feat: Complex Lorentz vector & Monoidal struct
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jstoobysmith
parent
37a6a23d1e
commit
f555bc6722
5 changed files with 291 additions and 18 deletions
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@ -9,6 +9,7 @@ import Mathlib.CategoryTheory.Monoidal.Category
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import Mathlib.CategoryTheory.Comma.Over
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import Mathlib.CategoryTheory.Core
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import HepLean.SpaceTime.WeylFermion.Basic
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import HepLean.SpaceTime.LorentzVector.Complex
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/-!
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## Category over color
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@ -49,8 +50,87 @@ lemma toEquiv_symm_apply (m : f ⟶ g) (i : g.left) :
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f.hom ((toEquiv m).symm i) = g.hom i := by
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simpa [toEquiv, types_comp] using congrFun m.inv.w i
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lemma toEquiv_comp_hom (m : f ⟶ g) : g.hom ∘ (toEquiv m) = f.hom := by
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ext x
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simpa [types_comp, toEquiv] using congrFun m.hom.w x
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end Hom
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instance (C : Type) : MonoidalCategoryStruct (OverColor C) where
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tensorObj f g := Over.mk (Sum.elim f.hom g.hom)
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tensorUnit := Over.mk Empty.elim
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whiskerLeft X Y1 Y2 m := Over.isoMk (Equiv.sumCongr (Equiv.refl X.left) (Hom.toEquiv m)).toIso
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(by
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ext x
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simp only [Functor.id_obj, Functor.const_obj_obj, Over.mk_left, Equiv.toIso_hom, Over.mk_hom,
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types_comp_apply, Equiv.sumCongr_apply, Equiv.coe_refl]
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rw [Sum.elim_map, Hom.toEquiv_comp_hom]
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rfl)
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whiskerRight m X := Over.isoMk (Equiv.sumCongr (Hom.toEquiv m) (Equiv.refl X.left)).toIso
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(by
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ext x
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simp only [Functor.id_obj, Functor.const_obj_obj, Over.mk_left, Equiv.toIso_hom, Over.mk_hom,
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types_comp_apply, Equiv.sumCongr_apply, Equiv.coe_refl]
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rw [Sum.elim_map, Hom.toEquiv_comp_hom]
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rfl)
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associator X Y Z := {
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hom := Over.isoMk (Equiv.sumAssoc X.left Y.left Z.left).toIso (by
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simp only [Functor.id_obj, Over.mk_left, Over.mk_hom, Functor.const_obj_obj, Equiv.sumAssoc,
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Equiv.toIso_hom, Equiv.coe_fn_mk, types_comp]
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ext x
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simp only [Function.comp_apply]
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cases x with
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| inl val =>
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cases val with
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| inl val_1 => simp_all only [Sum.elim_inl]
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| inr val_2 => simp_all only [Sum.elim_inl, Sum.elim_inr, Function.comp_apply]
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| inr val_1 => simp_all only [Sum.elim_inr, Function.comp_apply]),
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inv := (Over.isoMk (Equiv.sumAssoc X.left Y.left Z.left).toIso (by
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simp only [Functor.id_obj, Over.mk_left, Over.mk_hom, Functor.const_obj_obj, Equiv.sumAssoc,
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Equiv.toIso_hom, Equiv.coe_fn_mk, types_comp]
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ext x
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simp only [Function.comp_apply]
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cases x with
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| inl val =>
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cases val with
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| inl val_1 => simp_all only [Sum.elim_inl]
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| inr val_2 => simp_all only [Sum.elim_inl, Sum.elim_inr, Function.comp_apply]
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| inr val_1 => simp_all only [Sum.elim_inr, Function.comp_apply])).symm,
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hom_inv_id := by
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apply CategoryTheory.Iso.ext
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erw [CategoryTheory.Iso.trans_hom]
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simp only [Functor.id_obj, Over.mk_left, Over.mk_hom, Iso.symm_hom, Iso.hom_inv_id]
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rfl,
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inv_hom_id := by
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apply CategoryTheory.Iso.ext
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erw [CategoryTheory.Iso.trans_hom]
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simp only [Functor.id_obj, Over.mk_left, Over.mk_hom, Iso.symm_hom, Iso.inv_hom_id]
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rfl}
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leftUnitor X := {
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hom := Over.isoMk (Equiv.emptySum Empty X.left).toIso
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inv := (Over.isoMk (Equiv.emptySum Empty X.left).toIso).symm
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hom_inv_id := by
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apply CategoryTheory.Iso.ext
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erw [CategoryTheory.Iso.trans_hom]
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simp only [Functor.id_obj, Over.mk_left, Over.mk_hom, Iso.symm_hom, Iso.hom_inv_id]
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rfl,
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inv_hom_id := by
