refactor: Change structure of SM file
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jstoobysmith
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5 changed files with 300 additions and 264 deletions
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@ -55,4 +55,6 @@ import HepLean.FlavorPhysics.CKMMatrix.Rows
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import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic
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import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters
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import HepLean.StandardModel.Basic
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import HepLean.StandardModel.HiggsField
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import HepLean.StandardModel.HiggsBoson.Basic
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import HepLean.StandardModel.HiggsBoson.TargetSpace
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import HepLean.StandardModel.Representations
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@ -27,59 +27,8 @@ open ComplexConjugate
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/-- The space-time (TODO: Change to Minkowski.) -/
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abbrev spaceTime := EuclideanSpace ℝ (Fin 4)
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/-- The global gauge group of the standard model. -/
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abbrev guageGroup : Type := specialUnitaryGroup (Fin 3) ℂ ×
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specialUnitaryGroup (Fin 2) ℂ × unitary ℂ
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-- TODO: Move to MathLib.
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lemma star_specialUnitary (g : specialUnitaryGroup (Fin 2) ℂ) :
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star g.1 ∈ specialUnitaryGroup (Fin 2) ℂ := by
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have hg := mem_specialUnitaryGroup_iff.mp g.prop
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rw [@mem_specialUnitaryGroup_iff]
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apply And.intro
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rw [@mem_unitaryGroup_iff, star_star]
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exact mem_unitaryGroup_iff'.mp hg.1
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rw [star_eq_conjTranspose, det_conjTranspose, hg.2, star_one]
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-- TOMOVE
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@[simps!]
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noncomputable def repU1Map (g : unitary ℂ) : unitaryGroup (Fin 2) ℂ :=
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⟨g ^ 3 • 1, by
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rw [mem_unitaryGroup_iff, smul_one_mul, show g = ⟨g.1, g.prop⟩ from rfl]
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simp only [SubmonoidClass.mk_pow, Submonoid.mk_smul, star_smul, star_pow, RCLike.star_def,
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star_one]
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rw [smul_smul, ← mul_pow]
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erw [(unitary.mem_iff.mp g.prop).2]
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simp only [one_pow, one_smul]⟩
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@[simps!]
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noncomputable def repU1 : unitary ℂ →* unitaryGroup (Fin 2) ℂ where
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toFun g := repU1Map g
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map_mul' g h := by
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simp only [repU1Map, Submonoid.mk_mul_mk, mul_smul_one, smul_smul, mul_comm, ← mul_pow]
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map_one' := by
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simp only [repU1Map, one_pow, one_smul, Submonoid.mk_eq_one]
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@[simps!]
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def fundamentalSU2 : specialUnitaryGroup (Fin 2) ℂ →* unitaryGroup (Fin 2) ℂ where
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toFun g := ⟨g.1, g.prop.1⟩
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map_mul' _ _ := Subtype.ext rfl
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map_one' := Subtype.ext rfl
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lemma repU1_fundamentalSU2_commute (u1 : unitary ℂ) (g : specialUnitaryGroup (Fin 2) ℂ) :
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repU1 u1 * fundamentalSU2 g = fundamentalSU2 g * repU1 u1 := by
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apply Subtype.ext
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simp
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noncomputable def higgsRepUnitary : guageGroup →* 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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change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
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rw [repU1.map_mul, fundamentalSU2.map_mul]
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rw [mul_assoc, mul_assoc, ← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
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repeat rw [mul_assoc]
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map_one' := by
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simp
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/-- The global gauge group of the standard model. TODO: Generalize to quotient. -/
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abbrev guageGroup : Type :=
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specialUnitaryGroup (Fin 3) ℂ × specialUnitaryGroup (Fin 2) ℂ × unitary ℂ
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end StandardModel
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200
HepLean/StandardModel/HiggsBoson/Basic.lean
Normal file
200
HepLean/StandardModel/HiggsBoson/Basic.lean
Normal file
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@ -0,0 +1,200 @@
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.StandardModel.Basic
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import HepLean.StandardModel.HiggsBoson.TargetSpace
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Tactic.Polyrith
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import Mathlib.Geometry.Manifold.VectorBundle.Basic
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import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
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import Mathlib.Geometry.Manifold.Instances.Real
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import Mathlib.RepresentationTheory.Basic
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import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Analysis.InnerProductSpace.Adjoint
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import Mathlib.Geometry.Manifold.ContMDiff.Product
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import Mathlib.Analysis.Complex.RealDeriv
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import Mathlib.Algebra.QuadraticDiscriminant
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/-!
