feat: Add informal def for SM gauge group.
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3 changed files with 82 additions and 17 deletions
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@ -6,6 +6,7 @@ Authors: Joseph Tooby-Smith
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Geometry.Manifold.Instances.Real
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import Mathlib.LinearAlgebra.Matrix.ToLin
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import HepLean.Meta.Informal
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/-!
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# The Standard Model
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@ -21,8 +22,65 @@ open Matrix
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open Complex
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open ComplexConjugate
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/-- The global gauge group of the standard model. -/
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abbrev GaugeGroup : Type :=
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/-- The global gauge group of the Standard Model with no discrete quotients.
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The `I` in the Name is an indication of the statement that this has no discrete quotients. -/
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abbrev GaugeGroupI : Type :=
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specialUnitaryGroup (Fin 3) ℂ × specialUnitaryGroup (Fin 2) ℂ × unitary ℂ
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informal_definition gaugeGroupℤ₆SubGroup where
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physics :≈ "The subgroup of the un-quotiented gauge group which acts trivially on
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all particles in the standard model. "
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math :≈ "The ℤ₆-subgroup of ``GaugeGroupI with elements (α^2 * I₃, α^(-3) * I₂, α), where `α`
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is a sixth complex root of unity."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI]
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informal_definition GaugeGroupℤ₆ where
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physics :≈ "The smallest possible gauge group of the Standard Model."
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math :≈ "The quotient of ``GaugeGroupI by the ℤ₆-subgroup `gaugeGroupℤ₆SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₆SubGroup]
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informal_definition gaugeGroupℤ₂SubGroup where
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physics :≈ "The ℤ₂subgroup of the un-quotiented gauge group which acts trivially on
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all particles in the standard model. "
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math :≈ "The ℤ₂-subgroup of ``GaugeGroupI derived from the ℤ₂ subgroup of `gaugeGroupℤ₆SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₆SubGroup]
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informal_definition GaugeGroupℤ₂ where
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physics :≈ "The guage group of the Standard Model with a ℤ₂ quotient."
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math :≈ "The quotient of ``GaugeGroupI by the ℤ₂-subgroup `gaugeGroupℤ₂SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₂SubGroup]
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informal_definition gaugeGroupℤ₃SubGroup where
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physics :≈ "The ℤ₃-subgroup of the un-quotiented gauge group which acts trivially on
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all particles in the standard model. "
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math :≈ "The ℤ₃-subgroup of ``GaugeGroupI derived from the ℤ₃ subgroup of `gaugeGroupℤ₆SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₆SubGroup]
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informal_definition GaugeGroupℤ₃ where
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physics :≈ "The guage group of the Standard Model with a ℤ₃-quotient."
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math :≈ "The quotient of ``GaugeGroupI by the ℤ₃-subgroup `gaugeGroupℤ₃SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₃SubGroup]
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/-- Specifies the allowed quotients of `SU(3) x SU(2) x U(1)` which give a valid
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gauge group of the Standard Model. -/
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inductive GaugeGroupQuot : Type
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| ℤ₆ : GaugeGroupQuot
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| ℤ₂ : GaugeGroupQuot
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| ℤ₃ : GaugeGroupQuot
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| I : GaugeGroupQuot
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informal_definition GaugeGroup where
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physics :≈ "The (global) gauge group of the Standard Model given a choice of quotient."
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math :≈ "The map from `GaugeGroupQuot` to `Type` which gives the gauge group of the Standard Model
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for a given choice of quotient."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup, ``gaugeGroupℤ₃SubGroup,
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``GaugeGroupQuot]
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end StandardModel
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@ -32,7 +32,7 @@ open ComplexConjugate
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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 : GaugeGroupI →* 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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@ -75,14 +75,14 @@ def unitToLinear : unitary (HiggsVec →L[ℂ] HiggsVec) →* HiggsVec →ₗ[
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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 GaugeGroupI 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 : GaugeGroupI) (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 : GaugeGroupI) (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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/-!
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@ -131,7 +131,7 @@ lemma rotateMatrix_specialUnitary {φ : HiggsVec} (hφ : φ ≠ 0) :
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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 number. -/
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def rotateGuageGroup {φ : HiggsVec} (hφ : φ ≠ 0) : GaugeGroup :=
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def rotateGuageGroup {φ : HiggsVec} (hφ : φ ≠ 0) : GaugeGroupI :=
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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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@ -155,7 +155,7 @@ lemma rotateGuageGroup_apply {φ : HiggsVec} (hφ : φ ≠ 0) :
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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, Complex.ofReal ‖φ‖] := by
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∃ (g : GaugeGroupI), rep g φ = ![0, Complex.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 only [Prod.mk_one_one, _root_.map_one, h, map_zero, Nat.succ_eq_add_one, Nat.reduceAdd,
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@ -166,9 +166,9 @@ theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
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exact rotateGuageGroup_apply h
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theorem rotate_fst_real_snd_zero (φ : HiggsVec) :
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∃ (g : GaugeGroup), rep g φ = ![Complex.ofReal ‖φ‖, 0] := by
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∃ (g : GaugeGroupI), rep g φ = ![Complex.ofReal ‖φ‖, 0] := by
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obtain ⟨g, h⟩ := rotate_fst_zero_snd_real φ
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let P : GaugeGroup := ⟨1, ⟨!![0, 1; -1, 0], by
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let P : GaugeGroupI := ⟨1, ⟨!![0, 1; -1, 0], by
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rw [mem_specialUnitaryGroup_iff]
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apply And.intro
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· rw [mem_unitaryGroup_iff, star_eq_conjTranspose]
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@ -190,13 +190,20 @@ theorem rotate_fst_real_snd_zero (φ : HiggsVec) :
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· simp only [Fin.mk_one, Fin.isValue, Pi.smul_apply, Function.comp_apply, cons_val_one, head_cons,
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tail_cons, smul_zero]
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informal_lemma stablity_group where
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informal_lemma stability_group_single where
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physics :≈ "The Higgs boson breaks electroweak symmetry down to the electromagnetic force."
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math :≈ "The stablity group of the action of `rep` on `![0, Complex.ofReal ‖φ‖]`,
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for non-zero `‖φ‖` is the `SU(3) x U(1)` subgroup of
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`gaugeGroup := SU(3) x SU(2) x U(1)` with the embedding given by
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`(g, e^{i θ}) ↦ (g, diag (e ^ {3 * i θ}, e ^ {- 3 * i θ}), e^{i θ})`."
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deps :≈ [`StandardModel.HiggsVec, `StandardModel.HiggsVec.rep]
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deps :≈ [``StandardModel.HiggsVec, ``StandardModel.HiggsVec.rep]
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informal_lemma stability_group where
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math :≈ "The subgroup of `gaugeGroup := SU(3) x SU(2) x U(1)` which preserves every `HiggsVec`
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by the action of ``StandardModel.HiggsVec.rep is given by `SU(3) x ℤ₆` where ℤ₆
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is the subgroup of `SU(2) x U(1)` with elements `(α^(-3) * I₂, α)` where
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α is a sixth root of unity."
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deps :≈ [``StandardModel.HiggsVec, ``StandardModel.HiggsVec.rep]
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end HiggsVec
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