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feat: contr and prod for basis

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jstoobysmith 2024-10-23 10:50:14 +00:00
parent 865164ca81
commit b70e9bd005
2 changed files with 76 additions and 8 deletions
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54
HepLean/Tensors/ComplexLorentz/Basis.lean
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@ -79,21 +79,63 @@ lemma perm_basisVector {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, LinearEquiv.ofLinear_apply]
rw [basis_eq_FDiscrete]
lemma contr_basisVector {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C} {i : Fin n.succ.succ} {j : Fin n.succ}
{h : c (i.succAbove j) = complexLorentzTensor.τ (c i)} (b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
(contr i j h (tensorNode (basisVector c b))).tensor = (if (b i).val = ((b (i.succAbove j))).val then (1 : ℂ) else 0) •
basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(fun k => b (i.succAbove (j.succAbove k))) := by
lemma contr_basisVector {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
{i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
(contr i j h (tensorNode (basisVector c b))).tensor = (if (b i).val = (b (i.succAbove j)).val
then (1 : ℂ) else 0) • basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(fun k => b (i.succAbove (j.succAbove k))) := by
rw [contr_tensor]
simp only [Nat.succ_eq_add_one, tensorNode_tensor]
rw [basisVector]
erw [TensorSpecies.contrMap_tprod]
congr 1
rw [basis_eq_FDiscrete]
simp
simp only [Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V,
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, Functor.comp_obj, Discrete.functor_obj_eq_as,
Function.comp_apply]
erw [basis_contr]
rfl
def prodBasisVecEquiv {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
{c1 : Fin m → complexLorentzTensor.C} (i : Fin n ⊕ Fin m) :
Sum.elim (fun i => Fin (complexLorentzTensor.repDim (c i))) (fun i => Fin (complexLorentzTensor.repDim (c1 i)))
i ≃ Fin (complexLorentzTensor.repDim ((Sum.elim c c1 i))) :=
match i with
| Sum.inl _ => Equiv.refl _
| Sum.inr _ => Equiv.refl _
lemma prod_basisVector {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
{c1 : Fin m → complexLorentzTensor.C}
(b : Π k, Fin (complexLorentzTensor.repDim (c k)))
(b1 : Π k, Fin (complexLorentzTensor.repDim (c1 k))) :
(prod (tensorNode (basisVector c b)) (tensorNode (basisVector c1 b1))).tensor =
basisVector (Sum.elim c c1 ∘ finSumFinEquiv.symm) (fun i =>
prodBasisVecEquiv (finSumFinEquiv.symm i)
((HepLean.PiTensorProduct.elimPureTensor b b1) (finSumFinEquiv.symm i))) := by
rw [prod_tensor]
simp
rw [basisVector, basisVector]
simp [TensorSpecies.F_def]
have h1 := OverColor.lift.μ_tmul_tprod_mk complexLorentzTensor.FDiscrete
(fun i => (complexLorentzTensor.basis (c i)) (b i))
(fun i => (complexLorentzTensor.basis (c1 i)) (b1 i))
erw [h1]
erw [OverColor.lift.map_tprod]
apply congrArg
funext i
obtain ⟨k, hk⟩ := finSumFinEquiv.surjective i
subst hk
simp only [Functor.id_obj, OverColor.mk_hom, Function.comp_apply,
OverColor.lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl, Iso.refl_hom,
Action.id_hom, Iso.refl_inv, LinearEquiv.ofLinear_apply]
erw [← (Equiv.apply_eq_iff_eq_symm_apply finSumFinEquiv).mp rfl]
match k with
| Sum.inl k => rfl
| Sum.inr k => rfl
/-!
## Useful expansions.