| import Mathlib |
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| namespace OAI |
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| /-! |
| # Differentiating Haar rotation invariance |
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| Infinitesimal Haar invariance gives the integral identity for spectral Ward |
| identities under explicit domination assumptions. |
| -/ |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology |
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| namespace InvariantIsing |
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| abbrev Orthogonal (N : ℕ) := Matrix.orthogonalGroup (Fin N) ℝ |
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| /-- A dominated infinitesimal left rotation has Haar mean zero. -/ |
| lemma integral_rotation_derivative_eq_zero {N : ℕ} (μ : Measure (Orthogonal N)) |
| [μ.IsMulLeftInvariant] (R : ℝ → Orthogonal N) (hR : R 0 = 1) |
| (f : Orthogonal N → ℝ) (hf : Measurable f) (hfi : Integrable f μ) |
| (f' : ℝ → Orthogonal N → ℝ) (hf' : AEStronglyMeasurable (f' 0) μ) |
| (B : Orthogonal N → ℝ) (hB : Integrable B μ) |
| (hbound : ∀ᵐ U ∂μ, ∀ t ∈ Set.Ioo (-1 : ℝ) 1, ‖f' t U‖ ≤ B U) |
| (hderiv : ∀ᵐ U ∂μ, ∀ t ∈ Set.Ioo (-1 : ℝ) 1, |
| HasDerivAt (fun s => f (R s * U)) (f' t U) t) : |
| ∫ U, f' 0 U ∂μ = 0 := by |
| have hs : Set.Ioo (-1 : ℝ) 1 ∈ 𝓝 (0 : ℝ) := Ioo_mem_nhds (by norm_num) (by norm_num) |
| have hm : ∀ᶠ t in 𝓝 (0 : ℝ), AEStronglyMeasurable (fun U => f (R t * U)) μ := by |
| apply Filter.Eventually.of_forall |
| intro t |
| have hmul : Measurable (fun U : Orthogonal N => R t * U) := by |
| apply Measurable.subtype_mk |
| change Measurable (fun U : Orthogonal N => |
| (R t : Matrix (Fin N) (Fin N) ℝ) * (U : Matrix (Fin N) (Fin N) ℝ)) |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro j |
| simp only [Matrix.mul_apply] |
| exact Finset.measurable_sum _ fun k _ => |
| measurable_const.mul ((measurable_pi_apply j).comp |
| ((measurable_pi_apply k).comp measurable_subtype_coe)) |
| exact (hf.comp hmul).aestronglyMeasurable |
| have hi : Integrable (fun U => f (R 0 * U)) μ := by simpa only [hR, one_mul] using hfi |
| obtain ⟨_, hd⟩ := hasDerivAt_integral_of_dominated_loc_of_deriv_le hs hm hi hf' |
| hbound hB hderiv |
| have he : (fun t => ∫ U, f (R t * U) ∂μ) = fun _ : ℝ => ∫ U, f U ∂μ := by |
| funext t |
| exact integral_mul_left_eq_self f (R t) |
| rw [he] at hd |
| exact hd.unique (hasDerivAt_const (0 : ℝ) (∫ U, f U ∂μ)) |
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| end InvariantIsing |
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| end |
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| end OAI |
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