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import OAI.MathematicalPhysics.BFSS.FermionQuantization
namespace OAI
/-! Quantized-frequency commutators and self-adjointness. -/
namespace BFSSQuantum.AlgebraData
noncomputable section
open Matrix Finset
variable {N : ℕ} (M : AlgebraData N)
def thetaPair (r : FrequencyIndex N) : Fermion N →L[ℂ] Fermion N := M.theta r.1 r.2
lemma thetaPair_CAR (r s : FrequencyIndex N) :
M.thetaPair r*M.thetaPair s+M.thetaPair s*M.thetaPair r =
if r = s then 1 else 0 := by
rw [thetaPair,thetaPair,M.theta_CAR]
by_cases h : r = s
· subst s
simp
· have hh : ¬(r.1 = s.1 ∧ r.2 = s.2) := fun hh => h (Prod.ext hh.1 hh.2)
simp only [ite_eq_right h,ite_eq_right hh]
exact zero_smul ℕ (1 : Fermion N →L[ℂ] Fermion N)
lemma thetaPair_product_commutator (r s t : FrequencyIndex N) :
(M.thetaPair r*M.thetaPair s)*M.thetaPair t-
M.thetaPair t*(M.thetaPair r*M.thetaPair s) =
(if s = t then M.thetaPair r else 0) -
(if r = t then M.thetaPair s else 0) := by
calc
_ = M.thetaPair r*(M.thetaPair s*M.thetaPair t+M.thetaPair t*M.thetaPair s)-
(M.thetaPair r*M.thetaPair t+M.thetaPair t*M.thetaPair r)*M.thetaPair s := by
noncomm_ring
_ = _ := by
rw [M.thetaPair_CAR,M.thetaPair_CAR]
split_ifs <;> simp_all only [mul_one,one_mul,mul_zero,zero_mul,sub_self]
theorem quantizeFrequency_commutator (K : FrequencyMatrix N) (hK : K.transpose = -K)
(t : FrequencyIndex N) :
M.quantizeFrequency K*M.thetaPair t-M.thetaPair t*M.quantizeFrequency K =
Complex.I • ∑ r : FrequencyIndex N, (K r t : ℂ) • M.thetaPair r := by
have hdist (c : ℂ) (A B : Fermion N →L[ℂ] Fermion N) :
(c • A)*B-B*(c • A) = c • (A*B-B*A) := by
calc
_ = c • (A*B)-c • (B*A) := by rw [smul_mul_assoc,mul_smul_comm]
_ = _ := (smul_sub c (A*B) (B*A)).symm
have he : K t = fun r => -K r t := by
funext r
have h := congrArg (fun A : FrequencyMatrix N => A r t) hK
exact h
have hd (r s : FrequencyIndex N) :
((K r s : ℂ) • (M.thetaPair r*M.thetaPair s))*M.thetaPair t-
M.thetaPair t*((K r s : ℂ) • (M.thetaPair r*M.thetaPair s)) =
(K r s : ℂ) • ((if s = t then M.thetaPair r else 0)-
(if r = t then M.thetaPair s else 0)) := by
rw [hdist,M.thetaPair_product_commutator]
have hs : (∑ r : FrequencyIndex N, ∑ s : FrequencyIndex N,
(K r s : ℂ) • ((if s = t then M.thetaPair r else 0)-
(if r = t then M.thetaPair s else 0))) =
(∑ r : FrequencyIndex N, (K r t : ℂ) • M.thetaPair r)-
(∑ r : FrequencyIndex N, (K t r : ℂ) • M.thetaPair r) := by
simp only [smul_sub,Finset.sum_sub_distrib]
congr 1
· apply Finset.sum_congr rfl
intro r _
rw [Finset.sum_eq_single t]
· rw [ite_eq_left rfl]
· intro s _ hst
simp only [ite_eq_right hst,smul_zero]
· simp
· rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro s _
rw [Finset.sum_eq_single t]
· rw [ite_eq_left rfl]
· intro r _ hrt
simp only [ite_eq_right hrt,smul_zero]
· simp
change ((Complex.I/2) • (∑ r, ∑ s, (K r s : ℂ) •
(M.thetaPair r*M.thetaPair s)))*M.thetaPair t-
M.thetaPair t*((Complex.I/2) • (∑ r, ∑ s, (K r s : ℂ) •
(M.thetaPair r*M.thetaPair s))) = _
rw [hdist]
simp only [Finset.sum_mul,Finset.mul_sum,←Finset.sum_sub_distrib,hd]
have hneg (c : ℂ) (A : Fermion N →L[ℂ] Fermion N) :
(-c) • A = -(c • A) := neg_smul c A
have hn : (∑ r : FrequencyIndex N, (K t r : ℂ) • M.thetaPair r) =
-(∑ r : FrequencyIndex N, (K r t : ℂ) • M.thetaPair r) := by
rw [he]
simp only [Complex.ofReal_neg,hneg,Finset.sum_neg_distrib]
rw [hs,hn,sub_neg_eq_add]
module
end
end BFSSQuantum.AlgebraData
namespace BFSSQuantum.AlgebraData
noncomputable section
open Matrix Finset
variable {N : ℕ} (M : AlgebraData N)
lemma thetaPair_star (r : FrequencyIndex N) : star (M.thetaPair r) = M.thetaPair r :=
M.theta_selfAdjoint r.1 r.2
theorem quantizeFrequency_selfAdjoint (K : FrequencyMatrix N) (hK : K.transpose = -K) :
IsSelfAdjoint (M.quantizeFrequency K) := by
have hn (r s : FrequencyIndex N) : K s r = -K r s :=
congrArg (fun A : FrequencyMatrix N => A r s) hK
have hstar (c : ℂ) (A : Fermion N →L[ℂ] Fermion N) :
star (c • A) = star c • star A := star_smul c A
have hss (f : FrequencyIndex N → (Fermion N →L[ℂ] Fermion N)) :
star (∑ r, f r) = ∑ r, star (f r) := star_sum Finset.univ f
have hp (A B : Fermion N →L[ℂ] Fermion N) : star (A*B) = star B*star A := star_mul A B
have hneg (c : ℂ) (A : Fermion N →L[ℂ] Fermion N) :
(-c) • A = -(c • A) := neg_smul c A
have hr (c : ℝ) : star (c : ℂ) = (c : ℂ) := by simp
have hi : star (Complex.I/2) = -(Complex.I/2) := by simp; ring
have hs : (∑ r : FrequencyIndex N, ∑ s : FrequencyIndex N,
(K r s : ℂ) • (M.thetaPair s*M.thetaPair r)) =
-(∑ r : FrequencyIndex N, ∑ s : FrequencyIndex N,
(K r s : ℂ) • (M.thetaPair r*M.thetaPair s)) := by
rw [Finset.sum_comm]
rw [←Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro r _
rw [←Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro s _
rw [hn r s,Complex.ofReal_neg,hneg]
change star ((Complex.I/2) • (∑ r : FrequencyIndex N, ∑ s : FrequencyIndex N,
(K r s : ℂ) • (M.thetaPair r*M.thetaPair s))) = _
simp only [hstar,hss,hp,M.thetaPair_star,hr,hi]
rw [hs]
change -(Complex.I/2) • (-_) = (Complex.I/2) • _
exact neg_smul_neg _ _
end
end BFSSQuantum.AlgebraData
end OAI
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