Contr Co Contraction apply metric
Lorentz.contrCoContraction_apply_metric
Project documentation
The metric ηᵢᵢ as a morphism 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo, making its invariance under the action of SL(2,ℂ). -/ def coMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap (CoℂModule.SL2CRep.tprod CoℂModule.SL2CRep) where toFun := fun a => let a' : ℂ := a a' • coMetricVal map_add' := fun x y => by simp only [add_smul] map_smu...
Exact Lean statement
lemma contrCoContraction_apply_metric :
(TensorProduct.comm ℂ _ _ <|
(TensorProduct.lid ℂ _).lTensor _ <|
(contrCoContraction.toLinearMap.rTensor _).lTensor _ <|
(TensorProduct.assoc ℂ _ _ _).symm.toLinearMap.lTensor _<|
TensorProduct.assoc ℂ _ _ (_ ⊗[ℂ] _) <|
(contrMetric 1) ⊗ₜ[ℂ] (coMetric 1)) = coContrUnit (1 : ℝ)Formal artifact
Lean source
lemma contrCoContraction_apply_metric : (TensorProduct.comm ℂ _ _ <| (TensorProduct.lid ℂ _).lTensor _ <| (contrCoContraction.toLinearMap.rTensor _).lTensor _ <| (TensorProduct.assoc ℂ _ _ _).symm.toLinearMap.lTensor _<| TensorProduct.assoc ℂ _ _ (_ ⊗[ℂ] _) <| (contrMetric 1) ⊗ₜ[ℂ] (coMetric 1)) = coContrUnit (1 : ℝ) := by rw [contrMetric_apply_one, coMetric_apply_one] rw [contrMetricVal_expand_tmul, coMetricVal_expand_tmul] simp [Fin.isValue, tmul_sub, sub_tmul, map_sub] simp only [← Representation.IntertwiningMap.toLinearMap_apply] repeat erw [contrCoContraction_basis'] simp only [Fin.isValue, ↓reduceIte, one_smul, reduceCtorEq, zero_smul, sub_zero, zero_sub, Sum.inr.injEq, one_ne_zero, Fin.reduceEq, sub_neg_eq_add, zero_ne_one, sub_self] erw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/ComplexTensor/Metrics/Pre.lean:134-148
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