Update TargetSpace.lean
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Pietro Monticone
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7427ce4207
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ea6c61eb29
1 changed files with 58 additions and 104 deletions
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@ -46,10 +46,8 @@ section higgsVec
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casting vectors. -/
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def higgsVecToFin2ℂ : higgsVec →L[ℝ] (Fin 2 → ℂ) where
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toFun x := x
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map_add' x y := by
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simp
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map_smul' a x := by
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simp
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map_add' x y := by simp
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map_smul' a x := by simp
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lemma smooth_higgsVecToFin2ℂ : Smooth 𝓘(ℝ, higgsVec) 𝓘(ℝ, Fin 2 → ℂ) higgsVecToFin2ℂ :=
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ContinuousLinearMap.smooth higgsVecToFin2ℂ
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@ -63,11 +61,10 @@ noncomputable def higgsRepUnitary : gaugeGroup →* unitaryGroup (Fin 2) ℂ whe
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map_mul' := by
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intro ⟨_, a2, a3⟩ ⟨_, b2, b3⟩
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change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
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rw [repU1.map_mul, fundamentalSU2.map_mul]
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rw [mul_assoc, mul_assoc, ← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
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rw [repU1.map_mul, fundamentalSU2.map_mul, mul_assoc, mul_assoc,
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← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
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repeat rw [mul_assoc]
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map_one' := by
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simp only [Prod.snd_one, _root_.map_one, Prod.fst_one, mul_one]
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map_one' := by simp
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/-- An orthonormal basis of higgsVec. -/
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noncomputable def orthonormBasis : OrthonormalBasis (Fin 2) ℂ higgsVec :=
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@ -87,20 +84,16 @@ lemma matrixToLin_star (g : Matrix (Fin 2) (Fin 2) ℂ) :
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lemma matrixToLin_unitary (g : unitaryGroup (Fin 2) ℂ) :
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matrixToLin g ∈ unitary (higgsVec →L[ℂ] higgsVec) := by
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rw [@unitary.mem_iff, ← matrixToLin_star, ← matrixToLin.map_mul, ← matrixToLin.map_mul]
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rw [mem_unitaryGroup_iff.mp g.prop, mem_unitaryGroup_iff'.mp g.prop, matrixToLin.map_one]
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rw [@unitary.mem_iff, ← matrixToLin_star, ← matrixToLin.map_mul, ← matrixToLin.map_mul,
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mem_unitaryGroup_iff.mp g.prop, mem_unitaryGroup_iff'.mp g.prop, matrixToLin.map_one]
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simp
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/-- The natural homomorphism from unitary `2×2` complex matrices to unitary transformations
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of `higgsVec`. -/
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noncomputable def unitaryToLin : unitaryGroup (Fin 2) ℂ →* unitary (higgsVec →L[ℂ] higgsVec) where
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toFun g := ⟨matrixToLin g, matrixToLin_unitary g⟩
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map_mul' g h := by
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ext
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simp
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map_one' := by
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ext
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simp
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map_mul' g h := by simp
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map_one' := by simp
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/-- The inclusion of unitary transformations on `higgsVec` into all linear transformations. -/
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@[simps!]
