| import Mathlib |
| import OAI.Analysis.CoulombIonization.Localization.SpinCorrelated |
| import OAI.Analysis.CoulombIonization.Fermionic.Pauli |
| import OAI.Analysis.CoulombIonization.Ionization.PullAdjoint |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Complex TopologicalSpace |
| open scoped Topology InnerProductSpace ENNReal |
| open MeasureTheory Filter Complex |
| open scoped Topology BigOperators ComplexConjugate FourierTransform SchwartzMap ENNReal |
| namespace CoulombPackets |
| open CoulombPauli |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| section SpinIntegral |
| variable {B : Type*} [MeasurableSpace B] {ν : Measure B} [SigmaFinite ν] |
| [SeparableSpace (Lp ℂ 2 ν)] |
| lemma spinPartialOccupation_integrable (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (ψ : Lp ℂ 2 ((spinSpaceMeasure (V := V)).prod ν)) (s : Fin 2) : |
| Integrable (spinPartialOccupation g ψ s) ((volume : Measure V).prod volume) := by |
| have hp (p : V × V) : 0 ≤ spinPartialOccupation g ψ s p := sq_nonneg _ |
| have hm := spinPartialOccupation_measurable g hg ψ s |
| have hm' : AEStronglyMeasurable (spinPartialOccupation g ψ s) ((volume : Measure V).prod volume) := by |
| have h := hm.ennreal_toReal.aestronglyMeasurable |
| simpa only [Function.comp_def, ENNReal.toReal_ofReal (hp _)] using h |
| refine ⟨hm', ?_⟩ |
| rw [HasFiniteIntegral] |
| simp only [← ofReal_norm, Real.norm_eq_abs, abs_of_nonneg (hp _)] |
| apply lt_of_le_of_lt (b := ENNReal.ofReal (‖ψ‖^2)) _ ENNReal.ofReal_lt_top |
| calc |
| _ ≤ ∑ t : Fin 2, ∫⁻ p, ENNReal.ofReal (spinPartialOccupation g ψ t p) |
| ∂((volume : Measure V).prod volume) := |
| Finset.single_le_sum (fun _ _ => bot_le) (Finset.mem_univ s) |
| _ = _ := spinPartialOccupation_mass g hg ψ |
|
|
| lemma spinPartialOccupation_total_integral (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (ψ : Lp ℂ 2 ((spinSpaceMeasure (V := V)).prod ν)) : |
| (∫ p, ∑ s : Fin 2, spinPartialOccupation g ψ s p |
| ∂((volume : Measure V).prod volume)) = ‖ψ‖^2 := by |
| rw [integral_finsetSum Finset.univ (fun s _ => spinPartialOccupation_integrable g hg ψ s)] |
| apply (ENNReal.ofReal_eq_ofReal_iff |
| (Finset.sum_nonneg fun _ _ => integral_nonneg fun _ => sq_nonneg _) (sq_nonneg _)).1 |
| rw [ENNReal.ofReal_sum_of_nonneg (fun _ _ => integral_nonneg fun _ => sq_nonneg _)] |
| calc |
| _ = ∑ s : Fin 2, ∫⁻ p, ENNReal.ofReal (spinPartialOccupation g ψ s p) |
| ∂((volume : Measure V).prod volume) := by |
| apply Finset.sum_congr rfl |
| intro s hs |
| exact ofReal_integral_eq_lintegral_ofReal (spinPartialOccupation_integrable g hg ψ s) |
| (Filter.Eventually.of_forall fun _ => sq_nonneg _) |
| _ = _ := spinPartialOccupation_mass g hg ψ |
| end SpinIntegral |
|
|
| local instance : Fact ((2 : ℝ≥0∞) ≠ ⊤) := ⟨by norm_num⟩ |
|
|
| abbrev fermionL2 (N : ℕ) := sectorL2 N (spinSpaceMeasure (V := V)) |
|
|
| def splitFermion {N : ℕ} (i : Fin (N+1)) (ψ : fermionL2 (V := V) (N+1)) : |
