refactor: Higgs field
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jstoobysmith
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2 changed files with 178 additions and 42 deletions
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@ -35,6 +35,11 @@ open Matrix
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open Complex
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open ComplexConjugate
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open SpaceTime
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/-!
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## Definition of the Higgs bundle
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-/
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/-- The trivial vector bundle 𝓡² × ℂ². (TODO: Make associated bundle.) -/
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abbrev HiggsBundle := Bundle.Trivial SpaceTime HiggsVec
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@ -60,6 +65,12 @@ noncomputable def HiggsVec.toField (φ : HiggsVec) : HiggsField where
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namespace HiggsField
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open Complex Real
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/-!
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## Relation to `HiggsVec`
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-/
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/-- Given a `higgsField`, the corresponding map from `spaceTime` to `higgsVec`. -/
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def toHiggsVec (φ : HiggsField) : SpaceTime → HiggsVec := φ
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@ -80,6 +91,88 @@ lemma toField_toHiggsVec_apply (φ : HiggsField) (x : SpaceTime) :
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lemma higgsVecToFin2ℂ_toHiggsVec (φ : HiggsField) :
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higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := rfl
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/-!
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## The inner product and norm of Higgs fields
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-/
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/-- Given two `HiggsField`, the map `spaceTime → ℂ` obtained by taking their inner product. -/
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def innerProd (φ1 φ2 : HiggsField) : SpaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ
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/-- Notation for the inner product of two Higgs fields. -/
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scoped[StandardModel.HiggsField] notation "⟪" φ1 "," φ2 "⟫_H" => innerProd φ1 φ2
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@[simp]
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lemma innerProd_left_zero (φ : HiggsField) : ⟪0, φ⟫_H = 0 := by
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funext x
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simp [innerProd]
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@[simp]
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lemma innerProd_right_zero (φ : HiggsField) : ⟪φ, 0⟫_H = 0 := by
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funext x
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simp [innerProd]
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lemma innerProd_expand (φ1 φ2 : HiggsField) :
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⟪φ1, φ2⟫_H = fun x => (conj (φ1 x 0) * (φ2 x 0) + conj (φ1 x 1) * (φ2 x 1) ) := by
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funext x
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simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two]
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/-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the
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higgs vector. -/
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@[simp]
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def normSq (φ : HiggsField) : SpaceTime → ℝ := fun x => ( ‖φ x‖ ^ 2)
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/-- Notation for the norm squared of a Higgs field. -/
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scoped[StandardModel.HiggsField] notation "‖" φ1 "‖_H ^ 2" => normSq φ1
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lemma innerProd_self_eq_normSq (φ : HiggsField) (x : SpaceTime) :
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⟪φ, φ⟫_H x = ‖φ‖_H ^ 2 x := by
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erw [normSq, @PiLp.norm_sq_eq_of_L2, Fin.sum_univ_two]
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simp only [ ofReal_add, ofReal_pow, innerProd, PiLp.inner_apply,
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RCLike.inner_apply, Fin.sum_univ_two, conj_mul']
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lemma normSq_eq_innerProd_self (φ : HiggsField) (x : SpaceTime) :
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‖φ x‖ ^ 2 = (⟪φ, φ⟫_H x).re := by
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rw [innerProd_self_eq_normSq]
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rfl
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lemma toHiggsVec_norm (φ : HiggsField) (x : SpaceTime) :
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‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
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lemma normSq_expand (φ : HiggsField) :
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φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by
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funext x
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simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]
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lemma normSq_nonneg (φ : HiggsField) (x : SpaceTime) : 0 ≤ φ.normSq x := by
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simp [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
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lemma normSq_zero (φ : HiggsField) (x : SpaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
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simp [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
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/-!
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## The Higgs potential
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-/
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/-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/
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@[simp]
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def potential (φ : HiggsField) (μSq lambda : ℝ ) (x : SpaceTime) : ℝ :=
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- μSq * φ.normSq x + lambda * φ.normSq x * φ.normSq x
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lemma potential_apply (φ : HiggsField) (μSq lambda : ℝ) (x : SpaceTime) :
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(φ.potential μSq lambda) x = HiggsVec.potential μSq lambda (φ.toHiggsVec x) := by
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simp [HiggsVec.potential, toHiggsVec_norm]
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ring
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/-!
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## Smoothness
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-/
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lemma toVec_smooth (φ : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ :=
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smooth_higgsVecToFin2ℂ.comp φ.toHiggsVec_smooth
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@ -95,22 +188,6 @@ lemma apply_im_smooth (φ : HiggsField) (i : Fin 2):
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Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
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(ContinuousLinearMap.smooth imCLM).comp (φ.apply_smooth i)
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/-- Given two `higgsField`, the map `spaceTime → ℂ` obtained by taking their inner product. -/
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def innerProd (φ1 φ2 : HiggsField) : SpaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ
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/-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the
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higgs vector. -/
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@[simp]
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def normSq (φ : HiggsField) : SpaceTime → ℝ := fun x => ( ‖φ x‖ ^ 2)
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lemma toHiggsVec_norm (φ : HiggsField) (x : SpaceTime) :
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‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
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lemma normSq_expand (φ : HiggsField) :
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φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by
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funext x
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simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]
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lemma normSq_smooth (φ : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
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rw [normSq_expand]
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refine Smooth.add ?_ ?_
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@ -131,27 +208,17 @@ lemma normSq_smooth (φ : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ,
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exact φ.apply_im_smooth 1
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exact φ.apply_im_smooth 1
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lemma normSq_nonneg (φ : HiggsField) (x : SpaceTime) : 0 ≤ φ.normSq x := by
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simp [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
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lemma normSq_zero (φ : HiggsField) (x : SpaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
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simp [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
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/-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/
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@[simp]
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def potential (φ : HiggsField) (μSq lambda : ℝ ) (x : SpaceTime) : ℝ :=
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- μSq * φ.normSq x + lambda * φ.normSq x * φ.normSq x
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lemma potential_smooth (φ : HiggsField) (μSq lambda : ℝ) :
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Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by
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simp only [potential, normSq, neg_mul]
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exact (smooth_const.smul φ.normSq_smooth).neg.add
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((smooth_const.smul φ.normSq_smooth).smul φ.normSq_smooth)
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lemma potential_apply (φ : HiggsField) (μSq lambda : ℝ) (x : SpaceTime) :
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(φ.potential μSq lambda) x = HiggsVec.potential μSq lambda (φ.toHiggsVec x) := by
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simp [HiggsVec.potential, toHiggsVec_norm]
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ring
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/-!
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## Constant higgs fields
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-/
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/-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/
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def IsConst (Φ : HiggsField) : Prop := ∀ x y, Φ x = Φ y
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