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@ -114,8 +107,7 @@ def rep : Representation ℂ gaugeGroup higgsVec :=
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lemma higgsRepUnitary_mul (g : gaugeGroup) (φ : higgsVec) :
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(higgsRepUnitary g).1 *ᵥ φ = g.2.2 ^ 3 • (g.2.1.1 *ᵥ φ) := by
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simp only [higgsRepUnitary_apply_coe]
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exact smul_mulVec_assoc (g.2.2 ^ 3) (g.2.1.1) φ
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simp [higgsRepUnitary_apply_coe, smul_mulVec_assoc]
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lemma rep_apply (g : gaugeGroup) (φ : higgsVec) : rep g φ = g.2.2 ^ 3 • (g.2.1.1 *ᵥ φ) :=
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higgsRepUnitary_mul g φ
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@ -133,13 +125,11 @@ def potential (φ : higgsVec) : ℝ := - μSq * ‖φ‖ ^ 2 + λ * ‖φ‖ ^
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lemma potential_invariant (φ : higgsVec) (g : gaugeGroup) :
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potential μSq (λ) (rep g φ) = potential μSq (λ) φ := by
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simp only [potential, neg_mul]
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rw [norm_invariant]
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simp only [potential, neg_mul, norm_invariant]
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lemma potential_as_quad (φ : higgsVec) :
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λ * ‖φ‖ ^ 2 * ‖φ‖ ^ 2 + (- μSq ) * ‖φ‖ ^ 2 + (- potential μSq (λ) φ) = 0 := by
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simp [potential]
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ring
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simp [potential]; ring
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end potentialDefn
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section potentialProp
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@ -161,11 +151,9 @@ lemma zero_le_potential_discrim (φ : higgsVec) :
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0 ≤ discrim (λ) (- μSq ) (- potential μSq (λ) φ) := by
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have h1 := potential_as_quad μSq (λ) φ
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rw [quadratic_eq_zero_iff_discrim_eq_sq] at h1
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rw [h1]
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exact sq_nonneg (2 * (lambda ) * ‖φ‖ ^ 2 + -μSq)
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simp only [ne_eq, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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exact ne_of_gt hLam
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· simp only [h1, ne_eq, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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exact sq_nonneg (2 * lambda * ‖φ‖ ^ 2 + -μSq)
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· exact ne_of_gt hLam
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lemma potential_eq_zero_sol (φ : higgsVec)
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(hV : potential μSq (λ) φ = 0) : φ = 0 ∨ ‖φ‖ ^ 2 = μSq / λ := by
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@ -209,19 +197,14 @@ lemma potential_bounded_below (φ : higgsVec) :
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lemma potential_bounded_below_of_μSq_nonpos {μSq : ℝ}
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(hμSq : μSq ≤ 0) (φ : higgsVec) : 0 ≤ potential μSq (λ) φ := by
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simp only [potential, neg_mul, add_zero]
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refine add_nonneg ?_ (potential_snd_term_nonneg hLam φ)
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field_simp
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rw [@mul_nonpos_iff]
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simp_all only [ge_iff_le, norm_nonneg, pow_nonneg, and_self, or_true]
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field_simp [mul_nonpos_iff]
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simp_all [ge_iff_le, norm_nonneg, pow_nonneg, and_self, or_true]
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lemma potential_eq_bound_discrim_zero (φ : higgsVec)
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(hV : potential μSq (λ) φ = - μSq ^ 2 / (4 * λ)) :
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discrim (λ) (- μSq) (- potential μSq (λ) φ) = 0 := by
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simp [discrim, hV]
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field_simp
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ring
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field_simp [discrim, hV]
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lemma potential_eq_bound_higgsVec_sq (φ : higgsVec)
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(hV : potential μSq (λ) φ = - μSq ^ 2 / (4 * (λ))) :
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@ -229,81 +212,59 @@ lemma potential_eq_bound_higgsVec_sq (φ : higgsVec)
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have h1 := potential_as_quad μSq (λ) φ
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rw [quadratic_eq_zero_iff_of_discrim_eq_zero _
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(potential_eq_bound_discrim_zero μSq hLam φ hV)] at h1
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rw [h1]
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field_simp
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ring_nf
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simp only [ne_eq, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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simp_rw [h1, neg_neg]
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exact ne_of_gt hLam
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lemma potential_eq_bound_iff (φ : higgsVec) :
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potential μSq (λ) φ = - μSq ^ 2 / (4 * (λ)) ↔ ‖φ‖ ^ 2 = μSq / (2 * (λ)) := by
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apply Iff.intro
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· intro h
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exact potential_eq_bound_higgsVec_sq μSq hLam φ h
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· intro h
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have hv : ‖φ‖ ^ 4 = ‖φ‖ ^ 2 * ‖φ‖ ^ 2 := by
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ring_nf
