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refactor: Major refactor of Lorentz vecs

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jstoobysmith 2024-11-09 08:06:16 +00:00
parent 3eb5da875f
commit 5cc188146f
20 changed files with 494 additions and 1005 deletions
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41
HepLean/SpaceTime/LorentzVector/Complex/Basic.lean
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@ -94,57 +94,62 @@ def complexCoBasisFin4 : Basis (Fin 4) ℂ complexCo :=
/-- The semilinear map including real Lorentz vectors into complex contravariant
lorentz vectors. -/
def inclCongrRealLorentz : LorentzVector 3 →ₛₗ[Complex.ofRealHom] complexContr where
toFun v := {val := ofReal ∘ v}
def inclCongrRealLorentz : ContrMod 3 →ₛₗ[Complex.ofRealHom] complexContr where
toFun v := {val := ofReal ∘ v.toFin1dℝ}
map_add' x y := by
apply Lorentz.ContrℂModule.ext
rw [Lorentz.ContrℂModule.val_add]
funext i
simp only [Function.comp_apply, ofRealHom_eq_coe, Pi.add_apply]
change ofReal (x i + y i) = _
simp only [Function.comp_apply, ofRealHom_eq_coe, Pi.add_apply, map_add]
simp only [ofRealHom_eq_coe, ofReal_add]
map_smul' c x := by
apply Lorentz.ContrℂModule.ext
rw [Lorentz.ContrℂModule.val_smul]
funext i
simp only [Function.comp_apply, ofRealHom_eq_coe, Pi.smul_apply]
change ofReal (c • x i) = _
simp only [Function.comp_apply, ofRealHom_eq_coe, Pi.smul_apply, _root_.map_smul]
simp only [smul_eq_mul, ofRealHom_eq_coe, ofReal_mul]
lemma inclCongrRealLorentz_val (v : LorentzVector 3) :
(inclCongrRealLorentz v).val = ofRealHom ∘ v := rfl
lemma inclCongrRealLorentz_val (v : ContrMod 3) :
(inclCongrRealLorentz v).val = ofRealHom ∘ v.toFin1dℝ := rfl
lemma complexContrBasis_of_real (i : Fin 1 ⊕ Fin 3) :
(complexContrBasis i) = inclCongrRealLorentz (LorentzVector.stdBasis i) := by
(complexContrBasis i) = inclCongrRealLorentz (ContrMod.stdBasis i) := by
apply Lorentz.ContrℂModule.ext
simp only [complexContrBasis, Basis.coe_ofEquivFun, inclCongrRealLorentz, LorentzVector.stdBasis,
simp only [complexContrBasis, Basis.coe_ofEquivFun, inclCongrRealLorentz,
LinearMap.coe_mk, AddHom.coe_mk]
ext j
simp only [Function.comp_apply, ofRealHom_eq_coe]
erw [Pi.basisFun_apply]
simp
change (Pi.single i 1) j = _
exact Eq.symm (Pi.apply_single (fun _ => ofRealHom) (congrFun rfl) i 1 j)
by_cases h : i = j
· subst h
rw [ContrMod.toFin1dℝ, ContrMod.stdBasis_toFin1dℝEquiv_apply_same]
simp
· rw [ContrMod.toFin1dℝ, ContrMod.stdBasis_toFin1dℝEquiv_apply_ne h]
simp [h]
lemma inclCongrRealLorentz_ρ (M : SL(2, ℂ)) (v : LorentzVector 3) :
lemma inclCongrRealLorentz_ρ (M : SL(2, ℂ)) (v : ContrMod 3) :
(complexContr.ρ M) (inclCongrRealLorentz v) =
inclCongrRealLorentz (SL2C.repLorentzVector M v) := by
inclCongrRealLorentz ((Contr 3).ρ (SL2C.toLorentzGroup M) v) := by
apply Lorentz.ContrℂModule.ext
rw [complexContrBasis_ρ_val, inclCongrRealLorentz_val, inclCongrRealLorentz_val]
rw [LorentzGroup.toComplex_mulVec_ofReal]
apply congrArg
simp only [SL2C.toLorentzGroup_apply_coe]
rw [SL2C.repLorentzVector_apply_eq_mulVec]
rfl
change _ = ContrMod.toFin1dℝ ((SL2C.toLorentzGroup M) *ᵥ v)
simp only [SL2C.toLorentzGroup_apply_coe, ContrMod.mulVec_toFin1dℝ]
/-! TODO: Rename.-/
lemma SL2CRep_ρ_basis (M : SL(2, ℂ)) (i : Fin 1 ⊕ Fin 3) :
(complexContr.ρ M) (complexContrBasis i) =
∑ j, (SL2C.toLorentzGroup M).1 j i •
complexContrBasis j := by
rw [complexContrBasis_of_real, inclCongrRealLorentz_ρ, SL2C.repLorentzVector_stdBasis, map_sum]
rw [complexContrBasis_of_real, inclCongrRealLorentz_ρ]
rw [Contr.ρ_stdBasis, map_sum]
apply congrArg
funext j
simp only [LinearMap.map_smulₛₗ, ofRealHom_eq_coe, coe_smul]
rw [complexContrBasis_of_real]
/-! TODO: Include relation to real Lorentz vectors.-/
end Lorentz
end

2
HepLean/SpaceTime/LorentzVector/Complex/Metric.lean
View file

@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Complex.Two
import HepLean.SpaceTime.MinkowskiMetric
import HepLean.SpaceTime.MinkowskiMatrix
import HepLean.SpaceTime.LorentzVector.Complex.Contraction
import HepLean.SpaceTime.LorentzVector.Complex.Unit
/-!