feat: Add expansion of Pauli matrices as basis vect
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jstoobysmith
parent
1148234929
commit
74d4b2c2c0
4 changed files with 162 additions and 4 deletions
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@ -81,8 +81,7 @@ lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
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- basisVector ![Color.up, Color.up] (fun _ => 2)
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- basisVector ![Color.up, Color.up] (fun _ => 3) := by
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Functor.id_obj, Fin.isValue]
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V]
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erw [Lorentz.contrMetric_apply_one, Lorentz.contrMetricVal_expand_tmul]
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simp only [Fin.isValue, map_sub]
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congr 1
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@ -100,6 +99,45 @@ lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
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simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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/-- The expansion of the Pauli matrices `σ^μ^a^{dot a}` in terms of basis vectors. -/
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lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
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basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 0 | 2 => 0)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 1 | 2 => 1)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 0 | 2 => 1)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 1 | 2 => 0)
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- I • basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 0 | 2 => 1)
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+ I • basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 1 | 2 => 0)
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+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 0 | 2 => 0)
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- basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constThreeNode_tensor,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
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erw [PauliMatrix.asConsTensor_apply_one, PauliMatrix.asTensor_expand]
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simp only [Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, Action.instMonoidalCategory_tensorObj_V,
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Fin.isValue, map_sub, map_add, _root_.map_smul]
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congr 1
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congr 1
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congr 1
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congr 1
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congr 1
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congr 1
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congr 1
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all_goals
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erw [tripleIsoSep_tmul, basisVector]
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apply congrArg
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try apply congrArg
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funext i
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match i with
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| (0 : Fin 3) =>
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
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cons_val_zero, Fin.cases_zero]
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change _ = Lorentz.complexContrBasisFin4 _
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simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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| (1 : Fin 3) => rfl
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| (2 : Fin 3) => rfl
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end complexLorentzTensor
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end Fermion
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end
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@ -91,8 +91,7 @@ lemma pairIsoSep_tmul {c1 c2 : C} (x : F.obj (Discrete.mk c1)) (y : F.obj (Discr
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(((lift.obj F).map ((mkIso _).hom ⊗ (mkIso _).hom)).hom
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((PiTensorProduct.tprod k) _)) = _
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rw [lift.map_tprod]
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change ((lift.obj F).map fin2Iso.inv).hom ((PiTensorProduct.tprod k) fun i => _) = _
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rw [lift.map_tprod]
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erw [lift.map_tprod]
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congr
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funext i
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match i with
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@ -122,6 +121,44 @@ def tripleIsoSep {c1 c2 c3 : C} :
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((lift.obj F).μIso _ _).trans <|
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(lift.obj F).mapIso fin3Iso'.symm
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lemma tripleIsoSep_tmul {c1 c2 c3 : C} (x : F.obj (Discrete.mk c1)) (y : F.obj (Discrete.mk c2))
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(z : F.obj (Discrete.mk c3)) :
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(tripleIsoSep F).hom.hom (x ⊗ₜ[k] y ⊗ₜ[k] z) = PiTensorProduct.tprod k
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(fun | (0 : Fin 3) => x | (1 : Fin 3) => y | (2 : Fin 3) => z) := by
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Action.instMonoidalCategory_tensorObj_V,
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tripleIsoSep, Functor.mapIso_symm, Iso.trans_hom, whiskerLeftIso_hom, whiskerRightIso_hom,
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Iso.symm_hom, MonoidalFunctor.μIso_hom, Functor.mapIso_inv, Action.comp_hom,
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Action.instMonoidalCategory_whiskerLeft_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, ModuleCat.coe_comp, Function.comp_apply,
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ModuleCat.MonoidalCategory.whiskerLeft_apply, ModuleCat.MonoidalCategory.whiskerRight_apply,
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Functor.id_obj, mk_hom]
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erw [pairIsoSep_tmul F y z]
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erw [forgetLiftAppV_symm_apply F c1]
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erw [lift.obj_μ_tprod_tmul F _ _]
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erw [lift.map_tprod]
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apply congrArg
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funext i
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match i with
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| (0 : Fin 3) =>
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simp only [mk_hom, Fin.isValue, Matrix.cons_val_zero, instMonoidalCategoryStruct_tensorObj_left,
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instMonoidalCategoryStruct_tensorObj_hom, lift.discreteFunctorMapEqIso, eqToIso_refl,
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Functor.mapIso_refl, Iso.refl_hom, Action.id_hom, Iso.refl_inv, Functor.id_obj, Nat.reduceAdd,
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Nat.succ_eq_add_one, LinearEquiv.ofLinear_apply]
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rfl
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| (1 : Fin 3) =>
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simp only [mk_hom, Fin.isValue, Matrix.cons_val_one, Matrix.head_cons,
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instMonoidalCategoryStruct_tensorObj_left, instMonoidalCategoryStruct_tensorObj_hom,
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lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl, Iso.refl_hom, Action.id_hom,
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Iso.refl_inv, Functor.id_obj, Nat.reduceAdd, Nat.succ_eq_add_one, LinearEquiv.ofLinear_apply]
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rfl
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| (2 : Fin 3) =>
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simp only [mk_hom, Fin.isValue, Matrix.cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd,
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instMonoidalCategoryStruct_tensorObj_left, instMonoidalCategoryStruct_tensorObj_hom,
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lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl, Iso.refl_hom, Action.id_hom,
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Iso.refl_inv, Functor.id_obj, LinearEquiv.ofLinear_apply]
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rfl
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/-- The functor taking `c` to `F c ⊗ F (τ c)`. -/
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def pairτ (τ : C → C) : Discrete C ⥤ Rep k G :=
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F ⊗ ((Discrete.functor (Discrete.mk ∘ τ) : Discrete C ⥤ Discrete C) ⋙ F)
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@ -652,6 +652,14 @@ lemma constTwoNode_tensor {c1 c2 : S.C}
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(OverColor.Discrete.pairIsoSep S.FDiscrete).hom.hom (v.hom (1 : S.k)) :=
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rfl
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@[simp]
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lemma constThreeNode_tensor {c1 c2 c3 : S.C}
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(v : 𝟙_ (Rep S.k S.G) ⟶ S.FDiscrete.obj (Discrete.mk c1) ⊗ S.FDiscrete.obj (Discrete.mk c2) ⊗
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S.FDiscrete.obj (Discrete.mk c3)) :
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(constThreeNode v).tensor =
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(OverColor.Discrete.tripleIsoSep S.FDiscrete).hom.hom (v.hom (1 : S.k)) :=
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rfl
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lemma prod_tensor {c1 : Fin n → S.C} {c2 : Fin m → S.C} (t1 : TensorTree S c1)
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(t2 : TensorTree S c2) :
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(prod t1 t2).tensor = (S.F.map (OverColor.equivToIso finSumFinEquiv).hom).hom
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