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9b6e014 | 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 | import OAI.Probability.ClassicalON.CylindricalLaw
namespace OAI
universe uE uV
noncomputable section
open MeasureTheory
open scoped BigOperators InnerProductSpace
namespace ClassicalON
variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E]
def freeSpinReference (n : ℕ) : Measure (V → Spin n) := Measure.pi (fun _ => sphereProbability n)
instance freeSpinReference_isProbability (n : ℕ) [NeZero n] :
IsProbabilityMeasure (freeSpinReference (V := V) n) := by unfold freeSpinReference; infer_instance
def freeSpinEnergy (n : ℕ) (left right : E → V) (b : E → ℝ) (s : V → Spin n) : ℝ :=
∑ e,b e*⟪(s (left e)).val,(s (right e)).val⟫_ℝ
omit [Fintype V] in
theorem continuous_freeSpinEnergy (n : ℕ) (left right : E → V) (b : E → ℝ) :
Continuous (freeSpinEnergy n left right b) := by unfold freeSpinEnergy; fun_prop
def freeSpinMean (n : ℕ) (left right : E → V) (b : E → ℝ) (f : (V → Spin n) → ℝ) : ℝ :=
weightedMean (freeSpinReference n) (fun s => Real.exp (freeSpinEnergy n left right b s)) f
def cylindricalConfiguration (r : V → Amplitude) (τ : V → Bool) (θ : V → PlanarAngle) : V → Spin 3 :=
fun v => cylindricalSpin (r v) (τ v) (θ v)
omit [Fintype V] [Fintype E] in
theorem continuous_cylindricalConfiguration :
Continuous (fun p : (V → Amplitude)×(V → Bool)×(V → PlanarAngle) =>
cylindricalConfiguration p.1 p.2.1 p.2.2) := by
apply continuous_pi
intro v
exact continuous_cylindricalSpin.comp
(f := fun p : (V → Amplitude)×(V → Bool)×(V → PlanarAngle) => (p.1 v,p.2.1 v,p.2.2 v))
(((continuous_apply v).comp continuous_fst).prodMk
(((continuous_apply v).comp (continuous_fst.comp continuous_snd)).prodMk
((continuous_apply v).comp (continuous_snd.comp continuous_snd))))
omit [Fintype E] in
theorem cylindricalConfiguration_preserves : MeasurePreserving
(fun p : (V → Amplitude)×(V → Bool)×(V → PlanarAngle) => cylindricalConfiguration p.1 p.2.1 p.2.2)
((Measure.pi fun _ : V => sphericalAmplitudeLaw).prod ((isingReference (V := V)).prod planarReference))
(freeSpinReference 3) := by
let A := MeasurableEquiv.arrowProdEquivProdArrow Amplitude (Bool×PlanarAngle) V
let B := MeasurableEquiv.arrowProdEquivProdArrow Bool PlanarAngle V
have hA := (measurePreserving_arrowProdEquivProdArrow Amplitude (Bool×PlanarAngle) V
(fun _ => sphericalAmplitudeLaw) (fun _ => cylinderLaw)).symm A
have hB := (measurePreserving_arrowProdEquivProdArrow Bool PlanarAngle V
(fun _ => signLaw) (fun _ => angleLaw)).symm B
have h1 := (MeasurePreserving.id (Measure.pi fun _ : V => sphericalAmplitudeLaw)).prod hB
have h2 := hA.comp h1
have h3 := measurePreserving_pi (fun _ : V => sphericalAmplitudeLaw.prod cylinderLaw)
(fun _ => sphereProbability 3) (fun _ => cylindricalSpin_preserves)
exact h3.comp h2
omit [Fintype E] in
theorem integral_freeSpinReference_cylinder (f : (V → Spin 3) → ℝ) (hf : Continuous f) :
(∫ s,f s ∂freeSpinReference 3)=
∫ r,∫ τ,∫ θ,f (cylindricalConfiguration r τ θ) ∂planarReference ∂isingReference
∂Measure.pi (fun _ : V => sphericalAmplitudeLaw) := by
let : BorelSpace ((V → Amplitude) × (V → Bool) × (V → PlanarAngle)) := Prod.borelSpace
have hc : Continuous (fun p : (V → Amplitude)×(V → Bool)×(V → PlanarAngle) =>
f (cylindricalConfiguration p.1 p.2.1 p.2.2)) := hf.comp continuous_cylindricalConfiguration
rw [← cylindricalConfiguration_preserves.map_eq,
integral_map continuous_cylindricalConfiguration.measurable.aemeasurable hf.aestronglyMeasurable,
integral_prod _ (compact_integrable hc)]
apply integral_congr_ae
filter_upwards with r
have hr : Continuous (fun p : (V → Bool)×(V → PlanarAngle) => f (cylindricalConfiguration r p.1 p.2)) :=
hc.comp (continuous_const.prodMk continuous_id)
exact integral_prod _ (compact_integrable hr)
omit [Fintype V] [Fintype E] in
theorem cylindricalSpin_inner (r q : Amplitude) (τ υ : Bool) (θ φ : PlanarAngle) :
⟪(cylindricalSpin r τ θ).val,(cylindricalSpin q υ φ).val⟫_ℝ=
(r:ℝ)*(q:ℝ)*(signValue τ*signValue υ)+
transverseAmplitude r*transverseAmplitude q*planarCos (θ-φ) := by
rw [EuclideanSpace.inner_eq_star_dotProduct]
simp only [dotProduct,Fin.sum_univ_succ,Fin.isValue,cylindricalSpin,
WithLp.ofLp_toLp,Matrix.cons_val_zero,Matrix.cons_val_succ,star_trivial,Finset.univ_eq_empty,
Finset.sum_empty,add_zero]
simp only [planarCos,planarSin,sub_eq_add_neg,planarCharacter_add,planarCharacter_neg,
Complex.mul_re,Complex.conj_re,Complex.conj_im]
ring
omit [Fintype V] in
theorem cylindricalConfiguration_energy (left right : E → V) (b : E → ℝ)
(r : V → Amplitude) (τ : V → Bool) (θ : V → PlanarAngle) :
freeSpinEnergy 3 left right b (cylindricalConfiguration r τ θ)=
edgeHamiltonian (isingEnergy left right) (amplitudeCoupling left right b (fun v => (r v:ℝ))) τ+
edgeHamiltonian (planarEnergy left right) (amplitudeCoupling left right b (fun v => transverseAmplitude (r v))) θ := by
unfold freeSpinEnergy edgeHamiltonian cylindricalConfiguration
simp_rw [cylindricalSpin_inner]
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro e _
unfold amplitudeCoupling isingEnergy planarEnergy
ring
end ClassicalON
end
end OAI
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