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refactor: pass at removing double spaces

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jstoobysmith 2024-07-12 10:36:39 -04:00
parent 1fe51b2e04
commit 1133b883f3
19 changed files with 121 additions and 121 deletions
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22
HepLean/StandardModel/HiggsBoson/GaugeAction.lean
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@ -34,9 +34,9 @@ open ComplexConjugate
@[simps!]
noncomputable def higgsRepUnitary : GaugeGroup →* unitaryGroup (Fin 2) ℂ where
toFun g := repU1 g.2.2 * fundamentalSU2 g.2.1
map_mul' := by
map_mul' := by
intro ⟨_, a2, a3⟩ ⟨_, b2, b3⟩
change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
rw [repU1.map_mul, fundamentalSU2.map_mul, mul_assoc, mul_assoc,
← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
repeat rw [mul_assoc]
@ -97,31 +97,31 @@ def rotateMatrix (φ : HiggsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
lemma rotateMatrix_star (φ : HiggsVec) :
star φ.rotateMatrix =
![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
simp_rw [star, rotateMatrix, conjTranspose]
ext i j
fin_cases i <;> fin_cases j <;> simp [conj_ofReal]
lemma rotateMatrix_det {φ : HiggsVec} (hφ : φ ≠ 0) : (rotateMatrix φ).det = 1 := by
have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
field_simp [rotateMatrix, det_fin_two]
rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
lemma rotateMatrix_unitary {φ : HiggsVec} (hφ : φ ≠ 0) :
(rotateMatrix φ) ∈ unitaryGroup (Fin 2) ℂ := by
rw [mem_unitaryGroup_iff', rotateMatrix_star, rotateMatrix]
erw [mul_fin_two, one_fin_two]
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
ext i j
fin_cases i <;> fin_cases j <;> field_simp
<;> rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
· ring_nf
· ring_nf
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
lemma rotateMatrix_specialUnitary {φ : HiggsVec} (hφ : φ ≠ 0) :
@ -147,10 +147,10 @@ lemma rotateGuageGroup_apply {φ : HiggsVec} (hφ : φ ≠ 0) :
· simp only [Fin.mk_one, Fin.isValue, cons_val_one, head_cons, mulVec, Fin.isValue,
cons_val', empty_val', cons_val_fin_one, vecHead, cons_dotProduct, vecTail, Nat.succ_eq_add_one,
Nat.reduceAdd, Function.comp_apply, Fin.succ_zero_eq_one, dotProduct_empty, add_zero]
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
field_simp
rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
@ -166,7 +166,7 @@ theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
end HiggsVec
/-! TODO: Define the global gauge action on HiggsField. -/
/-! TODO: Prove `⟪φ1, φ2⟫_H` invariant under the global gauge action. (norm_map_of_mem_unitary) -/
/-! TODO: Prove `⟪φ1, φ2⟫_H` invariant under the global gauge action. (norm_map_of_mem_unitary) -/
/-! TODO: Prove invariance of potential under global gauge action. -/
end StandardModel