Feat: SMNu basics
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HepLean/AnomalyCancellation/SMNu/FamilyMaps.lean
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HepLean/AnomalyCancellation/SMNu/FamilyMaps.lean
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.SMNu.Basic
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/-!
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# Family maps for the Standard Model for RHN ACCs
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We define the a series of maps between the charges for different numbers of famlies.
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-/
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universe v u
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namespace SMRHN
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open SMνCharges
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open SMνACCs
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open BigOperators
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/-- Given a map of for a generic species, the corresponding map for charges. -/
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@[simps!]
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def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).charges →ₗ[ℚ] (SMνSpecies m).charges) :
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(SMνCharges n).charges →ₗ[ℚ] (SMνCharges m).charges where
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toFun S := toSpeciesEquiv.symm (fun i => (LinearMap.comp f (toSpecies i)) S)
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map_add' S T := by
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rw [charges_eq_toSpecies_eq]
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intro i
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rw [map_add]
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rw [toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
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rw [map_add]
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map_smul' a S := by
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rw [charges_eq_toSpecies_eq]
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intro i
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rw [map_smul]
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rw [toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
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rw [map_smul]
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rfl
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lemma chargesMapOfSpeciesMap_toSpecies {n m : ℕ}
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(f : (SMνSpecies n).charges →ₗ[ℚ] (SMνSpecies m).charges)
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(S : (SMνCharges n).charges) (j : Fin 6) :
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toSpecies j (chargesMapOfSpeciesMap f S) = (LinearMap.comp f (toSpecies j)) S := by
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erw [toSMSpecies_toSpecies_inv]
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/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
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@[simps!]
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def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
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(SMνSpecies m).charges →ₗ[ℚ] (SMνSpecies n).charges where
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toFun S := S ∘ Fin.castLE h
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map_add' S T := by
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funext i
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simp
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map_smul' a S := by
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funext i
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simp [HSMul.hSMul]
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-- rw [show Rat.cast a = a from rfl]
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/-- The projection of the `m`-family charges onto the first `n`-family charges. -/
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def familyProjection {m n : ℕ} (h : n ≤ m) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
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chargesMapOfSpeciesMap (speciesFamilyProj h)
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/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all
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other charges zero. -/
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@[simps!]
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def speciesEmbed (m n : ℕ) :
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(SMνSpecies m).charges →ₗ[ℚ] (SMνSpecies n).charges where
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toFun S := fun i =>
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if hi : i.val < m then
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S ⟨i, hi⟩
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else
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0
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map_add' S T := by
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funext i
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simp
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by_cases hi : i.val < m
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erw [dif_pos hi, dif_pos hi, dif_pos hi]
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erw [dif_neg hi, dif_neg hi, dif_neg hi]
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rfl
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map_smul' a S := by
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funext i
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simp [HSMul.hSMul]
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by_cases hi : i.val < m
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erw [dif_pos hi, dif_pos hi]
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erw [dif_neg hi, dif_neg hi]
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simp
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/-- The embedding of the `m`-family charges onto the `n`-family charges, with all
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other charges zero. -/
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def familyEmbedding (m n : ℕ) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
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chargesMapOfSpeciesMap (speciesEmbed m n)
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/-- For species, the embeddding of the `1`-family charges into the `n`-family charges in
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a universal manor. -/
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@[simps!]
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def speciesFamilyUniversial (n : ℕ) :
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(SMνSpecies 1).charges →ₗ[ℚ] (SMνSpecies n).charges where
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toFun S _ := S ⟨0, by simp⟩
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map_add' S T := by
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funext _
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simp
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map_smul' a S := by
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funext i
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simp [HSMul.hSMul]
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-- rw [show Rat.cast a = a from rfl]
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/-- The embeddding of the `1`-family charges into the `n`-family charges in
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a universal manor. -/
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def familyUniversal (n : ℕ) : (SMνCharges 1).charges →ₗ[ℚ] (SMνCharges n).charges :=
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chargesMapOfSpeciesMap (speciesFamilyUniversial n)
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lemma toSpecies_familyUniversal {n : ℕ} (j : Fin 6) (S : (SMνCharges 1).charges)
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(i : Fin n) : toSpecies j (familyUniversal n S) i = toSpecies j S ⟨0, by simp⟩ := by
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erw [chargesMapOfSpeciesMap_toSpecies]
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rfl
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lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).charges) (j : Fin 6) :
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∑ i, ((fun a => a ^ m) ∘ toSpecies j (familyUniversal n S)) i =
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n * (toSpecies j S ⟨0, by simp⟩) ^ m := by
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simp
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have h1 : (n : ℚ) * (toSpecies j S ⟨0, by simp⟩) ^ m = ∑ _i : Fin n, (toSpecies j S ⟨0, by simp⟩) ^ m:= by
