Joseph Tooby-Smith
GitHub
commit
1d76774ff5
57 changed files with 350 additions and 228 deletions
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@ -63,7 +63,7 @@ def repCharges {n : ℕ} : Representation ℚ (PermGroup n) (SMCharges n).Charge
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erw [toSMSpecies_toSpecies_inv]
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rfl
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/-- The species chages of a set of charges acted on by a family permutation is the permutation
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/-- The species charges of a set of charges acted on by a family permutation is the permutation
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of those species charges with the corresponding part of the family permutation. -/
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lemma repCharges_toSpecies (f : PermGroup n) (S : (SMCharges n).Charges) (j : Fin 5) :
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toSpecies j (repCharges f S) = toSpecies j S ∘ f⁻¹ j := by
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@ -78,7 +78,7 @@ lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMCharges n).Cha
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exact Fintype.sum_equiv (f⁻¹ j) (fun x => ((fun a => a ^ m) ∘ (toSpecies j) S ∘ ⇑(f⁻¹ j)) x)
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(fun x => ((fun a => a ^ m) ∘ (toSpecies j) S) x) (congrFun rfl)
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/-- The gravitional anomaly equations is invariant under family permutations. -/
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/-- The gravitational anomaly equations is invariant under family permutations. -/
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lemma accGrav_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
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accGrav (repCharges f S) = accGrav S := accGrav_ext
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(by simpa using toSpecies_sum_invariant 1 f S)
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@ -38,7 +38,7 @@ informal_lemma inclSM_ker where
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deps := [``inclSM, ``StandardModel.gaugeGroupℤ₆SubGroup]
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/-- The group embedding from `StandardModel.GaugeGroupℤ₆` to `GaugeGroupI` induced by `inclSM` by
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quotienting by the kernal `inclSM_ker`.
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quotienting by the kernel `inclSM_ker`.
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-/
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informal_definition embedSMℤ₆ where
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deps := [``inclSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupI, ``inclSM_ker]
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@ -43,7 +43,7 @@ informal_lemma inclSM_ker where
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deps := [``inclSM, ``StandardModel.gaugeGroupℤ₃SubGroup]
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/-- The group embedding from `StandardModel.GaugeGroupℤ₃` to `GaugeGroupI` induced by `inclSM` by
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quotienting by the kernal `inclSM_ker`.
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quotienting by the kernel `inclSM_ker`.
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-/
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informal_definition embedSMℤ₃ where
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deps := [``inclSM, ``StandardModel.GaugeGroupℤ₃, ``GaugeGroupI, ``inclSM_ker]
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@ -21,7 +21,7 @@ open ComplexConjugate
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section rows
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/-- The absolute value squared of any rwo of a CKM matrix is `1`, in terms of `Vabs`. -/
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/-- The absolute value squared of any row of a CKM matrix is `1`, in terms of `Vabs`. -/
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lemma VAbs_sum_sq_row_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
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(VAbs i 0 V) ^ 2 + (VAbs i 1 V) ^ 2 + (VAbs i 2 V) ^ 2 = 1 := by
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obtain ⟨V, hV⟩ := Quot.exists_rep V
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@ -36,31 +36,31 @@ lemma VAbs_sum_sq_row_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
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rw [← ofReal_inj]
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simpa using ht
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/-- The absolute value squared of the first rwo of a CKM matrix is `1`, in terms of `abs`. -/
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/-- The absolute value squared of the first row of a CKM matrix is `1`, in terms of `abs`. -/
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lemma fst_row_normalized_abs (V : CKMMatrix) : abs [V]ud ^ 2 + abs [V]us ^ 2 + abs [V]ub ^ 2 = 1 :=
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VAbs_sum_sq_row_eq_one ⟦V⟧ 0
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/-- The absolute value squared of the second rwo of a CKM matrix is `1`, in terms of `abs`. -/
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/-- The absolute value squared of the second row of a CKM matrix is `1`, in terms of `abs`. -/
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lemma snd_row_normalized_abs (V : CKMMatrix) : abs [V]cd ^ 2 + abs [V]cs ^ 2 + abs [V]cb ^ 2 = 1 :=
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VAbs_sum_sq_row_eq_one ⟦V⟧ 1
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/-- The absolute value squared of the third rwo of a CKM matrix is `1`, in terms of `abs`. -/
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/-- The absolute value squared of the third row of a CKM matrix is `1`, in terms of `abs`. -/
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lemma thd_row_normalized_abs (V : CKMMatrix) : abs [V]td ^ 2 + abs [V]ts ^ 2 + abs [V]tb ^ 2 = 1 :=
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VAbs_sum_sq_row_eq_one ⟦V⟧ 2
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/-- The absolute value squared of the first rwo of a CKM matrix is `1`, in terms of `nomSq`. -/
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/-- The absolute value squared of the first row of a CKM matrix is `1`, in terms of `nomSq`. -/
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lemma fst_row_normalized_normSq (V : CKMMatrix) :
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normSq [V]ud + normSq [V]us + normSq [V]ub = 1 := by
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repeat rw [← Complex.sq_abs]
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exact V.fst_row_normalized_abs
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/-- The absolute value squared of the second rwo of a CKM matrix is `1`, in terms of `nomSq`. -/
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/-- The absolute value squared of the second row of a CKM matrix is `1`, in terms of `nomSq`. -/
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lemma snd_row_normalized_normSq (V : CKMMatrix) :
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normSq [V]cd + normSq [V]cs + normSq [V]cb = 1 := by
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repeat rw [← Complex.sq_abs]
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exact V.snd_row_normalized_abs
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/-- The absolute value squared of the third rwo of a CKM matrix is `1`, in terms of `nomSq`. -/
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/-- The absolute value squared of the third row of a CKM matrix is `1`, in terms of `nomSq`. -/
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lemma thd_row_normalized_normSq (V : CKMMatrix) :
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normSq [V]td + normSq [V]ts + normSq [V]tb = 1 := by
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repeat rw [← Complex.sq_abs]
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@ -61,7 +61,7 @@ lemma coMetric_basis_expand_tree : {η' | μ ν}ᵀ.tensor =
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(smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 3))))).tensor :=
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coMetric_basis_expand
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/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors. -/
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/-- The expansion of the Lorentz contravariant metric in terms of basis vectors. -/
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lemma contrMatrix_basis_expand : {η | μ ν}ᵀ.tensor =
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basisVector ![Color.up, Color.up] (fun _ => 0)
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- basisVector ![Color.up, Color.up] (fun _ => 1)
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@ -87,7 +87,7 @@ lemma contrMatrix_basis_expand : {η | μ ν}ᵀ.tensor =
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simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
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rfl
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/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors as
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/-- The expansion of the Lorentz contravariant metric in terms of basis vectors as
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a structured tensor tree. -/
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lemma contrMatrix_basis_expand_tree : {η | μ ν}ᵀ.tensor =
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(TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
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@ -216,7 +216,7 @@ lemma basis_contr_pauliMatrix_basis_tree_expand' {n : ℕ} {c : Fin n → comple
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rfl
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/-- The map to color which appears when contracting a basis vector with
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puali matrices. -/
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Pauli matrices. -/
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def pauliMatrixBasisProdMap
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{n : ℕ} {c : Fin n → complexLorentzTensor.C}
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(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (i1 i2 i3 : Fin 4) :
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@ -43,7 +43,7 @@ def genBoostAux₁ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d
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smul_tmul, tmul_smul, map_smul, smul_eq_mul, RingHom.id_apply]
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rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
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/-- An auxiliary linear map used in the definition of a genearlised boost. -/
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/-- An auxiliary linear map used in the definition of a generalised boost. -/
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def genBoostAux₂ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d where
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toFun x := - (⟪x, u.1.1 + v.1.1⟫ₘ / (1 + ⟪u.1.1, v.1.1⟫ₘ)) • (u.1.1 + v.1.1)
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map_add' x y := by
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@ -113,7 +113,7 @@ lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
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local notation "ℤ₂" => Multiplicative (ZMod 2)
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/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
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/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernel are the Orthochronous elements. -/
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def orthchroMap : C(LorentzGroup d, ℤ₂) :=
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ContinuousMap.comp coeForℤ₂ {
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toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
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@ -189,7 +189,7 @@ lemma IsProper_iff (Λ : LorentzGroup d) : IsProper Λ ↔ detRep Λ = 1 := by
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lemma id_IsProper : @IsProper d 1 := by
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simp [IsProper]
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/-- If two Lorentz transformations are in the same connected componenet, and one is proper then
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/-- If two Lorentz transformations are in the same connected component, and one is proper then
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the other is also proper. -/
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lemma isProper_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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IsProper Λ ↔ IsProper Λ' := by
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@ -118,7 +118,7 @@ def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
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lemma dual_id : @dual d 1 = 1 := by
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simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
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/-- The Minkowski dual swaps multiplications (acts contrvariantly). -/
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/-- The Minkowski dual swaps multiplications (acts contravariantly). -/
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@[simp]
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lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
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simp only [dual, transpose_mul]
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@ -155,7 +155,7 @@ lemma det_dual : (dual Λ).det = Λ.det := by
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norm_cast
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simp
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/-- Expansion of the components of the Minkowski dual interms of the components
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/-- Expansion of the components of the Minkowski dual in terms of the components
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of the original matrix. -/
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lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
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dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by
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@ -163,7 +163,7 @@ lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
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diagonal_mul, transpose_apply, diagonal_apply_eq]
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/-- The components of the Minkowski dual of a matrix multiplied by the Minkowski matrix
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in tems of the original matrix. -/
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in terms of the original matrix. -/
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lemma dual_apply_minkowskiMatrix (μ ν : Fin 1 ⊕ Fin d) :
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dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by
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rw [dual_apply, mul_assoc]
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@ -338,7 +338,7 @@ lemma σSAL_decomp (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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ring
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/-- The component of a self-adjoint matrix in the direction `σ0` under
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the basis formed by the covaraiant Pauli matrices. -/
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the basis formed by the covariant Pauli matrices. -/
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@[simp]
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lemma σSAL_repr_inl_0 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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σSAL.repr M (Sum.inl 0) = 1 / 2 * Matrix.trace (σ0 * M.1) := by
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@ -355,7 +355,7 @@ lemma σSAL_repr_inl_0 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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simp [σSAL]
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/-- The component of a self-adjoint matrix in the direction `-σ1` under
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the basis formed by the covaraiant Pauli matrices. -/
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the basis formed by the covariant Pauli matrices. -/
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@[simp]
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lemma σSAL_repr_inr_0 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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σSAL.repr M (Sum.inr 0) = - 1 / 2 * Matrix.trace (σ1 * M.1) := by
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@ -372,7 +372,7 @@ lemma σSAL_repr_inr_0 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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simp [σSAL]
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/-- The component of a self-adjoint matrix in the direction `-σ2` under
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the basis formed by the covaraiant Pauli matrices. -/
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the basis formed by the covariant Pauli matrices. -/
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@[simp]
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lemma σSAL_repr_inr_1 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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σSAL.repr M (Sum.inr 1) = - 1 / 2 * Matrix.trace (σ2 * M.1) := by
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@ -389,7 +389,7 @@ lemma σSAL_repr_inr_1 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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simp [σSAL]
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/-- The component of a self-adjoint matrix in the direction `-σ3` under
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the basis formed by the covaraiant Pauli matrices. -/
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the basis formed by the covariant Pauli matrices. -/
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@[simp]
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lemma σSAL_repr_inr_2 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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σSAL.repr M (Sum.inr 2) = - 1 / 2 * Matrix.trace (σ3 * M.1) := by
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@ -405,7 +405,7 @@ lemma σSAL_repr_inr_2 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
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linear_combination (norm := ring_nf) -h0
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simp only [σSAL, Basis.mk_repr, Fin.isValue, sub_self]
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/-- The relationship between the basis `σSA` of contrvariant Pauli-matrices and the basis
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/-- The relationship between the basis `σSA` of contravariant Pauli-matrices and the basis
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`σSAL` of covariant Pauli matrices is by multiplication by the Minkowski matrix. -/
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lemma σSA_minkowskiMetric_σSAL (i : Fin 1 ⊕ Fin 3) :
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σSA i = minkowskiMatrix i i • σSAL i := by
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@ -25,7 +25,7 @@ open minkowskiMatrix
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Lorentz vectors. In index notation these have an up index `ψⁱ`. -/
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def Contr (d : ℕ) : Rep ℝ (LorentzGroup d) := Rep.of ContrMod.rep
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/-- The representation of contrvariant Lorentz vectors forms a topological space, induced
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/-- The representation of contravariant Lorentz vectors forms a topological space, induced
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by its equivalence to `Fin 1 ⊕ Fin d → ℝ`. -/
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instance : TopologicalSpace (Contr d) := TopologicalSpace.induced
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ContrMod.toFin1dℝEquiv (Pi.topologicalSpace)
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@ -113,7 +113,7 @@ lemma stdBasis_apply (μ ν : Fin 1 ⊕ Fin d) : (stdBasis μ).val ν = if μ =
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refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
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exact Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a)
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/-- Decomposition of a contrvariant Lorentz vector into the standard basis. -/
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/-- Decomposition of a contravariant Lorentz vector into the standard basis. -/
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lemma stdBasis_decomp (v : ContrMod d) : v = ∑ i, v.toFin1dℝ i • stdBasis i := by
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apply toFin1dℝEquiv.injective
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simp only [map_sum, _root_.map_smul]
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@ -755,7 +755,7 @@ lemma eraseIdx_insertionSort_fin {I : Type} (le1 : I → I → Prop) [DecidableR
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= List.insertionSort le1 (r.eraseIdx n) :=
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eraseIdx_insertionSort le1 n.val r (Fin.prop n)
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/-- Given a list `i :: l` the left-most minimial position `a` of `i :: l` wrt `r`.
