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feat: Finish proof of anti-symm symm tensor contract

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jstoobysmith 2024-10-21 06:47:51 +00:00
parent cb0197e127
commit c820a90939
3 changed files with 48 additions and 29 deletions
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57
HepLean/Tensors/ComplexLorentz/Lemmas.lean
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@ -73,36 +73,37 @@ lemma coMetric_symm : {Lorentz.coMetric | μ ν = Lorentz.coMetric | ν μ}ᵀ :
| (1 : Fin 2) => rfl
set_option maxRecDepth 20000 in
/-- Contracting a rank-2 anti-symmetric tensor with a rank-2 symmetric tensor gives zero. -/
lemma symm_contract_antiSymm (A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V)
(S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V)
(hA : (twoNodeE complexLorentzTensor Color.up Color.up A).tensor =
(TensorTree.neg (perm
(OverColor.equivToHomEq (Equality.finMapToEquiv ![1, 0] ![1, 0]) (by decide))
(twoNodeE complexLorentzTensor Color.up Color.up A))).tensor)
(hs : {S | μ ν = S | ν μ}ᵀ) : {A | μ ν ⊗ S | μ ν}ᵀ.tensor = 0 := by
have h1 : {A | μ ν ⊗ S | μ ν}ᵀ.tensor = - {A | μ ν ⊗ S | μ ν}ᵀ.tensor := by
nth_rewrite 1 [contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_fst hA))]
nth_rewrite 1 [(contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_snd hs)))]
rw [contr_tensor_eq (contr_tensor_eq (neg_fst_prod _ _))]
rw [contr_tensor_eq (neg_contr _)]
rw [neg_contr]
rw [neg_tensor]
apply congrArg
rw [contr_tensor_eq (contr_tensor_eq (prod_perm_left _ _ _ _))]
rw [contr_tensor_eq (perm_contr _ _)]
rw [perm_contr]
rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_perm_right _ _ _ _)))]
rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
rw [perm_tensor_eq (perm_contr _ _)]
rw [perm_perm]
nth_rewrite 1 [perm_tensor_eq (contr_contr _ _ _)]
rw [perm_perm]
rw [perm_eq_id]
· rfl
· apply OverColor.Hom.ext
rfl
(hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
{A | μ ν ⊗ S | μ ν}ᵀ.tensor = 0 := by
have hn {M : Type} [AddCommGroup M] [Module ℂ M] {x : M} (h : x = - x) : x = 0 := by
rw [eq_neg_iff_add_eq_zero, ← two_smul ℂ x] at h
simpa using h
refine hn ?_
rw [← neg_tensor]
rw [neg_perm] at hA
nth_rewrite 1 [contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_fst hA))]
nth_rewrite 1 [(contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_snd hs)))]
rw [contr_tensor_eq (contr_tensor_eq (neg_fst_prod _ _))]
rw [contr_tensor_eq (neg_contr _)]
rw [neg_contr]
rw [neg_tensor]
apply congrArg
rw [contr_tensor_eq (contr_tensor_eq (prod_perm_left _ _ _ _))]
rw [contr_tensor_eq (perm_contr _ _)]
rw [perm_contr]
rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_perm_right _ _ _ _)))]
rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
rw [perm_tensor_eq (perm_contr _ _)]
rw [perm_perm]
nth_rewrite 1 [perm_tensor_eq (contr_contr _ _ _)]
rw [perm_perm]
rw [perm_eq_id]
· rfl
· apply OverColor.Hom.ext
rfl
end Fermion