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# The Higgs field
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This file defines the basic properties for the higgs field in the standard model.
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## References
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- We use conventions given in: https://pdg.lbl.gov/2019/reviews/rpp2019-rev-higgs-boson.pdf
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-/
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universe v u
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namespace StandardModel
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noncomputable section
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open Manifold
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open Matrix
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open Complex
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open ComplexConjugate
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/-- The trivial vector bundle 𝓡² × ℂ². (TODO: Make associated bundle.) -/
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abbrev higgsBundle := Bundle.Trivial spaceTime higgsVec
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instance : SmoothVectorBundle higgsVec higgsBundle (𝓡 4) :=
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Bundle.Trivial.smoothVectorBundle higgsVec 𝓘(ℝ, spaceTime)
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/-- A higgs field is a smooth section of the higgs bundle. -/
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abbrev higgsField : Type := SmoothSection (𝓡 4) higgsVec higgsBundle
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instance : NormedAddCommGroup (Fin 2 → ℂ) := by
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exact Pi.normedAddCommGroup
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/-- Given a vector `ℂ²` the constant higgs field with value equal to that
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section. -/
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noncomputable def higgsVec.toField (φ : higgsVec) : higgsField where
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toFun := fun _ => φ
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contMDiff_toFun := by
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intro x
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rw [Bundle.contMDiffAt_section]
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exact smoothAt_const
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namespace higgsField
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open Complex Real
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/-- Given a `higgsField`, the corresponding map from `spaceTime` to `higgsVec`. -/
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def toHiggsVec (φ : higgsField) : spaceTime → higgsVec := φ
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lemma toHiggsVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, higgsVec) φ.toHiggsVec := by
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intro x0
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have h1 := φ.contMDiff x0
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rw [Bundle.contMDiffAt_section] at h1
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have h2 :
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(fun x => ((trivializationAt higgsVec (Bundle.Trivial spaceTime higgsVec) x0)
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{ proj := x, snd := φ x }).2) = φ := by
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rfl
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simp only [h2] at h1
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exact h1
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lemma toField_toHiggsVec_apply (φ : higgsField) (x : spaceTime) :
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(φ.toHiggsVec x).toField x = φ x := by
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rfl
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lemma higgsVecToFin2ℂ_toHiggsVec (φ : higgsField) : higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := by
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ext x
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rfl
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lemma toVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ := by
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rw [← φ.higgsVecToFin2ℂ_toHiggsVec]
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exact Smooth.comp smooth_higgsVecToFin2ℂ (φ.toHiggsVec_smooth)
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lemma apply_smooth (φ : higgsField) :
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∀ i, Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℂ) (fun (x : spaceTime) => (φ x i)) := by
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rw [← smooth_pi_space]
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exact φ.toVec_smooth
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lemma apply_re_smooth (φ : higgsField) (i : Fin 2):
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Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : spaceTime) => (φ x i))) :=
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Smooth.comp (ContinuousLinearMap.smooth reCLM) (φ.apply_smooth i)
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lemma apply_im_smooth (φ : higgsField) (i : Fin 2):
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Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : spaceTime) => (φ x i))) :=
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Smooth.comp (ContinuousLinearMap.smooth imCLM) (φ.apply_smooth i)
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/-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the
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higgs vector. -/
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@[simp]
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def normSq (φ : higgsField) : spaceTime → ℝ := fun x => ( ‖φ x‖ ^ 2)
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lemma toHiggsVec_norm (φ : higgsField) (x : spaceTime) :
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‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
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lemma normSq_expand (φ : higgsField) :
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φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by
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funext x
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simp only [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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rw [@norm_sq_eq_inner ℂ]
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simp
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lemma normSq_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
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rw [normSq_expand]
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refine Smooth.add ?_ ?_
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simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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refine Smooth.add ?_ ?_
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refine Smooth.smul ?_ ?_
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exact φ.apply_re_smooth 0
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exact φ.apply_re_smooth 0
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refine Smooth.smul ?_ ?_
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exact φ.apply_im_smooth 0
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exact φ.apply_im_smooth 0
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simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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refine Smooth.add ?_ ?_
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refine Smooth.smul ?_ ?_
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exact φ.apply_re_smooth 1
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exact φ.apply_re_smooth 1
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refine Smooth.smul ?_ ?_
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exact φ.apply_im_smooth 1
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exact φ.apply_im_smooth 1
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lemma normSq_nonneg (φ : higgsField) (x : spaceTime) : 0 ≤ φ.normSq x := by