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/-- Given a list `i :: l` the left-most minimal position `a` of `i :: l` wrt `r`.
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That is the first position
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of `l` such that for every element `(i :: l)[b]` before that position
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`r ((i :: l)[b]) ((i :: l)[a])` is not true. The use of `i :: l` here
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@ -23,7 +23,7 @@ lemma insertionSortMin_lt_length_succ {α : Type} (r : α → α → Prop) [Deci
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rw [eraseIdx_length']
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simp
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/-- Given a list `i :: l` the left-most minimial position `a` of `i :: l` wrt `r`
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/-- Given a list `i :: l` the left-most minimal position `a` of `i :: l` wrt `r`
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as an element of `Fin (insertionSortDropMinPos r i l).length.succ`. -/
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def insertionSortMinPosFin {α : Type} (r : α → α → Prop) [DecidableRel r] (i : α) (l : List α) :
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Fin (insertionSortDropMinPos r i l).length.succ :=
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@ -25,7 +25,7 @@ def RemarkInfo.toFullName (r : RemarkInfo) : Name :=
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else
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r.name
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/-- A Bool which is true if a name correponds to a remark. -/
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/-- A Bool which is true if a name corresponds to a remark. -/
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def RemarkInfo.IsRemark (n : Name) : m Bool := do
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let allRemarks ← allRemarkInfo
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let r := allRemarks.find? fun r => r.toFullName == n
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@ -11,8 +11,8 @@ import Mathlib.Algebra.BigOperators.Group.Finset
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-/
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/-- The type `CreateAnnihilate` is the type containing two elements `create` and `annihilate`.
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This type is used to specify if an operator is a creation or annihilation operator
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or the sum thereof or intergral thereover etc. -/
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This type is used to specify if an operator is a creation, or annihilation, operator
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or the sum thereof or integral thereover etc. -/
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inductive CreateAnnihilate where
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| create : CreateAnnihilate
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| annihilate : CreateAnnihilate
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@ -48,7 +48,7 @@ We define the direct sum of the edge and vertex momentum spaces.
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Corresponding to that spanned by its momentum. -/
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def HalfEdgeMomenta : Type := F.𝓱𝓔 → ℝ
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/-- The half momenta carries the structure of an addative commutative group. -/
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/-- The half momenta carries the structure of an additive commutative group. -/
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instance : AddCommGroup F.HalfEdgeMomenta := Pi.addCommGroup
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/-- The half momenta carries the structure of a module over `ℝ`. Defined via its target. -/
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@ -83,17 +83,17 @@ def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ]
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Corresponding to that spanned by its total outflowing momentum. -/
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def EdgeMomenta : Type := F.𝓔 → ℝ
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/-- The edge momenta form an addative commuative group. -/
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/-- The edge momenta form an additive commutative group. -/
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instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
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/-- The edge momenta form a module over `ℝ`. -/
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instance : Module ℝ F.EdgeMomenta := Pi.module _ _ _
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/-- The type which associates to each ege a `1`-dimensional vector space.
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/-- The type which associates to each edge a `1`-dimensional vector space.
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Corresponding to that spanned by its total inflowing momentum. -/
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def VertexMomenta : Type := F.𝓥 → ℝ
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/-- The vertex momenta carries the structure of an addative commuative group. -/
|
||||
/-- The vertex momenta carries the structure of an additive commutative group. -/
|
||||
instance : AddCommGroup F.VertexMomenta := Pi.addCommGroup
|
||||
|
||||
/-- The vertex momenta carries the structure of a module over `ℝ`. -/
|
||||
|
|
@ -106,7 +106,7 @@ def EdgeVertexMomentaMap : Fin 2 → Type := fun i =>
|
|||
| 1 => F.VertexMomenta
|
||||
|
||||
/-- The target of the map `EdgeVertexMomentaMap` is either the type of edge momenta
|
||||
or vertex momenta and thus carries the structure of an addative commuative group. -/
|
||||
or vertex momenta and thus carries the structure of an additive commutative group. -/
|
||||
instance (i : Fin 2) : AddCommGroup (EdgeVertexMomentaMap F i) :=
|
||||
match i with
|
||||
| 0 => instAddCommGroupEdgeMomenta F
|
||||
|
|
@ -122,7 +122,7 @@ instance (i : Fin 2) : Module ℝ (EdgeVertexMomentaMap F i) :=
|
|||
/-- The direct sum of `EdgeMomenta` and `VertexMomenta`. -/
|
||||
def EdgeVertexMomenta : Type := DirectSum (Fin 2) (EdgeVertexMomentaMap F)
|
||||
|
||||
/-- The structure of a addative commutative group on `EdgeVertexMomenta` for a
|
||||
/-- The structure of a additive commutative group on `EdgeVertexMomenta` for a
|
||||
Feynman diagram `F`. -/
|
||||
instance : AddCommGroup F.EdgeVertexMomenta := DirectSum.instAddCommGroup _
|
||||
|
||||
|
|
|
|||
|
|
@ -19,8 +19,8 @@ open FieldStatistic
|
|||
|
||||
variable (𝓕 : FieldSpecification)
|
||||
|
||||
/-- The set contains the super-commutors equal to zero in the operator algebra.
|
||||
This contains e.g. the super-commutor of two creation operators. -/
|
||||
/-- The set contains the super-commutators equal to zero in the operator algebra.
|
||||
This contains e.g. the super-commutator of two creation operators. -/
|
||||
def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
|
||||
{ x |
|
||||
(∃ (φ1 φ2 φ3 : 𝓕.CrAnFieldOp),
|
||||
|
|
@ -33,15 +33,18 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
|
|||
x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
|
||||
|
||||
/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
|
||||
of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by elements of the form
|
||||
- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`, which (with the other conditions)
|
||||
gives us that super-commutors are in the center.
|
||||
- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` creation operators. I.e two creation
|
||||
operators always super-commute.
|
||||
- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` annihilation operators. I.e two
|
||||
annihilation operators always super-commute.
|
||||
- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` field operators with different statistics.
|
||||
I.e. Fermions super-commute with bosons. -/
|
||||
of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by
|
||||
- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` field creation operators.
|
||||
This corresponds to the condition that two creation operators always super-commute.
|
||||
- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` field annihilation operators.
|
||||
This corresponds to the condition that two annihilation operators always super-commute.
|
||||
- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` operators with different statistics.
|
||||
This corresponds to the condition that two operators with different statistics always
|
||||
super-commute. In other words, fermions and bosons always super-commute.
|
||||
- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`. This corresponds to the condition,
|
||||
when combined with the conditions above, that the super-commutator is in the center of the
|
||||
of the algebra.
|
||||
-/
|
||||
abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
|
||||
|
||||
namespace FieldOpAlgebra
|
||||
|
|
@ -215,7 +218,7 @@ lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnFieldOp)
|
|||
|
||||
/-!
|
||||
|
||||
## The kernal of ι
|
||||
## The kernel of ι
|
||||
-/
|
||||
|
||||
lemma ι_eq_zero_iff_mem_ideal (x : FieldOpFreeAlgebra 𝓕) :
|
||||
|
|
@ -457,13 +460,15 @@ lemma ι_eq_zero_iff_ι_bosonicProjF_fermonicProj_zero (x : FieldOpFreeAlgebra
|
|||
|
||||
-/
|
||||
|
||||
/-- For a field specification `𝓕` and an element `φ` of `𝓕.FieldOp`, the element of
|
||||
/-- For a field specification `𝓕` and an element `φ` of `𝓕.FieldOp`,
|
||||
`ofFieldOp φ` is defined as the element of
|
||||
`𝓕.FieldOpAlgebra` given by `ι (ofFieldOpF φ)`. -/
|
||||
def ofFieldOp (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpF φ)
|
||||
|
||||
lemma ofFieldOp_eq_ι_ofFieldOpF (φ : 𝓕.FieldOp) : ofFieldOp φ = ι (ofFieldOpF φ) := rfl
|
||||
|
||||
/-- For a field specification `𝓕` and a list `φs` of `𝓕.FieldOp`, the element of
|
||||
/-- For a field specification `𝓕` and a list `φs` of `𝓕.FieldOp`,
|
||||
`ofFieldOpList φs` is defined as the element of
|
||||
`𝓕.FieldOpAlgebra` given by `ι (ofFieldOpListF φ)`. -/
|
||||
def ofFieldOpList (φs : List 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpListF φs)
|
||||
|
||||
|
|
@ -487,7 +492,8 @@ lemma ofFieldOpList_singleton (φ : 𝓕.FieldOp) :
|
|||
ofFieldOpList [φ] = ofFieldOp φ := by
|
||||
simp only [ofFieldOpList, ofFieldOp, ofFieldOpListF_singleton]
|
||||
|
||||
/-- For a field specification `𝓕` and an element `φ` of `𝓕.CrAnFieldOp`, the element of
|
||||
/-- For a field specification `𝓕` and an element `φ` of `𝓕.CrAnFieldOp`,
|
||||
`ofCrAnOp φ` is defined as the element of
|
||||
`𝓕.FieldOpAlgebra` given by `ι (ofCrAnOpF φ)`. -/
|
||||
def ofCrAnOp (φ : 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnOpF φ)
|
||||
|
||||
|
|
@ -500,7 +506,8 @@ lemma ofFieldOp_eq_sum (φ : 𝓕.FieldOp) :
|
|||
simp only [map_sum]
|
||||
rfl
|
||||
|
||||
/-- For a field specification `𝓕` and a list `φs` of `𝓕.CrAnFieldOp`, the element of
|
||||
/-- For a field specification `𝓕` and a list `φs` of `𝓕.CrAnFieldOp`,
|
||||
`ofCrAnList φs` is defined as the element of
|
||||
`𝓕.FieldOpAlgebra` given by `ι (ofCrAnListF φ)`. -/
|
||||
def ofCrAnList (φs : List 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnListF φs)
|
||||
|
||||
|
|
@ -528,8 +535,10 @@ remark notation_drop := "In doc-strings we will often drop explicit applications
|
|||
|
||||
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
|
||||
annihilation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
|
||||
If `φ` is an incoming asymptotic state this is zero by definition, otherwise
|
||||
it is of the form `ofCrAnOp _`. -/
|
||||
Thus for `φ`
|
||||
- an incoming asymptotic state this is `0`.