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field_simp [potential, hv, h]
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ring
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potential μSq (λ) φ = - μSq ^ 2 / (4 * (λ)) ↔ ‖φ‖ ^ 2 = μSq / (2 * (λ)) :=
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Iff.intro (potential_eq_bound_higgsVec_sq μSq hLam φ)
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(fun h ↦ by
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have hv : ‖φ‖ ^ 4 = ‖φ‖ ^ 2 * ‖φ‖ ^ 2 := by ring_nf
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field_simp [potential, hv, h]
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ring_nf)
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lemma potential_eq_bound_iff_of_μSq_nonpos {μSq : ℝ}
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(hμSq : μSq ≤ 0) (φ : higgsVec) : potential μSq (λ) φ = 0 ↔ φ = 0 := by
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apply Iff.intro
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· intro h
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exact potential_eq_zero_sol_of_μSq_nonpos μSq hLam hμSq φ h
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· intro h
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simp [potential, h]
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(hμSq : μSq ≤ 0) (φ : higgsVec) : potential μSq (λ) φ = 0 ↔ φ = 0 :=
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Iff.intro (fun h ↦ potential_eq_zero_sol_of_μSq_nonpos μSq hLam hμSq φ h) (fun h ↦ by simp [potential, h])
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lemma potential_eq_bound_IsMinOn (φ : higgsVec)
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lemma potential_eq_bound_IsMinOn (φ : higgsVec)
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(hv : potential μSq lambda φ = - μSq ^ 2 / (4 * lambda)) :
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IsMinOn (potential μSq lambda) Set.univ φ := by
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rw [isMinOn_univ_iff]
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intro x
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rw [hv]
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exact potential_bounded_below μSq hLam x
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rw [isMinOn_univ_iff, hv]
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exact fun x ↦ potential_bounded_below μSq hLam x
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lemma potential_eq_bound_IsMinOn_of_μSq_nonpos {μSq : ℝ}
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(hμSq : μSq ≤ 0) (φ : higgsVec) (hv : potential μSq lambda φ = 0) :
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IsMinOn (potential μSq lambda) Set.univ φ := by
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rw [isMinOn_univ_iff]
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intro x
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rw [hv]
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exact potential_bounded_below_of_μSq_nonpos hLam hμSq x
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rw [isMinOn_univ_iff, hv]
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exact fun x ↦ potential_bounded_below_of_μSq_nonpos hLam hμSq x
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lemma potential_bound_reached_of_μSq_nonneg {μSq : ℝ} (hμSq : 0 ≤ μSq) :
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∃ (φ : higgsVec), potential μSq lambda φ = - μSq ^ 2 / (4 * lambda) := by
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use ![√(μSq/(2 * lambda)), 0]
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refine (potential_eq_bound_iff μSq hLam _).mpr ?_
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simp [@PiLp.norm_sq_eq_of_L2, Fin.sum_univ_two]
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simp [PiLp.norm_sq_eq_of_L2]
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field_simp [mul_pow]
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lemma IsMinOn_potential_iff_of_μSq_nonneg {μSq : ℝ} (hμSq : 0 ≤ μSq) :
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IsMinOn (potential μSq lambda) Set.univ φ ↔ ‖φ‖ ^ 2 = μSq /(2 * lambda) := by
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apply Iff.intro
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apply Iff.intro <;> rw [← potential_eq_bound_iff μSq hLam φ]
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· intro h
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obtain ⟨φm, hφ⟩ := potential_bound_reached_of_μSq_nonneg hLam hμSq
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have hm := isMinOn_univ_iff.mp h φm
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rw [hφ] at hm
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have h1 := potential_bounded_below μSq hLam φ
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rw [← potential_eq_bound_iff μSq hLam φ]
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exact (Real.partialOrder.le_antisymm _ _ h1 hm).symm
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· intro h
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rw [← potential_eq_bound_iff μSq hLam φ] at h
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exact potential_eq_bound_IsMinOn μSq hLam φ h
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exact (Real.partialOrder.le_antisymm _ _ (potential_bounded_below μSq hLam φ) hm).symm
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· exact potential_eq_bound_IsMinOn μSq hLam φ
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lemma IsMinOn_potential_iff_of_μSq_nonpos {μSq : ℝ} (hμSq : μSq ≤ 0) :
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IsMinOn (potential μSq lambda) Set.univ φ ↔ φ = 0 := by
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apply Iff.intro
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apply Iff.intro <;> rw [← potential_eq_bound_iff_of_μSq_nonpos hLam hμSq φ]
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· intro h
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have h0 := isMinOn_univ_iff.mp h 0
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rw [(potential_eq_bound_iff_of_μSq_nonpos hLam hμSq 0).mpr (by rfl)] at h0
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have h1 := potential_bounded_below_of_μSq_nonpos hLam hμSq φ
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rw [← (potential_eq_bound_iff_of_μSq_nonpos hLam hμSq φ)]
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rw [(potential_eq_bound_iff_of_μSq_nonpos hLam hμSq 0).mpr (by rfl)] at h0
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exact (Real.partialOrder.le_antisymm _ _ h1 h0).symm