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simp only [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
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lemma normSq_zero (φ : higgsField) (x : spaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
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simp only [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
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/-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/
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@[simp]
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def potential (φ : higgsField) (μSq lambda : ℝ ) (x : spaceTime) : ℝ :=
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- μSq * φ.normSq x + lambda * φ.normSq x * φ.normSq x
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lemma potential_smooth (φ : higgsField) (μSq lambda : ℝ) :
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Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by
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simp only [potential, normSq, neg_mul]
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exact Smooth.add
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(Smooth.neg (Smooth.smul smooth_const φ.normSq_smooth))
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(Smooth.smul (Smooth.smul smooth_const φ.normSq_smooth) φ.normSq_smooth)
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lemma potential_apply (φ : higgsField) (μSq lambda : ℝ) (x : spaceTime) :
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(φ.potential μSq lambda) x = higgsVec.potential μSq lambda (φ.toHiggsVec x) := by
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simp [higgsVec.potential, toHiggsVec_norm]
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ring
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/-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/
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def isConst (Φ : higgsField) : Prop := ∀ x y, Φ x = Φ y
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lemma isConst_of_higgsVec (φ : higgsVec) : φ.toField.isConst := by
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intro x _
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simp [higgsVec.toField]
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lemma isConst_iff_of_higgsVec (Φ : higgsField) : Φ.isConst ↔ ∃ (φ : higgsVec), Φ = φ.toField := by
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apply Iff.intro
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intro h
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use Φ 0
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ext x y
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rw [← h x 0]
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rfl
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intro h
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intro x y
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obtain ⟨φ, hφ⟩ := h
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subst hφ
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rfl
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lemma normSq_of_higgsVec (φ : higgsVec) : φ.toField.normSq = fun x => (norm φ) ^ 2 := by
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simp only [normSq, higgsVec.toField]
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funext x
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simp
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lemma potential_of_higgsVec (φ : higgsVec) (μSq lambda : ℝ) :
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φ.toField.potential μSq lambda = fun _ => higgsVec.potential μSq lambda φ := by
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simp [higgsVec.potential]
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unfold potential
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rw [normSq_of_higgsVec]
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funext x
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simp only [neg_mul, add_right_inj]
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ring_nf
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end higgsField
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end
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end StandardModel
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@ -4,6 +4,7 @@ Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.StandardModel.Basic
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import HepLean.StandardModel.Representations
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Tactic.Polyrith
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import Mathlib.Geometry.Manifold.VectorBundle.Basic
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@ -13,14 +14,14 @@ import Mathlib.RepresentationTheory.Basic
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import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Analysis.InnerProductSpace.Adjoint
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import Mathlib.Geometry.Manifold.ContMDiff.Product
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import Mathlib.Analysis.Complex.RealDeriv
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import Mathlib.Analysis.Calculus.Deriv.Add
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import Mathlib.Analysis.Calculus.Deriv.Pow
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import Mathlib.Algebra.QuadraticDiscriminant
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/-!
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# The Higgs field
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# The Higgs vector space
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This file defines the basic properties for the higgs field in the standard model.
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This file defines the target vector space of the Higgs boson, the potential on it,
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and the representation of the SM gauge group acting on it.
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This file is a import of `SM.HiggsBoson.Basic`.
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## References
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@ -39,24 +40,8 @@ open ComplexConjugate
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/-- The complex vector space in which the Higgs field takes values. -/
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abbrev higgsVec := EuclideanSpace ℂ (Fin 2)
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/-- The trivial vector bundle 𝓡² × ℂ². (TODO: Make associated bundle.) -/
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abbrev higgsBundle := Bundle.Trivial spaceTime higgsVec
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instance : SmoothVectorBundle higgsVec higgsBundle (𝓡 4) :=
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Bundle.Trivial.smoothVectorBundle higgsVec 𝓘(ℝ, spaceTime)
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/-- A higgs field is a smooth section of the higgs bundle. -/
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abbrev higgsField : Type := SmoothSection (𝓡 4) higgsVec higgsBundle
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instance : NormedAddCommGroup (Fin 2 → ℂ) := by
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exact Pi.normedAddCommGroup
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section higgsVec
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/-- The continous linear map from the vector space `higgsVec` to `(Fin 2 → ℂ)` acheived by
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casting vectors. -/
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def higgsVecToFin2ℂ : higgsVec →L[ℝ] (Fin 2 → ℂ) where
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@ -69,41 +54,20 @@ def higgsVecToFin2ℂ : higgsVec →L[ℝ] (Fin 2 → ℂ) where
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lemma smooth_higgsVecToFin2ℂ : Smooth 𝓘(ℝ, higgsVec) 𝓘(ℝ, Fin 2 → ℂ) higgsVecToFin2ℂ :=
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ContinuousLinearMap.smooth higgsVecToFin2ℂ
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/-- Given an element of `gaugeGroup` the linear automorphism of `higgsVec` it gets taken to. -/
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@[simps!]