|
||||
- a position based state this is `ofCrAnOp ⟨φ, .create⟩`.
|
||||
- an outgoing asymptotic state this is `ofCrAnOp ⟨φ, ()⟩`. -/
|
||||
def anPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (anPartF φ)
|
||||
|
||||
lemma anPart_eq_ι_anPartF (φ : 𝓕.FieldOp) : anPart φ = ι (anPartF φ) := rfl
|
||||
|
|
@ -552,8 +561,10 @@ lemma anPart_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
|
|||
|
||||
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
|
||||
creation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
|
||||
If `φ` is an outgoing asymptotic state this is zero by definition, otherwise
|
||||
it is of the form `ofCrAnOp _`. -/
|
||||
Thus for `φ`
|
||||
- an incoming asymptotic state this is `ofCrAnOp ⟨φ, ()⟩`.
|
||||
- a position based state this is `ofCrAnOp ⟨φ, .create⟩`.
|
||||
- an outgoing asymptotic state this is `0`. -/
|
||||
def crPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (crPartF φ)
|
||||
|
||||
lemma crPart_eq_ι_crPartF (φ : 𝓕.FieldOp) : crPart φ = ι (crPartF φ) := rfl
|
||||
|
|
|
|||
|
|
@ -384,9 +384,10 @@ lemma directSum_eq_bosonic_plus_fermionic
|
|||
abel
|
||||
|
||||
/-- For a field statistic `𝓕`, the algebra `𝓕.FieldOpAlgebra` is graded by `FieldStatistic`.
|
||||
Those `ofCrAnList φs` for which `φs` has an overall `bosonic` statistic span `bosonic`
|
||||
submodule, whilst those `ofCrAnList φs` for which `φs` has an overall `fermionic` statistic span
|
||||
the `fermionic` submodule. -/
|
||||
Those `ofCrAnList φs` for which `φs` has an overall `bosonic` statistic
|
||||
(i.e. `𝓕 |>ₛ φs = bosonic`) span `bosonic`
|
||||
submodule, whilst those `ofCrAnList φs` for which `φs` has an overall `fermionic` statistic
|
||||
(i.e. `𝓕 |>ₛ φs = fermionic`) span the `fermionic` submodule. -/
|
||||
instance fieldOpAlgebraGrade : GradedAlgebra (A := 𝓕.FieldOpAlgebra) statSubmodule where
|
||||
one_mem := by
|
||||
simp only [statSubmodule]
|
||||
|
|
|
|||
|
|
@ -223,10 +223,11 @@ lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b)
|
|||
|
||||
`FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
|
||||
|
||||
defined as the decent of `ι ∘ₗ normalOrderF` from `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
|
||||
defined as the decent of `ι ∘ₗ normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
|
||||
from `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
|
||||
This decent exists because `ι ∘ₗ normalOrderF` is well-defined on equivalence classes.
|
||||
|
||||
The notation `𝓝(a)` is used for `normalOrder a`. -/
|
||||
The notation `𝓝(a)` is used for `normalOrder a` for `a` an element of `FieldOpAlgebra 𝓕`. -/
|
||||
noncomputable def normalOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ normalOrderF) ι_normalOrderF_eq_of_equiv
|
||||
map_add' x y := by
|
||||
|
|
|
|||
|
|
@ -229,9 +229,10 @@ The proof of this result ultimately goes as follows
|
|||
a `ofCrAnList φsn` where `φsn` is the normal ordering of `φ₀…φₙ`.
|
||||
- `superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum` is used to rewrite the super commutator of `φ`
|
||||
(considered as a list with one element) with
|
||||
`ofCrAnList φsn` as a sum of supercommutors, one for each element of `φsn`.
|
||||
- The fact that super-commutors are in the center of `𝓕.FieldOpAlgebra` is used to rearrange terms.
|
||||
- Properties of ordered lists, and `normalOrderSign_eraseIdx` is then used to complete the proof.
|
||||
`ofCrAnList φsn` as a sum of super commutators, one for each element of `φsn`.
|
||||
- The fact that super-commutators are in the center of `𝓕.FieldOpAlgebra` is used to rearrange
|
||||
terms.
|
||||
- Properties of ordered lists, and `normalOrderSign_eraseIdx` are then used to complete the proof.
|
||||
-/
|
||||
lemma ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum (φ : 𝓕.CrAnFieldOp)
|
||||
(φs : List 𝓕.CrAnFieldOp) : [ofCrAnOp φ, 𝓝(ofCrAnList φs)]ₛ = ∑ n : Fin φs.length,
|
||||
|
|
@ -281,8 +282,8 @@ lemma ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnFieldOp
|
|||
rw [← Finset.sum_mul, ← map_sum, ← map_sum, ← ofFieldOp_eq_sum, ← ofFieldOpList_eq_sum]
|
||||
|
||||
/--
|
||||
The commutor of the annihilation part of a field operator with a normal ordered list of field
|
||||
operators can be decomponsed into the sum of the commutators of the annihilation part with each
|
||||
The commutator of the annihilation part of a field operator with a normal ordered list of field
|
||||
operators can be decomposed into the sum of the commutators of the annihilation part with each
|
||||
element of the list of field operators, i.e.
|
||||
`[anPart φ, 𝓝(φ₀…φₙ)]ₛ= ∑ i, 𝓢(φ, φ₀…φᵢ₋₁) • [anPart φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ)`.
|
||||
-/
|
||||
|
|
|
|||
|
|
@ -26,7 +26,7 @@ variable {𝓕 : FieldSpecification}
|
|||
-/
|
||||
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, and a `i ≤ φs.length` the following relation holds
|
||||
`𝓕.FieldOp`, and a `i ≤ φs.length`, then the following relation holds:
|
||||
|
||||
`𝓝([φsΛ ↩Λ φ i none]ᵘᶜ) = s • 𝓝(φ :: [φsΛ]ᵘᶜ)`
|
||||
|
||||
|
|
@ -101,7 +101,7 @@ lemma normalOrder_uncontracted_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp
|
|||
/--
|
||||
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted`, then
|
||||
`𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `FieldOp`
|
||||
`𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `𝓕.FieldOp`
|
||||
corresponding to `k` removed.
|
||||
|
||||
The proof of this result ultimately a consequence of definitions.
|
||||
|
|
|
|||
|
|
@ -31,8 +31,8 @@ noncomputable section
|
|||
def staticWickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
|
||||
φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
|
||||
|
||||
/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the empty Wick contraction
|
||||
`empty` of `[]` (its only Wick contraction) is `1`. -/
|
||||
/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the Wick contraction
|
||||
corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
|
||||
@[simp]
|
||||
lemma staticWickTerm_empty_nil :
|
||||
staticWickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
|
||||
|
|
@ -68,7 +68,7 @@ lemma staticWickTerm_insert_zero_none (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
|
|||
uncontracted fields in `φ₀…φₖ₋₁`
|
||||
- the normal ordering `𝓝([φsΛ]ᵘᶜ.erase (uncontractedFieldOpEquiv φs φsΛ k))`.
|
||||
|
||||
The proof of this result ultimitley relies on
|
||||
The proof of this result ultimately relies on
|
||||
- `staticContract_insert_some` to rewrite static contractions.
|
||||
- `normalOrder_uncontracted_some` to rewrite normal orderings.
|
||||
- `sign_insert_some_zero` to rewrite signs.
|
||||
|
|
@ -116,9 +116,9 @@ lemma staticWickTerm_insert_zero_some (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
|
|||
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, the following relation
|
||||
holds
|
||||
|
||||
`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
|
||||
`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ 0 k).staticWickTerm`
|
||||
|
||||
where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
|
||||
where the sum is over all `k` in `Option φsΛ.uncontracted`, so `k` is either `none` or `some k`.
|
||||
|
||||
The proof of proceeds as follows:
|
||||
- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
|
||||
|
|
|
|||
|
|
@ -22,7 +22,7 @@ namespace FieldOpAlgebra
|
|||
/--
|
||||
For a list `φs` of `𝓕.FieldOp`, the static version of Wick's theorem states that
|
||||
|
||||
`φs =∑ φsΛ, φsΛ.staticWickTerm`
|
||||
`φs = ∑ φsΛ, φsΛ.staticWickTerm`
|
||||
|
||||
where the sum is over all Wick contraction `φsΛ`.
|
||||
|
||||
|
|
|
|||
|
|
@ -49,7 +49,7 @@ lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.FieldOpFreeAlgebra) (h
|
|||
simp only [LinearMap.mem_ker, ← map_sub]
|
||||
exact ι_superCommuteF_right_zero_of_mem_ideal a (b1 - b2) h
|
||||
|
||||
/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
|
||||
/-- The super commutator on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
|
||||
noncomputable def superCommuteRight (a : 𝓕.FieldOpFreeAlgebra) :
|
||||
FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ superCommuteF a)
|
||||
|
|
@ -367,7 +367,7 @@ lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.FieldOp) :
|
|||
## Mul equal superCommute
|
||||
|
||||
Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
|
||||
multiplication with a sign plus the super commutor.
|
||||
multiplication with a sign plus the super commutator.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -446,7 +446,7 @@ lemma anPart_mul_anPart_swap (φ φ' : 𝓕.FieldOp) :
|
|||
|
||||
/-!
|
||||
|
||||
## Symmetry of the super commutor.
|
||||
## Symmetry of the super commutator.
|
||||
|
||||
-/
|
||||
|
||||
|
|
|
|||
|
|
@ -24,7 +24,7 @@ namespace FieldOpAlgebra
|
|||
open FieldStatistic
|
||||
|
||||
/-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, the element of
|
||||
`𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ)- 𝓝(φψ)`. -/
|
||||
`𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ) - 𝓝(φψ)`. -/
|
||||
def timeContract (φ ψ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra :=
|
||||
𝓣(ofFieldOp φ * ofFieldOp ψ) - 𝓝(ofFieldOp φ * ofFieldOp ψ)
|
||||
|
||||
|
|
|
|||
|
|
@ -371,7 +371,8 @@ lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
|
|||
|
||||
`FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
|
||||
|
||||
defined as the decent of `ι ∘ₗ timeOrderF` from `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
|
||||
defined as the decent of `ι ∘ₗ timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕` from
|
||||
`FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
|
||||
This decent exists because `ι ∘ₗ timeOrderF` is well-defined on equivalence classes.
|
||||
|
||||
The notation `𝓣(a)` is used for `timeOrder a`. -/
|
||||
|
|
|
|||
|
|
@ -78,8 +78,9 @@ lemma universalLift_ι {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFie
|
|||
For a field specification, `𝓕`, the algebra `𝓕.FieldOpAlgebra` satisfies the following universal
|
||||
property. Let `f : 𝓕.CrAnFieldOp → A` be a function and `g : 𝓕.FieldOpFreeAlgebra →ₐ[ℂ] A`
|
||||
the universal lift of that function associated with the free algebra `𝓕.FieldOpFreeAlgebra`.
|
||||
If `g` is zero on the ideal defining `𝓕.FieldOpAlgebra`, then there is a unique
|
||||
algebra map `g' : FieldOpAlgebra 𝓕 →ₐ[ℂ] A` such that `g' ∘ ι = g`.
|
||||
If `g` is zero on the ideal defining `𝓕.FieldOpAlgebra`, then there exists
|
||||
algebra map `g' : FieldOpAlgebra 𝓕 →ₐ[ℂ] A` such that `g' ∘ ι = g`, and furthermore this
|
||||
algebra map is unique.
