PhysLean/HepLean/AnomalyCancellation/SMNu/FamilyMaps.lean

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license.
Authors: Joseph Tooby-Smith
-/
import HepLean.AnomalyCancellation.SMNu.Basic
/-!
# Family maps for the Standard Model  for RHN ACCs

We define the a series of maps between the charges for different numbers of families.

-/

universe v u
namespace SMRHN
open SMνCharges
open SMνACCs
open BigOperators

/-- Given a map of for a generic species, the corresponding map for charges. -/
@[simps!]
def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).charges →ₗ[ℚ] (SMνSpecies m).charges) :
    (SMνCharges n).charges →ₗ[ℚ] (SMνCharges m).charges where
  toFun S := toSpeciesEquiv.symm (fun i => (LinearMap.comp f (toSpecies i)) S)
  map_add' S T := by
    rw [charges_eq_toSpecies_eq]
    intro i
    rw [map_add]
    rw [toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
    rw [map_add]
  map_smul' a S := by
    rw [charges_eq_toSpecies_eq]
    intro i
    rw [map_smul]
    rw [toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
    rw [map_smul]
    rfl

lemma chargesMapOfSpeciesMap_toSpecies {n m : ℕ}
    (f : (SMνSpecies n).charges →ₗ[ℚ] (SMνSpecies m).charges)
    (S : (SMνCharges n).charges) (j : Fin 6) :
    toSpecies j (chargesMapOfSpeciesMap f S) =  (LinearMap.comp f (toSpecies j)) S := by
  erw [toSMSpecies_toSpecies_inv]

/-- The projection of the `m`-family charges onto the first `n`-family charges for species.  -/
@[simps!]
def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
    (SMνSpecies m).charges →ₗ[ℚ] (SMνSpecies n).charges where
  toFun S := S ∘ Fin.castLE h
  map_add' S T := by
    funext i
    simp
  map_smul' a S := by
    funext i
    simp [HSMul.hSMul]
    -- rw [show Rat.cast a = a from rfl]

/-- The projection of the `m`-family charges onto the first `n`-family charges.  -/
def familyProjection {m n : ℕ} (h : n ≤ m) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
  chargesMapOfSpeciesMap (speciesFamilyProj h)

/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero.  -/
@[simps!]
def speciesEmbed (m n : ℕ) :
    (SMνSpecies m).charges →ₗ[ℚ] (SMνSpecies n).charges where
  toFun S := fun i =>
    if hi : i.val < m then
      S ⟨i, hi⟩
    else
      0
  map_add' S T := by
    funext i
    simp only [SMνSpecies_numberCharges, ACCSystemCharges.chargesAddCommMonoid_add]
    by_cases hi : i.val < m
    erw [dif_pos hi, dif_pos hi, dif_pos hi]
    erw [dif_neg hi, dif_neg hi, dif_neg hi]
    rfl
  map_smul' a S := by
    funext i
    simp [HSMul.hSMul]
    by_cases hi : i.val < m
    erw [dif_pos hi, dif_pos hi]
    erw [dif_neg hi, dif_neg hi]
    simp

/-- The embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero.  -/
def familyEmbedding (m n : ℕ) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
  chargesMapOfSpeciesMap (speciesEmbed m n)

/-- For species, the embedding of the `1`-family charges into the `n`-family charges in
a universal manor. -/
@[simps!]
def speciesFamilyUniversial (n : ℕ) :
    (SMνSpecies 1).charges →ₗ[ℚ] (SMνSpecies n).charges where
  toFun S _ := S ⟨0, by simp⟩
  map_add' S T := by
    funext _
    simp
  map_smul' a S := by
    funext i
    simp [HSMul.hSMul]
    -- rw [show Rat.cast a = a from rfl]

/-- The embedding of the `1`-family charges into the `n`-family charges in
a universal manor. -/
def familyUniversal (n : ℕ) : (SMνCharges 1).charges →ₗ[ℚ] (SMνCharges n).charges :=
  chargesMapOfSpeciesMap (speciesFamilyUniversial n)

lemma toSpecies_familyUniversal {n : ℕ} (j : Fin 6) (S : (SMνCharges 1).charges)
    (i : Fin n) : toSpecies j (familyUniversal n S) i = toSpecies j S ⟨0, by simp⟩ := by
  erw [chargesMapOfSpeciesMap_toSpecies]
  rfl

lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).charges) (j : Fin 6) :
    ∑ i, ((fun a => a ^ m) ∘ toSpecies j (familyUniversal n S)) i =
    n * (toSpecies j S ⟨0, by simp⟩) ^ m := by
  simp only [SMνSpecies_numberCharges, Function.comp_apply, toSpecies_apply, Fin.zero_eta,
    Fin.isValue]
  have h1 : (n : ℚ) * (toSpecies j S ⟨0, by simp⟩) ^ m =
      ∑ _i : Fin n, (toSpecies j S ⟨0, by simp⟩) ^ m := by
    rw [Fin.sum_const]
    simp
  erw [h1]
  apply Finset.sum_congr
  simp only
  intro i _
  erw [toSpecies_familyUniversal]

