| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.MassiveMoments |
| import OAI.MathematicalPhysics.BFSS.GaugeGeometry |
|
|
| namespace OAI |
|
|
| /-! Uniform potential and energy bounds by distance to the zero set. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Finset |
| variable {N : ℕ} (M : BFSSQuantum.AlgebraData N) |
|
|
| lemma potential_uniform_normal_lower (hN : 2 ≤ N) : |
| ∃ c δ : ℝ, 0 < c ∧ 0 < δ ∧ ∀ q ∈ M.massiveZeros, ∀ z : Boson N, |
| z ∈ M.normalSlice q → ‖z‖ ≤ δ → |
| c * ‖z‖^2 ≤ M.deformedBosonicPotential 1 1 (q+z) := by |
| obtain ⟨c,hc,hcQ⟩ := M.normalQuadratic_uniform_coercive hN |
| obtain ⟨C,hC,hCR⟩ := M.quarticRemainder_bound |
| refine ⟨c/4,min 1 (c/(4*C)),by positivity,lt_min zero_lt_one (by positivity),?_⟩ |
| intro q hq z hz hsmall |
| have h1 : ‖z‖ ≤ 1 := hsmall.trans (min_le_left _ _) |
| have hsq : ‖z‖^2 ≤ ‖z‖ := by nlinarith [norm_nonneg z] |
| have h2 : ‖z‖^2 ≤ c/(4*C) := |
| hsq.trans (hsmall.trans (min_le_right _ _)) |
| have h3 : C * ‖z‖^2 ≤ c/4 := by |
| have hh := (le_div_iff₀ (by positivity : 0 < 4*C)).mp h2 |
| nlinarith |
| have h4 : C * ‖z‖^4 ≤ (c/4)*‖z‖^2 := by |
| have hh := mul_le_mul_of_nonneg_right h3 (sq_nonneg ‖z‖) |
| nlinarith [hh] |
| have ht := M.potential_taylor_lower q z hq |
| have hr := hCR z |
| have hq' := hcQ q hq z hz |
| linarith |
|
|
| lemma nearest_massive_zero (hN : 2 ≤ N) (x : Boson N) : |
| ∃ q ∈ M.massiveZeros, ‖x-q‖ = Metric.infDist x M.massiveZeros ∧ |
| x-q ∈ M.normalSlice q := by |
| obtain ⟨q,hq,he⟩ := (M.massiveZeros_isCompact hN).exists_infDist_eq_dist |
| ⟨0,M.zero_mem_massiveZeros⟩ x |
| have he' : ‖x-q‖ = Metric.infDist x M.massiveZeros := by simpa only [dist_eq_norm] using he.symm |
| refine ⟨q,hq,he',M.nearest_zero_normal x q hq ?_⟩ |
| intro p hp |
| rw [he'] |
| simpa only [dist_eq_norm] using Metric.infDist_le_dist_of_mem (x := x) hp |
|
|
| lemma infDist_massiveZeros_le_norm (x : Boson N) : Metric.infDist x M.massiveZeros ≤ ‖x‖ := by |
| simpa only [dist_zero_right] using |
| Metric.infDist_le_dist_of_mem (x := x) M.zero_mem_massiveZeros |
|
|
| theorem potential_global_distance_lower (hN : 2 ≤ N) : |
| ∃ c : ℝ, 0 < c ∧ ∀ x : Boson N, |
| c * (Metric.infDist x M.massiveZeros)^2 ≤ M.deformedBosonicPotential 1 1 x := by |
| obtain ⟨c₀,δ,hc₀,hδ,hlocal⟩ := M.potential_uniform_normal_lower hN |
| obtain ⟨cI,hcI,R₀,hI⟩ := M.deformedBosonicPotential_coercive hN |
| let R := max R₀ 1 |
| have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _) |
| let K : Set (Boson N) := Metric.closedBall 0 R ∩ {x | δ ≤ Metric.infDist x M.massiveZeros} |
| have hK : IsCompact K := (isCompact_closedBall (0:Boson N) R).inter_right |