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noncomputable def higgsRepMap (g : guageGroup) : higgsVec →L[ℂ] higgsVec where
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toFun S := (g.2.2 ^ 3) • (g.2.1.1 *ᵥ S)
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map_add' S T := by
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simp [Matrix.mulVec_add, smul_add]
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rw [Matrix.mulVec_add, smul_add]
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map_smul' a S := by
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simp [Matrix.mulVec_smul]
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rw [Matrix.mulVec_smul]
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exact smul_comm _ _ _
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cont := by
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exact (continuous_const_smul_iff _).mpr (Continuous.matrix_mulVec continuous_const
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(Pi.continuous_precomp fun x => x))
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/-- The representation of the SM guage group acting on `ℂ²`. -/
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noncomputable def higgsRep : Representation ℂ guageGroup higgsVec where
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toFun g := (higgsRepMap g).toLinearMap
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map_mul' U V := by
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apply LinearMap.ext
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intro S
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simp only [higgsRepMap, Prod.snd_mul, Submonoid.coe_inf, Prod.fst_mul, Submonoid.coe_mul,
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LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply, LinearMap.map_smul_of_tower,
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mulVec_mulVec]
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simp [mul_pow, smul_smul, mul_comm]
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map_one' := by
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apply LinearMap.ext
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intro S
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simp only [higgsRepMap, LinearMap.mul_apply, AddHom.coe_mk, LinearMap.coe_mk]
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change 1 ^ 3 • (1 *ᵥ _) = _
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rw [one_pow, Matrix.one_mulVec]
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simp only [one_smul, LinearMap.one_apply]
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namespace higgsVec
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@[simps!]
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noncomputable def higgsRepUnitary : guageGroup →* 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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change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
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rw [repU1.map_mul, fundamentalSU2.map_mul]
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rw [mul_assoc, mul_assoc, ← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
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repeat rw [mul_assoc]
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map_one' := by
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simp only [Prod.snd_one, _root_.map_one, Prod.fst_one, mul_one]
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/-- An orthonomral basis of higgsVec. -/
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noncomputable def orthonormBasis : OrthonormalBasis (Fin 2) ℂ higgsVec :=
|
||||
EuclideanSpace.basisFun (Fin 2) ℂ
|
||||
|
|
@ -119,7 +83,7 @@ noncomputable def matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* (higgsVec →L[
|
|||
|
||||
lemma matrixToLin_star (g : Matrix (Fin 2) (Fin 2) ℂ) :
|
||||
matrixToLin (star g) = star (matrixToLin g) :=
|
||||
ContinuousLinearMap.coe_inj.mp (Matrix.toLin_conjTranspose orthonormBasis orthonormBasis g)
|
||||
ContinuousLinearMap.coe_inj.mp $ Matrix.toLin_conjTranspose orthonormBasis orthonormBasis g
|
||||
|
||||
lemma matrixToLin_unitary (g : unitaryGroup (Fin 2) ℂ) :
|
||||
matrixToLin g ∈ unitary (higgsVec →L[ℂ] higgsVec) := by
|
||||
|
|
@ -127,6 +91,7 @@ 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!]