|
||||
-/
|
||||
lemma universality {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp → A)
|
||||
(h1 : ∀ a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet, FreeAlgebra.lift ℂ f a = 0) :
|
||||
|
|
|
|||
|
|
@ -33,8 +33,8 @@ noncomputable section
|
|||
def wickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
|
||||
φsΛ.sign • φsΛ.timeContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
|
||||
|
||||
/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the empty Wick contraction
|
||||
`empty` of `[]` (its only Wick contraction) is `1`. -/
|
||||
/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the Wick contraction
|
||||
corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
|
||||
@[simp]
|
||||
lemma wickTerm_empty_nil :
|
||||
wickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
|
||||
|
|
@ -95,16 +95,16 @@ lemma wickTerm_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
|||
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, `i ≤ φs.length` and a `k` in `φsΛ.uncontracted`,
|
||||
such that all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
|
||||
such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
|
||||
`φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
|
||||
`(φsΛ ↩Λ φ i (some k)).wickTerm`
|
||||
`(φsΛ ↩Λ φ i (some k)).staticWickTerm`
|
||||
is equal the product of
|
||||
- the sign `𝓢(φ, φ₀…φᵢ₋₁) `
|
||||
- the sign `φsΛ.sign`
|
||||
- `φsΛ.timeContract`
|
||||
- `s • [anPart φ, ofFieldOp φs[k]]ₛ` where `s` is the sign associated with moving `φ` through
|
||||
uncontracted fields in `φ₀…φₖ₋₁`
|
||||
- the normal ordering `[φsΛ]ᵘᶜ` with the field corresonding to `k` removed.
|
||||
- the normal ordering `[φsΛ]ᵘᶜ` with the field corresponding to `k` removed.
|
||||
|
||||
The proof of this result relies on
|
||||
- `timeContract_insert_some_of_not_lt`
|
||||
|
|
@ -175,17 +175,18 @@ lemma wickTerm_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
|||
exact hg'
|
||||
|
||||
/--
|
||||
Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Let `φ` be a field with time
|
||||
greater then or equal to all the `FieldOp` in `φs`. Let `i ≤ φs.length` be such that
|
||||
all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ`. Then
|
||||
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, and `i ≤ φs.length`
|
||||
such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
|
||||
`φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
|
||||
|
||||
`φ * φsΛ.wickTerm = 𝓢(φ, φ₀…φᵢ₋₁) • ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
|
||||
|
||||
where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
|
||||
where the sum is over all `k` in `Option φsΛ.uncontracted`, so `k` is either `none` or `some k`.
|
||||
|
||||
The proof of proceeds as follows:
|
||||
- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
|
||||
a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`..
|
||||
a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`.
|
||||
- Then `wickTerm_insert_none` and `wickTerm_insert_some` are used to equate terms.
|
||||
-/
|
||||
lemma mul_wickTerm_eq_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (i : Fin φs.length.succ)
|
||||
|
|
|
|||
|
|
@ -47,13 +47,13 @@ lemma wicks_theorem_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') :
|
|||
rfl
|
||||
|
||||
remark wicks_theorem_context := "
|
||||
In perturbation quantum field theory, Wick's theorem allows
|
||||
In perturbative quantum field theory, Wick's theorem allows
|
||||
us to expand expectation values of time-ordered products of fields in terms of normal-orders
|
||||
and time contractions.
|
||||
The theorem is used to simplify the calculation of scattering amplitudes, and is the precurser
|
||||
to Feynman diagrams.
|
||||
|
||||
There is are actually three different versions of Wick's theorem used.
|
||||
There are actually three different versions of Wick's theorem used.
|
||||
The static version, the time-dependent version, and the normal-ordered time-dependent version.
|
||||
HepLean contains a formalization of all three of these theorems in complete generality for
|
||||
mixtures of bosonic and fermionic fields.
|
||||
|
|
@ -75,11 +75,11 @@ The inductive step works as follows:
|
|||
|
||||
For the LHS:
|
||||
1. `timeOrder_eq_maxTimeField_mul_finset` is used to write
|
||||
`𝓣(φ₀…φₙ)` as ` 𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
|
||||
`𝓣(φ₀…φₙ)` as `𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
|
||||
the maximal time field in `φ₀…φₙ`
|
||||
2. The induction hypothesis is then used on `𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` to expand it as a sum over
|
||||
Wick contractions of `φ₀…φᵢ₋₁φᵢ₊₁φₙ`.
|
||||
3. This gives terms of the form `φᵢ * φsΛ.timeContract` on which
|
||||
3. This gives terms of the form `φᵢ * φsΛ.wickTerm` on which
|
||||
`mul_wickTerm_eq_sum` is used where `φsΛ` is a Wick contraction of `φ₀…φᵢ₋₁φᵢ₊₁φ`,
|
||||
to rewrite terms as a sum over optional uncontracted elements of `φsΛ`
|
||||
|
||||
|
|
|
|||
|
|
@ -31,9 +31,10 @@ This result follows from
|
|||
- `static_wick_theorem` to rewrite `𝓣(φs)` on the left hand side as a sum of
|
||||
`𝓣(φsΛ.staticWickTerm)`.
|
||||
- `EqTimeOnly.timeOrder_staticContract_of_not_mem` and `timeOrder_timeOrder_mid` to set to
|
||||
zero those terms in which the contracted elements do not have equal time.
|
||||
- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract as a time contract
|
||||
for those terms which have equal time.
|
||||
those `𝓣(φsΛ.staticWickTerm)` for which `φsΛ` has a contracted pair which are not
|
||||
equal time to zero.
|
||||
- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract
|
||||
in the remaining `𝓣(φsΛ.staticWickTerm)` as a time contract.
|
||||
- `timeOrder_timeContract_mul_of_eqTimeOnly_left` to move the time contracts out of the time
|
||||
ordering.
|
||||
-/
|
||||
|
|
@ -111,20 +112,20 @@ For a list `φs` of `𝓕.FieldOp`, then `𝓣(φs)` is equal to the sum of
|
|||
|
||||
- `∑ φsΛ, φsΛ.wickTerm` where the sum is over all Wick contraction `φsΛ` which have
|
||||
no contractions of equal time.
|
||||
- `∑ φsΛ, sign φs ↑φsΛ • (φsΛ.1).timeContract ∑ φssucΛ, φssucΛ.wickTerm`, where
|
||||
- `∑ φsΛ, φsΛ.sign • φsΛ.timeContract * (∑ φssucΛ, φssucΛ.wickTerm)`, where
|
||||
the first sum is over all Wick contraction `φsΛ` which only have equal time contractions
|
||||
and the second sum is over all Wick contraction `φssucΛ` of the uncontracted elements of `φsΛ`
|
||||
which do not have any equal time contractions.
|
||||
|
||||
The proof of this result relies on `wicks_theorem` to rewrite `𝓣(φs)` as a sum over
|
||||
all Wick contractions.
|
||||
The sum over all Wick contractions is then split additively into two parts using based on having or
|
||||
not having equal time contractions.
|
||||
The sum over Wick contractions which do have equal time contractions is turned into two sums
|
||||
one over the Wick contractions which only have equal time contractions and the other over the
|
||||
uncontracted elements of the Wick contraction which do not have equal time contractions using
|
||||
`join`.
|
||||
The properties of `join_sign_timeContract` is then used to equate terms.
|
||||
The proof of proceeds as follows
|
||||
- `wicks_theorem` is used to rewrite `𝓣(φs)` as a sum over all Wick contractions.
|
||||
- The sum over all Wick contractions is then split additively into two parts using based on having
|
||||
or not having an equal time contractions.
|
||||
- Using `join`, the sum `∑ φsΛ, _` over Wick contractions which do have equal time contractions
|
||||
is split into two sums `∑ φsΛ, ∑ φsucΛ, _`, the first over non-zero elements
|
||||
which only have equal time contractions and the second over Wick contractions `φsucΛ` of
|
||||
`[φsΛ]ᵘᶜ` which do not have equal time contractions.
|
||||
- `join_sign_timeContract` is then used to equate terms.
|
||||
-/
|
||||
lemma timeOrder_haveEqTime_split (φs : List 𝓕.FieldOp) :
|
||||
𝓣(ofFieldOpList φs) = (∑ (φsΛ : {φsΛ : WickContraction φs.length // ¬ HaveEqTime φsΛ}),
|
||||
|
|
|
|||
|
|
@ -41,13 +41,14 @@ abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ
|
|||
namespace FieldOpFreeAlgebra
|
||||
|
||||
remark naming_convention := "
|
||||
For mathematicial objects defined in relation to `FieldOpFreeAlgebra` we will often postfix
|
||||
their names with an `F` to indicate that they are related to the free algebra.
|
||||
For mathematicial objects defined in relation to `FieldOpFreeAlgebra` the postfix `F`
|
||||
may be given to
|
||||
their names to indicate that they are related to the free algebra.
|
||||
This is to avoid confusion when working within the context of `FieldOpAlgebra` which is defined
|
||||
as a quotient of `FieldOpFreeAlgebra`."
|
||||
|
||||
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
|
||||
single `𝓕.CrAnFieldOp`. -/
|
||||
/-- For a field specification `𝓕`, and a element `φ` of `𝓕.CrAnFieldOp`,
|
||||
`ofCrAnOpF φ` is defined as the element of `𝓕.FieldOpFreeAlgebra` formed by `φ`. -/
|
||||
def ofCrAnOpF (φ : 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 :=
|
||||
FreeAlgebra.ι ℂ φ
|
||||
|
||||
|
|
@ -70,9 +71,11 @@ lemma universality {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp
|
|||
ext x
|
||||
simpa using congrFun hg x
|
||||
|
||||
/-- For a field specification `𝓕`, `ofCrAnListF φs` of `𝓕.FieldOpFreeAlgebra` formed by a
|
||||
list `φs` of `𝓕.CrAnFieldOp`. For example for the list `[φ₁ᶜ, φ₂ᵃ, φ₃ᶜ]` we schematically
|
||||
get `φ₁ᶜφ₂ᵃφ₃ᶜ`. The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
|
||||
/-- For a field specification `𝓕`, and a list `φs` of `𝓕.CrAnFieldOp`,
|
||||
`ofCrAnListF φs` is defined as the element of `𝓕.FieldOpFreeAlgebra`
|
||||
obtained by the product of `ofCrAnListF φ` for each `φ` in `φs`.
|
||||
For example `ofCrAnListF [φ₁, φ₂, φ₃] = ofCrAnOpF φ₁ * ofCrAnOpF φ₂ * ofCrAnOpF φ₃`.
|
||||
The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
|
||||
def ofCrAnListF (φs : List 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
|
||||
|
||||
@[simp]
|
||||
|
|
@ -88,22 +91,25 @@ lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnFieldOp) :
|
|||
lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
|
||||
ofCrAnListF [φ] = ofCrAnOpF φ := by simp [ofCrAnListF]
|
||||
|
||||
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
|
||||
`𝓕.FieldOp` by summing over the creation and annihilation components of `𝓕.FieldOp`.
|
||||
For example for `φ₁` an incoming asymptotic field operator we get
|
||||
`ofCrAnOpF φ₁ᶜ`, and for `φ₁` a
|
||||
position field operator we get `ofCrAnOpF φ₁ᶜ + ofCrAnOpF φ₁ᵃ`. -/
|
||||
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`,
|
||||
`ofFieldOpF φ` is the element of `𝓕.FieldOpFreeAlgebra` formed by summing over
|
||||
`ofCrAnOpF` of the
|
||||
creation and annihilation parts of `φ`.
|
||||
|
||||
For example for `φ` an incoming asymptotic field operator we get
|
||||
`ofCrAnOpF ⟨φ, ()⟩`, and for `φ` a
|
||||
position field operator we get `ofCrAnOpF ⟨φ, .create⟩ + ofCrAnOpF ⟨φ, .annihilate⟩`. -/
|
||||
def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
|
||||
∑ (i : 𝓕.fieldOpToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
|
||||
|
||||
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
|
||||
list of `𝓕.FieldOp` by summing over the creation and annihilation components.
|
||||
For example, `φ₁` and `φ₂` position states `[φ1, φ2]` gets sent to
|
||||
`(ofCrAnOpF φ1ᶜ + ofCrAnOpF φ1ᵃ) * (ofCrAnOpF φ2ᶜ + ofCrAnOpF φ2ᵃ)`. -/
|
||||
/-- For a field specification `𝓕`, and a list `φs` of `𝓕.FieldOp`,
|
||||
`𝓕.ofFieldOpListF φs` is defined as the element of `𝓕.FieldOpFreeAlgebra`
|
||||
obtained by the product of `ofFieldOpF φ` for each `φ` in `φs`.
