174 lines
5.7 KiB
Text
174 lines
5.7 KiB
Text
/-
|
||
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.Meta.Informal
|
||
import HepLean.SpaceTime.SL2C.Basic
|
||
import Mathlib.RepresentationTheory.Rep
|
||
import Mathlib.Logic.Equiv.TransferInstance
|
||
/-!
|
||
|
||
## Modules associated with Lorentz vectors
|
||
|
||
These have not yet been fully-implmented.
|
||
|
||
We define these modules to prevent casting between different types of Lorentz vectors.
|
||
|
||
-/
|
||
|
||
namespace Lorentz
|
||
|
||
noncomputable section
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
|
||
/-- The module for contravariant (up-index) complex Lorentz vectors. -/
|
||
structure ContrℂModule where
|
||
/-- The underlying value as a vector `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
val : Fin 1 ⊕ Fin 3 → ℂ
|
||
|
||
namespace ContrℂModule
|
||
|
||
/-- The equivalence between `ContrℂModule` and `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
def toFin13ℂFun : ContrℂModule ≃ (Fin 1 ⊕ Fin 3 → ℂ) where
|
||
toFun v := v.val
|
||
invFun f := ⟨f⟩
|
||
left_inv _ := rfl
|
||
right_inv _ := rfl
|
||
|
||
/-- The instance of `AddCommMonoid` on `ContrℂModule` defined via its equivalence
|
||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
instance : AddCommMonoid ContrℂModule := Equiv.addCommMonoid toFin13ℂFun
|
||
|
||
/-- The instance of `AddCommGroup` on `ContrℂModule` defined via its equivalence
|
||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
instance : AddCommGroup ContrℂModule := Equiv.addCommGroup toFin13ℂFun
|
||
|
||
/-- The instance of `Module` on `ContrℂModule` defined via its equivalence
|
||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
instance : Module ℂ ContrℂModule := Equiv.module ℂ toFin13ℂFun
|
||
|
||
@[ext]
|
||
lemma ext (ψ ψ' : ContrℂModule) (h : ψ.val = ψ'.val) : ψ = ψ' := by
|
||
cases ψ
|
||
cases ψ'
|
||
subst h
|
||
simp_all only
|
||
|
||
@[simp]
|
||
lemma val_add (ψ ψ' : ContrℂModule) : (ψ + ψ').val = ψ.val + ψ'.val := rfl
|
||
|
||
@[simp]
|
||
lemma val_smul (r : ℂ) (ψ : ContrℂModule) : (r • ψ).val = r • ψ.val := rfl
|
||
|
||
/-- The linear equivalence between `ContrℂModule` and `(Fin 1 ⊕ Fin 3 → ℂ)`. -/
|
||
@[simps!]
|
||
def toFin13ℂEquiv : ContrℂModule ≃ₗ[ℂ] (Fin 1 ⊕ Fin 3 → ℂ) where
|
||
toFun := toFin13ℂFun
|
||
map_add' := fun _ _ => rfl
|
||
map_smul' := fun _ _ => rfl
|
||
invFun := toFin13ℂFun.symm
|
||
left_inv := fun _ => rfl
|
||
right_inv := fun _ => rfl
|
||
|
||
/-- The underlying element of `Fin 1 ⊕ Fin 3 → ℂ` of a element in `ContrℂModule` defined
|
||
through the linear equivalence `toFin13ℂEquiv`. -/
|
||
abbrev toFin13ℂ (ψ : ContrℂModule) := toFin13ℂEquiv ψ
|
||
|
||
/-- The representation of the Lorentz group on `ContrℂModule`. -/
|
||
def lorentzGroupRep : Representation ℂ (LorentzGroup 3) ContrℂModule where
|
||
toFun M := {
|
||
toFun := fun v => toFin13ℂEquiv.symm (LorentzGroup.toComplex M *ᵥ v.toFin13ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp [mulVec_add]
|
||
map_smul' := by
|
||
intro r ψ
|
||
simp [mulVec_smul]}
|
||
map_one' := by
