128 lines
6.3 KiB
Text
128 lines
6.3 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.Tensors.OverColor.Basic
|
||
import HepLean.Mathematics.PiTensorProduct
|
||
import HepLean.SpaceTime.LorentzVector.Complex.Basic
|
||
import HepLean.SpaceTime.WeylFermion.Two
|
||
import HepLean.SpaceTime.PauliMatrices.Basic
|
||
/-!
|
||
|
||
## Pauli matrices
|
||
|
||
-/
|
||
|
||
namespace PauliMatrix
|
||
|
||
open Complex
|
||
open Lorentz
|
||
open Fermion
|
||
open TensorProduct
|
||
open CategoryTheory.MonoidalCategory
|
||
|
||
noncomputable section
|
||
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
open TensorProduct
|
||
open SpaceTime
|
||
|
||
/-- The tensor `σ^μ^a^{dot a}` based on the Pauli-matrices as an element of
|
||
`complexContr ⊗ leftHanded ⊗ rightHanded`. -/
|
||
def asTensor : (complexContr ⊗ leftHanded ⊗ rightHanded).V :=
|
||
∑ i, complexContrBasis i ⊗ₜ leftRightToMatrix.symm (σSA i)
|
||
|
||
/-- The tensor `σ^μ^a^{dot a}` based on the Pauli-matrices as a morphism,
|
||
`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ leftHanded ⊗ rightHanded` manifesting
|
||
the invariance under the `SL(2,ℂ)` action. -/
|
||
def asConsTensor : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ leftHanded ⊗ rightHanded where
|
||
hom := {
|
||
toFun := fun a =>
|
||
let a' : ℂ := a
|
||
a' • asTensor,
|
||
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' • asTensor =
|
||
(TensorProduct.map (complexContr.ρ M) (
|
||
TensorProduct.map (leftHanded.ρ M) (rightHanded.ρ M))) (x' • asTensor)
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, _root_.map_smul]
|
||
apply congrArg
|
||
nth_rewrite 2 [asTensor]
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, CategoryTheory.Equivalence.symm_inverse,
|
||
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
|
||
map_sum, map_tmul]
|
||
symm
|
||
calc _ = ∑ x, ((complexContr.ρ M) (complexContrBasis x) ⊗ₜ[ℂ]
|
||
leftRightToMatrix.symm (SL2C.repSelfAdjointMatrix M (σSA x))) := by
|
||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||
rw [← leftRightToMatrix_ρ_symm_selfAdjoint]
|
||
rfl
|
||
_ = ∑ x, ((∑ i, (SL2C.toLorentzGroup M).1 i x • (complexContrBasis i)) ⊗ₜ[ℂ]
|
||
∑ j, leftRightToMatrix.symm ((SL2C.toLorentzGroup M⁻¹).1 x j • (σSA j))) := by
|
||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||
rw [SL2CRep_ρ_basis, SL2C.repSelfAdjointMatrix_σSA]
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, SL2C.toLorentzGroup_apply_coe,
|
||
Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
|
||
Finset.sum_singleton, map_inv, lorentzGroupIsGroup_inv, AddSubgroup.coe_add,
|
||
selfAdjoint.val_smul, AddSubgroup.val_finset_sum, map_add, map_sum]
|
||
_ = ∑ x, ∑ i, ∑ j, ((SL2C.toLorentzGroup M).1 i x • (complexContrBasis i)) ⊗ₜ[ℂ]
|
||