|
||||
For example `ofFieldOpListF [φ₁, φ₂, φ₃] = ofFieldOpF φ₁ * ofFieldOpF φ₂ * ofFieldOpF φ₃`. -/
|
||||
def ofFieldOpListF (φs : List 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
|
||||
|
||||
remark notation_drop := "In doc-strings we will often drop explicit applications of `ofCrAnOpF`,
|
||||
`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF`"
|
||||
remark notation_drop := "In doc-strings explicit applications of `ofCrAnOpF`,
|
||||
`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF` will often be dropped."
|
||||
|
||||
/-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
|
||||
instance : Coe (List 𝓕.FieldOp) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
|
||||
|
|
|
|||
|
|
@ -237,8 +237,10 @@ lemma directSum_eq_bosonic_plus_fermionic
|
|||
abel
|
||||
|
||||
/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpFreeAlgebra` is graded by `FieldStatistic`.
|
||||
Those `ofCrAnListF φs` for which `φs` has an overall `bosonic` statistic span `bosonic`
|
||||
submodule, whilst those `ofCrAnListF φs` for which `φs` has an overall `fermionic` statistic span
|
||||
Those `ofCrAnListF φs` for which `φs` has an overall `bosonic` statistic
|
||||
(i.e. `𝓕 |>ₛ φs = bosonic`) span `bosonic`
|
||||
submodule, whilst those `ofCrAnListF φs` for which `φs` has an overall `fermionic` statistic
|
||||
(i.e. `𝓕 |>ₛ φs = fermionic`) span
|
||||
the `fermionic` submodule. -/
|
||||
instance fieldOpFreeAlgebraGrade :
|
||||
GradedAlgebra (A := 𝓕.FieldOpFreeAlgebra) statisticSubmodule where
|
||||
|
|
|
|||
|
|
@ -31,8 +31,11 @@ noncomputable section
|
|||
`FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕`
|
||||
defined by its action on the basis `ofCrAnListF φs`, taking `ofCrAnListF φs` to
|
||||
`normalOrderSign φs • ofCrAnListF (normalOrderList φs)`.
|
||||
That is, `normalOrderF` normal-orders the field operators and multiplies by the sign of the
|
||||
normal order.
|
||||
|
||||
The notation `𝓝ᶠ(a)` is used for `normalOrderF a`. -/
|
||||
The notation `𝓝ᶠ(a)` is used for `normalOrderF a` for `a` an element of
|
||||
`FieldOpFreeAlgebra 𝓕`. -/
|
||||
def normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
|
||||
Basis.constr ofCrAnListFBasis ℂ fun φs =>
|
||||
normalOrderSign φs • ofCrAnListF (normalOrderList φs)
|
||||
|
|
|
|||
|
|
@ -17,7 +17,7 @@ namespace FieldOpFreeAlgebra
|
|||
|
||||
/-!
|
||||
|
||||
## The super commutor on the FieldOpFreeAlgebra.
|
||||
## The super commutator on the FieldOpFreeAlgebra.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -40,7 +40,7 @@ scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca"
|
|||
|
||||
/-!
|
||||
|
||||
## The super commutor of different types of elements
|
||||
## The super commutator of different types of elements
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -257,7 +257,7 @@ lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
|
|||
## Mul equal superCommuteF
|
||||
|
||||
Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
|
||||
multiplication with a sign plus the super commutor.
|
||||
multiplication with a sign plus the super commutator.
|
||||
|
||||
-/
|
||||
lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
|
|
@ -328,7 +328,7 @@ lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnField
|
|||
|
||||
/-!
|
||||
|
||||
## Symmetry of the super commutor.
|
||||
## Symmetry of the super commutator.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -359,7 +359,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
|
|||
|
||||
/-!
|
||||
|
||||
## Splitting the super commutor on lists into sums.
|
||||
## Splitting the super commutator on lists into sums.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -405,7 +405,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : L
|
|||
simp [mul_comm]
|
||||
|
||||
/--
|
||||
For a field specification `𝓕`, and to lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
|
||||
For a field specification `𝓕`, and two lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
|
||||
the following super commutation relation holds:
|
||||
|
||||
`[φs', φ₀…φₙ]ₛca = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛca * φᵢ₊₁ … φₙ`
|
||||
|
|
|
|||
|
|
@ -30,6 +30,8 @@ open HepLean.List
|
|||
`FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕`
|
||||
defined by its action on the basis `ofCrAnListF φs`, taking
|
||||
`ofCrAnListF φs` to `crAnTimeOrderSign φs • ofCrAnListF (crAnTimeOrderList φs)`.
|
||||
That is, `timeOrderF` time-orders the field operators and multiplies by the sign of the
|
||||
time order.
|
||||
|
||||
The notation `𝓣ᶠ(a)` is used for `timeOrderF a` -/
|
||||
def timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
|
||||
|
|
|
|||
|
|
@ -42,13 +42,15 @@ The structure `FieldSpecification` is defined to have the following content:
|
|||
- For `f` a *Weyl fermion*, `PositionLabel f` will have four elements, one for each Lorentz
|
||||
index of the field and its conjugate.
|
||||
- For every field `f` in `Field`, a type `AsymptoticLabel f` whose elements label the different
|
||||
asymptotic based field operators associated with the field `f`. For example,
|
||||
types of incoming asymptotic field operators associated with the
|
||||
field `f` (this is also matches the types of outgoing asymptotic field operators).
|
||||
For example,
|
||||
- For `f` a *real-scalar field*, `AsymptoticLabel f` will have a unique element.
|
||||
- For `f` a *complex-scalar field*, `AsymptoticLabel f` will have two elements, one for the
|
||||
field operator and one for its conjugate.
|
||||
- For `f` a *Dirac fermion*, `AsymptoticLabel f` will have four elements, two for each spin.
|
||||
- For `f` a *Weyl fermion*, `AsymptoticLabel f` will have two elements, one for each spin.
|
||||
- For each field `f` in `Field`, a field statistic `statistic f` which classifying `f` as either
|
||||
- For each field `f` in `Field`, a field statistic `statistic f` which classifies `f` as either
|
||||
`bosonic` or `fermionic`.
|
||||
-/
|
||||
structure FieldSpecification where
|
||||
|
|
@ -67,20 +69,30 @@ structure FieldSpecification where
|
|||
namespace FieldSpecification
|
||||
variable (𝓕 : FieldSpecification)
|
||||
|
||||
/-- For a field specification `𝓕`, the type `𝓕.FieldOp` is defined such that every element of
|
||||
`FieldOp` corresponds either to:
|
||||
- an incoming asymptotic field operator `.inAsymp` which is specified by
|
||||
a field `f` in `𝓕.Field`, an element of `AsymptoticLabel f` (which specifies exactly
|
||||
which asymptotic field operator associated with `f`) and a `3`-momentum.
|
||||
- an position operator `.position` which is specified by
|
||||
a field `f` in `𝓕.Field`, an element of `PositionLabel f` (which specifies exactly
|
||||
which position field operator associated with `f`) and a element of `SpaceTime`.
|
||||
- an outgoing asymptotic field operator `.outAsymp` which is specified by
|
||||
a field `f` in `𝓕.Field`, an element of `AsymptoticLabel f` (which specifies exactly
|
||||
which asymptotic field operator associated with `f`) and a `3`-momentum.
|
||||
/-- For a field specification `𝓕`, the inductive type `𝓕.FieldOp` is defined
|
||||
to contain the following elements:
|
||||
- for every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
|
||||
element labelled `inAsymp f e p` corresponding to an incoming asymptotic field operator
|
||||
of the field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
|
||||
- for every `f` in `𝓕.Field`, element of `e` of `PositionLabel f` and space-time position `x`, an
|
||||
element labelled `position f e x` corresponding to a position field operator of the field `f`,
|
||||
of label `e` (e.g. specifying the Lorentz index), and position `x`.
|
||||
- for every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
|
||||
element labelled `outAsymp f e p` corresponding to an outgoing asymptotic field operator of the
|
||||
field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
|
||||
|
||||
Note the use of `3`-momentum here rather then `4`-momentum. This is because the asymptotic states
|
||||
have on-shell momenta.
|
||||
As some intuition, if `f` corresponds to a Weyl-fermion field, then
|
||||
- For `inAsymp f e p`, `e` would correspond to a spin `s`, and `inAsymp f e p` would, once
|
||||
represented in the operator algebra,
|
||||
be proportional to the creation operator `a(p, s)`.
|
||||
- `position f e x`, `e` would correspond to a Lorentz index `α`, and `position f e x` would,
|
||||
once represented in the operator algebra, be proportional to the operator
|
||||
`∑ s, ∫ d^3p/(…) (x_α(p,s) a(p, s) e^{-i p x} + y_α(p,s) a^†(p, s) e^{-i p x})`.
|
||||
- `outAsymp f e p`, `e` would correspond to a spin `s`, and `outAsymp f e p` would,
|
||||
once represented in the operator algebra, be proportional to the
|
||||
annihilation operator `a^†(p, s)`.
|
||||
|
||||
This type contains all operators which are related to a field.
|
||||
-/
|
||||
inductive FieldOp (𝓕 : FieldSpecification) where
|
||||
| inAsymp : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ) → 𝓕.FieldOp
|
||||
|
|
|
|||
|
|
@ -63,15 +63,31 @@ def fieldOpToCreateAnnihilateTypeCongr : {i j : 𝓕.FieldOp} → i = j →
|
|||
| _, _, rfl => Equiv.refl _
|
||||
|
||||
/--
|
||||
For a field specification `𝓕`, elements in `𝓕.CrAnFieldOp`, the type
|
||||
of creation and annihilation field operators, corresponds to
|
||||
- an incoming asymptotic field operator `.inAsymp` in `𝓕.FieldOp`.
|
||||
- a position operator `.position` in `𝓕.FieldOp` and an element of
|
||||
`CreateAnnihilate` specifying the creation or annihilation part of that position operator.
|
||||
- an outgoing asymptotic field operator `.outAsymp` in `𝓕.FieldOp`.
|
||||
For a field specification `𝓕`, the (sigma) type `𝓕.CrAnFieldOp`
|
||||
corresponds to the type of creation and annihilation parts of field operators.
|
||||
It formally defined to consist of the following elements:
|
||||
- for each in incoming asymptotic field operator `φ` in `𝓕.FieldOp` an element
|
||||
written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the creation part of `φ`.
|
||||
Here `φ` has no annihilation part. (Here `()` is the unique element of `Unit`.)
|
||||
- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
|
||||
written as `⟨φ, .create⟩`, corresponding to the creation part of `φ`.
|
||||
- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
|
||||
written as `⟨φ, .annihilate⟩`, corresponding to the annihilation part of `φ`.
|
||||
- for each out outgoing asymptotic field operator `φ` in `𝓕.FieldOp` an element
|
||||
written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the annihilation part of `φ`.
|
||||
Here `φ` has no creation part. (Here `()` is the unique element of `Unit`.)
|
||||
|
||||
As some intuition, if `f` corresponds to a Weyl-fermion field, it would contribute
|
||||
the following elements to `𝓕.CrAnFieldOp`
|
||||
- an element corresponding to incoming asymptotic operators for each spin `s`: `a(p, s)`.
|
||||
- an element corresponding to the creation parts of position operators for each each Lorentz
|
||||
index `α`:
|
||||
`∑ s, ∫ d^3p/(…) (x_α(p,s) a(p, s) e^{-i p x})`.
|
||||
- an element corresponding to annihilation parts of position operator,
|
||||
for each each Lorentz index `α`:
|
||||
`∑ s, ∫ d^3p/(…) (y_α(p,s) a^†(p, s) e^{-i p x})`.