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rw [Fin.sum_const]
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simp
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erw [h1]
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apply Finset.sum_congr
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simp
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intro i _
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erw [toSpecies_familyUniversal]
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lemma sum_familyUniversal_one {n : ℕ} (S : (SMνCharges 1).charges) (j : Fin 6) :
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∑ i, toSpecies j (familyUniversal n S) i = n * (toSpecies j S ⟨0, by simp⟩) := by
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simpa using @sum_familyUniversal n 1 S j
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lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).charges)
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(T : (SMνCharges n).charges) (j : Fin 6) :
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∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i) =
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(toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i := by
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simp
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rw [Finset.mul_sum]
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apply Finset.sum_congr
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simp
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intro i _
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erw [toSpecies_familyUniversal]
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rfl
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lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).charges)
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(T L : (SMνCharges n).charges) (j : Fin 6) :
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∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i * toSpecies j L i) =
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(toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i * toSpecies j L i := by
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simp
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rw [Finset.mul_sum]
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apply Finset.sum_congr
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simp
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intro i _
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erw [toSpecies_familyUniversal]
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simp
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ring
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lemma familyUniversal_accGrav (S : (SMνCharges 1).charges) :
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accGrav (familyUniversal n S) = n * (accGrav S) := by
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rw [accGrav_decomp, accGrav_decomp]
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repeat rw [sum_familyUniversal_one]
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simp
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ring
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lemma familyUniversal_accSU2 (S : (SMνCharges 1).charges) :
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accSU2 (familyUniversal n S) = n * (accSU2 S) := by
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rw [accSU2_decomp, accSU2_decomp]
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repeat rw [sum_familyUniversal_one]
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simp
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ring
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lemma familyUniversal_accSU3 (S : (SMνCharges 1).charges) :
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accSU3 (familyUniversal n S) = n * (accSU3 S) := by
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rw [accSU3_decomp, accSU3_decomp]
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repeat rw [sum_familyUniversal_one]
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simp
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ring
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lemma familyUniversal_accYY (S : (SMνCharges 1).charges) :
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accYY (familyUniversal n S) = n * (accYY S) := by
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rw [accYY_decomp, accYY_decomp]
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repeat rw [sum_familyUniversal_one]
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simp
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ring
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lemma familyUniversal_quadBiLin (S : (SMνCharges 1).charges) (T : (SMνCharges n).charges) :
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quadBiLin (familyUniversal n S, T) =
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S (0 : Fin 6) * ∑ i, Q T i - 2 * S (1 : Fin 6) * ∑ i, U T i + S (2 : Fin 6) *∑ i, D T i -
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S (3 : Fin 6) * ∑ i, L T i + S (4 : Fin 6) * ∑ i, E T i := by
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rw [quadBiLin_decomp]
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repeat rw [sum_familyUniversal_two]
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repeat rw [toSpecies_one]
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simp
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ring
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lemma familyUniversal_accQuad (S : (SMνCharges 1).charges) :
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accQuad (familyUniversal n S) = n * (accQuad S) := by
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rw [accQuad_decomp]
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repeat erw [sum_familyUniversal]
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rw [accQuad_decomp]
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simp
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ring
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lemma familyUniversal_cubeTriLin (S : (SMνCharges 1).charges) (T R : (SMνCharges n).charges) :
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cubeTriLin (familyUniversal n S, T, R) = 6 * S (0 : Fin 6) * ∑ i, (Q T i * Q R i) +
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3 * S (1 : Fin 6) * ∑ i, (U T i * U R i) + 3 * S (2 : Fin 6) * ∑ i, (D T i * D R i)
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+ 2 * S (3 : Fin 6) * ∑ i, (L T i * L R i) +
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S (4 : Fin 6) * ∑ i, (E T i * E R i) + S (5 : Fin 6) * ∑ i, (N T i * N R i) := by
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rw [cubeTriLin_decomp]
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repeat rw [sum_familyUniversal_three]
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repeat rw [toSpecies_one]
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simp
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ring
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lemma familyUniversal_cubeTriLin' (S T : (SMνCharges 1).charges) (R : (SMνCharges n).charges) :
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cubeTriLin (familyUniversal n S, familyUniversal n T, R) =
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6 * S (0 : Fin 6) * T (0 : Fin 6) * ∑ i, Q R i +
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3 * S (1 : Fin 6) * T (1 : Fin 6) * ∑ i, U R i
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+ 3 * S (2 : Fin 6) * T (2 : Fin 6) * ∑ i, D R i +
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2 * S (3 : Fin 6) * T (3 : Fin 6) * ∑ i, L R i +
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S (4 : Fin 6) * T (4 : Fin 6) * ∑ i, E R i + S (5 : Fin 6) * T (5 : Fin 6) * ∑ i, N R i := by
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rw [familyUniversal_cubeTriLin]
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repeat rw [sum_familyUniversal_two]
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repeat rw [toSpecies_one]
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simp
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ring
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lemma familyUniversal_accCube (S : (SMνCharges 1).charges) :
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accCube (familyUniversal n S) = n * (accCube S) := by
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rw [accCube_decomp]
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repeat erw [sum_familyUniversal]
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rw [accCube_decomp]
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simp
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ring
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end SMRHN
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