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feat: Add docs

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jstoobysmith 2024-11-11 16:30:02 +00:00
parent 278848247c
commit 7948160e53
3 changed files with 20 additions and 5 deletions
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12
HepLean/StandardModel/HiggsBoson/Potential.lean
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@ -87,11 +87,14 @@ lemma complete_square (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
field_simp
ring
/-- The quadratic equation satisfied by the Higgs potential at a spacetime point `x`. -/
lemma as_quad (φ : HiggsField) (x : SpaceTime) :
P.𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- P.μ2) * ‖φ‖_H ^ 2 x + (- P.toFun φ x) = 0 := by
simp only [normSq, neg_mul, toFun, neg_add_rev, neg_neg]
ring
/-- The Higgs potential is zero iff and only if the higgs field is zero, or the
higgs field has norm-squared `P.μ2 / P.𝓵`, assuming `P.𝓁 = 0`. -/
lemma toFun_eq_zero_iff (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
P.toFun φ x = 0 ↔ φ x = 0 ∨ ‖φ‖_H ^ 2 x = P.μ2 / P.𝓵 := by
refine Iff.intro (fun hV => ?_) (fun hD => ?_)
@ -173,6 +176,8 @@ lemma pos_𝓵_quadDiscrim_zero_bound (h : 0 < P.𝓵) (φ : HiggsField) (x : Sp
rw [neg_le, neg_div'] at h1
exact h1
/-- If `P.𝓵` is negative, then if `P.μ2` is greater then zero, for all space-time points,
the potential is negative `P.toFun φ x ≤ 0`. -/
lemma neg_𝓵_toFun_neg (h : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
(0 < P.μ2 ∧ P.toFun φ x ≤ 0) ∨ P.μ2 ≤ 0 := by
by_cases hμ2 : P.μ2 ≤ 0
@ -187,6 +192,8 @@ lemma neg_𝓵_toFun_neg (h : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
exact mul_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonpos_of_nonneg (le_of_lt h)
(sq_nonneg ‖φ x‖)) (sq_nonneg ‖φ x‖)
/-- If `P.𝓵` is bigger then zero, then if `P.μ2` is less then zero, for all space-time points,
the potential is positive `0 ≤ P.toFun φ x`. -/
lemma pos_𝓵_toFun_pos (h : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
(P.μ2 < 0 ∧ 0 ≤ P.toFun φ x) ∨ 0 ≤ P.μ2 := by
simpa using P.neg.neg_𝓵_toFun_neg (by simpa using h) φ x
@ -263,8 +270,8 @@ lemma pos_𝓵_sol_exists_iff (h𝓵 : 0 < P.𝓵) (c : ℝ) : (∃ φ x, P.toFu
def IsBounded : Prop :=
∃ c, ∀ Φ x, c ≤ P.toFun Φ x
lemma isBounded_𝓵_nonneg (h : P.IsBounded) :
0 ≤ P.𝓵 := by
/-- If the potential is bounded, then `P.𝓵` is non-negative. -/
lemma isBounded_𝓵_nonneg (h : P.IsBounded) : 0 ≤ P.𝓵 := by
by_contra hl
rw [not_le] at hl
obtain ⟨c, hc⟩ := h
@ -308,6 +315,7 @@ lemma isBounded_𝓵_nonneg (h : P.IsBounded) :
rw [hφ] at hc2
linarith
/-- If `P.𝓵` is positive, then the potential is bounded. -/
lemma isBounded_of_𝓵_pos (h : 0 < P.𝓵) : P.IsBounded := by
simp only [IsBounded]
have h2 := P.pos_𝓵_quadDiscrim_zero_bound h