|
||||
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
|
||||
|
|
@ -144,18 +109,18 @@ def unitToLinear : unitary (higgsVec →L[ℂ] higgsVec) →* higgsVec →ₗ[
|
|||
def rep : Representation ℂ guageGroup higgsVec :=
|
||||
unitToLinear.comp (unitaryToLin.comp higgsRepUnitary)
|
||||
|
||||
lemma higgsRepUnitary_mul (g : guageGroup) (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) :=
|
||||
higgsRepUnitary_mul g φ
|
||||
|
||||
|
||||
lemma norm_invariant (g : guageGroup) (φ : higgsVec) : ‖rep g φ‖ = ‖φ‖ :=
|
||||
ContinuousLinearMap.norm_map_of_mem_unitary (unitaryToLin (higgsRepUnitary g)).2 φ
|
||||
|
||||
/-- Given a vector `ℂ²` the constant higgs field with value equal to that
|
||||
section. -/
|
||||
noncomputable def toField (φ : higgsVec) : higgsField where
|
||||
toFun := fun _ => φ
|
||||
contMDiff_toFun := by
|
||||
intro x
|
||||
rw [Bundle.contMDiffAt_section]
|
||||
exact smoothAt_const
|
||||
|
||||
/-- 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
|
||||
|
|
@ -178,7 +143,7 @@ lemma potential_as_quad (μSq lambda : ℝ) (φ : higgsVec) :
|
|||
ring
|
||||
|
||||
lemma zero_le_potential_discrim (μSq lambda : ℝ) (φ : higgsVec) (hLam : 0 < lambda) :
|
||||
0 ≤ discrim (lambda ) (- μSq ) (- potential μSq lambda φ) := by
|
||||
0 ≤ discrim (lambda) (- μSq ) (- potential μSq lambda φ) := by
|
||||
have h1 := potential_as_quad μSq lambda φ
|
||||
rw [quadratic_eq_zero_iff_discrim_eq_sq] at h1
|
||||
rw [h1]
|
||||
|
|
@ -378,8 +343,9 @@ def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : guageGroup :=
|
|||
⟨1, ⟨(rotateMatrix φ), rotateMatrix_specialUnitary hφ⟩, 1⟩
|
||||
|
||||
lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
|
||||
higgsRep (rotateGuageGroup hφ) φ = ![0, ofReal ‖φ‖] := by
|
||||
simp [higgsRep, higgsRepMap, rotateGuageGroup, rotateMatrix, higgsRepMap]
|
||||
rep (rotateGuageGroup hφ) φ = ![0, ofReal ‖φ‖] := by
|
||||
rw [rep_apply]
|
||||
simp [rotateGuageGroup, rotateMatrix]
|
||||
ext i
|
||||
fin_cases i
|
||||
simp [mulVec, vecHead, vecTail]
|
||||
|
|
@ -394,10 +360,10 @@ lemma rotateGuageGroup_apply {φ : higgsVec} (hφ : φ ≠ 0) :
|
|||
rfl
|
||||
|
||||
theorem rotate_fst_zero_snd_real (φ : higgsVec) :
|
||||
∃ (g : guageGroup), higgsRep g φ = ![0, ofReal ‖φ‖] := by
|
||||
∃ (g : guageGroup), rep g φ = ![0, ofReal ‖φ‖] := by
|
||||
by_cases h : φ = 0
|
||||
· use ⟨1, 1, 1⟩
|
||||
simp [h, higgsRep, higgsRepMap]
|
||||
simp [h]
|
||||
ext i
|
||||
fin_cases i <;> rfl
|
||||
· use rotateGuageGroup h
|
||||
|
|
@ -406,145 +372,5 @@ theorem rotate_fst_zero_snd_real (φ : higgsVec) :
|
|||
end higgsVec
|
||||
end higgsVec
|
||||
|
||||
namespace higgsField
|
||||
open Complex Real
|
||||
|
||||
/-- Given a `higgsField`, the corresponding map from `spaceTime` to `higgsVec`. -/
|
||||
def toHiggsVec (φ : higgsField) : spaceTime → higgsVec := φ
|
||||
|
||||
|
||||
lemma toHiggsVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, higgsVec) φ.toHiggsVec := by
|
||||
intro x0
|
||||
have h1 := φ.contMDiff x0
|
||||
rw [Bundle.contMDiffAt_section] at h1
|
||||
have h2 :
|
||||
(fun x => ((trivializationAt higgsVec (Bundle.Trivial spaceTime higgsVec) x0)
|
||||
{ proj := x, snd := φ x }).2) = φ := by
|
||||
rfl
|
||||
simp only [h2] at h1
|
||||
exact h1
|
||||
|
||||
lemma toField_toHiggsVec_apply (φ : higgsField) (x : spaceTime) :
|
||||
(φ.toHiggsVec x).toField x = φ x := by
|
||||
rfl
|
||||
|
||||
