| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.PotentialSquares |
| import OAI.MathematicalPhysics.BFSS.ColorLie |
| import OAI.MathematicalPhysics.BFSS.NormalCocycle |
|
|
| namespace OAI |
|
|
| /-! Massive zero relations and positive normal quadratic forms. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Finset |
| variable {N : ℕ} (M : BFSSQuantum.AlgebraData N) |
|
|
| def spatialColor (x : Boson N) (i : SpaceIndex) : ColorVector N := |
| WithLp.toLp 2 (fun A => x (i,A)) |
|
|
| lemma spatialColor_add (x z : Boson N) (i : SpaceIndex) : |
| spatialColor (x+z) i = spatialColor x i + spatialColor z i := rfl |
| lemma spatialColor_smul (r : ℝ) (x : Boson N) (i : SpaceIndex) : |
| spatialColor (r • x) i = r • spatialColor x i := rfl |
|
|
| lemma colorEmbed_normSq (a : ColorVector N) : |
| BFSS.hsNormSq (M.colorEmbed a) = ‖a‖^2 := by |
| rw [show M.colorEmbed a = M.colorCombination a from rfl, M.colorCombination_normSq, |
| EuclideanSpace.real_norm_sq_eq] |
|
|
| lemma colorEmbed_spatial (x : Boson N) (i : SpaceIndex) : |
| M.colorEmbed (spatialColor x i) = (M.matrixConfiguration x i).val := rfl |
|
|
| def ZeroRelations (q : Boson N) : Prop := |
| M.colorLie (spatialColor q 0) (spatialColor q 1) = (2:ℝ) • spatialColor q 2 ∧ |
| M.colorLie (spatialColor q 1) (spatialColor q 2) = (2:ℝ) • spatialColor q 0 ∧ |
| M.colorLie (spatialColor q 2) (spatialColor q 0) = (2:ℝ) • spatialColor q 1 ∧ |
| ∀ d : SpaceIndex, 3 ≤ d.val → spatialColor q d = 0 |
|
|
| lemma residual_normSq (q : Boson N) (i j k : SpaceIndex) : |
| BFSS.hsNormSq (BFSS.realBracket (M.matrixConfiguration q i).val |
| (M.matrixConfiguration q j).val - (2:ℂ) • (M.matrixConfiguration q k).val) = |
| ‖M.colorLie (spatialColor q i) (spatialColor q j) - (2:ℝ) • spatialColor q k‖^2 := by |
| rw [←M.colorEmbed_normSq] |
| simp only [map_sub,map_smul,M.colorEmbed_lie,M.colorEmbed_spatial] |
| rfl |
|
|
| theorem potential_zero_relations (q : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) : M.ZeroRelations q := by |
| rw [M.deformedBosonicPotential_matrix] at hq |
| obtain ⟨_,h01,h12,h20,ht⟩ := BFSS.massivePotential_bounds (M.matrixConfiguration q) |
| rw [hq,mul_zero,M.residual_normSq] at h01 h12 h20 |
| have hz {v : ColorVector N} (h : ‖v‖^2 ≤ 0) : v = 0 := by |
| have : ‖v‖ = 0 := by nlinarith [sq_nonneg ‖v‖] |
| exact norm_eq_zero.mp this |
| refine ⟨sub_eq_zero.mp (hz h01),sub_eq_zero.mp (hz h12),sub_eq_zero.mp (hz h20),?_⟩ |
| intro d hd |
| have hh := ht d hd |
| rw [hq,mul_zero,←M.colorEmbed_spatial,M.colorEmbed_normSq] at hh |
| exact hz hh |
|
|
| def bracketLinearization (q z : Boson N) (i j : SpaceIndex) : ColorVector N := |
| M.colorLie (spatialColor q i) (spatialColor z j) + |
| M.colorLie (spatialColor z i) (spatialColor q j) |
|
|
| def cyclicLinearization (q z : Boson N) (i j k : SpaceIndex) : ColorVector N := |
| M.bracketLinearization q z i j - (2:ℝ) • spatialColor z k |
|
|
| def normalQuadratic (q z : Boson N) : ℝ := |
| (‖M.cyclicLinearization q z 0 1 2‖^2 + |
| ‖M.cyclicLinearization q z 1 2 0‖^2 + |
| ‖M.cyclicLinearization q z 2 0 1‖^2 + |
| (∑ i : SpaceIndex, ∑ j : SpaceIndex, |
| if i < j ∧ 3 ≤ j.val then ‖M.bracketLinearization q z i j‖^2 else 0) + |
| (∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor z d‖^2 else 0)) / 2 |
|
|
| lemma normalQuadratic_nonneg (q z : Boson N) : 0 ≤ M.normalQuadratic q z := by |
| unfold normalQuadratic |
| positivity |