|
||
ext i
|
||
simp
|
||
map_mul' M N := by
|
||
ext i
|
||
simp
|
||
|
||
/-- The representation of the SL(2, ℂ) on `ContrℂModule` induced by the representation of the
|
||
Lorentz group. -/
|
||
def SL2CRep : Representation ℂ SL(2, ℂ) ContrℂModule :=
|
||
MonoidHom.comp lorentzGroupRep SpaceTime.SL2C.toLorentzGroup
|
||
|
||
end ContrℂModule
|
||
|
||
/-- The module for covariant (up-index) complex Lorentz vectors. -/
|
||
structure CoℂModule where
|
||
/-- The underlying value as a vector `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
val : Fin 1 ⊕ Fin 3 → ℂ
|
||
|
||
namespace CoℂModule
|
||
|
||
/-- The equivalence between `CoℂModule` and `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
def toFin13ℂFun : CoℂModule ≃ (Fin 1 ⊕ Fin 3 → ℂ) where
|
||
toFun v := v.val
|
||
invFun f := ⟨f⟩
|
||
left_inv _ := rfl
|
||
right_inv _ := rfl
|
||
|
||
/-- The instance of `AddCommMonoid` on `CoℂModule` defined via its equivalence
|
||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
instance : AddCommMonoid CoℂModule := Equiv.addCommMonoid toFin13ℂFun
|
||
|
||
/-- The instance of `AddCommGroup` on `CoℂModule` defined via its equivalence
|
||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
instance : AddCommGroup CoℂModule := Equiv.addCommGroup toFin13ℂFun
|
||
|
||
/-- The instance of `Module` on `CoℂModule` defined via its equivalence
|
||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||
instance : Module ℂ CoℂModule := Equiv.module ℂ toFin13ℂFun
|
||
|
||
/-- The linear equivalence between `CoℂModule` and `(Fin 1 ⊕ Fin 3 → ℂ)`. -/
|
||
@[simps!]
|
||
def toFin13ℂEquiv : CoℂModule ≃ₗ[ℂ] (Fin 1 ⊕ Fin 3 → ℂ) where
|
||
toFun := toFin13ℂFun
|
||
map_add' := fun _ _ => rfl
|
||
map_smul' := fun _ _ => rfl
|
||
invFun := toFin13ℂFun.symm
|
||
left_inv := fun _ => rfl
|
||
right_inv := fun _ => rfl
|
||
|
||
/-- The underlying element of `Fin 1 ⊕ Fin 3 → ℂ` of a element in `CoℂModule` defined
|
||
through the linear equivalence `toFin13ℂEquiv`. -/
|
||
abbrev toFin13ℂ (ψ : CoℂModule) := toFin13ℂEquiv ψ
|
||
|
||
/-- The representation of the Lorentz group on `CoℂModule`. -/
|
||
def lorentzGroupRep : Representation ℂ (LorentzGroup 3) CoℂModule where
|
||
toFun M := {
|
||
toFun := fun v => toFin13ℂEquiv.symm ((LorentzGroup.toComplex M)⁻¹ᵀ *ᵥ v.toFin13ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp [mulVec_add]
|
||
map_smul' := by
|
||
intro r ψ
|
||
simp [mulVec_smul]}
|
||
map_one' := by
|
||
ext i
|
||
simp
|
||
map_mul' M N := by
|
||
ext1 x
|
||
simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply,
|
||
LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
|
||
refine (congrFun (congrArg _ ?_) _)
|
||
simp only [_root_.map_mul]
|
||
rw [Matrix.mul_inv_rev]
|
||
exact transpose_mul _ _
|
||
|
||
/-- The representation of the SL(2, ℂ) on `ContrℂModule` induced by the representation of the
|
||
Lorentz group. -/
|
||
def SL2CRep : Representation ℂ SL(2, ℂ) CoℂModule :=
|
||
MonoidHom.comp lorentzGroupRep SpaceTime.SL2C.toLorentzGroup
|
||
|
||
end CoℂModule
|
||
|
||
end
|
||
end Lorentz
|