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· intro h
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rw [← potential_eq_bound_iff_of_μSq_nonpos hLam hμSq φ] at h
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exact potential_eq_bound_IsMinOn_of_μSq_nonpos hLam hμSq φ h
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· exact potential_eq_bound_IsMinOn_of_μSq_nonpos hLam hμSq φ
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end potentialProp
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/-- Given a Higgs vector, a rotation matrix which puts the first component of the
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@ -314,40 +275,30 @@ def rotateMatrix (φ : higgsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
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lemma rotateMatrix_star (φ : higgsVec) :
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star φ.rotateMatrix =
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![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
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simp [star]
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rw [rotateMatrix, conjTranspose]
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simp_rw [star, rotateMatrix, conjTranspose]
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ext i j
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fin_cases i <;> fin_cases j <;> simp [conj_ofReal]
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lemma rotateMatrix_det {φ : higgsVec} (hφ : φ ≠ 0) : (rotateMatrix φ).det = 1 := by
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simp [rotateMatrix, det_fin_two]
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have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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field_simp
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field_simp [rotateMatrix, det_fin_two]
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rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
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rfl
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lemma rotateMatrix_unitary {φ : higgsVec} (hφ : φ ≠ 0) :
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(rotateMatrix φ) ∈ unitaryGroup (Fin 2) ℂ := by
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rw [mem_unitaryGroup_iff', rotateMatrix_star, rotateMatrix]
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erw [mul_fin_two, one_fin_two]
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have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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congr
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field_simp
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ext i j
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fin_cases i <;> fin_cases j <;> field_simp
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· rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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fin_cases i <;> fin_cases j <;> field_simp <;> rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
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rfl
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· ring_nf
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· ring_nf
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· rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
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rfl
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lemma rotateMatrix_specialUnitary {φ : higgsVec} (hφ : φ ≠ 0) :
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(rotateMatrix φ) ∈ specialUnitaryGroup (Fin 2) ℂ :=
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@ -361,19 +312,22 @@ def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : gaugeGroup :=
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lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
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rep (rotateGuageGroup hφ) φ = ![0, ofReal ‖φ‖] := by
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rw [rep_apply]
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simp [rotateGuageGroup, rotateMatrix]
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simp only [rotateGuageGroup, rotateMatrix, one_pow, one_smul,
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Nat.succ_eq_add_one, Nat.reduceAdd, ofReal_eq_coe]
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ext i
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fin_cases i
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simp [mulVec, vecHead, vecTail]
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ring_nf
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simp only [Fin.mk_one, Fin.isValue, cons_val_one, head_cons]
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simp [mulVec, vecHead, vecTail]
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have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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field_simp
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rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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· simp only [mulVec, Fin.zero_eta, Fin.isValue, cons_val', empty_val', cons_val_fin_one,
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cons_val_zero, cons_dotProduct, vecHead, vecTail, Nat.succ_eq_add_one, Nat.reduceAdd,
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Function.comp_apply, Fin.succ_zero_eq_one, dotProduct_empty, add_zero]
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ring_nf
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· simp only [Fin.mk_one, Fin.isValue, cons_val_one, head_cons, mulVec, Fin.isValue,
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cons_val', empty_val', cons_val_fin_one, vecHead, cons_dotProduct, vecTail, Nat.succ_eq_add_one,
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Nat.reduceAdd, Function.comp_apply, Fin.succ_zero_eq_one, dotProduct_empty, add_zero]
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have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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field_simp
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rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
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rfl
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theorem rotate_fst_zero_snd_real (φ : higgsVec) :
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∃ (g : gaugeGroup), rep g φ = ![0, ofReal ‖φ‖] := by
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