|
|
| lemma normalQuadratic_zero_components (q z : Boson N) |
| (hz : M.normalQuadratic q z = 0) : |
| M.cyclicLinearization q z 0 1 2 = 0 ∧ |
| M.cyclicLinearization q z 1 2 0 = 0 ∧ |
| M.cyclicLinearization q z 2 0 1 = 0 ∧ |
| (∀ d : SpaceIndex, 3 ≤ d.val → spatialColor z d = 0) := by |
| have hs : 0 ≤ (∑ i : SpaceIndex, ∑ j : SpaceIndex, |
| if i < j ∧ 3 ≤ j.val then ‖M.bracketLinearization q z i j‖^2 else 0) := by positivity |
| have ht : 0 ≤ (∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor z d‖^2 else 0) := by positivity |
| have h1 := sq_nonneg ‖M.cyclicLinearization q z 0 1 2‖ |
| have h2 := sq_nonneg ‖M.cyclicLinearization q z 1 2 0‖ |
| have h3 := sq_nonneg ‖M.cyclicLinearization q z 2 0 1‖ |
| unfold normalQuadratic at hz |
| have ha : ‖M.cyclicLinearization q z 0 1 2‖ = 0 := by nlinarith |
| have hb : ‖M.cyclicLinearization q z 1 2 0‖ = 0 := by nlinarith |
| have hc : ‖M.cyclicLinearization q z 2 0 1‖ = 0 := by nlinarith |
| refine ⟨norm_eq_zero.mp ha,norm_eq_zero.mp hb,norm_eq_zero.mp hc,?_⟩ |
| intro d hd |
| have hh := Finset.single_le_sum (s := Finset.univ) |
| (f := fun d : SpaceIndex => if 3 ≤ d.val then ‖spatialColor z d‖^2 else 0) |
| (fun i _ => by positivity) (Finset.mem_univ d) |
| rw [ite_eq_left hd] at hh |
| have h0 : ‖spatialColor z d‖ = 0 := by nlinarith [sq_nonneg ‖spatialColor z d‖] |
| exact norm_eq_zero.mp h0 |
|
|
| def gaugeDivergence (q z : Boson N) : ColorVector N := |
| ∑ i : SpaceIndex, M.colorAd (spatialColor q i) (spatialColor z i) |
|
|
| lemma colorLie_zero_left (b : ColorVector N) : M.colorLie 0 b = 0 := by |
| have h := M.colorLie_smul_left 0 (0:ColorVector N) b |
| simpa only [zero_smul] using h |
| lemma colorLie_zero_right (a : ColorVector N) : M.colorLie a 0 = 0 := by |
| have h := M.colorLie_smul_right 0 a (0:ColorVector N) |
| simpa only [zero_smul] using h |
|
|
| lemma gaugeDivergence_first_three (q z : Boson N) (hq : M.ZeroRelations q) : |
| M.gaugeDivergence q z = |
| M.colorAd (spatialColor q 0) (spatialColor z 0) + |
| M.colorAd (spatialColor q 1) (spatialColor z 1) + |
| M.colorAd (spatialColor q 2) (spatialColor z 2) := by |
| have ht (d : SpaceIndex) (hd : 3 ≤ d.val) : |
| M.colorAd (spatialColor q d) (spatialColor z d) = 0 := by |
| change M.colorLie (spatialColor q d) (spatialColor z d) = 0 |
| rw [hq.2.2.2 d hd, M.colorLie_zero_left] |
| unfold gaugeDivergence |
| simp only [Fin.sum_univ_succ] |
| have h3 := ht 3 (by decide) |
| have h4 := ht 4 (by decide) |
| have h5 := ht 5 (by decide) |
| have h6 := ht 6 (by decide) |
| have h7 := ht 7 (by decide) |
| have h8 := ht 8 (by decide) |
| change M.colorAd (spatialColor q 0) (spatialColor z 0) + |
| (M.colorAd (spatialColor q 1) (spatialColor z 1) + |
| (M.colorAd (spatialColor q 2) (spatialColor z 2) + |
| (M.colorAd (spatialColor q 3) (spatialColor z 3) + |
| (M.colorAd (spatialColor q 4) (spatialColor z 4) + |
| (M.colorAd (spatialColor q 5) (spatialColor z 5) + |
| (M.colorAd (spatialColor q 6) (spatialColor z 6) + |
| (M.colorAd (spatialColor q 7) (spatialColor z 7) + |
| (M.colorAd (spatialColor q 8) (spatialColor z 8) + 0)))))))) = _ |
| simp only [h3,h4,h5,h6,h7,h8,add_zero] |
| abel |
|
|
| theorem normalQuadratic_zero_of_normal (q z : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) |
| (hperp : M.gaugeDivergence q z = 0) |
| (hz : M.normalQuadratic q z = 0) : z = 0 := by |
| have hrel := M.potential_zero_relations q hq |
