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docs: For SO(3)

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jstoobysmith 2024-11-26 09:33:24 +00:00
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9
HepLean/Mathematics/SO3/Basic.lean
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@ -56,6 +56,7 @@ lemma subtype_val_eq_toGL : (Subtype.val : SO3 → Matrix (Fin 3) (Fin 3) ℝ) =
Units.val ∘ toGL.toFun :=
rfl
/-- The inclusino of `SO(3)` into `GL(3,ℝ)` is an injection. -/
lemma toGL_injective : Function.Injective toGL := by
refine fun A B h ↦ Subtype.eq ?_
rw [@Units.ext_iff] at h
@ -116,6 +117,7 @@ lemma toGL_embedding : IsEmbedding toGL.toFun where
instance : TopologicalGroup SO(3) :=
IsInducing.topologicalGroup toGL toGL_embedding.toIsInducing
/-- The determinant of an `SO(3)` matrix minus the identity is equal to zero. -/
lemma det_minus_id (A : SO(3)) : det (A.1 - 1) = 0 := by
have h1 : det (A.1 - 1) = - det (A.1 - 1) :=
calc
@ -129,6 +131,7 @@ lemma det_minus_id (A : SO(3)) : det (A.1 - 1) = 0 := by
_ = - det (A.1 - 1) := by simp [pow_three]
exact CharZero.eq_neg_self_iff.mp h1
/-- The determinant of the identity minus an `SO(3)` matrix is zero. -/
@[simp]
lemma det_id_minus (A : SO(3)) : det (1 - A.1) = 0 := by
have h1 : det (1 - A.1) = - det (A.1 - 1) := by
@ -139,6 +142,7 @@ lemma det_id_minus (A : SO(3)) : det (1 - A.1) = 0 := by
rw [h1, det_minus_id]
exact neg_zero
/-- For every matrix in `SO(3)`, the real number `1` is in its spectrum. -/
@[simp]
lemma one_in_spectrum (A : SO(3)) : 1 ∈ spectrum ℝ (A.1) := by
rw [spectrum.mem_iff]
@ -155,11 +159,14 @@ def toEnd (A : SO(3)) : End ℝ (EuclideanSpace ℝ (Fin 3)) :=
Matrix.toLin (EuclideanSpace.basisFun (Fin 3) ℝ).toBasis
(EuclideanSpace.basisFun (Fin 3) ℝ).toBasis A.1
/-- Every `SO(3)` matrix has an eigenvalue equal to `1`. -/
lemma one_is_eigenvalue (A : SO(3)) : A.toEnd.HasEigenvalue 1 := by
rw [End.hasEigenvalue_iff_mem_spectrum]
erw [AlgEquiv.spectrum_eq (Matrix.toLinAlgEquiv (EuclideanSpace.basisFun (Fin 3) ℝ).toBasis) A.1]
exact one_in_spectrum A
/-- For every element of `SO(3)` there exists a vector which remains unchanged under the
action of that `SO(3)` element. -/
lemma exists_stationary_vec (A : SO(3)) :
∃ (v : EuclideanSpace ℝ (Fin 3)),
Orthonormal ℝ (({0} : Set (Fin 3)).restrict (fun _ => v))
@ -185,6 +192,8 @@ lemma exists_stationary_vec (A : SO(3)) :
· rw [_root_.map_smul, End.mem_eigenspace_iff.mp hv.1]
simp
/-- For every element of `SO(3)` there exists a basis indexed by `Fin 3` under which the first
element remains invariant. -/
lemma exists_basis_preserved (A : SO(3)) :
∃ (b : OrthonormalBasis (Fin 3) ℝ (EuclideanSpace ℝ (Fin 3))), A.toEnd (b 0) = b 0 := by
obtain ⟨v, hv⟩ := exists_stationary_vec A

7
HepLean/StandardModel/Basic.lean
View file

@ -71,9 +71,16 @@ informal_definition GaugeGroupℤ₃ where
/-- Specifies the allowed quotients of `SU(3) x SU(2) x U(1)` which give a valid
gauge group of the Standard Model. -/
inductive GaugeGroupQuot : Type
/-- The element of `GaugeGroupQuot` corresponding to the quotient of the full SM gauge group
by the sub-group `ℤ₆`. -/
| ℤ₆ : GaugeGroupQuot
/-- The element of `GaugeGroupQuot` corresponding to the quotient of the full SM gauge group
by the sub-group `ℤ₂`. -/
| ℤ₂ : GaugeGroupQuot
/-- The element of `GaugeGroupQuot` corresponding to the quotient of the full SM gauge group
by the sub-group `ℤ₃`. -/
| ℤ₃ : GaugeGroupQuot
/-- The element of `GaugeGroupQuot` corresponding to the full SM gauge group. -/
| I : GaugeGroupQuot
informal_definition GaugeGroup where

2
HepLean/StandardModel/HiggsBoson/Basic.lean
View file

@ -103,6 +103,8 @@ def HiggsVec.toField (φ : HiggsVec) : HiggsField where
rw [Bundle.contMDiffAt_section]
exact smoothAt_const
/-- For all spacetime points, the constant Higgs field defined by a Higgs vector,
returns that Higgs Vector. -/
@[simp]
lemma HiggsVec.toField_apply (φ : HiggsVec) (x : SpaceTime) : (φ.toField x) = φ := rfl