openAI-math-dataset / lean /OAI /Combinatorics /PolynomialVacuum /Algebra /AnalyticNormalization.lean
| import OAI.Combinatorics.PolynomialVacuum.Algebra.AnalyticResolvent | |
| import OAI.Combinatorics.PolynomialVacuum.Algebra.EmptyNonzero | |
| import OAI.Combinatorics.PolynomialVacuum.Pairing.PairingTarget | |
| namespace OAI | |
| namespace PolynomialVacuumAlgebra | |
| namespace AnalyticVacuum | |
| open Classical Filter | |
| open scoped Topology | |
| theorem normalizedVacuum_homogeneous (N : ℕ) (s : Fin (N+1) → Spin) : | |
| normalizedVacuum (N+1) (fun _=>1) s=physicalProjection N s (fun _=>.zero) := by | |
| let v : (Fin (N+1) → Spin) → ℂ:=fun r=>physicalProjection N r (fun _=>.zero) | |
| have hv : (spectralMatrix (N+1) (fun _=>1)).mulVec v=rowDenominator (N+1) (fun _=>1) • v := by | |
| rw [spectralMatrix_one,rowDenominator_one,Matrix.smul_mulVec] | |
| congr 1 | |
| funext r | |
| exact congrFun (congrFun (physicalProjection_left_fixed N) r) (fun _=>.zero) | |
| have he:=normalizedVacuum_eq_ratio (N+1) (fun _=>1) v hv (vacuumResolvent_det_ne_zero _) (by simp [v]) s | |
| simpa [v] using he | |
| @[simp] theorem normalizedVacuum_empty (N : ℕ) : normalizedVacuum N (fun _=>1) (fun _=>.zero)=1 := by | |
| cases N with | |
| | zero => | |
| have hv : (spectralMatrix 0 (fun _=>1)).mulVec (fun _=>1)=rowDenominator 0 (fun _=>1) • (fun _=>1) := by | |
| funext s | |
| simp [spectralMatrix,transferMatrix,rowKernel,rowDenominator,trainDenominator, | |
| Matrix.mulVec,dotProduct,sum_spin,charge,PolynomialVacuumCylinder.zeta_eq_I] | |
| simpa using normalizedVacuum_eq_ratio 0 (fun _=>1) (fun _=>1) hv (vacuumResolvent_det_ne_zero 0) one_ne_zero (fun _=>.zero) | |
| | succ N => rw [normalizedVacuum_homogeneous,physicalProjection_empty] | |
| theorem normalizedVacuum_eventually_ratio (N : ℕ) (s : Fin N → Spin) : | |
| ∀ᶠ X in nhds (fun _ : Fin N => (1:ℂ)), | |
| MvPolynomial.eval X (polynomialVacuum N (fun _=>Spin.zero))≠0 → | |
| normalizedVacuum N X s=MvPolynomial.eval X (polynomialVacuum N s)/MvPolynomial.eval X (polynomialVacuum N (fun _=>Spin.zero)) := by | |
| filter_upwards [fusion_eventually_ne_zero N,vacuumResolvent_eventually_ne_zero N] with X hf hd | |
| intro h0 | |
| exact normalizedVacuum_eq_ratio N X (spectralPolynomial N X) (spectralPolynomial_eigen N X hf) hd h0 s | |
| theorem actual_analytic_normalization : | |
| PolynomialVacuumCylinder.AnalyticNormalization polynomialVacuum normalizedVacuum := | |
| ⟨polynomialVacuum_empty_ne_zero,analytic_normalizedVacuum, | |
| normalizedVacuum_eventually_ratio,normalizedVacuum_empty⟩ | |
| end AnalyticVacuum | |
| end PolynomialVacuumAlgebra | |
| end OAI | |