|
||||
- an element corresponding to outgoing asymptotic operators for each spin `s`: `a^†(p, s)`.
|
||||
|
||||
Note that the incoming and outgoing asymptotic field operators are implicitly creation and
|
||||
annihilation operators respectively.
|
||||
-/
|
||||
def CrAnFieldOp : Type := Σ (s : 𝓕.FieldOp), 𝓕.fieldOpToCrAnType s
|
||||
|
||||
|
|
@ -82,8 +98,13 @@ def crAnFieldOpToFieldOp : 𝓕.CrAnFieldOp → 𝓕.FieldOp := Sigma.fst
|
|||
lemma crAnFieldOpToFieldOp_prod (s : 𝓕.FieldOp) (t : 𝓕.fieldOpToCrAnType s) :
|
||||
𝓕.crAnFieldOpToFieldOp ⟨s, t⟩ = s := rfl
|
||||
|
||||
/-- The map from creation and annihilation states to the type `CreateAnnihilate`
|
||||
specifying if a state is a creation or an annihilation state. -/
|
||||
/-- For a field specification `𝓕`, `𝓕.crAnFieldOpToCreateAnnihilate` is the map from
|
||||
`𝓕.CrAnFieldOp` to `CreateAnnihilate` taking `φ` to `create` if
|
||||
- `φ` corresponds to an incoming asymptotic field operator or the creation part of a position based
|
||||
field operator.
|
||||
|
||||
otherwise it takes `φ` to `annihilate`.
|
||||
-/
|
||||
def crAnFieldOpToCreateAnnihilate : 𝓕.CrAnFieldOp → CreateAnnihilate
|
||||
| ⟨FieldOp.inAsymp _, _⟩ => CreateAnnihilate.create
|
||||
| ⟨FieldOp.position _, CreateAnnihilate.create⟩ => CreateAnnihilate.create
|
||||
|
|
@ -92,7 +113,7 @@ def crAnFieldOpToCreateAnnihilate : 𝓕.CrAnFieldOp → CreateAnnihilate
|
|||
|
||||
/-- For a field specification `𝓕`, and an element `φ` in `𝓕.CrAnFieldOp`, the field
|
||||
statistic `crAnStatistics φ` is defined to be the statistic associated with the field `𝓕.Field`
|
||||
(or `𝓕.FieldOp`) underlying `φ`.
|
||||
(or the `𝓕.FieldOp`) underlying `φ`.
|
||||
|
||||
The following notation is used in relation to `crAnStatistics`:
|
||||
- For `φ` an element of `𝓕.CrAnFieldOp`, `𝓕 |>ₛ φ` is `crAnStatistics φ`.
|
||||
|
|
@ -115,18 +136,18 @@ scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => FieldStatistic.ofList
|
|||
scoped[FieldSpecification] infixl:80 "|>ᶜ" =>
|
||||
crAnFieldOpToCreateAnnihilate
|
||||
|
||||
remark notation_remark := "When working with a field specification `𝓕` we will use
|
||||
some notation within doc-strings and in code. The main notation used is:
|
||||
- In doc-strings when field statistics occur in exchange signs we may drop the `𝓕 |>ₛ _`.
|
||||
- In doc-strings we will often write lists of `FieldOp` or `CrAnFieldOp` `φs` as e.g. `φ₀…φₙ`,
|
||||
remark notation_remark := "When working with a field specification `𝓕` the
|
||||
following notation will be used within doc-strings:
|
||||
- when field statistics occur in exchange signs the `𝓕 |>ₛ _` may be dropped.
|
||||
- lists of `FieldOp` or `CrAnFieldOp` `φs` may be written as `φ₀…φₙ`,
|
||||
which should be interpreted within the context in which it appears.
|
||||
- In doc-strings we may use e.g. `φᶜ` to indicate the creation part of an operator and
|
||||
- `φᶜ` may be used to indicate the creation part of an operator and
|
||||
`φᵃ` to indicate the annihilation part of an operator.
|
||||
|
||||
Some examples:
|
||||
- `𝓢(φ, φs)` corresponds to `𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs)`
|
||||
- `φ₀…φᵢ₋₁φᵢ₊₁…φₙ` corresponds to a (given) list `φs = φ₀…φₙ` with the element at the `i`th position
|
||||
removed.
|
||||
Some examples of these notation-conventions are:
|
||||
- `𝓢(φ, φs)` which corresponds to `𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs)`
|
||||
- `φ₀…φᵢ₋₁φᵢ₊₁…φₙ` which corresponds to a (given) list `φs = φ₀…φₙ` with the element at the
|
||||
`i`th position removed.
|
||||
"
|
||||
|
||||
end FieldSpecification
|
||||
|
|
|
|||
|
|
@ -126,7 +126,7 @@ def singletonEquiv {φ : 𝓕.FieldOp} : CrAnSection [φ] ≃
|
|||
simp only [head]
|
||||
rfl
|
||||
|
||||
/-- An equivalence seperating the head of a creation and annihilation section
|
||||
/-- An equivalence separating the head of a creation and annihilation section
|
||||
from the tail. -/
|
||||
def consEquiv {φ : 𝓕.FieldOp} {φs : List 𝓕.FieldOp} : CrAnSection (φ :: φs) ≃
|
||||
𝓕.fieldOpToCrAnType φ × CrAnSection φs where
|
||||
|
|
|
|||
|
|
@ -17,8 +17,8 @@ variable {𝓕 : FieldSpecification}
|
|||
/-- For a field specification `𝓕`, `𝓕.normalOrderRel` is a relation on `𝓕.CrAnFieldOp`
|
||||
representing normal ordering. It is defined such that `𝓕.normalOrderRel φ₀ φ₁`
|
||||
is true if one of the following is true
|
||||
- `φ₀` is a creation operator
|
||||
- `φ₁` is an annihilation.
|
||||
- `φ₀` is a field creation operator
|
||||
- `φ₁` is a field annihilation operator.
|
||||
|
||||
Thus, colloquially `𝓕.normalOrderRel φ₀ φ₁` says the creation operators are 'less then'
|
||||
annihilation operators. -/
|
||||
|
|
@ -344,11 +344,12 @@ lemma normalOrderList_eraseIdx_normalOrderEquiv {φs : List 𝓕.CrAnFieldOp} (n
|
|||
rw [HepLean.List.eraseIdx_insertionSort_fin]
|
||||
|
||||
/-- For a field specification `𝓕`, a list `φs = φ₀…φₙ` of `𝓕.CrAnFieldOp` and an `i < φs.length`,
|
||||
the following relation holds
|
||||
then
|
||||
`normalOrderSign (φ₀…φᵢ₋₁φᵢ₊₁…φₙ)` is equal to the product of
|
||||
- `normalOrderSign φ₀…φₙ`,
|
||||
- `𝓢(φᵢ, φ₀…φᵢ₋₁)` i.e. the sign needed to remove `φᵢ` from `φ₀…φₙ`,
|
||||
- `𝓢(φᵢ, _)` where `_` is the list of elements appearing before `φᵢ` after normal ordering. I.e.
|
||||
- `𝓢(φᵢ, _)` where `_` is the list of elements appearing before `φᵢ` after normal ordering,
|
||||
i.e.
|
||||
the sign needed to insert `φᵢ` back into the normal-ordered list at the correct place. -/
|
||||
lemma normalOrderSign_eraseIdx (φs : List 𝓕.CrAnFieldOp) (i : Fin φs.length) :
|
||||
normalOrderSign (φs.eraseIdx i) = normalOrderSign φs *
|
||||
|
|
|
|||
|
|
@ -32,7 +32,7 @@ def timeOrderRel : 𝓕.FieldOp → 𝓕.FieldOp → Prop
|
|||
| FieldOp.inAsymp _, FieldOp.position _ => False
|
||||
| FieldOp.inAsymp _, FieldOp.inAsymp _ => True
|
||||
|
||||
/-- The relation `timeOrderRel` is decidable, but not computablly so due to
|
||||
/-- The relation `timeOrderRel` is decidable, but not computable so due to
|
||||
`Real.decidableLE`. -/
|
||||
noncomputable instance : (φ φ' : 𝓕.FieldOp) → Decidable (timeOrderRel φ φ')
|
||||
| FieldOp.outAsymp _, _ => isTrue True.intro
|
||||
|
|
@ -193,7 +193,7 @@ lemma timeOrderList_eq_maxTimeField_timeOrderList (φ : 𝓕.FieldOp) (φs : Lis
|
|||
|
||||
/-- For a field specification `𝓕`, `𝓕.crAnTimeOrderRel` is a relation on
|
||||
`𝓕.CrAnFieldOp` representing time ordering.
|
||||
It is defined as such that `𝓕.crAnTimeOrderRel φ₀ φ₁` is true if and only if one of the following
|
||||
It is defined such that `𝓕.crAnTimeOrderRel φ₀ φ₁` is true if and only if one of the following
|
||||
holds
|
||||
- `φ₀` is an *outgoing* asymptotic operator
|
||||
- `φ₁` is an *incoming* asymptotic field operator
|
||||
|
|
@ -201,10 +201,12 @@ lemma timeOrderList_eq_maxTimeField_timeOrderList (φ : 𝓕.FieldOp) (φs : Lis
|
|||
the `SpaceTime` point of `φ₀` has a time *greater* then or equal to that of `φ₁`.
|
||||
|
||||
Thus, colloquially `𝓕.crAnTimeOrderRel φ₀ φ₁` if `φ₀` has time *greater* then or equal to `φ₁`.
|
||||
The use of *greater* then rather then *less* then is because on ordering lists of operators
|
||||
it is needed that the operator with the greatest time is to the left.
|
||||
-/
|
||||
def crAnTimeOrderRel (a b : 𝓕.CrAnFieldOp) : Prop := 𝓕.timeOrderRel a.1 b.1
|
||||
|
||||
/-- The relation `crAnTimeOrderRel` is decidable, but not computablly so due to
|
||||
/-- The relation `crAnTimeOrderRel` is decidable, but not computable so due to
|
||||
`Real.decidableLE`. -/
|
||||
noncomputable instance (φ φ' : 𝓕.CrAnFieldOp) : Decidable (crAnTimeOrderRel φ φ') :=
|
||||
inferInstanceAs (Decidable (𝓕.timeOrderRel φ.1 φ'.1))
|
||||
|
|
@ -506,7 +508,7 @@ lemma sum_crAnSections_timeOrder {φs : List 𝓕.FieldOp} [AddCommMonoid M]
|
|||
def normTimeOrderRel (a b : 𝓕.CrAnFieldOp) : Prop :=
|
||||
crAnTimeOrderRel a b ∧ (crAnTimeOrderRel b a → normalOrderRel a b)
|
||||
|
||||
/-- The relation `normTimeOrderRel` is decidable, but not computablly so due to
|
||||
/-- The relation `normTimeOrderRel` is decidable, but not computable so due to
|
||||
`Real.decidableLE`. -/
|
||||
noncomputable instance (φ φ' : 𝓕.CrAnFieldOp) : Decidable (normTimeOrderRel φ φ') :=
|
||||
instDecidableAnd
|
||||
|
|
|
|||
|
|
@ -26,7 +26,7 @@ namespace FieldStatistic
|
|||
|
||||
variable {𝓕 : Type}
|
||||
|
||||
/-- The type `FieldStatistic` carries an instance of a commuative group in which
|
||||
/-- The type `FieldStatistic` carries an instance of a commutative group in which
|
||||
- `bosonic * bosonic = bosonic`
|
||||
- `bosonic * fermionic = fermionic`
|
||||
- `fermionic * bosonic = fermionic`
|
||||
|
|
|
|||
|
|
@ -27,8 +27,8 @@ variable {𝓕 : Type}
|
|||
/-- The exchange sign, `exchangeSign`, is defined as the group homomorphism
|
||||
`FieldStatistic →* FieldStatistic →* ℂ`,
|
||||
for which `exchangeSign a b` is `-1` if both `a` and `b` are `fermionic` and `1` otherwise.