| Lp ℂ 2 ((spinSpaceMeasure (V := V)).prod (configMeasure N (spinSpaceMeasure (V := V)))) := |
| pull (splitAt i).symm ((splitAt_preserving (μ := spinSpaceMeasure (V := V)) i).symm (splitAt i)) ψ |
|
|
| lemma splitFermion_norm {N : ℕ} (i : Fin (N+1)) (ψ : fermionL2 (V := V) (N+1)) : |
| ‖splitFermion i ψ‖ = ‖ψ‖ := pull_norm _ _ _ |
|
|
| lemma insert_adjoint_split {N : ℕ} (i : Fin (N+1)) (u : Lp ℂ 2 (spinSpaceMeasure (V := V))) |
| (ψ : fermionL2 (V := V) (N+1)) : |
| (insertOrbital i u).adjoint ψ = (tensorLeft u).adjoint (splitFermion i ψ) := by |
| rw [insertOrbital, ContinuousLinearMap.adjoint_comp, ContinuousLinearMap.comp_apply, |
| pull_adjoint_symm] |
| rfl |
|
|
| def fermionHusimi {N : ℕ} (g : 𝓢(V, ℝ)) (i : Fin (N+1)) |
| (ψ : fermionL2 (V := V) (N+1)) (p : V × V) : ℝ := |
| (N+1 : ℝ) * ∑ s : Fin 2, ‖(insertOrbital i (spinPacket g s p)).adjoint ψ‖^2 |
|
|
| lemma fermionHusimi_eq {N : ℕ} (g : 𝓢(V, ℝ)) (i : Fin (N+1)) |
| (ψ : fermionL2 (V := V) (N+1)) (p : V × V) : |
| fermionHusimi g i ψ p = |
| (N+1 : ℝ) * ∑ s : Fin 2, spinPartialOccupation g (splitFermion i ψ) s p := by |
| simp only [fermionHusimi, spinPartialOccupation, insert_adjoint_split] |
|
|
| lemma fermionHusimi_integrable {N : ℕ} (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (i : Fin (N+1)) (ψ : fermionL2 (V := V) (N+1)) : |
| Integrable (fermionHusimi g i ψ) ((volume : Measure V).prod volume) := by |
| simp_rw [show fermionHusimi g i ψ = fun p => |
| (N+1 : ℝ) * ∑ s : Fin 2, spinPartialOccupation g (splitFermion i ψ) s p from |
| funext (fermionHusimi_eq g i ψ)] |
| exact (integrable_finsetSum Finset.univ |
| (fun s _ => spinPartialOccupation_integrable g hg (splitFermion i ψ) s)).const_mul _ |
|
|
| lemma fermionHusimi_mass {N : ℕ} (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (i : Fin (N+1)) (ψ : fermionL2 (V := V) (N+1)) : |
| (∫ p, fermionHusimi g i ψ p ∂((volume : Measure V).prod volume)) = |
| (N+1 : ℝ) * ‖ψ‖^2 := by |
| simp_rw [fermionHusimi_eq] |
| rw [integral_const_mul, spinPartialOccupation_total_integral g hg, splitFermion_norm] |
|
|
| lemma fermionHusimi_nonneg {N : ℕ} (g : 𝓢(V, ℝ)) (i : Fin (N+1)) |
| (ψ : fermionL2 (V := V) (N+1)) (p : V × V) : 0 ≤ fermionHusimi g i ψ p := |
| mul_nonneg (by positivity) (Finset.sum_nonneg fun _ _ => sq_nonneg _) |
|
|
| lemma fermionHusimi_le_two {N : ℕ} (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (i : Fin (N+1)) (ψ : fermionL2 (V := V) (N+1)) |
| (ha : ∀ i j : Fin (N+1), i ≠ j → permute (Equiv.swap i j) ψ = -ψ) |
| (p : V × V) : fermionHusimi g i ψ p ≤ 2 * ‖ψ‖^2 := by |
| unfold fermionHusimi |
| rw [Finset.mul_sum] |
| calc |
| _ ≤ ∑ _s : Fin 2, ‖ψ‖^2 := Finset.sum_le_sum (fun s _ => |
| occupation_le_one i (spinPacket g s p) (spinPacket_norm g hg s p) ψ ha) |
| _ = _ := by simp |
|
|
| end CoulombPackets |
|
|
| open MeasureTheory Filter |
| open scoped Topology ContDiff SchwartzMap FourierTransform ENNReal |
|
|
| end |
|
|
| end OAI |
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|