lemma sum_familyUniversal_one  {n : ℕ} (S : (SMνCharges 1).charges) (j : Fin 6) :
    ∑ i, toSpecies j (familyUniversal n S) i = n * (toSpecies j S ⟨0, by simp⟩) := by
  simpa using @sum_familyUniversal n 1 S j

lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).charges)
    (T : (SMνCharges n).charges) (j : Fin 6) :
    ∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i) =
    (toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i := by
  simp only [SMνSpecies_numberCharges, toSpecies_apply, Fin.zero_eta, Fin.isValue]
  rw [Finset.mul_sum]
  apply Finset.sum_congr
  simp only
  intro i _
  erw [toSpecies_familyUniversal]
  rfl

lemma sum_familyUniversal_three  {n : ℕ} (S : (SMνCharges 1).charges)
    (T L : (SMνCharges n).charges) (j : Fin 6) :
    ∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i * toSpecies j L i) =
    (toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i * toSpecies j L i := by
  simp only [SMνSpecies_numberCharges, toSpecies_apply, Fin.zero_eta, Fin.isValue]
  rw [Finset.mul_sum]
  apply Finset.sum_congr
  simp only
  intro i _
  erw [toSpecies_familyUniversal]
  simp only [SMνSpecies_numberCharges, Fin.zero_eta, Fin.isValue, toSpecies_apply]
  ring

lemma familyUniversal_accGrav (S : (SMνCharges 1).charges) :
    accGrav (familyUniversal n S)  = n * (accGrav S) := by
  rw [accGrav_decomp, accGrav_decomp]
  repeat rw [sum_familyUniversal_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
    Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
  ring

lemma familyUniversal_accSU2 (S : (SMνCharges 1).charges) :
    accSU2 (familyUniversal n S)  = n * (accSU2 S) := by
  rw [accSU2_decomp, accSU2_decomp]
  repeat rw [sum_familyUniversal_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
    Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
  ring

lemma familyUniversal_accSU3 (S : (SMνCharges 1).charges) :
    accSU3 (familyUniversal n S)  = n * (accSU3 S) := by
  rw [accSU3_decomp, accSU3_decomp]
  repeat rw [sum_familyUniversal_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
    Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
  ring

lemma familyUniversal_accYY (S : (SMνCharges 1).charges) :
    accYY (familyUniversal n S)  = n * (accYY S) := by
  rw [accYY_decomp, accYY_decomp]
  repeat rw [sum_familyUniversal_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
    Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
  ring

lemma familyUniversal_quadBiLin (S : (SMνCharges 1).charges) (T : (SMνCharges n).charges) :
    quadBiLin (familyUniversal n S) T =
    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 -
    S (3 : Fin 6) * ∑ i, L T i + S (4 : Fin 6) * ∑ i, E T i  := by
  rw [quadBiLin_decomp]
  repeat rw [sum_familyUniversal_two]
  repeat rw [toSpecies_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, toSpecies_apply, add_left_inj, sub_left_inj,
    sub_right_inj]
  ring

lemma familyUniversal_accQuad (S : (SMνCharges 1).charges) :
    accQuad (familyUniversal n S)  = n * (accQuad S) := by
  rw [accQuad_decomp]
  repeat erw [sum_familyUniversal]
  rw [accQuad_decomp]
  simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
    Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
  ring

lemma familyUniversal_cubeTriLin (S : (SMνCharges 1).charges) (T R : (SMνCharges n).charges) :
    cubeTriLin (familyUniversal n S) T R = 6 * S (0 : Fin 6) * ∑ i, (Q T i * Q R i) +
      3 * S (1 : Fin 6) * ∑ i,  (U T i * U R i) + 3 * S (2 : Fin 6) * ∑ i,  (D T i * D R i)
      + 2 * S (3 : Fin 6) * ∑ i, (L T i * L R i) +
      S (4 : Fin 6) * ∑ i, (E T i * E R i) + S (5 : Fin 6) * ∑ i, (N T i * N R i) := by
  rw [cubeTriLin_decomp]
  repeat rw [sum_familyUniversal_three]
  repeat rw [toSpecies_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, toSpecies_apply, add_left_inj]
  ring

lemma familyUniversal_cubeTriLin' (S T : (SMνCharges 1).charges) (R : (SMνCharges n).charges) :
    cubeTriLin (familyUniversal n S) (familyUniversal n T) R =
      6 * S (0 : Fin 6) * T (0 : Fin 6) * ∑ i, Q R i +
       3 * S (1 : Fin 6) * T (1 : Fin 6) * ∑ i, U R i
      + 3 * S (2 : Fin 6) * T (2 : Fin 6) * ∑ i, D R i +
      2 * S (3 : Fin 6) * T (3 : Fin 6) * ∑ i, L R i +
      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
  rw [familyUniversal_cubeTriLin]
  repeat rw [sum_familyUniversal_two]
  repeat rw [toSpecies_one]
  simp only [Fin.isValue, SMνSpecies_numberCharges, toSpecies_apply]
  ring

lemma familyUniversal_accCube (S : (SMνCharges 1).charges) :
    accCube (familyUniversal n S) = n * (accCube S) := by
  rw [accCube_decomp]
  repeat erw [sum_familyUniversal]
  rw [accCube_decomp]
  simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
    Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
  ring

end SMRHN