| (isClosed_le continuous_const (Metric.continuous_infDist_pt M.massiveZeros)) |
| have hpos (x : Boson N) (hx : x ∈ K) : 0 < M.deformedBosonicPotential 1 1 x := by |
| have hn : M.deformedBosonicPotential 1 1 x ≠ 0 := by |
| intro he |
| have hd := Metric.infDist_zero_of_mem (show x ∈ M.massiveZeros from he) |
| exact (not_lt_of_ge hx.2) (hd ▸ hδ) |
| exact lt_of_le_of_ne (M.potential_nonneg x) hn.symm |
| have hlower : ∃ κ : ℝ, 0 < κ ∧ ∀ x ∈ K, κ ≤ M.deformedBosonicPotential 1 1 x := by |
| rcases Set.eq_empty_or_nonempty K with he|hn |
| · refine ⟨1,zero_lt_one,?_⟩ |
| intro x hx |
| rw [he] at hx |
| exact hx.elim |
| · obtain ⟨x,hx,hmin⟩ := hK.exists_isMinOn hn M.potential_continuous.continuousOn |
| exact ⟨_,hpos x hx,fun y hy => hmin hy⟩ |
| obtain ⟨κ,hκ,hκbound⟩ := hlower |
| let c := min c₀ (min cI (κ/R^2)) |
| have hc : 0 < c := lt_min hc₀ (lt_min hcI (div_pos hκ (sq_pos_of_pos hR))) |
| refine ⟨c,hc,?_⟩ |
| intro x |
| have hd0 := Metric.infDist_nonneg (x := x) (s := M.massiveZeros) |
| have hdn := M.infDist_massiveZeros_le_norm x |
| have hd2 : (Metric.infDist x M.massiveZeros)^2 ≤ ‖x‖^2 := |
| (sq_le_sq₀ hd0 (norm_nonneg x)).mpr hdn |
| by_cases hnear : Metric.infDist x M.massiveZeros ≤ δ |
| · obtain ⟨q,hq,he,hnormal⟩ := M.nearest_massive_zero hN x |
| have hh := hlocal q hq (x-q) hnormal (he ▸ hnear) |
| rw [add_sub_cancel] at hh |
| rw [he] at hh |
| exact (mul_le_mul_of_nonneg_right (min_le_left _ _) (sq_nonneg _)).trans hh |
| · by_cases hfar : R ≤ ‖x‖ |
| · have hh := hI x ((le_max_left _ _).trans hfar) |
| have hc' : c ≤ cI := (min_le_right _ _).trans (min_le_left _ _) |
| exact (mul_le_mul_of_nonneg_right hc' (sq_nonneg _)).trans |
| ((mul_le_mul_of_nonneg_left hd2 hcI.le).trans hh) |
| · have hxn : ‖x‖ ≤ R := (lt_of_not_ge hfar).le |
| have hx : x ∈ K := ⟨by simpa only [Metric.mem_closedBall,dist_zero_right] using hxn, |
| (lt_of_not_ge hnear).le⟩ |
| have hdR : (Metric.infDist x M.massiveZeros)^2 ≤ R^2 := |
| hd2.trans ((sq_le_sq₀ (norm_nonneg x) hR.le).mpr hxn) |
| have hc' : c ≤ κ/R^2 := (min_le_right _ _).trans (min_le_right _ _) |
| calc |
| c * (Metric.infDist x M.massiveZeros)^2 ≤ (κ/R^2) * (Metric.infDist x M.massiveZeros)^2 := |
| mul_le_mul_of_nonneg_right hc' (sq_nonneg _) |
| _ ≤ (κ/R^2) * R^2 := mul_le_mul_of_nonneg_left hdR (div_nonneg hκ.le (sq_nonneg _)) |
| _ = κ := div_mul_cancel₀ _ (ne_of_gt (sq_pos_of_pos hR)) |
| _ ≤ M.deformedBosonicPotential 1 1 x := hκbound x hx |
| end |
| end BFSSQuantum.AlgebraData |
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open MeasureTheory Finset |
| open scoped Pointwise |
