refactor: Change case of type and props
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jstoobysmith
parent
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commit
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58 changed files with 695 additions and 696 deletions
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@ -37,25 +37,24 @@ open Complex
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open ComplexConjugate
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/-- The complex vector space in which the Higgs field takes values. -/
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abbrev higgsVec := EuclideanSpace ℂ (Fin 2)
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abbrev HiggsVec := EuclideanSpace ℂ (Fin 2)
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section higgsVec
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/-- The continuous linear map from the vector space `higgsVec` to `(Fin 2 → ℂ)` achieved by
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casting vectors. -/
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def higgsVecToFin2ℂ : higgsVec →L[ℝ] (Fin 2 → ℂ) where
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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 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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lemma smooth_higgsVecToFin2ℂ : Smooth 𝓘(ℝ, HiggsVec) 𝓘(ℝ, Fin 2 → ℂ) higgsVecToFin2ℂ :=
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ContinuousLinearMap.smooth higgsVecToFin2ℂ
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namespace higgsVec
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namespace HiggsVec
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/-- The Higgs representation as a homomorphism from the gauge group to unitary `2×2`-matrices. -/
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@[simps!]
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noncomputable def higgsRepUnitary : gaugeGroup →* unitaryGroup (Fin 2) ℂ where
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noncomputable def higgsRepUnitary : GaugeGroup →* unitaryGroup (Fin 2) ℂ where
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toFun g := repU1 g.2.2 * fundamentalSU2 g.2.1
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map_mul' := by
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intro ⟨_, a2, a3⟩ ⟨_, b2, b3⟩
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@ -66,11 +65,11 @@ noncomputable def higgsRepUnitary : gaugeGroup →* unitaryGroup (Fin 2) ℂ whe
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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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noncomputable def orthonormBasis : OrthonormalBasis (Fin 2) ℂ HiggsVec :=
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EuclideanSpace.basisFun (Fin 2) ℂ
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/-- Takes in a `2×2`-matrix and returns a linear map of `higgsVec`. -/
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noncomputable def matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* (higgsVec →L[ℂ] higgsVec) where
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noncomputable def matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* (HiggsVec →L[ℂ] HiggsVec) where
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toFun g := LinearMap.toContinuousLinearMap
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$ Matrix.toLin orthonormBasis.toBasis orthonormBasis.toBasis g
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map_mul' g h := ContinuousLinearMap.coe_inj.mp $
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@ -82,36 +81,36 @@ lemma matrixToLin_star (g : Matrix (Fin 2) (Fin 2) ℂ) :
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ContinuousLinearMap.coe_inj.mp $ Matrix.toLin_conjTranspose orthonormBasis orthonormBasis g
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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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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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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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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 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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def unitToLinear : unitary (higgsVec →L[ℂ] higgsVec) →* higgsVec →ₗ[ℂ] higgsVec :=
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DistribMulAction.toModuleEnd ℂ higgsVec
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def unitToLinear : unitary (HiggsVec →L[ℂ] HiggsVec) →* HiggsVec →ₗ[ℂ] HiggsVec :=
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DistribMulAction.toModuleEnd ℂ HiggsVec
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/-- The representation of the gauge group acting on `higgsVec`. -/
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@[simps!]
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def rep : Representation ℂ gaugeGroup higgsVec :=
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def rep : Representation ℂ GaugeGroup HiggsVec :=
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unitToLinear.comp (unitaryToLin.comp higgsRepUnitary)
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lemma higgsRepUnitary_mul (g : gaugeGroup) (φ : 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 [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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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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lemma norm_invariant (g : gaugeGroup) (φ : higgsVec) : ‖rep g φ‖ = ‖φ‖ :=
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lemma norm_invariant (g : GaugeGroup) (φ : HiggsVec) : ‖rep g φ‖ = ‖φ‖ :=
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ContinuousLinearMap.norm_map_of_mem_unitary (unitaryToLin (higgsRepUnitary g)).2 φ
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section potentialDefn
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@ -120,13 +119,13 @@ variable (μSq lambda : ℝ)
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local notation "λ" => lambda
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/-- The higgs potential for `higgsVec`, i.e. for constant higgs fields. -/
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def potential (φ : higgsVec) : ℝ := - μSq * ‖φ‖ ^ 2 + λ * ‖φ‖ ^ 4
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def potential (φ : HiggsVec) : ℝ := - μSq * ‖φ‖ ^ 2 + λ * ‖φ‖ ^ 4
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lemma potential_invariant (φ : higgsVec) (g : gaugeGroup) :
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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, norm_invariant]
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lemma potential_as_quad (φ : higgsVec) :
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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]; ring
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@ -138,7 +137,7 @@ variable (μSq : ℝ)
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variable (hLam : 0 < lambda)
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local notation "λ" => lambda
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lemma potential_snd_term_nonneg (φ : higgsVec) :
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lemma potential_snd_term_nonneg (φ : HiggsVec) :
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0 ≤ λ * ‖φ‖ ^ 4 := by
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rw [mul_nonneg_iff]
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apply Or.inl
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@ -146,7 +145,7 @@ lemma potential_snd_term_nonneg (φ : higgsVec) :
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exact le_of_lt hLam
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lemma zero_le_potential_discrim (φ : higgsVec) :
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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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@ -154,7 +153,7 @@ lemma zero_le_potential_discrim (φ : higgsVec) :
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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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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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have h1 := potential_as_quad μSq (λ) φ
