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feat: lemmas relating to index notation

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jstoobysmith 2024-10-17 11:43:33 +00:00
parent ac11a510cf
commit 672cc1ed8b
11 changed files with 510 additions and 17 deletions
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22
HepLean/SpaceTime/LorentzVector/Complex/Metric.lean
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@ -5,6 +5,7 @@ Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Complex.Two
import HepLean.SpaceTime.MinkowskiMetric
import LLMLean
/-!
# Metric for complex Lorentz vectors
@ -55,6 +56,22 @@ def contrMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexContr wh
def coMetricVal : (complexCo ⊗ complexCo).V :=
coCoToMatrix.symm ((@minkowskiMatrix 3).map ofReal)
lemma coMetricVal_expand_tmul : coMetricVal =
complexCoBasis (Sum.inl 0) ⊗ₜ[ℂ] complexCoBasis (Sum.inl 0)
- complexCoBasis (Sum.inr 0) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 0)
- complexCoBasis (Sum.inr 1) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 1)
- complexCoBasis (Sum.inr 2) ⊗ₜ[ℂ] complexCoBasis (Sum.inr 2) := by
simp only [Action.instMonoidalCategory_tensorObj_V, coMetricVal, Fin.isValue]
erw [coCoToMatrix_symm_expand_tmul]
simp only [map_apply, ofReal_eq_coe, coe_smul, Fintype.sum_sum_type, Finset.univ_unique,
Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, Fin.sum_univ_three, ne_eq, reduceCtorEq,
not_false_eq_true, minkowskiMatrix.off_diag_zero, zero_smul, add_zero, zero_add, Sum.inr.injEq,
zero_ne_one, Fin.reduceEq, one_ne_zero]
rw [minkowskiMatrix.inl_0_inl_0, minkowskiMatrix.inr_i_inr_i,
minkowskiMatrix.inr_i_inr_i, minkowskiMatrix.inr_i_inr_i]
simp only [Fin.isValue, one_smul, neg_smul]
rfl
/-- 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 : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo where
@ -84,5 +101,10 @@ def coMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexCo where
simp only [lorentzGroupIsGroup_inv, SL2C.toLorentzGroup_apply_coe,
LorentzGroup.toComplex_transpose_mul_minkowskiMatrix_mul_self]
lemma coMetric_apply_one : coMetric.hom (1 : ℂ) = coMetricVal := by
change coMetric.hom.toFun (1 : ℂ) = coMetricVal
simp [coMetric]
end Lorentz
end

13
HepLean/SpaceTime/LorentzVector/Complex/Two.lean
View file

@ -34,6 +34,19 @@ def coCoToMatrix : (complexCo ⊗ complexCo).V ≃ₗ[ℂ]
Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)) ≪≫ₗ
LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)
/-- Expanding `coCoToMatrix` in terms of the standard basis. -/
lemma coCoToMatrix_symm_expand_tmul (M : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ) :
coCoToMatrix.symm M = ∑ i, ∑ j, M i j • (complexCoBasis i ⊗ₜ[ℂ] complexCoBasis j) := by
simp only [Action.instMonoidalCategory_tensorObj_V, coCoToMatrix, LinearEquiv.trans_symm,
LinearEquiv.trans_apply, Basis.repr_symm_apply]
rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)]
· erw [Finset.sum_product]
refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
erw [Basis.tensorProduct_apply complexCoBasis complexCoBasis i j]
rfl
· simp
/-- Equivalence of `complexContr ⊗ complexCo` to `4 x 4` complex matrices. -/
def contrCoToMatrix : (complexContr ⊗ complexCo).V ≃ₗ[ℂ]
Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ :=