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refactor: Lint

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jstoobysmith 2024-05-09 15:16:38 -04:00
parent 5af2eb4d8d
commit 8667120377
3 changed files with 18 additions and 12 deletions
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23
HepLean/StandardModel/HiggsBoson/TargetSpace.lean
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@ -56,8 +56,9 @@ lemma smooth_higgsVecToFin2ℂ : Smooth 𝓘(ℝ, higgsVec) 𝓘(ℝ, Fin 2 →
namespace higgsVec
/-- The Higgs representation as a homomorphism from the gauge group to unitary `2×2`-matrices. -/
@[simps!]
noncomputable def higgsRepUnitary : guageGroup →* unitaryGroup (Fin 2) ℂ where
noncomputable def higgsRepUnitary : gaugeGroup →* unitaryGroup (Fin 2) ℂ where
toFun g := repU1 g.2.2 * fundamentalSU2 g.2.1
map_mul' := by
intro ⟨_, a2, a3⟩ ⟨_, b2, b3⟩
@ -73,7 +74,6 @@ noncomputable def orthonormBasis : OrthonormalBasis (Fin 2) ℂ higgsVec :=
EuclideanSpace.basisFun (Fin 2) ℂ
/-- Takes in a `2×2`-matrix and returns a linear map of `higgsVec`. -/
@[simps!]
noncomputable def matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* (higgsVec →L[ℂ] higgsVec) where
toFun g := LinearMap.toContinuousLinearMap
$ Matrix.toLin orthonormBasis.toBasis orthonormBasis.toBasis g
@ -91,7 +91,8 @@ lemma matrixToLin_unitary (g : unitaryGroup (Fin 2) ℂ) :
rw [mem_unitaryGroup_iff.mp g.prop, mem_unitaryGroup_iff'.mp g.prop, matrixToLin.map_one]
simp
@[simps!]
/-- The natural homomorphism from unitary `2×2` complex matrices to unitary transformations
of `higgsVec`. -/
noncomputable def unitaryToLin : unitaryGroup (Fin 2) ℂ →* unitary (higgsVec →L[ℂ] higgsVec) where
toFun g := ⟨matrixToLin g, matrixToLin_unitary g⟩
map_mul' g h := by
@ -101,31 +102,33 @@ noncomputable def unitaryToLin : unitaryGroup (Fin 2) ℂ →* unitary (higgsVec
ext
simp
/-- The inclusion of unitary transformations on `higgsVec` into all linear transformations. -/
@[simps!]
def unitToLinear : unitary (higgsVec →L[ℂ] higgsVec) →* higgsVec →ₗ[ℂ] higgsVec :=
DistribMulAction.toModuleEnd ℂ higgsVec
/-- The representation of the gauge group acting on `higgsVec`. -/
@[simps!]
def rep : Representation ℂ guageGroup higgsVec :=
def rep : Representation ℂ gaugeGroup higgsVec :=
unitToLinear.comp (unitaryToLin.comp higgsRepUnitary)
lemma higgsRepUnitary_mul (g : guageGroup) (φ : higgsVec) :
lemma higgsRepUnitary_mul (g : gaugeGroup) (φ : higgsVec) :
(higgsRepUnitary g).1 *ᵥ φ = g.2.2 ^ 3 • (g.2.1.1 *ᵥ φ) := by
simp only [higgsRepUnitary_apply_coe]
exact smul_mulVec_assoc (g.2.2 ^ 3) (g.2.1.1) φ
lemma rep_apply (g : guageGroup) (φ : higgsVec) : rep g φ = g.2.2 ^ 3 • (g.2.1.1 *ᵥ φ) :=
lemma rep_apply (g : gaugeGroup) (φ : higgsVec) : rep g φ = g.2.2 ^ 3 • (g.2.1.1 *ᵥ φ) :=
higgsRepUnitary_mul g φ
lemma norm_invariant (g : guageGroup) (φ : higgsVec) : ‖rep g φ‖ = ‖φ‖ :=
lemma norm_invariant (g : gaugeGroup) (φ : higgsVec) : ‖rep g φ‖ = ‖φ‖ :=
ContinuousLinearMap.norm_map_of_mem_unitary (unitaryToLin (higgsRepUnitary g)).2 φ
/-- The higgs potential for `higgsVec`, i.e. for constant higgs fields. -/
def potential (μSq lambda : ℝ) (φ : higgsVec) : ℝ := - μSq * ‖φ‖ ^ 2 +
lambda * ‖φ‖ ^ 4
lemma potential_invariant (μSq lambda : ℝ) (φ : higgsVec) (g : guageGroup) :
lemma potential_invariant (μSq lambda : ℝ) (φ : higgsVec) (g : gaugeGroup) :
potential μSq lambda (rep g φ) = potential μSq lambda φ := by
simp only [potential, neg_mul]
rw [norm_invariant]
@ -339,7 +342,7 @@ lemma rotateMatrix_specialUnitary {φ : higgsVec} (hφ : φ ≠ 0) :
/-- Given a Higgs vector, an element of the gauge group which puts the fst component of the
vector to zero, and the snd componenet to a real -/
def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : guageGroup :=
def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : gaugeGroup :=
⟨1, ⟨(rotateMatrix φ), rotateMatrix_specialUnitary hφ⟩, 1⟩
lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
@ -360,7 +363,7 @@ lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
rfl
theorem rotate_fst_zero_snd_real (φ : higgsVec) :
∃ (g : guageGroup), rep g φ = ![0, ofReal ‖φ‖] := by
∃ (g : gaugeGroup), rep g φ = ![0, ofReal ‖φ‖] := by
by_cases h : φ = 0
· use ⟨1, 1, 1⟩
simp [h]