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rw [hV] at h1
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@ -169,7 +168,7 @@ lemma potential_eq_zero_sol (φ : higgsVec)
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linear_combination h2
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lemma potential_eq_zero_sol_of_μSq_nonpos (hμSq : μSq ≤ 0)
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(φ : higgsVec) (hV : potential μSq (λ) φ = 0) : φ = 0 := by
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(φ : HiggsVec) (hV : potential μSq (λ) φ = 0) : φ = 0 := by
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cases' (potential_eq_zero_sol μSq hLam φ hV) with h1 h1
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exact h1
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by_cases hμSqZ : μSq = 0
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@ -181,7 +180,7 @@ lemma potential_eq_zero_sol_of_μSq_nonpos (hμSq : μSq ≤ 0)
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· rw [← h1]
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exact sq_nonneg ‖φ‖
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lemma potential_bounded_below (φ : higgsVec) :
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lemma potential_bounded_below (φ : HiggsVec) :
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- μSq ^ 2 / (4 * (λ)) ≤ potential μSq (λ) φ := by
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have h1 := zero_le_potential_discrim μSq hLam φ
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simp [discrim] at h1
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@ -195,17 +194,17 @@ lemma potential_bounded_below (φ : higgsVec) :
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exact h2
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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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(hμSq : μSq ≤ 0) (φ : HiggsVec) : 0 ≤ potential μSq (λ) φ := by
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refine add_nonneg ?_ (potential_snd_term_nonneg hLam φ)
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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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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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field_simp [discrim, hV]
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lemma potential_eq_bound_higgsVec_sq (φ : higgsVec)
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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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‖φ‖ ^ 2 = μSq / (2 * λ) := by
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have h1 := potential_as_quad μSq (λ) φ
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@ -214,7 +213,7 @@ lemma potential_eq_bound_higgsVec_sq (φ : higgsVec)
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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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lemma potential_eq_bound_iff (φ : HiggsVec) :
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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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@ -223,24 +222,24 @@ lemma potential_eq_bound_iff (φ : higgsVec) :
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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 :=
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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)
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(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, 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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(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, 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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∃ (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]
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@ -269,24 +268,24 @@ lemma IsMinOn_potential_iff_of_μSq_nonpos {μSq : ℝ} (hμSq : μSq ≤ 0) :
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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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vector to zero, and the second component to a real -/
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def rotateMatrix (φ : higgsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
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def rotateMatrix (φ : HiggsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
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![![φ 1 /‖φ‖ , - φ 0 /‖φ‖], ![conj (φ 0) / ‖φ‖ , conj (φ 1) / ‖φ‖] ]
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lemma rotateMatrix_star (φ : higgsVec) :
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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_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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lemma rotateMatrix_det {φ : HiggsVec} (hφ : φ ≠ 0) : (rotateMatrix φ).det = 1 := by
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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 [rotateMatrix, det_fin_two]
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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, add_comm]
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lemma rotateMatrix_unitary {φ : higgsVec} (hφ : φ ≠ 0) :
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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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@ -301,16 +300,16 @@ lemma rotateMatrix_unitary {φ : higgsVec} (hφ : φ ≠ 0) :
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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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lemma rotateMatrix_specialUnitary {φ : higgsVec} (hφ : φ ≠ 0) :
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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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mem_specialUnitaryGroup_iff.mpr ⟨rotateMatrix_unitary hφ, rotateMatrix_det hφ⟩
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/-- Given a Higgs vector, an element of the gauge group which puts the first component of the
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vector to zero, and the second component to a real -/
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def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : gaugeGroup :=
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def rotateGuageGroup {φ : HiggsVec} (hφ : φ ≠ 0) : GaugeGroup :=
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⟨1, ⟨(rotateMatrix φ), rotateMatrix_specialUnitary hφ⟩, 1⟩
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lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
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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 only [rotateGuageGroup, rotateMatrix, one_pow, one_smul,
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@ -330,8 +329,8 @@ lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
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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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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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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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by_cases h : φ = 0
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· use ⟨1, 1, 1⟩
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simp [h]
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@ -340,8 +339,7 @@ theorem rotate_fst_zero_snd_real (φ : higgsVec) :
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· use rotateGuageGroup h
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exact rotateGuageGroup_apply h
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end higgsVec
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end higgsVec
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end HiggsVec
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end
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end StandardModel
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