rfl
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/-- The underlying element of `Fin 2 → ℂ` of a element in `LeftHandedModule` defined
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through the linear equivalence `toFin2ℂEquiv`. -/
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abbrev toFin2ℂ (ψ : LeftHandedModule) := toFin2ℂEquiv ψ
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end LeftHandedModule
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end LeftHanded
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section AltLeftHanded
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/-- The module in which alt-left handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
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structure AltLeftHandedModule where
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/-- The underlying value in `Fin 2 → ℂ`. -/
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val : Fin 2 → ℂ
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namespace AltLeftHandedModule
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/-- The equivalence between `AltLeftHandedModule` and `Fin 2 → ℂ`. -/
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def toFin2ℂFun : AltLeftHandedModule ≃ (Fin 2 → ℂ) where
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toFun v := v.val
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invFun f := ⟨f⟩
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left_inv _ := rfl
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right_inv _ := rfl
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/-- The instance of `AddCommMonoid` on `AltLeftHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommMonoid AltLeftHandedModule := Equiv.addCommMonoid toFin2ℂFun
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/-- The instance of `AddCommGroup` on `AltLeftHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommGroup AltLeftHandedModule := Equiv.addCommGroup toFin2ℂFun
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/-- The instance of `Module` on `AltLeftHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : Module ℂ AltLeftHandedModule := Equiv.module ℂ toFin2ℂFun
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/-- The linear equivalence between `AltLeftHandedModule` and `(Fin 2 → ℂ)`. -/
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@[simps!]
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def toFin2ℂEquiv : AltLeftHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
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toFun := toFin2ℂFun
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map_add' := fun _ _ => rfl
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map_smul' := fun _ _ => rfl
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invFun := toFin2ℂFun.symm
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left_inv := fun _ => rfl
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right_inv := fun _ => rfl
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/-- The underlying element of `Fin 2 → ℂ` of a element in `AltLeftHandedModule` defined
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through the linear equivalence `toFin2ℂEquiv`. -/
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abbrev toFin2ℂ (ψ : AltLeftHandedModule) := toFin2ℂEquiv ψ
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end AltLeftHandedModule
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end AltLeftHanded
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section RightHanded
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/-- The module in which right handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
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structure RightHandedModule where
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/-- The underlying value in `Fin 2 → ℂ`. -/
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val : Fin 2 → ℂ
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namespace RightHandedModule
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/-- The equivalence between `RightHandedModule` and `Fin 2 → ℂ`. -/
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def toFin2ℂFun : RightHandedModule ≃ (Fin 2 → ℂ) where
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toFun v := v.val
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invFun f := ⟨f⟩
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left_inv _ := rfl
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right_inv _ := rfl
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/-- The instance of `AddCommMonoid` on `RightHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommMonoid RightHandedModule := Equiv.addCommMonoid toFin2ℂFun
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/-- The instance of `AddCommGroup` on `RightHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommGroup RightHandedModule := Equiv.addCommGroup toFin2ℂFun
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/-- The instance of `Module` on `RightHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : Module ℂ RightHandedModule := Equiv.module ℂ toFin2ℂFun
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/-- The linear equivalence between `RightHandedModule` and `(Fin 2 → ℂ)`. -/
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@[simps!]
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def toFin2ℂEquiv : RightHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
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toFun := toFin2ℂFun
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map_add' := fun _ _ => rfl
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map_smul' := fun _ _ => rfl
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invFun := toFin2ℂFun.symm
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left_inv := fun _ => rfl
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right_inv := fun _ => rfl
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/-- The underlying element of `Fin 2 → ℂ` of a element in `RightHandedModule` defined
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through the linear equivalence `toFin2ℂEquiv`. -/
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abbrev toFin2ℂ (ψ : RightHandedModule) := toFin2ℂEquiv ψ
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end RightHandedModule
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end RightHanded
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section AltRightHanded
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/-- The module in which alt-right handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
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structure AltRightHandedModule where
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/-- The underlying value in `Fin 2 → ℂ`. -/
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val : Fin 2 → ℂ
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namespace AltRightHandedModule
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/-- The equivalence between `AltRightHandedModule` and `Fin 2 → ℂ`. -/
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def toFin2ℂFun : AltRightHandedModule ≃ (Fin 2 → ℂ) where
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toFun v := v.val
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invFun f := ⟨f⟩
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left_inv _ := rfl
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right_inv _ := rfl
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/-- The instance of `AddCommMonoid` on `AltRightHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommMonoid AltRightHandedModule := Equiv.addCommMonoid toFin2ℂFun
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/-- The instance of `AddCommGroup` on `AltRightHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : AddCommGroup AltRightHandedModule := Equiv.addCommGroup toFin2ℂFun
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/-- The instance of `Module` on `AltRightHandedModule` defined via its equivalence
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with `Fin 2 → ℂ`. -/
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instance : Module ℂ AltRightHandedModule := Equiv.module ℂ toFin2ℂFun
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/-- The linear equivalence between `AltRightHandedModule` and `(Fin 2 → ℂ)`. -/
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@[simps!]
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def toFin2ℂEquiv : AltRightHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
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toFun := toFin2ℂFun
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map_add' := fun _ _ => rfl
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map_smul' := fun _ _ => rfl
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invFun := toFin2ℂFun.symm
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left_inv := fun _ => rfl
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right_inv := fun _ => rfl
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/-- The underlying element of `Fin 2 → ℂ` of a element in `AltRightHandedModule` defined
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through the linear equivalence `toFin2ℂEquiv`. -/
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abbrev toFin2ℂ (ψ : AltRightHandedModule) := toFin2ℂEquiv ψ
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end AltRightHandedModule
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end AltRightHanded
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end
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end Fermion
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