| obtain ⟨h01,h12,h20,ht⟩ := M.normalQuadratic_zero_components q z hz |
| have hcyc (i j k : SpaceIndex) (hh : M.cyclicLinearization q z i j k = 0) : |
| M.colorAd (spatialColor q i) (spatialColor z j) - |
| M.colorAd (spatialColor q j) (spatialColor z i) = (2:ℝ) • spatialColor z k := by |
| unfold cyclicLinearization bracketLinearization at hh |
| rw [M.colorLie_swap (spatialColor z i) (spatialColor q j)] at hh |
| exact sub_eq_zero.mp hh |
| have hcomm (i j k : SpaceIndex) |
| (hh : M.colorLie (spatialColor q i) (spatialColor q j) = (2:ℝ) • spatialColor q k) |
| (w : ColorVector N) : |
| M.colorAd (spatialColor q i) (M.colorAd (spatialColor q j) w) - |
| M.colorAd (spatialColor q j) (M.colorAd (spatialColor q i) w) = |
| (2:ℝ) • M.colorAd (spatialColor q k) w := by |
| rw [M.colorAd_commutator,hh] |
| exact M.colorLie_smul_left 2 _ _ |
| rw [M.gaugeDivergence_first_three q z hrel] at hperp |
| obtain ⟨ha,hb,hc⟩ := BFSSNormal.normal_cocycle_zero |
| (M.colorAd (spatialColor q 0)) (M.colorAd (spatialColor q 1)) (M.colorAd (spatialColor q 2)) |
| (M.colorAd_skew _) (M.colorAd_skew _) (M.colorAd_skew _) |
| (hcomm 0 1 2 hrel.1) (hcomm 1 2 0 hrel.2.1) (hcomm 2 0 1 hrel.2.2.1) |
| (spatialColor z 0) (spatialColor z 1) (spatialColor z 2) |
| (hcyc 0 1 2 h01) (hcyc 1 2 0 h12) (hcyc 2 0 1 h20) hperp |
| ext p |
| obtain ⟨i,A⟩ := p |
| have hi : spatialColor z i = 0 := by |
| by_cases hid : 3 ≤ i.val |
| · exact ht i hid |
| · have : i = 0 ∨ i = 1 ∨ i = 2 := by |
| simp only [Fin.ext_iff]; omega |
| rcases this with rfl|rfl|rfl |
| · exact ha |
| · exact hb |
| · exact hc |
| exact congrArg (fun a : ColorVector N => a A) hi |
| end |
| end BFSSQuantum.AlgebraData |
| namespace BFSSNormal |
| noncomputable section |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| lemma positive_quadratic_lower (Q : E → ℝ) (hcont : Continuous Q) |
| (hsmul : ∀ r x, Q (r • x) = r^2 * Q x) |
| (hpos : ∀ x, x ≠ 0 → 0 < Q x) : |
| ∃ c : ℝ, 0 < c ∧ ∀ x, c * ‖x‖^2 ≤ Q x := by |
| have hzero : Q 0 = 0 := by |
| simpa only [zero_smul,zero_pow (by decide : 2 ≠ 0),zero_mul] |
| using (hsmul 0 (0:E)) |
| rcases subsingleton_or_nontrivial E with hs|hn |
| · let := hs |
| refine ⟨1,zero_lt_one,?_⟩ |
| intro x |
| rw [Subsingleton.elim x 0,hzero,norm_zero] |
| norm_num |
| · let := hn |
| obtain ⟨a,ha,hmin⟩ := (isCompact_sphere (0:E) 1).exists_isMinOn |
| (NormedSpace.sphere_nonempty.mpr zero_le_one) hcont.continuousOn |
| have ha_norm : ‖a‖ = 1 := by simpa only [Metric.mem_sphere,dist_zero_right] using ha |
| have ha_ne : a ≠ 0 := by intro h; simp only [h,norm_zero] at ha_norm; exact zero_ne_one ha_norm |
| refine ⟨Q a,hpos a ha_ne,?_⟩ |
| intro x |
| by_cases hx : x = 0 |
| · rw [hx,hzero,norm_zero]; norm_num |
| let y := ‖x‖⁻¹ • x |
| have hy : y ∈ Metric.sphere (0:E) 1 := by |
| rw [Metric.mem_sphere,dist_zero_right] |
| dsimp [y] |
| rw [norm_smul,Real.norm_eq_abs,abs_inv,abs_of_nonneg (norm_nonneg x)] |
| exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx) |
| have he : x = ‖x‖ • y := by |
| dsimp [y] |
| rw [smul_smul,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hx),one_smul] |
| calc |
| Q a * ‖x‖^2 ≤ Q y * ‖x‖^2 := mul_le_mul_of_nonneg_right (hmin hy) (sq_nonneg _) |
| _ = Q (‖x‖ • y) := by rw [hsmul ‖x‖ y, mul_comm] |
| _ = Q x := congrArg Q he.symm |
| end |
| end BFSSNormal |
|
|
| end OAI |
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