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e5a97c6 | 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 | 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
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