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import OAI.Analysis.CoulombIonization.FormDomain.WeakChangeVariables
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace Convolution
namespace CoulombAtom
@[simp] lemma fermionPermutation_apply {N : ℕ} (π : Equiv.Perm (Fin N))
(F : SectorHilbert N) (s : Spins N) :
fermionPermutation π F s = (((Equiv.Perm.sign π : ℤ) : ℂ)) •
Lp.compMeasurePreserving (permuteConfiguration π) (permuteConfiguration_preserving π)
(F (s ∘ π)) := rfl
lemma permuteConfiguration_mul {N : ℕ} (π σ : Equiv.Perm (Fin N)) :
(permuteConfiguration (π * σ) : Configuration N → Configuration N) =
permuteConfiguration σ ∘ permuteConfiguration π := by
funext x i
simp only [Function.comp_apply, permuteConfiguration_apply, Equiv.Perm.coe_mul]
lemma fermionPermutation_mul {N : ℕ} (π σ : Equiv.Perm (Fin N)) (F : SectorHilbert N) :
fermionPermutation π (fermionPermutation σ F) = fermionPermutation (π * σ) F := by
apply PiLp.ext
intro s
simp only [fermionPermutation_apply]
change _ • (Lp.compMeasurePreservingₗ ℂ (permuteConfiguration π)
(permuteConfiguration_preserving π)) (_ • _) = _
rw [map_smul]
rw [smul_smul]
have hsign : (((Equiv.Perm.sign (π * σ) : ℤ) : ℂ)) =
(((Equiv.Perm.sign π : ℤ) : ℂ)) * (((Equiv.Perm.sign σ : ℤ) : ℂ)) := by
simp
rw [hsign]
congr 1
have hs : (s ∘ π) ∘ σ = s ∘ (π * σ) := rfl
rw [hs]
change Lp.compMeasurePreserving (permuteConfiguration π) _
(Lp.compMeasurePreserving (permuteConfiguration σ) _ _) = _
rw [← Lp.compMeasurePreserving_comp_apply]
simp only [← permuteConfiguration_mul]
lemma sectorDirections_permute {N : ℕ} (π : Equiv.Perm (Fin N)) (k : Fin N × Fin 3) :
permuteConfiguration π (sectorDirections N k) = sectorDirections N (π.symm k.1, k.2) := by
classical
ext i a
simp only [permuteConfiguration_apply, Function.comp_apply, sectorDirections, direction]
by_cases h : i = π.symm k.1
· subst i
simp
· have h' : π i ≠ k.1 := by
intro hh
apply h
exact π.injective (by simpa using hh)
simp [h, h']
def sectorGraphPermutation {N : ℕ} (π : Equiv.Perm (Fin N)) : SectorGraph N →L[ℂ] SectorGraph N :=
(PiLp.continuousLinearEquiv 2 ℂ (fun _ : Spins N => weakGraph (sectorDirections N))
).symm.toContinuousLinearMap.comp
(ContinuousLinearMap.pi fun s =>
((((Equiv.Perm.sign π : ℤ) : ℂ)) • weakGraphChange (sectorDirections N)
(permuteConfiguration π) (permuteConfiguration_preserving π)
(fun k => (π.symm k.1, k.2)) (sectorDirections_permute π)).comp
(PiLp.proj 2 (fun _ : Spins N => weakGraph (sectorDirections N)) (s ∘ π)))
lemma sectorGraphPermutation_value {N : ℕ} (π : Equiv.Perm (Fin N)) (F : SectorGraph N) :
sectorGraphValue N (sectorGraphPermutation π F) = fermionPermutation π (sectorGraphValue N F) := by
rfl
def fermionAverage (N : ℕ) : SectorHilbert N →L[ℂ] SectorHilbert N :=
(Fintype.card (Equiv.Perm (Fin N)) : ℂ)⁻¹ • ∑ π : Equiv.Perm (Fin N), fermionPermutation π
lemma fermionAverage_mem {N : ℕ} (F : SectorHilbert N) : fermionAverage N F ∈ fermionSpace N := by
classical
rw [mem_fermionSpace]
intro π
simp only [fermionAverage, smul_apply, sum_apply,
map_smul, map_sum, fermionPermutation_mul]
congr 1
exact Fintype.sum_equiv (Equiv.mulLeft π) _ _ (fun σ => rfl)
lemma fermionAverage_eq_self {N : ℕ} (F : fermionSpace N) : fermionAverage N F = F := by
classical
have hF := (mem_fermionSpace F.val).mp F.property
simp only [fermionAverage, smul_apply, sum_apply, hF]
simp [← Nat.cast_smul_eq_nsmul ℂ, smul_smul]
lemma sectorGraphValue_average (N : ℕ) (F : SectorGraph N) :
sectorGraphValue N (((Fintype.card (Equiv.Perm (Fin N)) : ℂ)⁻¹ •
∑ π : Equiv.Perm (Fin N), sectorGraphPermutation π) F) =
fermionAverage N (sectorGraphValue N F) := by
simp only [smul_apply, sum_apply, map_smul, map_sum, sectorGraphPermutation_value,
fermionAverage]
def fermionGraphAverage (N : ℕ) : SectorGraph N →L[ℂ] fermionGraph N :=
((Fintype.card (Equiv.Perm (Fin N)) : ℂ)⁻¹ • ∑ π : Equiv.Perm (Fin N), sectorGraphPermutation π
).codRestrict (fermionGraph N) (by
intro F
change sectorGraphValue N _ ∈ fermionSpace N
rw [sectorGraphValue_average]
exact fermionAverage_mem _)
lemma fermionGraphAverage_value {N : ℕ} (F : SectorGraph N) :
(fermionGraphValue N (fermionGraphAverage N F)).val = fermionAverage N (sectorGraphValue N F) :=
sectorGraphValue_average N F
lemma sectorGraphValue_denseRange (N : ℕ) : DenseRange (sectorGraphValue N) := by
let eV := PiLp.continuousLinearEquiv 2 ℂ (fun _ : Spins N => weakGraph (sectorDirections N))
let eH := PiLp.continuousLinearEquiv 2 ℂ
(fun _ : Spins N => Lp ℂ 2 (volume : Measure (Configuration N)))
have hd := DenseRange.piMap (fun _ : Spins N => weakGraphValue_denseRange (sectorDirections N))
have hc : Continuous (Pi.map (fun _ : Spins N => weakGraphValue (sectorDirections N))) := by
fun_prop
exact eH.symm.surjective.denseRange.comp (hd.comp eV.surjective.denseRange hc) eH.symm.continuous
theorem fermionGraphValue_denseRange (N : ℕ) : DenseRange (fermionGraphValue N) := by
let p : SectorHilbert N →L[ℂ] fermionSpace N :=
(fermionAverage N).codRestrict (fermionSpace N) fermionAverage_mem
have hp : Function.Surjective p := by
intro F
exact ⟨F.val, Subtype.ext (fermionAverage_eq_self F)⟩
have hd := hp.denseRange.comp (sectorGraphValue_denseRange N) p.continuous
have he : p ∘ sectorGraphValue N = fermionGraphValue N ∘ fermionGraphAverage N := by
funext F
exact Subtype.ext (fermionGraphAverage_value F).symm
rw [he] at hd
exact hd.of_comp
end CoulombAtom
end
end OAI
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