| variable {N : ℕ} (M : AlgebraData N) |
|
|
| def massiveDistance (h : ℝ) (x : Boson N) : ℝ := |
| Metric.infDist x (h⁻¹ • M.massiveZeros) |
|
|
| lemma massiveDistance_continuous (h : ℝ) : Continuous (M.massiveDistance h) := |
| Metric.continuous_infDist_pt _ |
|
|
| lemma massiveDistance_nonneg (h : ℝ) (x : Boson N) : 0 ≤ M.massiveDistance h x := |
| Metric.infDist_nonneg |
|
|
| lemma massiveDistance_rescale (h : ℝ) (hh : 0 < h) (x : Boson N) : |
| Metric.infDist (h • x) M.massiveZeros = h*M.massiveDistance h x := by |
| have he := infDist_smul₀ (inv_ne_zero (ne_of_gt hh)) M.massiveZeros (h • x) |
| rw [inv_smul_smul₀ (ne_of_gt hh),norm_inv,Real.norm_eq_abs,abs_of_pos hh] at he |
| change M.massiveDistance h x = h⁻¹ * Metric.infDist (h • x) M.massiveZeros at he |
| rw [he,mul_inv_cancel_left₀ (ne_of_gt hh)] |
|
|
| theorem potential_uniform_distance (hN : 2 ≤ N) : |
| ∃ c : ℝ, 0 < c ∧ ∀ h : ℝ, 0 < h → ∀ x : Boson N, |
| c*(M.massiveDistance h x)^2 ≤ M.deformedBosonicPotential h 1 x := by |
| obtain ⟨c,hc,hbound⟩ := M.potential_global_distance_lower hN |
| refine ⟨c,hc,?_⟩ |
| intro h hh x |
| have hd := mul_le_mul_of_nonneg_left (hbound (h • x)) |
| (show 0 ≤ 1/h^2 by positivity) |
| rw [M.massiveDistance_rescale h hh] at hd |
| rw [M.deformedBosonicPotential_scaling h 1 hh zero_lt_one] |
| simp only [one_pow,div_one] |
| have he : (1/h^2)*(c*(h*M.massiveDistance h x)^2) = c*(M.massiveDistance h x)^2 := by |
| field_simp |
| rw [he] at hd |
| exact hd |
|
|
| lemma massiveDistance_scaled_norm (hN : 2 ≤ N) : |
| ∃ R : ℝ, 0 ≤ R ∧ ∀ h : ℝ, 0 < h → ∀ x : Boson N, |
| h*‖x‖ ≤ h*M.massiveDistance h x + R := by |
| obtain ⟨R,hR⟩ := (M.massiveZeros_isCompact hN).isBounded.exists_norm_le |
| refine ⟨R,?_,?_⟩ |
| · simpa using hR 0 M.zero_mem_massiveZeros |
| intro h hh x |
| obtain ⟨q,hq,he,_⟩ := M.nearest_massive_zero hN (h • x) |
| have ht := norm_add_le (h • x - q) q |
| rw [sub_add_cancel, norm_smul,Real.norm_eq_abs,abs_of_pos hh,he, |
| M.massiveDistance_rescale h hh] at ht |
| linarith [hR q hq] |
|
|
| theorem deformedPotential_fermion_uniform_distance (hN : 2 ≤ N) : |
| ∃ c K : ℝ, 0 < c ∧ 0 ≤ K ∧ ∀ h : ℝ, 0 < h → h ≤ 1 → |
| ∀ (x : Boson N) (z : Fermion N), |
| (c/2*(M.massiveDistance h x)^2-K)*‖z‖^2 ≤ |
| M.deformedBosonicPotential h 1 x*‖z‖^2 + |
| inner ℝ z (M.deformedFermionField h 1 x z) := by |
| obtain ⟨c,hc,hu⟩ := M.potential_uniform_distance hN |
| obtain ⟨R,hR,hr⟩ := M.massiveDistance_scaled_norm hN |
| let A := M.fermionGrowthConstant |
| have hA : 0 ≤ A := M.fermionGrowthConstant_nonneg |
| let K := A*(R+1)+A^2/(2*c) |
| refine ⟨c,K,hc,by dsimp [K]; positivity,?_⟩ |
| intro h hh h1 x z |
