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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import Mathlib
import OAI.MathematicalPhysics.BFSS.FermionModes
import OAI.MathematicalPhysics.BFSS.FrequencySpectrum
import OAI.MathematicalPhysics.BFSS.Vacuum
namespace OAI
/-! Irreducibility of negative modes and the unique normal fermion ground line. -/
namespace BFSSQuantum.AlgebraData
noncomputable section
open Matrix Finset
open scoped ComplexConjugate
variable {N : ℕ} (M : AlgebraData N)
lemma thetaMode_zero : M.thetaMode (0 : FrequencyIndex N → ℂ) = 0 := by
unfold thetaMode
apply Finset.sum_eq_zero
intro r _
exact zero_smul ℂ (M.thetaPair r)
lemma thetaMode_add (v w : FrequencyIndex N → ℂ) :
M.thetaMode (v+w) = M.thetaMode v+M.thetaMode w := by
unfold thetaMode
have ha (a b : ℂ) (T : Fermion N →L[ℂ] Fermion N) :
(a+b) • T = a • T+b • T := add_smul a b T
simp only [Pi.add_apply,ha,Finset.sum_add_distrib]
lemma thetaMode_single (r : FrequencyIndex N) :
M.thetaMode (Pi.single r (1 : ℂ)) = M.thetaPair r := by
classical
unfold thetaMode
rw [Finset.sum_eq_single r]
· rw [Pi.single_eq_same]
exact one_smul ℂ (M.thetaPair r)
· intro s _ hsr
rw [Pi.single_eq_of_ne hsr]
exact zero_smul ℂ (M.thetaPair s)
· simp
def NegativeFrequencyMode (K : FrequencyMatrix N) :=
{v : FrequencyIndex N → ℂ // ∃ ω : ℝ, ω < 0 ∧
∀ r, Complex.I*(∑ s, (K r s : ℂ)*v s) = (ω : ℂ)*v r}
theorem normal_negative_modes_irreducible (q : Boson N)
(hq : M.deformedBosonicPotential 1 1 q = 0) (W : Submodule ℂ (Fermion N))
(ha : ∀ i : NegativeFrequencyMode (M.normalFermionFrequency q),
∀ x ∈ W, M.thetaMode i.1 x ∈ W)
(hc : ∀ i : NegativeFrequencyMode (M.normalFermionFrequency q),
∀ x ∈ W, M.thetaMode (fun r => conj (i.1 r)) x ∈ W) : W = ⊥ ∨ W = ⊤ := by
classical
let K := M.normalFermionFrequency q
have hH := complexFrequency_hermitian K (M.normalFermionFrequency_skew q)
let b := hH.eigenvectorBasis
let V : Submodule ℂ (EuclideanSpace ℂ (FrequencyIndex N)) := {
carrier := {v | ∀ x ∈ W, M.thetaMode v x ∈ W}
zero_mem' := by
intro x hx
change M.thetaMode (0 : FrequencyIndex N → ℂ) x ∈ W
rw [M.thetaMode_zero]
exact W.zero_mem
add_mem' := by
intro v w hv hw x hx
change M.thetaMode ((v : FrequencyIndex N → ℂ)+(w : FrequencyIndex N → ℂ)) x ∈ W
rw [M.thetaMode_add]
exact W.add_mem (hv x hx) (hw x hx)
smul_mem' := by
intro c v hv x hx
change M.thetaMode (c • (v : FrequencyIndex N → ℂ)) x ∈ W
rw [M.thetaMode_smul]
exact W.smul_mem c (hv x hx)
}
have hb (j : FrequencyIndex N) : b j ∈ V := by
have hm := complexFrequency_eigenmode K (M.normalFermionFrequency_skew q) j
by_cases hj : hH.eigenvalues j < 0
· exact ha ⟨b j,hH.eigenvalues j,hj,hm⟩
· have hp : 0 < hH.eigenvalues j := lt_of_le_of_ne (not_lt.mp hj)