|
||||
The exchange sign is sign one picks up on exchanging an operator or field `φ₁` of statistic `a`
|
||||
with one `φ₂` of statistic `b`, i.e. `φ₁φ₂ → φ₂φ₁`.
|
||||
The exchange sign is the sign one picks up on exchanging an operator or field `φ₁` of statistic
|
||||
`a` with an operator or field `φ₂` of statistic `b`, i.e. `φ₁φ₂ → φ₂φ₁`.
|
||||
|
||||
The notation `𝓢(a, b)` is used for the exchange sign of `a` and `b`. -/
|
||||
def exchangeSign : FieldStatistic →* FieldStatistic →* ℂ where
|
||||
|
|
|
|||
|
|
@ -19,20 +19,6 @@ contracting, a Wick contraction
|
|||
is a finite set of pairs of `Fin n` (numbers `0`, …, `n-1`), such that no
|
||||
element of `Fin n` occurs in more then one pair. The pairs are the positions of fields we
|
||||
'contract' together.
|
||||
|
||||
For example for `n = 3` there are `4` Wick contractions:
|
||||
- `∅`, corresponding to the case where no fields are contracted.
|
||||
- `{{0, 1}}`, corresponding to the case where the field at position `0` and `1` are contracted.
|
||||
- `{{0, 2}}`, corresponding to the case where the field at position `0` and `2` are contracted.
|
||||
- `{{1, 2}}`, corresponding to the case where the field at position `1` and `2` are contracted.
|
||||
|
||||
For `n=4` some possible Wick contractions are
|
||||
- `∅`, corresponding to the case where no fields are contracted.
|
||||
- `{{0, 1}, {2, 3}}`, corresponding to the case where the field at position `0` and `1` are
|
||||
contracted and the field at position `2` and `3` are contracted.
|
||||
- `{{0, 2}, {1, 3}}`, corresponding to the case where the field at position `0` and `2` are
|
||||
contracted and the field at position `1` and `3` are contracted.
|
||||
etc.
|
||||
-/
|
||||
def WickContraction (n : ℕ) : Type :=
|
||||
{f : Finset ((Finset (Fin n))) // (∀ a ∈ f, a.card = 2) ∧
|
||||
|
|
@ -534,8 +520,8 @@ lemma prod_finset_eq_mul_fst_snd (c : WickContraction n) (a : c.1)
|
|||
/-- For a field specification `𝓕`, `φs` a list of `𝓕.FieldOp` and a Wick contraction
|
||||
`φsΛ` of `φs`, the Wick contraction `φsΛ` is said to be `GradingCompliant` if
|
||||
for every pair in `φsΛ` the contracted fields are either both `fermionic` or both `bosonic`.
|
||||
I.e. in a `GradingCompliant` Wick contraction no contractions occur between `fermionic` and
|
||||
`bosonic` fields. -/
|
||||
In other words, in a `GradingCompliant` Wick contraction no contractions occur between
|
||||
`fermionic` and `bosonic` fields. -/
|
||||
def GradingCompliant (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :=
|
||||
∀ (a : φsΛ.1), (𝓕 |>ₛ φs[φsΛ.fstFieldOfContract a]) = (𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a])
|
||||
|
||||
|
|
|
|||
|
|
@ -107,4 +107,52 @@ lemma sum_extractEquiv_congr [AddCommMonoid M] {n m : ℕ} (i : Fin n) (f : Wick
|
|||
rw [Finset.sum_sigma']
|
||||
rfl
|
||||
|
||||
/-- For `n = 3` there are `4` possible Wick contractions:
|
||||
|
||||
- `∅`, corresponding to the case where no fields are contracted.
|
||||
- `{{0, 1}}`, corresponding to the case where the field at position `0` and `1` are contracted.
|
||||
- `{{0, 2}}`, corresponding to the case where the field at position `0` and `2` are contracted.
|
||||
- `{{1, 2}}`, corresponding to the case where the field at position `1` and `2` are contracted.
|
||||
|
||||
The proof of this result uses the fact that Lean is an executable programming language
|
||||
and can calculate all Wick contractions for a given `n`. -/
|
||||
lemma mem_three (c : WickContraction 3) : c.1 ∈ ({∅, {{0, 1}}, {{0, 2}}, {{1, 2}}} :
|
||||
Finset (Finset (Finset (Fin 3)))) := by
|
||||
fin_cases c <;>
|
||||
simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Function.Embedding.coeFn_mk,
|
||||
Finset.mem_insert, Finset.mem_singleton]
|
||||
· exact Or.inl rfl
|
||||
· exact Or.inr (Or.inl rfl)
|
||||
· exact Or.inr (Or.inr (Or.inl rfl))
|
||||
· exact Or.inr (Or.inr (Or.inr rfl))
|
||||
|
||||
/-- For `n = 4` there are `10` possible Wick contractions including e.g.
|
||||
|
||||
- `∅`, corresponding to the case where no fields are contracted.
|
||||
- `{{0, 1}, {2, 3}}`, corresponding to the case where the fields at position `0` and `1` are
|
||||
contracted, and the fields at position `2` and `3` are contracted.
|
||||
- `{{0, 2}, {1, 3}}`, corresponding to the case where the fields at position `0` and `2` are
|
||||
contracted, and the fields at position `1` and `3` are contracted.
|
||||
|
||||
The proof of this result uses the fact that Lean is an executable programming language
|
||||
and can calculate all Wick contractions for a given `n`.
|
||||
-/
|
||||
lemma mem_four (c : WickContraction 4) : c.1 ∈ ({∅,
|
||||
{{0, 1}}, {{0, 2}}, {{0, 3}}, {{1, 2}}, {{1, 3}}, {{2,3}},
|
||||
{{0, 1}, {2, 3}}, {{0, 2}, {1, 3}}, {{0, 3}, {1, 2}}} :
|
||||
Finset (Finset (Finset (Fin 4)))) := by
|
||||
fin_cases c <;>
|
||||
simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Function.Embedding.coeFn_mk,
|
||||
Finset.mem_insert, Finset.mem_singleton]
|
||||
· exact Or.inl rfl -- ∅
|
||||
· exact Or.inr (Or.inl rfl) -- {{0, 1}}
|
||||
· exact Or.inr (Or.inr (Or.inl rfl)) -- {{0, 2}}
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inl rfl))) -- {{0, 3}}
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl)))) -- {{1, 2}}
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr rfl))))))))
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl))))) -- {{1, 3}}
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl))))))))
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl)))))) -- {{2, 3 }}
|
||||
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl)))))))
|
||||
|
||||
end WickContraction
|
||||
|
|
|
|||
|
|
@ -25,7 +25,7 @@ open HepLean.Fin
|
|||
-/
|
||||
|
||||
/-- Given a Wick contraction `φsΛ` for a list `φs` of `𝓕.FieldOp`,
|
||||
a `𝓕.FieldOp` `φ`, an `i ≤ φs.length` and a `j` which is either `none` or
|
||||
an element `φ` of `𝓕.FieldOp`, an `i ≤ φs.length` and a `j` which is either `none` or
|
||||
some element of `φsΛ.uncontracted`, the new Wick contraction
|
||||
`φsΛ.insertAndContract φ i j` is defined by inserting `φ` into `φs` after
|
||||
the first `i`-elements and moving the values representing the contracted pairs in `φsΛ`
|
||||
|
|
@ -35,6 +35,8 @@ open HepLean.Fin
|
|||
|
||||
In other words, `φsΛ.insertAndContract φ i j` is formed by adding `φ` to `φs` at position `i`,
|
||||
and contracting `φ` with the field originally at position `j` if `j` is not none.
|
||||
It is a Wick contraction of `φs.insertIdx φ i`, the list `φs` with `φ` inserted at
|
||||
position `i`.
|
||||
|
||||
The notation `φsΛ ↩Λ φ i j` is used to denote `φsΛ.insertAndContract φ i j`. -/
|
||||
def insertAndContract {φs : List 𝓕.FieldOp} (φ : 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
|
||||
|
|
@ -285,11 +287,11 @@ lemma insert_fin_eq_self (φ : 𝓕.FieldOp) {φs : List 𝓕.FieldOp}
|
|||
rfl
|
||||
|
||||
/-- For a list `φs` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, a `i ≤ φs.length` a sum over
|
||||
`𝓕.FieldOp` and a `i ≤ φs.length` then a sum over
|
||||
Wick contractions of `φs` with `φ` inserted at `i` is equal to the sum over Wick contractions
|
||||
`φsΛ` of just `φs` and the sum over optional uncontracted elements of the `φsΛ`.
|
||||
|
||||
I.e. `∑ (φsΛ : WickContraction (φs.insertIdx i φ).length), f φsΛ` is equal to
|
||||
In other words, `∑ (φsΛ : WickContraction (φs.insertIdx i φ).length), f φsΛ` is equal to
|
||||
`∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) `.
|
||||
where `(φs.insertIdx i φ)` is `φs` with `φ` inserted at position `i`. -/
|
||||
lemma insertLift_sum (φ : 𝓕.FieldOp) {φs : List 𝓕.FieldOp}
|
||||
|
|
|
|||
|
|
@ -25,7 +25,13 @@ open FieldOpAlgebra
|
|||
|
||||
/-- Given a list `φs` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs` and a Wick contraction
|
||||
`φsucΛ` of `[φsΛ]ᵘᶜ`, `join φsΛ φsucΛ` is defined as the Wick contraction of `φs` consisting of
|
||||
the contractions in `φsΛ` and those in `φsucΛ`. -/
|
||||
the contractions in `φsΛ` and those in `φsucΛ`.
|
||||
|
||||
As an example, for `φs = [φ1, φ2, φ3, φ4]`,
|
||||
`φsΛ = {{0, 1}}` corresponding to the contraction of `φ1` and `φ2` in `φs` and
|
||||
`φsucΛ = {{0, 1}}`
|
||||
corresponding to the contraction of `φ3` and `φ4` in `[φsΛ]ᵘᶜ = [φ3, φ4]`, then
|
||||
`join φsΛ φsucΛ` is the contraction `{{0, 1}, {2, 3}}` of `φs`. -/
|
||||
def join {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) : WickContraction φs.length :=
|
||||
⟨φsΛ.1 ∪ φsucΛ.1.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding, by
|
||||
|
|
|
|||
|
|
@ -21,7 +21,7 @@ open FieldStatistic
|
|||
|
||||
/-- Given a Wick contraction `c : WickContraction n` and `i1 i2 : Fin n` the finite set
|
||||
of elements of `Fin n` between `i1` and `i2` which are either uncontracted
|
||||
or are contracted but are contracted with an element occuring after `i1`.
|
||||
or are contracted but are contracted with an element occurring after `i1`.
|
||||
I.e. the elements of `Fin n` between `i1` and `i2` which are not contracted with before `i1`.
|
||||
One should assume `i1 < i2` otherwise this finite set is empty. -/
|
||||
def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
|
||||
|
|
@ -30,8 +30,9 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
|
|||
|
||||
/-- For a list `φs` of `𝓕.FieldOp`, and a Wick contraction `φsΛ` of `φs`,
|
||||
the complex number `φsΛ.sign` is defined to be the sign (`1` or `-1`) corresponding
|
||||
to the number of `fermionic`-`fermionic` exchanges that must done to put
|
||||
contracted pairs with `φsΛ` next to one another, starting from the contracted pair
|
||||
to the number of `fermionic`-`fermionic` exchanges that must be done to put
|
||||
contracted pairs within `φsΛ` next to one another, starting recursively
|
||||
from the contracted pair
|
||||
whose first element occurs at the left-most position. -/
|
||||
def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
|
||||
∏ (a : φsΛ.1), 𝓢(𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a],
|
||||
|
|
|
|||
|
|
@ -239,7 +239,7 @@ lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
|||
· exact hG
|
||||
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
|
||||
an `i ≤ φs.length` and a `φ` in `𝓕.FieldOp`, the following relation holds
|
||||
an `i ≤ φs.length`, and a `φ` in `𝓕.FieldOp`, then
|
||||
`(φsΛ ↩Λ φ i none).sign = s * φsΛ.sign`
|
||||
where `s` is the sign got by moving `φ` through the elements of `φ₀…φᵢ₋₁` which
|
||||
are contracted with some element.