lemma higgsVecToFin2ℂ_toHiggsVec (φ : higgsField) : higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := by
|
||||
ext x
|
||||
rfl
|
||||
|
||||
lemma toVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ := by
|
||||
rw [← φ.higgsVecToFin2ℂ_toHiggsVec]
|
||||
exact Smooth.comp smooth_higgsVecToFin2ℂ (φ.toHiggsVec_smooth)
|
||||
|
||||
lemma apply_smooth (φ : higgsField) :
|
||||
∀ i, Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℂ) (fun (x : spaceTime) => (φ x i)) := by
|
||||
rw [← smooth_pi_space]
|
||||
exact φ.toVec_smooth
|
||||
|
||||
lemma apply_re_smooth (φ : higgsField) (i : Fin 2):
|
||||
Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : spaceTime) => (φ x i))) :=
|
||||
Smooth.comp (ContinuousLinearMap.smooth reCLM) (φ.apply_smooth i)
|
||||
|
||||
lemma apply_im_smooth (φ : higgsField) (i : Fin 2):
|
||||
Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : spaceTime) => (φ x i))) :=
|
||||
Smooth.comp (ContinuousLinearMap.smooth imCLM) (φ.apply_smooth i)
|
||||
|
||||
/-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the
|
||||
higgs vector. -/
|
||||
@[simp]
|
||||
def normSq (φ : higgsField) : spaceTime → ℝ := fun x => ( ‖φ x‖ ^ 2)
|
||||
|
||||
lemma toHiggsVec_norm (φ : higgsField) (x : spaceTime) :
|
||||
‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
|
||||
|
||||
lemma normSq_expand (φ : higgsField) :
|
||||
φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by
|
||||
funext x
|
||||
simp only [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
|
||||
rw [@norm_sq_eq_inner ℂ]
|
||||
simp
|
||||
|
||||
lemma normSq_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
|
||||
rw [normSq_expand]
|
||||
refine Smooth.add ?_ ?_
|
||||
simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
|
||||
refine Smooth.add ?_ ?_
|
||||
refine Smooth.smul ?_ ?_
|
||||
exact φ.apply_re_smooth 0
|
||||
exact φ.apply_re_smooth 0
|
||||
refine Smooth.smul ?_ ?_
|
||||
exact φ.apply_im_smooth 0
|
||||
exact φ.apply_im_smooth 0
|
||||
simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
|
||||
refine Smooth.add ?_ ?_
|
||||
refine Smooth.smul ?_ ?_
|
||||
exact φ.apply_re_smooth 1
|
||||
exact φ.apply_re_smooth 1
|
||||
refine Smooth.smul ?_ ?_
|
||||
exact φ.apply_im_smooth 1
|
||||
exact φ.apply_im_smooth 1
|
||||
|
||||
lemma normSq_nonneg (φ : higgsField) (x : spaceTime) : 0 ≤ φ.normSq x := by
|
||||
simp only [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
|
||||
|
||||
lemma normSq_zero (φ : higgsField) (x : spaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
|
||||
simp only [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
|
||||
|
||||
/-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/
|
||||
@[simp]
|
||||
def potential (φ : higgsField) (μSq lambda : ℝ ) (x : spaceTime) : ℝ :=
|
||||
- μSq * φ.normSq x + lambda * φ.normSq x * φ.normSq x
|
||||
|
||||
lemma potential_smooth (φ : higgsField) (μSq lambda : ℝ) :
|
||||
Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by
|
||||
simp only [potential, normSq, neg_mul]
|
||||
exact Smooth.add
|
||||
(Smooth.neg (Smooth.smul smooth_const φ.normSq_smooth))
|
||||
(Smooth.smul (Smooth.smul smooth_const φ.normSq_smooth) φ.normSq_smooth)
|
||||
|
||||
|
||||
lemma potential_apply (φ : higgsField) (μSq lambda : ℝ) (x : spaceTime) :
|
||||
(φ.potential μSq lambda) x = higgsVec.potential μSq lambda (φ.toHiggsVec x) := by
|
||||
simp [higgsVec.potential, toHiggsVec_norm]