(Ne.symm (M.normal_frequency_eigenvalues_ne_zero q hq j))
let i : NegativeFrequencyMode K := ⟨fun r => conj (b j r),
-hH.eigenvalues j,neg_neg_of_pos hp,complexFrequency_conjugate_mode K (b j)
(hH.eigenvalues j) hm⟩
have hi := hc i
have he : (fun r => conj (i.1 r)) = (b j : FrequencyIndex N → ℂ) := by
funext r
exact star_star _
rw [he] at hi
exact hi
have hV : V = ⊤ := by
apply top_unique
rw [← b.toBasis.span_eq]
apply Submodule.span_le.mpr
rintro v ⟨j,rfl⟩
exact hb j
apply M.theta_irreducible W
intro α A x hx
have hv : WithLp.toLp 2 (Pi.single (α,A) (1 : ℂ)) ∈ V := by rw [hV]; trivial
have h := hv x hx
change M.thetaMode (Pi.single (α,A) (1 : ℂ)) x ∈ W at h
rw [M.thetaMode_single] at h
exact h
theorem normal_fermion_ground_line (q : Boson N)
(hq : M.deformedBosonicPotential 1 1 q = 0) :
∃ (E : ℝ) (u : Fermion N), u ≠ 0 ∧
M.quantizeFrequency (M.normalFermionFrequency q) u = (E : ℂ) • u ∧
(∀ (e : ℝ) (w : Fermion N), w ≠ 0 →
M.quantizeFrequency (M.normalFermionFrequency q) w = (e : ℂ) • w → E ≤ e) ∧
∀ z : Fermion N, M.quantizeFrequency (M.normalFermionFrequency q) z = (E : ℂ) • z ↔
z ∈ Submodule.span ℂ {u} := by
classical
let K := M.normalFermionFrequency q
have hs := M.normalFermionFrequency_skew q
obtain ⟨E,u,hu,he,hm⟩ := M.quantizeFrequency_ground_exists K hs
let a (i : NegativeFrequencyMode K) := (M.thetaMode i.1).toLinearMap
let c (i : NegativeFrequencyMode K) := (M.thetaMode (fun r => conj (i.1 r))).toLinearMap
let κ (i j : NegativeFrequencyMode K) : ℂ := ∑ r, i.1 r*conj (j.1 r)
have hcar : ∀ i j x, a i (c j x)+c j (a i x) = κ i j • x := by
intro i j x
exact congrArg (fun T : Fermion N →L[ℂ] Fermion N => T x)
(M.thetaMode_CAR i.1 (fun r => conj (j.1 r)))
have hadj : ∀ i x y, inner ℂ x (c i y) = inner ℂ (a i x) y := by
intro i x y
change inner ℂ x (M.thetaMode (fun r => conj (i.1 r)) y) =
inner ℂ (M.thetaMode i.1 x) y
rw [←M.thetaMode_star]
exact (M.thetaMode i.1).adjoint_inner_right x y
have hirr : ∀ W : Submodule ℂ (Fermion N),
(∀ i, ∀ x ∈ W, a i x ∈ W) → (∀ i, ∀ x ∈ W, c i x ∈ W) → W = ⊥ ∨ W = ⊤ := by
intro W ha hc
exact M.normal_negative_modes_irreducible q hq W ha hc
have killed (z : Fermion N) (hz : M.quantizeFrequency K z = (E : ℂ) • z) :
∀ i, a i z = 0 := by
intro i
obtain ⟨ω,hω,hv⟩ := i.2
exact M.quantizeFrequency_ground_killed K hs E z hz hm i.1 ω hω hv
have hline := BFSSScalarVacuum.vacuum_line_of_exists a c κ hcar hadj hirr u hu (killed u he)
refine ⟨E,u,hu,he,hm,?_⟩
intro z
constructor
· intro hz
exact (hline z).mp (killed z hz)
· intro hz
obtain ⟨r,rfl⟩ := Submodule.mem_span_singleton.mp hz
rw [map_smul,he,smul_comm]
end
end BFSSQuantum.AlgebraData
end OAI
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