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9655094 | 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 | import Mathlib
import OAI.Analysis.FiniteFactor.CrossedProduct.VacuumFunctional
namespace OAI
section
open MeasureTheory Filter InnerProductSpace
open scoped ComplexInnerProductSpace Topology ENNReal CStarAlgebra
noncomputable section
namespace FiniteFactor
local instance : IsometricContinuousFunctionalCalculus ℝ
(SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) IsSelfAdjoint :=
IsSelfAdjoint.instIsometricContinuousFunctionalCalculus
instance productHilbertNontrivial : Nontrivial (SourceHilbert SourceCoordinate) := by
have hv : sourceVacuum≠0 := by
intro hz
have h := sourceVacuum_norm
rw [hz,norm_zero] at h
norm_num at h
exact ⟨⟨sourceVacuum,0,hv⟩⟩
end FiniteFactor
end
end
section
open MeasureTheory Filter InnerProductSpace
open scoped ComplexInnerProductSpace Topology
noncomputable section
namespace FiniteFactor
local notation "productMeasurable" =>
@Prod.instMeasurableSpace (SourceCoordinate → ℝ) ℤ
(@MeasurableSpace.pi SourceCoordinate (fun _ => ℝ)
(fun _ => Real.measurableSpace)) Int.instMeasurableSpace
local notation "productSemiring" =>
Complex.instNormedField.toField.toCommRing.toRing.toSemiring
local notation "productIdentity" =>
@RingHom.id ℂ (@Semiring.toNonAssocSemiring ℂ productSemiring)
local notation "productExponent" =>
@Nat.cast ENNReal
(ENNReal.instAddCommMonoidWithOne.toAddMonoidWithOne.toNatCast)
(Nat.succ (Nat.succ Nat.zero))
local notation "productHilbertGroup" =>
@MeasureTheory.Lp.instNormedAddCommGroup
((SourceCoordinate → ℝ) × ℤ) ℂ productMeasurable
productExponent
(sourceMeasure SourceCoordinate) Complex.instNormedAddCommGroup
fact_one_le_two_ennreal
local notation "productHilbertInner" =>
@MeasureTheory.L2.innerProductSpace
((SourceCoordinate → ℝ) × ℤ) ℂ ℂ Complex.instRCLike
productMeasurable (sourceMeasure SourceCoordinate)
Complex.instNormedAddCommGroup (@RCLike.innerProductSpace ℂ Complex.instRCLike)
local notation "productHilbertTopology" =>
@UniformSpace.toTopologicalSpace (SourceHilbert SourceCoordinate)
(@PseudoMetricSpace.toUniformSpace (SourceHilbert SourceCoordinate)
(@MetricSpace.toPseudoMetricSpace (SourceHilbert SourceCoordinate)
(@NormedAddCommGroup.toMetricSpace (SourceHilbert SourceCoordinate) productHilbertGroup)))
local notation "productHilbertAddition" =>
@AddCommGroup.toAddCommMonoid (SourceHilbert SourceCoordinate)
(@NormedAddCommGroup.toAddCommGroup (SourceHilbert SourceCoordinate) productHilbertGroup)
local notation "productHilbertModule" =>
@NormedSpace.toModule ℂ (SourceHilbert SourceCoordinate)
(@DenselyNormedField.toNormedField ℂ
(@RCLike.toDenselyNormedField ℂ Complex.instRCLike))
(@NormedAddCommGroup.toSeminormedAddCommGroup
(SourceHilbert SourceCoordinate) productHilbertGroup)
(@InnerProductSpace.toNormedSpace ℂ (SourceHilbert SourceCoordinate) Complex.instRCLike
(@NormedAddCommGroup.toSeminormedAddCommGroup
(SourceHilbert SourceCoordinate) productHilbertGroup)
productHilbertInner)
abbrev ProductOp :=
@ContinuousLinearMap ℂ ℂ productSemiring productSemiring productIdentity