|
||||
ring
|
||||
|
||||
|
||||
/-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/
|
||||
def isConst (Φ : higgsField) : Prop := ∀ x y, Φ x = Φ y
|
||||
|
||||
lemma isConst_of_higgsVec (φ : higgsVec) : φ.toField.isConst := by
|
||||
intro x _
|
||||
simp [higgsVec.toField]
|
||||
|
||||
lemma isConst_iff_of_higgsVec (Φ : higgsField) : Φ.isConst ↔ ∃ (φ : higgsVec), Φ = φ.toField := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
use Φ 0
|
||||
ext x y
|
||||
rw [← h x 0]
|
||||
rfl
|
||||
intro h
|
||||
intro x y
|
||||
obtain ⟨φ, hφ⟩ := h
|
||||
subst hφ
|
||||
rfl
|
||||
|
||||
lemma normSq_of_higgsVec (φ : higgsVec) : φ.toField.normSq = fun x => (norm φ) ^ 2 := by
|
||||
simp only [normSq, higgsVec.toField]
|
||||
funext x
|
||||
simp
|
||||
|
||||
lemma potential_of_higgsVec (φ : higgsVec) (μSq lambda : ℝ) :
|
||||
φ.toField.potential μSq lambda = fun _ => higgsVec.potential μSq lambda φ := by
|
||||
simp [higgsVec.potential]
|
||||
unfold potential
|
||||
rw [normSq_of_higgsVec]
|
||||
funext x
|
||||
simp only [neg_mul, add_right_inj]
|
||||
ring_nf
|
||||
|
||||
|
||||
|
||||
end higgsField
|
||||
end
|
||||
end StandardModel
|
||||
59
HepLean/StandardModel/Representations.lean
Normal file
59
HepLean/StandardModel/Representations.lean
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.StandardModel.Basic
|
||||
import Mathlib.Data.Complex.Exponential
|
||||
import Mathlib.Geometry.Manifold.VectorBundle.Basic
|
||||
import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
|
||||
import Mathlib.Geometry.Manifold.Instances.Real
|
||||
import Mathlib.RepresentationTheory.Basic
|
||||
import Mathlib.LinearAlgebra.Matrix.ToLin
|
||||
import Mathlib.Analysis.InnerProductSpace.Adjoint
|
||||
/-!
|
||||
# Representations appearing in the Standard Model
|
||||
|
||||
This file defines the basic representations which appear in the Standard Model.
|
||||
|
||||
-/
|
||||
universe v u
|
||||
namespace StandardModel
|
||||
|
||||
open Manifold
|
||||
open Matrix
|
||||
open Complex
|
||||
open ComplexConjugate
|
||||
|
||||
|
||||
@[simps!]
|
||||
noncomputable def repU1Map (g : unitary ℂ) : unitaryGroup (Fin 2) ℂ :=
|
||||
⟨g ^ 3 • 1, by
|
||||
rw [mem_unitaryGroup_iff, smul_one_mul, show g = ⟨g.1, g.prop⟩ from rfl]
|
||||
simp only [SubmonoidClass.mk_pow, Submonoid.mk_smul, star_smul, star_pow, RCLike.star_def,
|
||||
star_one]
|
||||
rw [smul_smul, ← mul_pow]
|
||||
erw [(unitary.mem_iff.mp g.prop).2]
|
||||
simp only [one_pow, one_smul]⟩
|
||||
|
||||
@[simps!]
|
||||
noncomputable def repU1 : unitary ℂ →* unitaryGroup (Fin 2) ℂ where
|
||||
toFun g := repU1Map g
|
||||
map_mul' g h := by
|
||||
simp only [repU1Map, Submonoid.mk_mul_mk, mul_smul_one, smul_smul, mul_comm, ← mul_pow]
|
||||
map_one' := by
|
||||
simp only [repU1Map, one_pow, one_smul, Submonoid.mk_eq_one]
|
||||
|
||||
@[simps!]
|
||||
def fundamentalSU2 : specialUnitaryGroup (Fin 2) ℂ →* unitaryGroup (Fin 2) ℂ where
|
||||
toFun g := ⟨g.1, g.prop.1⟩
|
||||
map_mul' _ _ := Subtype.ext rfl
|
||||
map_one' := Subtype.ext rfl
|
||||
|
||||
lemma repU1_fundamentalSU2_commute (u1 : unitary ℂ) (g : specialUnitaryGroup (Fin 2) ℂ) :
|
||||
repU1 u1 * fundamentalSU2 g = fundamentalSU2 g * repU1 u1 := by
|
||||
apply Subtype.ext
|
||||
simp
|
||||
|
||||
|
||||
end StandardModel
|
||||
Loading…
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Reference in a new issue