| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.Multiplier |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open Set Filter Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators Pointwise |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open scoped Convolution |
| open scoped RealInnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open Filter |
| open scoped Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff BigOperators |
| namespace CoulombAtom |
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| lemma graph_value_components_ext {N : ℕ} {F G : fermionGraph N} |
| (h : ∀ s, graphComponent s none F = graphComponent s none G) : F = G := by |
| apply fermionGraphValue_injective N |
| apply Subtype.ext |
| apply PiLp.ext |
| intro s |
| exact h s |
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| lemma complex_square_partition (p q : ℝ) (h : p ^ 2 + q ^ 2 = 1) (z : ℂ) : |
| (p : ℂ) * ((p : ℂ) * z) + (q : ℂ) * ((q : ℂ) * z) = z := by |
| calc |
| _ = ((p ^ 2 + q ^ 2 : ℝ) : ℂ) * z := by push_cast; ring |
| _ = z := by rw [h]; simp |
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| attribute [local irreducible] FermionMultiplier.apply graphComponent |
| fermionGraphValue coulombFormOperator graphFormVector formEnergy |
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| lemma FermionMultiplier.partition_apply {N : ℕ} (p q : FermionMultiplier N) |
| (h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) : |
| p.apply (p.apply F) + q.apply (q.apply F) = F := by |
| apply graph_value_components_ext |
| intro s |
| rw [map_add] |
| apply Lp.ext |
| filter_upwards [Lp.coeFn_add (graphComponent s none (p.apply (p.apply F))) |
| (graphComponent s none (q.apply (q.apply F))), |
| p.apply_value (p.apply F) s, p.apply_value F s, |
| q.apply_value (q.apply F) s, q.apply_value F s] with x ha hb hc hd he |
| exact ha.trans ((congrArg₂ (fun a b : ℂ => a + b) |
| (hb.trans (congrArg (fun z : ℂ => (p.value x : ℂ) * z) hc)) |
| (hd.trans (congrArg (fun z : ℂ => (q.value x : ℂ) * z) he))).trans |
| (complex_square_partition (p.value x) (q.value x) (h x) _)) |
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| lemma FermionMultiplier.partition_mass {N : ℕ} (p q : FermionMultiplier N) |
| (h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) : |
| ‖fermionGraphValue N (p.apply F)‖ ^ 2 + |
| ‖fermionGraphValue N (q.apply F)‖ ^ 2 = ‖fermionGraphValue N F‖ ^ 2 := by |
| have he := congrArg (fermionGraphValue N) (p.partition_apply q h F) |
| have hi := congrArg (fun U => (⟪U, fermionGraphValue N F⟫_ℂ).re) he |
| simp only [map_add, inner_add_left, Complex.add_re] at hi |
| calc |
| _ = _ := congrArg₂ (fun a b : ℝ => a + b) |
| (p.value_pairing F).symm (q.value_pairing F).symm |
| _ = _ := hi |
| _ = _ := (norm_sq_eq_re_inner (𝕜 := ℂ) (fermionGraphValue N F)).symm |
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| def multiplierError {N : ℕ} (p : FermionMultiplier N) (F : fermionGraph N) : ℝ := |
| (1 / 2 : ℝ) * (∑ s : Spins N, ∑ i : Fin N, ∑ a : Fin 3, |
| ∫ x, (lineDeriv ℝ p.value x (direction i a)) ^ 2 * |
| ‖graphComponent s none F x‖ ^ 2) |
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| lemma multiplierError_nonneg {N : ℕ} (p : FermionMultiplier N) (F : fermionGraph N) : |
| 0 ≤ multiplierError p F := by |
| apply mul_nonneg (by norm_num) |
| apply Finset.sum_nonneg; intro s _ |
| apply Finset.sum_nonneg; intro i _ |
| apply Finset.sum_nonneg; intro a _ |
| exact integral_nonneg fun x => mul_nonneg (sq_nonneg _) (sq_nonneg _) |
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| theorem quantum_partition_ims {N : ℕ} (Z : ℝ) (p q : FermionMultiplier N) |
| (h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) : |
| formEnergy Z (graphFormVector (p.apply F)) + |
| formEnergy Z (graphFormVector (q.apply F)) = |
| formEnergy Z (graphFormVector F) + multiplierError p F + multiplierError q F := by |
| have hp := p.form_ims Z F |
| have hq := q.form_ims Z F |
| have hi : (⟪p.apply (p.apply F), coulombFormOperator Z N F⟫_ℂ).re + |
| (⟪q.apply (q.apply F), coulombFormOperator Z N F⟫_ℂ).re = |
| formEnergy Z (graphFormVector F) := by |
| rw [← Complex.add_re, ← inner_add_left, p.partition_apply q h F, |
| coulombFormOperator_inner] |
| change _ = multiplierError p F at hp |
| change _ = multiplierError q F at hq |
| linarith |
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| lemma quantum_form_above_sector {Z : ℝ} (hZ : 0 ≤ Z) {N : ℕ} (F : fermionGraph N) : |
| energy Z N * ‖fermionGraphValue N F‖ ^ 2 ≤ formEnergy Z (graphFormVector F) := by |
| rw [← coulombFormOperator_inner] |
| exact form_lower_of_unit (fermionGraphValue N) (fermionGraphValue_injective N) |
| (coulombFormOperator Z N) (fun U hU => energy_le_graphForm hZ U hU) F |
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| theorem quantum_partition_excess {Z b : ℝ} (hZ : 0 ≤ Z) {N : ℕ} |
| (hb : b ≤ energy Z N) (p q : FermionMultiplier N) |
| (h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) : |
| formEnergy Z (graphFormVector (p.apply F)) - |
| b * ‖fermionGraphValue N (p.apply F)‖ ^ 2 ≤ |
| formEnergy Z (graphFormVector F) - b * ‖fermionGraphValue N F‖ ^ 2 + |
| multiplierError p F + multiplierError q F := by |
| have hl := quantum_form_above_sector hZ (q.apply F) |
| have hbq := mul_le_mul_of_nonneg_right hb (sq_nonneg ‖fermionGraphValue N (q.apply F)‖) |
| have hmass := p.partition_mass q h F |
| have henergy := quantum_partition_ims Z p q h F |
| have hm := congrArg (fun x : ℝ => b * x) hmass |
| simp only [mul_add] at hm |
| linarith |
|
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| end CoulombAtom |
|
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| open MeasureTheory Filter Set |
| open scoped Topology BigOperators |
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| end |
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| end OAI |
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