198 lines
9.3 KiB
Text
198 lines
9.3 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.SpaceTime.LorentzVector.Complex.Two
|
||
import HepLean.SpaceTime.LorentzVector.Complex.Contraction
|
||
/-!
|
||
|
||
# Unit for complex Lorentz vectors
|
||
|
||
-/
|
||
noncomputable section
|
||
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
open TensorProduct
|
||
open SpaceTime
|
||
open CategoryTheory.MonoidalCategory
|
||
namespace Lorentz
|
||
|
||
/-- The contra-co unit for complex lorentz vectors. Usually denoted `δⁱᵢ`. -/
|
||
def contrCoUnitVal : (complexContr ⊗ complexCo).V :=
|
||
contrCoToMatrix.symm 1
|
||
|
||
/-- Expansion of `contrCoUnitVal` into basis. -/
|
||
lemma contrCoUnitVal_expand_tmul : contrCoUnitVal =
|
||
complexContrBasis (Sum.inl 0) ⊗ₜ[ℂ] complexCoBasis (Sum.inl 0)
|
||
+ complexContrBasis (Sum.inr 0) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 0)
|
||
+ complexContrBasis (Sum.inr 1) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 1)
|
||
+ complexContrBasis (Sum.inr 2) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 2) := by
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, contrCoUnitVal, Fin.isValue]
|
||
erw [contrCoToMatrix_symm_expand_tmul]
|
||
simp only [Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
|
||
Finset.sum_singleton, Fin.sum_univ_three, ne_eq, reduceCtorEq, not_false_eq_true, one_apply_ne,
|
||
zero_smul, add_zero, one_apply_eq, one_smul, zero_add, Sum.inr.injEq, zero_ne_one, Fin.reduceEq,
|
||
one_ne_zero]
|
||
rfl
|
||
|
||
/-- The contra-co unit for complex lorentz vectors as a morphism
|
||
`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo`, manifesting the invaraince under
|
||
the `SL(2, ℂ)` action. -/
|
||
def contrCoUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo where
|
||
hom := {
|
||
toFun := fun a =>
|
||
let a' : ℂ := a
|
||
a' • contrCoUnitVal,
|
||
map_add' := fun x y => by
|
||
simp only [add_smul],
|
||
map_smul' := fun m x => by
|
||
simp only [smul_smul]
|
||
rfl}
|
||
comm M := by
|
||
ext x : 2
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
|
||
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
|
||
Function.comp_apply]
|
||
let x' : ℂ := x
|
||
change x' • contrCoUnitVal =
|
||
(TensorProduct.map (complexContr.ρ M) (complexCo.ρ M)) (x' • contrCoUnitVal)
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
|
||
apply congrArg
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, contrCoUnitVal]
|
||
erw [contrCoToMatrix_ρ_symm]
|
||
apply congrArg
|
||
simp
|
||
|
||
lemma contrCoUnit_apply_one : contrCoUnit.hom (1 : ℂ) = contrCoUnitVal := by
|
||
change contrCoUnit.hom.toFun (1 : ℂ) = contrCoUnitVal
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
|
||
contrCoUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
|
||
|
||
/-- The co-contra unit for complex lorentz vectors. Usually denoted `δᵢⁱ`. -/
|
||
def coContrUnitVal : (complexCo ⊗ complexContr).V :=
|
||
coContrToMatrix.symm 1
|
||
|
||
/-- Expansion of `coContrUnitVal` into basis. -/
|
||
lemma coContrUnitVal_expand_tmul : coContrUnitVal =
|
||
complexCoBasis (Sum.inl 0) ⊗ₜ[ℂ] complexContrBasis (Sum.inl 0)
|
||
+ complexCoBasis (Sum.inr 0) ⊗ₜ[ℂ] complexContrBasis (Sum.inr 0)
|
||
+ complexCoBasis (Sum.inr 1) ⊗ₜ[ℂ] complexContrBasis (Sum.inr 1)
|
||
+ complexCoBasis (Sum.inr 2) ⊗ₜ[ℂ] complexContrBasis (Sum.inr 2) := by
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, coContrUnitVal, Fin.isValue]
|
||
erw [coContrToMatrix_symm_expand_tmul]
|
||
simp only [Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
|
||
Finset.sum_singleton, Fin.sum_univ_three, ne_eq, reduceCtorEq, not_false_eq_true, one_apply_ne,
|
||
zero_smul, add_zero, one_apply_eq, one_smul, zero_add, Sum.inr.injEq, zero_ne_one, Fin.reduceEq,
|
||
one_ne_zero]
|
||
rfl
|
||
|
||
/-- The co-contra unit for complex lorentz vectors as a morphism
|
||
`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr`, manifesting the invaraince under
|
||
the `SL(2, ℂ)` action. -/
|
||
def coContrUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr where
|
||
hom := {
|
||
toFun := fun a =>
|
||
let a' : ℂ := a
|
||
a' • coContrUnitVal,
|
||
map_add' := fun x y => by
|
||
simp only [add_smul],
|
||
map_smul' := fun m x => by
|
||
simp only [smul_smul]
|
||
rfl}
|
||
comm M := by
|
||
ext x : 2
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
|
||
Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.coe_comp,
|
||
Function.comp_apply]
|
||
let x' : ℂ := x
|
||
change x' • coContrUnitVal =
|
||
(TensorProduct.map (complexCo.ρ M) (complexContr.ρ M)) (x' • coContrUnitVal)
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
|
||
apply congrArg
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, coContrUnitVal]
|
||
erw [coContrToMatrix_ρ_symm]
|
||
apply congrArg
|
||
symm
|
||
refine transpose_eq_one.mp ?h.h.h.a
|
||
simp
|
||
|
||
lemma coContrUnit_apply_one : coContrUnit.hom (1 : ℂ) = coContrUnitVal := by
|
||
change coContrUnit.hom.toFun (1 : ℂ) = coContrUnitVal
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
|
||
coContrUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
|
||
/-!
