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fix: ofReal -> ofRealHom

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jstoobysmith 2024-11-02 08:18:20 +00:00
parent 32ca614942
commit 4df8663cbc
1 changed files with 20 additions and 18 deletions
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38
HepLean/SpaceTime/LorentzGroup/Basic.lean
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@ -231,23 +231,23 @@ lemma toProd_continuous : Continuous (@toProd d) := by
/-- The embedding from the Lorentz Group into the monoid of matrices times the opposite of
the monoid of matrices. -/
lemma toProd_embedding : Embedding (@toProd d) where
lemma toProd_embedding : IsEmbedding (@toProd d) where
inj := toProd_injective
induced :=
(inducing_iff ⇑toProd).mp (inducing_of_inducing_compose toProd_continuous continuous_fst
((inducing_iff (Prod.fst ∘ ⇑toProd)).mpr rfl))
eq_induced :=
(isInducing_iff ⇑toProd).mp (IsInducing.of_comp toProd_continuous continuous_fst
((isInducing_iff (Prod.fst ∘ ⇑toProd)).mpr rfl))
/-- The embedding from the Lorentz Group into `GL (Fin 4) ℝ`. -/
lemma toGL_embedding : Embedding (@toGL d).toFun where
lemma toGL_embedding : IsEmbedding (@toGL d).toFun where
inj := toGL_injective
induced := by
eq_induced := by
refine ((fun {X} {t t'} => TopologicalSpace.ext_iff.mpr) fun _ ↦ ?_).symm
rw [TopologicalSpace.ext_iff.mp toProd_embedding.induced _, isOpen_induced_iff,
rw [TopologicalSpace.ext_iff.mp toProd_embedding.eq_induced _, isOpen_induced_iff,
isOpen_induced_iff]
exact exists_exists_and_eq_and
instance : TopologicalGroup (LorentzGroup d) :=
Inducing.topologicalGroup toGL toGL_embedding.toInducing
IsInducing.topologicalGroup toGL toGL_embedding.toIsInducing
section
open LorentzVector
@ -294,17 +294,17 @@ lemma timeComp_mul (Λ Λ' : LorentzGroup d) : timeComp (Λ * Λ') =
/-- The monoid homomorphisms taking the lorentz group to complex matrices. -/
def toComplex : LorentzGroup d →* Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℂ where
toFun Λ := Λ.1.map ofReal
toFun Λ := Λ.1.map ofRealHom
map_one' := by
ext i j
simp only [lorentzGroupIsGroup_one_coe, map_apply, ofReal_eq_coe]
simp only [lorentzGroupIsGroup_one_coe, map_apply, ofRealHom_eq_coe]
simp only [Matrix.one_apply, ofReal_one, ofReal_zero]
split_ifs
· rfl
· rfl
map_mul' Λ Λ' := by
ext i j
simp only [lorentzGroupIsGroup_mul_coe, map_apply, ofReal_eq_coe]
simp only [lorentzGroupIsGroup_mul_coe, map_apply, ofRealHom_eq_coe]
simp only [← Matrix.map_mul, RingHom.map_matrix_mul]
rfl
@ -325,26 +325,28 @@ lemma toComplex_inv (Λ : LorentzGroup d) : (toComplex Λ)⁻¹ = toComplex Λ
@[simp]
lemma toComplex_mul_minkowskiMatrix_mul_transpose (Λ : LorentzGroup d) :
toComplex Λ * minkowskiMatrix.map ofReal * (toComplex Λ)ᵀ = minkowskiMatrix.map ofReal := by
toComplex Λ * minkowskiMatrix.map ofRealHom * (toComplex Λ)ᵀ =
minkowskiMatrix.map ofRealHom := by
simp only [toComplex, MonoidHom.coe_mk, OneHom.coe_mk]
have h1 : ((Λ.1).map ⇑ofReal)ᵀ = (Λ.1ᵀ).map ofReal := rfl
have h1 : ((Λ.1).map ⇑ofRealHom)ᵀ = (Λ.1ᵀ).map ofRealHom := rfl
rw [h1]
trans (Λ.1 * minkowskiMatrix * Λ.1ᵀ).map ofReal
trans (Λ.1 * minkowskiMatrix * Λ.1ᵀ).map ofRealHom
· simp only [Matrix.map_mul]
simp only [mul_minkowskiMatrix_mul_transpose]
@[simp]
lemma toComplex_transpose_mul_minkowskiMatrix_mul_self (Λ : LorentzGroup d) :
(toComplex Λ)ᵀ * minkowskiMatrix.map ofReal * toComplex Λ = minkowskiMatrix.map ofReal := by
(toComplex Λ)ᵀ * minkowskiMatrix.map ofRealHom * toComplex Λ =
minkowskiMatrix.map ofRealHom := by
simp only [toComplex, MonoidHom.coe_mk, OneHom.coe_mk]
have h1 : ((Λ.1).map ⇑ofReal)ᵀ = (Λ.1ᵀ).map ofReal := rfl
have h1 : ((Λ.1).map ofRealHom)ᵀ = (Λ.1ᵀ).map ofRealHom := rfl
rw [h1]
trans (Λ.1ᵀ * minkowskiMatrix * Λ.1).map ofReal
trans (Λ.1ᵀ * minkowskiMatrix * Λ.1).map ofRealHom
· simp only [Matrix.map_mul]
simp only [transpose_mul_minkowskiMatrix_mul_self]
lemma toComplex_mulVec_ofReal (v : Fin 1 ⊕ Fin d → ℝ) (Λ : LorentzGroup d) :
toComplex Λ *ᵥ (ofReal ∘ v) = ofReal ∘ (Λ *ᵥ v) := by
toComplex Λ *ᵥ (ofRealHom ∘ v) = ofRealHom ∘ (Λ *ᵥ v) := by
simp only [toComplex, MonoidHom.coe_mk, OneHom.coe_mk]
funext i
rw [← RingHom.map_mulVec]