| have hd := M.massiveDistance_nonneg h x |
| have hx : h*‖x‖+1 ≤ M.massiveDistance h x+R+1 := by |
| have he := mul_le_mul_of_nonneg_right h1 hd |
| linarith [hr h hh x] |
| have hy : c/2*(M.massiveDistance h x)^2-K ≤ |
| c*(M.massiveDistance h x)^2-A*(M.massiveDistance h x+R+1) := by |
| have he : (c*M.massiveDistance h x-A)^2/(2*c) = |
| (c*(M.massiveDistance h x)^2-A*(M.massiveDistance h x+R+1)) - |
| (c/2*(M.massiveDistance h x)^2-K) := by |
| dsimp [K] |
| field_simp |
| ring |
| have hs : 0 ≤ (c*M.massiveDistance h x-A)^2/(2*c) := by positivity |
| rw [he] at hs |
| linarith |
| have hl : c/2*(M.massiveDistance h x)^2-K ≤ |
| M.deformedBosonicPotential h 1 x - A*(h*‖x‖+1) := by |
| have hax := mul_le_mul_of_nonneg_left hx hA |
| linarith [hu h hh x] |
| have hf := M.deformedFermionField_inner_lower h 1 x z |
| rw [abs_of_pos hh,abs_one] at hf |
| have hz := mul_le_mul_of_nonneg_right hl (sq_nonneg ‖z‖) |
| change -(A*(h*‖x‖+1))*‖z‖^2 ≤ _ at hf |
| nlinarith |
|
|
| theorem massive_core_uniform_distance (hN : 2 ≤ N) : |
| ∃ c K : ℝ, 0 < c ∧ 0 ≤ K ∧ ∀ h : ℝ, 0 < h → h ≤ 1 → |
| ∀ f : SmoothCore N, |
| (1/2 : ℝ)*∑ p : SpaceIndex × ColorIndex N, |
| ‖coreToL2 (coordinateDerivative p.1 p.2 f)‖^2 + |
| c/2*(∫ x : Boson N, (M.massiveDistance h x)^2*‖f x‖^2) ≤ |
| M.deformedCoreEnergy h 1 f + K*‖coreToL2 f‖^2 := by |
| obtain ⟨c,K,hc,hK,hbound⟩ := M.deformedPotential_fermion_uniform_distance hN |
| refine ⟨c,K,hc,hK,?_⟩ |
| intro h hh h1 f |
| have hi : Integrable (fun x : Boson N => (c/2*(M.massiveDistance h x)^2-K)*‖f x‖^2) volume := |
| weighted_core_square_integrable _ ((continuous_const.mul ((M.massiveDistance_continuous h).pow 2)).sub continuous_const) f |
| have hmom := weighted_core_square_integrable |
| (fun x : Boson N => (M.massiveDistance h x)^2) ((M.massiveDistance_continuous h).pow 2) f |
| have hn := coreNormSq_integrable f |
| have hr := (M.bosonic_core_integrable h 1 f).add (M.fermion_core_integrable h 1 f) |
| have hint := integral_mono hi hr (fun x => hbound h hh h1 x (f x)) |
| have he : (∫ x : Boson N, (c/2*(M.massiveDistance h x)^2-K)*‖f x‖^2) = |
| c/2*(∫ x : Boson N, (M.massiveDistance h x)^2*‖f x‖^2) - K*‖coreToL2 f‖^2 := by |
| simp_rw [sub_mul,mul_assoc] |
| rw [integral_sub (hmom.const_mul _) (hn.const_mul K),integral_const_mul, |
| integral_const_mul,←coreToL2_norm_sq] |
| rw [he] at hint |
| simp only [Pi.add_apply] at hint |
| unfold deformedCoreEnergy |
| rw [M.averaged_core_energy] |
| linarith |
| end |
| end BFSSQuantum.AlgebraData |
|
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| end OAI |
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