|
||
|
||
## Contraction of the units
|
||
|
||
-/
|
||
|
||
/-- Contraction on the right with `contrCoUnit` does nothing. -/
|
||
lemma contr_contrCoUnit (x : complexCo) :
|
||
(λ_ complexCo).hom.hom
|
||
((coContrContraction ▷ complexCo).hom
|
||
((α_ _ _ complexCo).inv.hom
|
||
(x ⊗ₜ[ℂ] contrCoUnit.hom (1 : ℂ)))) = x := by
|
||
obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span complexCoBasis x)
|
||
subst hc
|
||
rw [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
|
||
Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
|
||
Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
|
||
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
|
||
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
|
||
Finset.sum_singleton, Fin.sum_univ_three, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
|
||
_root_.map_smul]
|
||
have h1' (x y : CoeSort.coe complexCo) (z : CoeSort.coe complexContr) :
|
||
(α_ complexCo.V complexContr.V complexCo.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y) = (x ⊗ₜ[ℂ] z) ⊗ₜ[ℂ] y := rfl
|
||
repeat rw [h1']
|
||
have h1'' ( y : CoeSort.coe complexCo) (z : CoeSort.coe complexCo ⊗[ℂ] CoeSort.coe complexContr) :
|
||
(coContrContraction.hom ▷ complexCo.V) (z ⊗ₜ[ℂ] y) = (coContrContraction.hom z) ⊗ₜ[ℂ] y := rfl
|
||
repeat rw (config := { transparency := .instances }) [h1'']
|
||
repeat rw [coContrContraction_basis']
|
||
simp only [Fin.isValue, leftUnitor, ModuleCat.MonoidalCategory.leftUnitor, ModuleCat.of_coe,
|
||
CategoryTheory.Iso.trans_hom, LinearEquiv.toModuleIso_hom, ModuleCat.ofSelfIso_hom,
|
||
CategoryTheory.Category.comp_id, Action.instMonoidalCategory_tensorUnit_V, ↓reduceIte,
|
||
reduceCtorEq, zero_tmul, map_zero, smul_zero, add_zero, Sum.inr.injEq, one_ne_zero,
|
||
Fin.reduceEq, zero_add, zero_ne_one]
|
||
erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul, TensorProduct.lid_tmul,
|
||
TensorProduct.lid_tmul]
|
||
simp only [Fin.isValue, one_smul]
|
||
repeat rw [add_assoc]
|
||
|
||
/-- Contraction on the right with `coContrUnit`. -/
|
||
lemma contr_coContrUnit (x : complexContr) :
|
||
(λ_ complexContr).hom.hom
|
||
((contrCoContraction ▷ complexContr).hom
|
||
((α_ _ _ complexContr).inv.hom
|
||
(x ⊗ₜ[ℂ] coContrUnit.hom (1 : ℂ)))) = x := by
|
||
obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span complexContrBasis x)
|
||
subst hc
|
||
rw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
|
||
Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
|
||
Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
|
||
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
|
||
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
|
||
Finset.sum_singleton, Fin.sum_univ_three, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
|
||
_root_.map_smul]
|
||
have h1' (x y : CoeSort.coe complexContr) (z : CoeSort.coe complexCo) :
|
||
(α_ complexContr.V complexCo.V complexContr.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y) = (x ⊗ₜ[ℂ] z) ⊗ₜ[ℂ] y := rfl
|
||
repeat rw [h1']
|
||
have h1'' ( y : CoeSort.coe complexContr) (z : CoeSort.coe complexContr ⊗[ℂ] CoeSort.coe complexCo) :
|
||
(contrCoContraction.hom ▷ complexContr.V) (z ⊗ₜ[ℂ] y) = (contrCoContraction.hom z) ⊗ₜ[ℂ] y := rfl
|
||
repeat rw (config := { transparency := .instances }) [h1'']
|
||
repeat rw [contrCoContraction_basis']
|
||
simp only [Fin.isValue, Action.instMonoidalCategory_tensorUnit_V, ↓reduceIte, reduceCtorEq,
|
||
zero_tmul, map_zero, smul_zero, add_zero, Sum.inr.injEq, one_ne_zero, Fin.reduceEq, zero_add,
|
||
zero_ne_one]
|
||
erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul, TensorProduct.lid_tmul,
|
||
TensorProduct.lid_tmul]
|
||
simp only [Fin.isValue, one_smul]
|
||
repeat rw [add_assoc]
|
||
|
||
|
||
end Lorentz
|
||
end
|