PhysLean/HepLean/Lorentz/PauliMatrices/AsTensor.lean

/-
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.Lorentz.Weyl.Two
/-!

## 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

/-- 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 expansion of `asTensor` into complexContrBasis basis vectors . -/
lemma asTensor_expand_complexContrBasis : asTensor =
    complexContrBasis (Sum.inl 0) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inl 0))
    + complexContrBasis (Sum.inr 0) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inr 0))
    + complexContrBasis (Sum.inr 1) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inr 1))
    + complexContrBasis (Sum.inr 2) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inr 2)) := by
  rfl

/-- The expansion of the pauli matrix `σ₀` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inl_0_expand : leftRightToMatrix.symm (σSA (Sum.inl 0)) =
    leftBasis 0 ⊗ₜ rightBasis 0 + leftBasis 1 ⊗ₜ rightBasis 1 := by
  simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
  erw [leftRightToMatrix_symm_expand_tmul]
  simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ0, of_apply, cons_val', empty_val',
    cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, one_smul, zero_smul,
    add_zero, head_fin_const, zero_add, CategoryTheory.Equivalence.symm_inverse,
    Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]

  /-- The expansion of the pauli matrix `σ₁` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inr_0_expand : leftRightToMatrix.symm (σSA (Sum.inr 0)) =
    leftBasis 0 ⊗ₜ rightBasis 1 + leftBasis 1 ⊗ₜ rightBasis 0:= by
  simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
  erw [leftRightToMatrix_symm_expand_tmul]
  simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ1, of_apply, cons_val', empty_val',
    cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, zero_smul, one_smul,
    zero_add, head_fin_const, add_zero, CategoryTheory.Equivalence.symm_inverse,
    Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]

/-- The expansion of the pauli matrix `σ₂` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inr_1_expand : leftRightToMatrix.symm (σSA (Sum.inr 1)) =
    -(I • leftBasis 0 ⊗ₜ[ℂ] rightBasis 1) + I • leftBasis 1 ⊗ₜ[ℂ] rightBasis 0 := by
  simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
  erw [leftRightToMatrix_symm_expand_tmul]
  simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ2, of_apply, cons_val', empty_val',
    cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, zero_smul, neg_smul,
    zero_add, head_fin_const, add_zero]

/-- The expansion of the pauli matrix `σ₃` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inr_2_expand : leftRightToMatrix.symm (σSA (Sum.inr 2)) =
    leftBasis 0 ⊗ₜ rightBasis 0 - leftBasis 1 ⊗ₜ rightBasis 1 := by
  simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
  erw [leftRightToMatrix_symm_expand_tmul]
  simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ3, of_apply, cons_val', empty_val',
    cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, one_smul, zero_smul,
    add_zero, head_fin_const, neg_smul, zero_add, CategoryTheory.Equivalence.symm_inverse,
    Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]
  rfl

/-- The expansion of `asTensor` into complexContrBasis basis of tensor product vectors. -/
lemma asTensor_expand : asTensor =
    complexContrBasis (Sum.inl 0) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 0)
    + complexContrBasis (Sum.inl 0) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 1)
    + complexContrBasis (Sum.inr 0) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 1)
    + complexContrBasis (Sum.inr 0) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 0)
    - I • complexContrBasis (Sum.inr 1) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 1)
    + I • complexContrBasis (Sum.inr 1) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 0)
    + complexContrBasis (Sum.inr 2) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 0)
    - complexContrBasis (Sum.inr 2) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 1) := by
  rw [asTensor_expand_complexContrBasis]
  rw [leftRightToMatrix_σSA_inl_0_expand, leftRightToMatrix_σSA_inr_0_expand,
    leftRightToMatrix_σSA_inr_1_expand, leftRightToMatrix_σSA_inr_2_expand]
  simp only [Action.instMonoidalCategory_tensorObj_V, CategoryTheory.Equivalence.symm_inverse,
    Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
    Fin.isValue, tmul_add, tmul_neg, tmul_smul, tmul_sub]
  rfl

/-- 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 := ModuleCat.ofHom {
    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
    refine ModuleCat.hom_ext ?_
    refine LinearMap.ext fun x : ℂ => ?_
    simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
      Action.tensorUnit_ρ', CategoryTheory.Category.id_comp, Action.tensor_ρ', ModuleCat.hom_comp,
      Function.comp_apply]
    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.toSelfAdjointMap 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.toSelfAdjointMap_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]

/-- The map `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ leftHanded ⊗ rightHanded` corresponding
  to Pauli matrices, when evaluated on `1` corresponds to the tensor `PauliMatrix.asTensor`. -/
lemma asConsTensor_apply_one : asConsTensor.hom (1 : ℂ) = asTensor := by
  change asConsTensor.hom.hom.toFun (1 : ℂ) = asTensor
  simp only [asConsTensor, one_smul]

end
end PauliMatrix