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apply CategoryTheory.Iso.ext
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erw [CategoryTheory.Iso.trans_hom]}
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rightUnitor X := {
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hom := Over.isoMk (Equiv.sumEmpty X.left Empty).toIso
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inv := (Over.isoMk (Equiv.sumEmpty X.left Empty).toIso).symm
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hom_inv_id := by
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apply CategoryTheory.Iso.ext
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erw [CategoryTheory.Iso.trans_hom]
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simp only [Functor.id_obj, Over.mk_left, Over.mk_hom, Iso.symm_hom, Iso.hom_inv_id]
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rfl,
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inv_hom_id := by
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apply CategoryTheory.Iso.ext
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erw [CategoryTheory.Iso.trans_hom]}
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end OverColor
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end IndexNotation
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@ -72,6 +152,8 @@ inductive Color
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| downL : Color
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| upR : Color
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| downR : Color
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| up : Color
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| down : Color
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/-- The corresponding representations associated with a color. -/
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def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
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@ -80,6 +162,8 @@ def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
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| Color.downL => leftHanded
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| Color.upR => altRightHanded
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| Color.downR => rightHanded
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| Color.up => Lorentz.complexContr
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| Color.down => Lorentz.complexCo
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/-- The linear equivalence between `colorToRep c1` and `colorToRep c2` when `c1 = c2`. -/
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def colorToRepCongr {c1 c2 : Color} (h : c1 = c2) : colorToRep c1 ≃ₗ[ℂ] colorToRep c2 where
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@ -172,44 +256,34 @@ def colorFun : OverColor Color ⥤ Rep ℂ SL(2, ℂ) where
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obj := colorFun.obj'
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map := colorFun.map'
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map_id f := by
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simp only
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ext x
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refine PiTensorProduct.induction_on' x ?_ (by
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intro x y hx hy
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refine PiTensorProduct.induction_on' x (fun r x => ?_) (fun x y hx hy => by
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simp only [CategoryTheory.Functor.id_obj, map_add, hx, ModuleCat.coe_comp,
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Function.comp_apply, hy])
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intro r x
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simp only [CategoryTheory.Functor.id_obj, PiTensorProduct.tprodCoeff_eq_smul_tprod,
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_root_.map_smul, Action.id_hom, ModuleCat.id_apply]
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apply congrArg
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rw [colorFun.map']
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erw [colorFun.mapToLinearEquiv'_tprod]
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apply congrArg
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funext i
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rfl
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exact congrArg _ (funext (fun i => rfl))
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map_comp {X Y Z} f g := by
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simp only
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ext x
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refine PiTensorProduct.induction_on' x ?_ (by
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intro x y hx hy
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refine PiTensorProduct.induction_on' x (fun r x => ?_) (fun x y hx hy => by
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simp only [CategoryTheory.Functor.id_obj, map_add, hx, ModuleCat.coe_comp,
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Function.comp_apply, hy])
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intro r x
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simp only [Functor.id_obj, PiTensorProduct.tprodCoeff_eq_smul_tprod, _root_.map_smul,
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Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply]
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apply congrArg
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rw [colorFun.map', colorFun.map', colorFun.map']
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simp only
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change (colorFun.mapToLinearEquiv' (CategoryTheory.CategoryStruct.comp f g))
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((PiTensorProduct.tprod ℂ) x) =
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(colorFun.mapToLinearEquiv' g) ((colorFun.mapToLinearEquiv' f) ((PiTensorProduct.tprod ℂ) x))
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rw [colorFun.mapToLinearEquiv'_tprod, colorFun.mapToLinearEquiv'_tprod]
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erw [colorFun.mapToLinearEquiv'_tprod]
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apply congrArg
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funext i
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refine congrArg _ (funext (fun i => ?_))
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simp only [colorToRepCongr, Function.comp_apply, Equiv.cast_symm, LinearEquiv.coe_mk,
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Equiv.cast_apply, cast_cast, cast_inj]
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rfl
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end
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end Fermion
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