(SourceHilbert SourceCoordinate) productHilbertTopology productHilbertAddition
(SourceHilbert SourceCoordinate) productHilbertTopology productHilbertAddition
productHilbertModule productHilbertModule
section
open MeasureTheory Filter InnerProductSpace
open scoped ComplexInnerProductSpace Topology ENNReal
local instance : Module ℝ ProductOp := Module.complexToReal ProductOp
local instance : IsometricContinuousFunctionalCalculus ℝ ProductOp IsSelfAdjoint :=
IsSelfAdjoint.instIsometricContinuousFunctionalCalculus
def logarithmCutoff (n : ℕ) : C(ℝ,ℝ) where
toFun t := Real.log (max (Real.exp (-(n:ℝ))) t)
continuous_toFun := (continuous_const.max continuous_id).log
(fun _=>ne_of_gt ((Real.exp_pos _).trans_le (le_max_left _ _)))
run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}
def FugledeKadisonDeterminant (τ : ProductOp → ℂ) (T : ProductOp) : ℝ≥0∞ :=
let determinant := fun (Hilbert : Type) [NormedAddCommGroup Hilbert]
[InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert] [Nontrivial Hilbert]
(trace : (Hilbert →L[ℂ] Hilbert) → ℂ) (operator : Hilbert →L[ℂ] Hilbert) =>
letI : Module ℝ (Hilbert →L[ℂ] Hilbert) := Module.complexToReal _
letI : IsometricContinuousFunctionalCalculus ℝ (Hilbert →L[ℂ] Hilbert)
IsSelfAdjoint := IsSelfAdjoint.instIsometricContinuousFunctionalCalculus
⨅ n : ℕ, ENNReal.ofReal
(Real.exp ((trace (cfc (logarithmCutoff n) (CFC.abs operator))).re))
let domain := (SourceCoordinate → ℝ) × ℤ
let measurable : MeasurableSpace domain := productMeasurable
let exponent : ℝ≥0∞ := productExponent
let measure : @Measure domain measurable := sourceMeasure SourceCoordinate
let complete := @MeasureTheory.Lp.instCompleteSpace domain measurable exponent measure ℂ
Complex.instNormedAddCommGroup Complex.instCompleteSpace fact_one_le_two_ennreal
@determinant (SourceHilbert SourceCoordinate) productHilbertGroup productHilbertInner
complete productHilbertNontrivial τ T
end
section
open MeasureTheory Filter
open scoped ENNReal Topology
def expLogPotential (μ : Measure ℂ) (z : ℂ) : ℝ≥0∞ :=
⨅ n : ℕ, ENNReal.ofReal (Real.exp (∫ w,logarithmCutoff n ‖z-w‖ ∂μ))
def IsBrownMeasure (τ : ProductOp → ℂ) (T : ProductOp) (μ : Measure ℂ) : Prop :=
let translate := fun (Hilbert : Type) [NormedAddCommGroup Hilbert]
[NormedSpace ℂ Hilbert] (operator : Hilbert →L[ℂ] Hilbert) (scalar : ℂ) =>
operator - algebraMap ℂ (Hilbert →L[ℂ] Hilbert) scalar
IsProbabilityMeasure μ ∧ IsCompact μ.support ∧
∀ z : ℂ, FugledeKadisonDeterminant τ
(@translate (SourceHilbert SourceCoordinate) productHilbertGroup
(@InnerProductSpace.toNormedSpace ℂ (SourceHilbert SourceCoordinate) Complex.instRCLike
(@NormedAddCommGroup.toSeminormedAddCommGroup
(SourceHilbert SourceCoordinate) productHilbertGroup)
productHilbertInner) T z) = expLogPotential μ z
def ProductBrownClaim : Prop :=
∀ (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ),
IsBrownMeasure sourceVacuumFunctional
(sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg) (Measure.dirac 0)
end
end FiniteFactor
end
end
end OAI
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