| import OAI.Analysis.FiniteFactor.Matrix.Entries | |
| namespace OAI | |
| namespace FiniteFactor | |
| open MeasureTheory Filter | |
| open scoped ENNReal ComplexConjugate | |
| noncomputable section | |
| theorem source_fiber_restrict (A : Set (SourceCoordinate → ℝ)) (n : ℤ) : | |
| (sourceMeasure SourceCoordinate).restrict (A ×ˢ {n}) = | |
| (sourceBaseMeasure.restrict A).map (fun x => (x,n)) := by | |
| rw [← Measure.prod_restrict] | |
| simp only [Measure.restrict_singleton, Measure.count_singleton, one_smul] | |
| exact Measure.prod_dirac _ | |
| theorem source_fiber_integral (f : SourceHilbert SourceCoordinate) | |
| (A : Set (SourceCoordinate → ℝ)) (n : ℤ) : | |
| ∫ z in A ×ˢ {n}, f z ∂sourceMeasure SourceCoordinate = | |
| ∫ x in A, f (x,n) ∂sourceBaseMeasure := by | |
| rw [source_fiber_restrict] | |
| exact integral_map (measurable_id.prodMk measurable_const).aemeasurable | |
| (Lp.stronglyMeasurable f).aestronglyMeasurable | |
| theorem source_fiber_integrable (f : SourceHilbert SourceCoordinate) (n : ℤ) : | |
| Integrable (fun x => f (x,n)) sourceBaseMeasure := by | |
| have hi := integrableOn_Lp_of_measure_ne_top f (by norm_num : (1 : ℝ≥0∞) ≤ 2) | |
| (sourceFiberMeasure_ne_top Set.univ n) | |
| rw [IntegrableOn, source_fiber_restrict, Measure.restrict_univ] at hi | |
| exact (integrable_map_measure (Lp.stronglyMeasurable f).aestronglyMeasurable | |
| (measurable_id.prodMk measurable_const).aemeasurable).mp hi | |
| theorem sourceMatrixEntry_measurable | |
| (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) (k n : ℤ) : | |
| Measurable (sourceMatrixEntry H k n) := | |
| (Lp.stronglyMeasurable (H (sourceBasis n))).measurable.comp | |
| (measurable_id.prodMk measurable_const) | |
| theorem sourceMatrixEntry_integrable | |
| (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) (k n : ℤ) : | |
| Integrable (sourceMatrixEntry H k n) sourceBaseMeasure := | |
| source_fiber_integrable _ _ | |
| theorem sourceMatrixEntry_setIntegral | |
| (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) | |
| (A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A) (k n : ℤ) : | |
| ∫ x in A, sourceMatrixEntry H k n x ∂sourceBaseMeasure = | |
| inner ℂ (sourceFiberIndicator A hA k) (H (sourceBasis n)) := by | |
| rw [sourceFiberIndicator, L2.inner_indicatorConstLp_one, source_fiber_integral] | |
| rfl | |
| /-- The displayed trace integral is the actual previously proved normal state. -/ | |
| theorem sourceVacuumFunctional_matrixIntegral | |
| (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) : | |
| sourceVacuumFunctional H = ∫ x, sourceMatrixEntry H 0 0 x ∂sourceBaseMeasure := by | |
| rw [← setIntegral_univ, sourceMatrixEntry_setIntegral H Set.univ MeasurableSet.univ] | |
| rfl | |
| /-- Adjoint matrix coefficients agree almost everywhere, on the actual source base. -/ | |
| theorem sourceMatrixEntry_star (θ : SourceCoordinate → ℝ) | |
| {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate} | |
| (hH : H ∈ sourceAlgebra θ) (k n : ℤ) : | |
| sourceMatrixEntry (star H) k n =ᵐ[sourceBaseMeasure] | |
| fun x => star (sourceMatrixEntry H n k x) := by | |
| apply Integrable.ae_eq_of_forall_setIntegral_eq _ _ | |
| (sourceMatrixEntry_integrable _ _ _) | |
| (Complex.conjCLE.integrable_comp_iff.mpr (sourceMatrixEntry_integrable H n k)) | |
| intro A hA _ | |
| change (∫ x in A, sourceMatrixEntry (star H) k n x ∂sourceBaseMeasure) = | |
| ∫ x in A, conj (sourceMatrixEntry H n k x) ∂sourceBaseMeasure | |
| rw [sourceMatrixEntry_setIntegral _ A hA, integral_conj, sourceMatrixEntry_setIntegral _ A hA, | |
| inner_conj_symm] | |
| rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right] | |
| have he : H (sourceFiberIndicator A hA k) = | |
| sourceScalar (sourceBaseIndicator A hA) (H (sourceBasis k)) := by | |
| rw [← sourceScalar_indicator_basis] | |
| exact (congrArg (fun T : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate => | |
| T (sourceBasis k)) (sourceAlgebra_commutes_scalar θ hH (sourceBaseIndicator A hA))).symm | |
| rw [he] | |
| have hs : star (sourceScalar (sourceBaseIndicator A hA)) = | |
| sourceScalar (sourceBaseIndicator A hA) := by | |
| rw [sourceScalar_star] | |
| congr 1 | |
| unfold sourceBaseStar sourceBaseIndicator | |
| congr 1 | |
| funext x | |
| by_cases hx : x ∈ A <;> simp [hx] | |
| rw [← ContinuousLinearMap.adjoint_inner_right, ← ContinuousLinearMap.star_eq_adjoint, | |
| hs, sourceScalar_indicator_basis] | |
| end | |
| end FiniteFactor | |
| end OAI | |