leftRightToMatrix.symm.toLinearMap ((SL2C.toLorentzGroup M⁻¹).1 x j • (σSA j)) := by
|
||
refine Finset.sum_congr rfl (fun x _ => ?_)
|
||
rw [sum_tmul]
|
||
refine Finset.sum_congr rfl (fun i _ => ?_)
|
||
rw [tmul_sum]
|
||
rfl
|
||
_ = ∑ x, ∑ i, ∑ j, ((SL2C.toLorentzGroup M).1 i x • (complexContrBasis i)) ⊗ₜ[ℂ]
|
||
((SL2C.toLorentzGroup M⁻¹).1 x j • leftRightToMatrix.symm ((σSA j))) := by
|
||
refine Finset.sum_congr rfl (fun x _ => (Finset.sum_congr rfl (fun i _ =>
|
||
(Finset.sum_congr rfl (fun j _ => ?_)))))
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, SL2C.toLorentzGroup_apply_coe,
|
||
map_inv, lorentzGroupIsGroup_inv, LinearMap.map_smul_of_tower, LinearEquiv.coe_coe,
|
||
tmul_smul]
|
||
_ = ∑ x, ∑ i, ∑ j, ((SL2C.toLorentzGroup M).1 i x * (SL2C.toLorentzGroup M⁻¹).1 x j)
|
||
• ((complexContrBasis i)) ⊗ₜ[ℂ] leftRightToMatrix.symm ((σSA j)) := by
|
||
refine Finset.sum_congr rfl (fun x _ => (Finset.sum_congr rfl (fun i _ =>
|
||
(Finset.sum_congr rfl (fun j _ => ?_)))))
|
||
rw [smul_tmul, smul_smul, tmul_smul]
|
||
_ = ∑ i, ∑ x, ∑ j, ((SL2C.toLorentzGroup M).1 i x * (SL2C.toLorentzGroup M⁻¹).1 x j)
|
||
• ((complexContrBasis i)) ⊗ₜ[ℂ] leftRightToMatrix.symm ((σSA j)) := Finset.sum_comm
|
||
_ = ∑ i, ∑ j, ∑ x, ((SL2C.toLorentzGroup M).1 i x * (SL2C.toLorentzGroup M⁻¹).1 x j)
|
||
• ((complexContrBasis i)) ⊗ₜ[ℂ] leftRightToMatrix.symm ((σSA j)) :=
|
||
Finset.sum_congr rfl (fun x _ => Finset.sum_comm)
|
||
_ = ∑ i, ∑ j, (∑ x, (SL2C.toLorentzGroup M).1 i x * (SL2C.toLorentzGroup M⁻¹).1 x j)
|
||
• ((complexContrBasis i)) ⊗ₜ[ℂ] leftRightToMatrix.symm ((σSA j)) := by
|
||
refine Finset.sum_congr rfl (fun i _ => (Finset.sum_congr rfl (fun j _ => ?_)))
|
||
rw [Finset.sum_smul]
|
||
_ = ∑ i, ∑ j, ((1 : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) i j)
|
||
• ((complexContrBasis i)) ⊗ₜ[ℂ] leftRightToMatrix.symm ((σSA j)) := by
|
||
refine Finset.sum_congr rfl (fun i _ => (Finset.sum_congr rfl (fun j _ => ?_)))
|
||
congr
|
||
change ((SL2C.toLorentzGroup M) * (SL2C.toLorentzGroup M⁻¹)).1 i j = _
|
||
rw [← SL2C.toLorentzGroup.map_mul]
|
||
simp only [mul_inv_cancel, _root_.map_one, lorentzGroupIsGroup_one_coe]
|
||
_ = ∑ i, ((1 : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) i i)
|
||
• ((complexContrBasis i)) ⊗ₜ[ℂ] leftRightToMatrix.symm ((σSA i)) := by
|
||
refine Finset.sum_congr rfl (fun i _ => ?_)
|
||
refine Finset.sum_eq_single i (fun b _ hb => ?_) (fun hb => ?_)
|
||
· simp [one_apply_ne' hb]
|
||
· simp only [Finset.mem_univ, not_true_eq_false] at hb
|
||
_ = asTensor := by
|
||
refine Finset.sum_congr rfl (fun i _ => ?_)
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, one_apply_eq, one_smul,
|
||
CategoryTheory.Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
|
||
Action.FunctorCategoryEquivalence.functor_obj_obj]
|
||
|
||
end
|
||
end PauliMatrix
|