|
||||
|
|
@ -255,7 +255,7 @@ lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
|||
exact hG
|
||||
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
|
||||
and a `φ` in `𝓕.FieldOp`, the following relation holds `(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
|
||||
and a `φ` in `𝓕.FieldOp`, then `(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
|
||||
|
||||
This is a direct corollary of `sign_insert_none`. -/
|
||||
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
||||
|
|
|
|||
|
|
@ -422,7 +422,7 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
|
|||
`(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign`.
|
||||
|
||||
In `φsΛ.sign` the sign is determined by starting with the contracted pair
|
||||
whose first element occurs at the left-most position. This lemma manifests that
|
||||
whose first element occurs at the left-most position. This lemma manifests that this
|
||||
choice does not matter, and that contracted pairs can be brought together in any order. -/
|
||||
lemma join_sign {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :
|
||||
|
|
|
|||
|
|
@ -19,9 +19,9 @@ variable {n : ℕ} (c : WickContraction n)
|
|||
open HepLean.List
|
||||
open FieldOpAlgebra
|
||||
|
||||
/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ` the
|
||||
/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ`, the
|
||||
element of the center of `𝓕.FieldOpAlgebra`, `φsΛ.staticContract` is defined as the product
|
||||
of `[anPart φs[j], φs[k]]ₛ` over contracted pairs `{j, k}` (both indices of `φs`) in `φsΛ`
|
||||
of `[anPart φs[j], φs[k]]ₛ` over contracted pairs `{j, k}` in `φsΛ`
|
||||
with `j < k`. -/
|
||||
noncomputable def staticContract {φs : List 𝓕.FieldOp}
|
||||
(φsΛ : WickContraction φs.length) :
|
||||
|
|
@ -31,7 +31,7 @@ noncomputable def staticContract {φs : List 𝓕.FieldOp}
|
|||
superCommute_anPart_ofFieldOp_mem_center _ _⟩
|
||||
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, and a `i ≤ φs.length` the following relation holds
|
||||
`𝓕.FieldOp`, and a `i ≤ φs.length`, then the following relation holds:
|
||||
|
||||
`(φsΛ ↩Λ φ i none).staticContract = φsΛ.staticContract`
|
||||
|
||||
|
|
|
|||
|
|
@ -97,7 +97,7 @@ lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly :
|
|||
simp [empty]
|
||||
|
||||
/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` with
|
||||
in which every contraction involves two `FieldOp`s that have the same time. Then
|
||||
in which every contraction involves two `𝓕FieldOp`s that have the same time, then
|
||||
`φsΛ.staticContract = φsΛ.timeContract`. -/
|
||||
lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
|
||||
φsΛ.staticContract = φsΛ.timeContract := by
|
||||
|
|
@ -194,7 +194,7 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid {φs : List 𝓕.FieldOp}
|
|||
exact timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsΛ hl a b φsΛ.1.card rfl
|
||||
|
||||
/-- Let `φs` be a list of `𝓕.FieldOp`, `φsΛ` a `WickContraction` of `φs` with
|
||||
in which every contraction involves two `FieldOp`s that have the same time and
|
||||
in which every contraction involves two `𝓕.FieldOp`s that have the same time and
|
||||
`b` a general element in `𝓕.FieldOpAlgebra`. Then
|
||||
`𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b)`.
|
||||
|
||||
|
|
@ -248,7 +248,7 @@ lemma timeOrder_timeContract_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
|
|||
simp_all
|
||||
|
||||
/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` with
|
||||
at least one contraction between `FieldOp` that do not have the same time. Then
|
||||
at least one contraction between `𝓕.FieldOp` that do not have the same time. Then
|
||||
`𝓣(φsΛ.staticContract.1) = 0`. -/
|
||||
lemma timeOrder_staticContract_of_not_mem {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
|
||||
(hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.staticContract.1) = 0 := by
|
||||
|
|
@ -506,7 +506,7 @@ lemma hasEqTimeEquiv_ext_sigma {φs : List 𝓕.FieldOp} {x1 x2 :
|
|||
simp only [ne_eq, congr_refl] at h2
|
||||
simp [h2]
|
||||
|
||||
/-- The equivalence which seperates a Wick contraction which has an equal time contraction
|
||||
/-- The equivalence which separates a Wick contraction which has an equal time contraction
|
||||
into a non-empty contraction only between equal-time fields and a Wick contraction which
|
||||
does not have equal time contractions. -/
|
||||
def hasEqTimeEquiv (φs : List 𝓕.FieldOp) :
|
||||
|
|
|
|||
|
|
@ -21,7 +21,7 @@ open FieldOpAlgebra
|
|||
|
||||
/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ` the
|
||||
element of the center of `𝓕.FieldOpAlgebra`, `φsΛ.timeContract` is defined as the product
|
||||
of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` (both indices of `φs`) in `φsΛ`
|
||||
of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` in `φsΛ`
|
||||
with `j < k`. -/
|
||||
noncomputable def timeContract {φs : List 𝓕.FieldOp}
|
||||
(φsΛ : WickContraction φs.length) :
|
||||
|
|
@ -86,7 +86,7 @@ open FieldStatistic
|
|||
- `[anPart φ, φs[k]]ₛ`
|
||||
- `φsΛ.timeContract`
|
||||
- two copies of the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
|
||||
These two exchange signs cancle each other out but are included for convenience.
|
||||
These two exchange signs cancel each other out but are included for convenience.
|
||||
|
||||
The proof of this result ultimately a consequence of definitions and
|
||||
`timeContract_of_timeOrderRel`. -/
|
||||
|
|
@ -125,15 +125,13 @@ lemma timeContract_insert_some_of_lt
|
|||
|
||||
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
|
||||
`𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted` such that `k < i`, with the
|
||||
condition that `φs[k]` does not have has greater or equal time to `φ`, then
|
||||
condition that `φs[k]` does not have time greater or equal to `φ`, then
|
||||
`(φsΛ ↩Λ φ i (some k)).timeContract` is equal to the product of
|
||||
- `[anPart φ, φs[k]]ₛ`
|
||||
- `φsΛ.timeContract`
|
||||
- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
|
||||
- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ`.
|
||||
|
||||
Most of the contributes to the exchange signs cancle.
|
||||
|
||||
The proof of this result ultimately a consequence of definitions and
|
||||
`timeContract_of_not_timeOrderRel_expand`. -/
|
||||
lemma timeContract_insert_some_of_not_lt
|
||||
|
|
|
|||
|
|
@ -43,7 +43,7 @@ noncomputable def higgsRepUnitary : GaugeGroupI →* unitaryGroup (Fin 2) ℂ wh
|
|||
repeat rw [mul_assoc]
|
||||
map_one' := by simp
|
||||
|
||||
/-- Using the orthonormal basis of `HiggsVec`, turns a `2×2`-matrix intoa a linear map
|
||||
/-- Using the orthonormal basis of `HiggsVec`, turns a `2×2`-matrix into a linear map
|
||||
of `HiggsVec`. -/
|
||||
noncomputable def matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* (HiggsVec →L[ℂ] HiggsVec) where
|
||||
toFun g := LinearMap.toContinuousLinearMap
|
||||
|
|
@ -220,7 +220,7 @@ informal_lemma guage_orbit where
|
|||
deps := [``rotate_fst_zero_snd_real]
|
||||
|
||||
/-- The Higgs boson breaks electroweak symmetry down to the electromagnetic force, i.e., the
|
||||
stablity group of the action of `rep` on `![0, Complex.ofReal ‖φ‖]`, for non-zero `‖φ‖`, is the
|
||||
stability group of the action of `rep` on `![0, Complex.ofReal ‖φ‖]`, for non-zero `‖φ‖`, is the
|
||||
`SU(3) × U(1)` subgroup of `gaugeGroup := SU(3) × SU(2) × U(1)` with the embedding given by
|
||||
`(g, e^{i θ}) ↦ (g, diag (e ^ {3 * i θ}, e ^ {- 3 * i θ}), e^{i θ})`.
|
||||
-/
|
||||
|
|
|
|||
|
|
@ -77,7 +77,7 @@ syntax num : indexExpr
|
|||
/-- Notation to describe the jiggle of a tensor index. -/
|
||||
syntax "τ(" ident ")" : indexExpr
|
||||
|
||||
/-- Bool which is ture if an index is a num. -/
|
||||
/-- Bool which is true if an index is a num. -/
|
||||
def indexExprIsNum (stx : Syntax) : Bool :=
|
||||
match stx with
|
||||
| `(indexExpr|$_:num) => true
|
||||
|
|
|
|||
|
|
@ -143,10 +143,21 @@ def perturbationTheory : Note where
|
|||
parts := [
|
||||
.h1 "Introduction",
|
||||
.name `FieldSpecification.wicks_theorem_context .incomplete,
|
||||
.p "In this note we walk through the important parts of the proof of Wick's theorem
|
||||
for both fermions and bosons,
|
||||
.p "In this note we walk through the important parts of the proof of the three versions of
|
||||
Wick's theorem for field operators containing carrying both fermionic and bosonic statitics,
|
||||
as it appears in HepLean. Not every lemma or definition is covered here.
|
||||
The aim is to give just enough that the story can be understood.",
|
||||
.p "
|
||||
Before proceeding with the steps in the proof, we review some basic terminology
|
||||
related to Lean and type theory. The most important notion is that of a type.
|
||||
We don't give any formal definition here, except to say that a type `T`, like a set, has
|
||||
elements `x` which we say are of type `T` and write `x : T`. Examples of types include,
|
||||
the type of natural numbers `ℕ`, the type of real numbers `ℝ`, the type of numbers
|
||||
`0, …, n-1` denoted `Fin n`. Given two types `T` and `S`, we can form the product type `T × S`,
|
||||
and the function type `T → S`.
|
||||
|
||||
Types form the foundation of Lean and the theory behind them will be used both explicitly and
|
||||
implicitly throughout this note.",
|
||||
.h1 "Field operators",
|
||||
.h2 "Field statistics",
|
||||
.name ``FieldStatistic .complete,
|
||||
|
|
@ -159,6 +170,7 @@ def perturbationTheory : Note where
|
|||
.name ``FieldSpecification.fieldOpStatistic .complete,
|
||||
.name ``CreateAnnihilate .complete,
|
||||
.name ``FieldSpecification.CrAnFieldOp .complete,
|
||||
.name ``FieldSpecification.crAnFieldOpToCreateAnnihilate .complete,
|
||||
.name ``FieldSpecification.crAnStatistics .complete,
|
||||
.name `FieldSpecification.notation_remark .complete,
|
||||
.h2 "Field-operator free algebra",
|
||||
|
|
@ -208,6 +220,8 @@ def perturbationTheory : Note where
|
|||
.h1 "Wick Contractions",
|
||||
.h2 "Definition",
|
||||
.name ``WickContraction .complete,
|
||||
.name ``WickContraction.mem_three .complete,
|
||||
.name ``WickContraction.mem_four .complete,
|
||||
.name `WickContraction.contraction_notation .complete,
|
||||
.name ``WickContraction.GradingCompliant .complete,
|
||||
.h2 "Aside: Cardinality",
|
||||
|
|
@ -235,7 +249,7 @@ def perturbationTheory : Note where
|
|||
.name ``WickContraction.staticContract .complete,
|
||||
.name ``WickContraction.staticContract_insert_none .complete,
|
||||
.name ``WickContraction.staticContract_insert_some .complete,
|
||||
.h2 "Static wick terms",
|
||||
.h2 "Static Wick terms",
|
||||
.name ``WickContraction.staticWickTerm .complete,
|
||||
.name ``WickContraction.staticWickTerm_empty_nil .complete,
|
||||
.name ``WickContraction.staticWickTerm_insert_zero_none .complete,
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue