376 lines
14 KiB
Text
376 lines
14 KiB
Text
/-
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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.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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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.Algebra.QuadraticDiscriminant
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/-!
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# The Higgs vector space
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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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- 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 complex vector space in which the Higgs field takes values. -/
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abbrev higgsVec := EuclideanSpace ℂ (Fin 2)
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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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toFun x := x
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map_add' x y := by
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simp
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map_smul' a x := by
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simp
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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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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 :=
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EuclideanSpace.basisFun (Fin 2) ℂ
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/-- Takes in a `2×2`-matrix and returns a linear map of `higgsVec`. -/
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@[simps!]
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noncomputable def matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* (higgsVec →L[ℂ] higgsVec) where
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toFun g := LinearMap.toContinuousLinearMap
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$ Matrix.toLin orthonormBasis.toBasis orthonormBasis.toBasis g
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map_mul' g h := ContinuousLinearMap.coe_inj.mp $
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Matrix.toLin_mul orthonormBasis.toBasis orthonormBasis.toBasis orthonormBasis.toBasis g h
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map_one' := ContinuousLinearMap.coe_inj.mp $ Matrix.toLin_one orthonormBasis.toBasis
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lemma matrixToLin_star (g : Matrix (Fin 2) (Fin 2) ℂ) :
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matrixToLin (star g) = star (matrixToLin g) :=
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ContinuousLinearMap.coe_inj.mp $ Matrix.toLin_conjTranspose orthonormBasis orthonormBasis g
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lemma matrixToLin_unitary (g : unitaryGroup (Fin 2) ℂ) :
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matrixToLin g ∈ unitary (higgsVec →L[ℂ] higgsVec) := by
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rw [@unitary.mem_iff, ← matrixToLin_star, ← matrixToLin.map_mul, ← matrixToLin.map_mul]
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rw [mem_unitaryGroup_iff.mp g.prop, mem_unitaryGroup_iff'.mp g.prop, matrixToLin.map_one]
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simp
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@[simps!]
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noncomputable def unitaryToLin : unitaryGroup (Fin 2) ℂ →* unitary (higgsVec →L[ℂ] higgsVec) where
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toFun g := ⟨matrixToLin g, matrixToLin_unitary g⟩
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map_mul' g h := by
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ext
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simp
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map_one' := by
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ext
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simp
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@[simps!]
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def unitToLinear : unitary (higgsVec →L[ℂ] higgsVec) →* higgsVec →ₗ[ℂ] higgsVec :=
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DistribMulAction.toModuleEnd ℂ higgsVec
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@[simps!]
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def rep : Representation ℂ guageGroup higgsVec :=
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unitToLinear.comp (unitaryToLin.comp higgsRepUnitary)
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lemma higgsRepUnitary_mul (g : guageGroup) (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 only [higgsRepUnitary_apply_coe]
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exact smul_mulVec_assoc (g.2.2 ^ 3) (g.2.1.1) φ
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lemma rep_apply (g : guageGroup) (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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lemma norm_invariant (g : guageGroup) (φ : higgsVec) : ‖rep g φ‖ = ‖φ‖ :=
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ContinuousLinearMap.norm_map_of_mem_unitary (unitaryToLin (higgsRepUnitary g)).2 φ
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/-- The higgs potential for `higgsVec`, i.e. for constant higgs fields. -/
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def potential (μSq lambda : ℝ) (φ : higgsVec) : ℝ := - μSq * ‖φ‖ ^ 2 +
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lambda * ‖φ‖ ^ 4
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lemma potential_invariant (μSq lambda : ℝ) (φ : higgsVec) (g : guageGroup) :
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potential μSq lambda (rep g φ) = potential μSq lambda φ := by
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simp only [potential, neg_mul]
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rw [norm_invariant]
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lemma potential_snd_term_nonneg {lambda : ℝ} (hLam : 0 < lambda) (φ : higgsVec) :
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0 ≤ lambda * ‖φ‖ ^ 4 := by
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rw [mul_nonneg_iff]
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apply Or.inl
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simp_all only [ge_iff_le, norm_nonneg, pow_nonneg, and_true]
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exact le_of_lt hLam
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lemma potential_as_quad (μSq lambda : ℝ) (φ : higgsVec) :
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lambda * ‖φ‖ ^ 2 * ‖φ‖ ^ 2 + (- μSq ) * ‖φ‖ ^ 2 + (- potential μSq lambda φ ) = 0 := by
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simp [potential]
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ring
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lemma zero_le_potential_discrim (μSq lambda : ℝ) (φ : higgsVec) (hLam : 0 < lambda) :
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0 ≤ discrim (lambda) (- μSq ) (- potential μSq lambda φ) := by
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have h1 := potential_as_quad μSq lambda φ
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rw [quadratic_eq_zero_iff_discrim_eq_sq] at h1
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rw [h1]
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exact sq_nonneg (2 * (lambda ) * ‖φ‖ ^ 2 + -μSq)
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simp only [ne_eq, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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exact ne_of_gt hLam
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lemma potential_eq_zero_sol (μSq lambda : ℝ) (hLam : 0 < lambda)(φ : higgsVec)
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(hV : potential μSq lambda φ = 0) : φ = 0 ∨ ‖φ‖ ^ 2 = μSq / lambda := by
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have h1 := potential_as_quad μSq lambda φ
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rw [hV] at h1
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have h2 : ‖φ‖ ^ 2 * (lambda * ‖φ‖ ^ 2 + -μSq ) = 0 := by
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linear_combination h1
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simp at h2
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cases' h2 with h2 h2
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simp_all
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apply Or.inr
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field_simp at h2 ⊢
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ring_nf
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linear_combination h2
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lemma potential_eq_zero_sol_of_μSq_nonpos {μSq lambda : ℝ} (hLam : 0 < lambda) (hμSq : μSq ≤ 0)
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(φ : higgsVec) (hV : potential μSq lambda φ = 0) : φ = 0 := by
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cases' (potential_eq_zero_sol μSq lambda hLam φ hV) with h1 h1
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exact h1
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by_cases hμSqZ : μSq = 0
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simpa [hμSqZ] using h1
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refine ((?_ : ¬ 0 ≤ μSq / lambda) (?_)).elim
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· simp_all [div_nonneg_iff]
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intro h
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exact lt_imp_lt_of_le_imp_le (fun _ => h) (lt_of_le_of_ne hμSq hμSqZ)
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· rw [← h1]
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exact sq_nonneg ‖φ‖
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lemma potential_bounded_below (μSq lambda : ℝ) (hLam : 0 < lambda) (φ : higgsVec) :
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- μSq ^ 2 / (4 * lambda) ≤ potential μSq lambda φ := by
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have h1 := zero_le_potential_discrim μSq lambda φ hLam
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simp [discrim] at h1
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ring_nf at h1
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rw [← neg_le_iff_add_nonneg'] at h1
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have h3 : lambda * potential μSq lambda φ * 4 = (4 * lambda) * potential μSq lambda φ := by
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ring
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rw [h3] at h1
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have h2 := (div_le_iff' (by simp [hLam] : 0 < 4 * lambda )).mpr h1
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ring_nf at h2 ⊢
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exact h2
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lemma potential_bounded_below_of_μSq_nonpos {μSq lambda : ℝ} (hLam : 0 < lambda)
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(hμSq : μSq ≤ 0) (φ : higgsVec) : 0 ≤ potential μSq lambda φ := by
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simp only [potential, neg_mul, add_zero]
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refine add_nonneg ?_ (potential_snd_term_nonneg hLam φ)
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field_simp
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rw [@mul_nonpos_iff]
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simp_all only [ge_iff_le, norm_nonneg, pow_nonneg, and_self, or_true]
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lemma potential_eq_bound_discrim_zero (μSq lambda : ℝ) (hLam : 0 < lambda)(φ : higgsVec)
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(hV : potential μSq lambda φ = - μSq ^ 2 / (4 * lambda)) :
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discrim (lambda) (- μSq) (- potential μSq lambda φ) = 0 := by
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simp [discrim, hV]
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field_simp
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ring
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lemma potential_eq_bound_higgsVec_sq (μSq lambda : ℝ) (hLam : 0 < lambda)(φ : higgsVec)
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(hV : potential μSq lambda φ = - μSq ^ 2 / (4 * lambda)) :
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‖φ‖ ^ 2 = μSq / (2 * lambda) := by
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have h1 := potential_as_quad μSq lambda φ
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rw [quadratic_eq_zero_iff_of_discrim_eq_zero _
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(potential_eq_bound_discrim_zero μSq lambda hLam φ hV)] at h1
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rw [h1]
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field_simp
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ring_nf
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simp only [ne_eq, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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exact ne_of_gt hLam
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lemma potential_eq_bound_iff (μSq lambda : ℝ) (hLam : 0 < lambda)(φ : higgsVec) :
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potential μSq lambda φ = - μSq ^ 2 / (4 * lambda) ↔ ‖φ‖ ^ 2 = μSq / (2 * lambda) := by
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apply Iff.intro
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· intro h
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exact potential_eq_bound_higgsVec_sq μSq lambda hLam φ h
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· intro h
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have hv : ‖φ‖ ^ 4 = ‖φ‖ ^ 2 * ‖φ‖ ^ 2 := by
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ring_nf
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field_simp [potential, hv, h]
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ring
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lemma potential_eq_bound_iff_of_μSq_nonpos {μSq lambda : ℝ} (hLam : 0 < lambda)
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(hμSq : μSq ≤ 0) (φ : higgsVec) : potential μSq lambda φ = 0 ↔ φ = 0 := by
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apply Iff.intro
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· intro h
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exact potential_eq_zero_sol_of_μSq_nonpos hLam hμSq φ h
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· intro h
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simp [potential, h]
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lemma potential_eq_bound_IsMinOn (μSq lambda : ℝ) (hLam : 0 < lambda) (φ : higgsVec)
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(hv : potential μSq lambda φ = - μSq ^ 2 / (4 * lambda)) :
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IsMinOn (potential μSq lambda) Set.univ φ := by
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rw [isMinOn_univ_iff]
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intro x
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rw [hv]
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exact potential_bounded_below μSq lambda hLam x
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lemma potential_eq_bound_IsMinOn_of_μSq_nonpos {μSq lambda : ℝ} (hLam : 0 < lambda)
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(hμSq : μSq ≤ 0) (φ : higgsVec) (hv : potential μSq lambda φ = 0) :
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IsMinOn (potential μSq lambda) Set.univ φ := by
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rw [isMinOn_univ_iff]
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intro x
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rw [hv]
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exact potential_bounded_below_of_μSq_nonpos hLam hμSq x
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lemma potential_bound_reached_of_μSq_nonneg {μSq lambda : ℝ} (hLam : 0 < lambda) (hμSq : 0 ≤ μSq) :
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∃ (φ : higgsVec), potential μSq lambda φ = - μSq ^ 2 / (4 * lambda) := by
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use ![√(μSq/(2 * lambda)), 0]
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refine (potential_eq_bound_iff μSq lambda hLam _).mpr ?_
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simp [@PiLp.norm_sq_eq_of_L2, Fin.sum_univ_two]
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field_simp [mul_pow]
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lemma IsMinOn_potential_iff_of_μSq_nonneg {μSq lambda : ℝ} (hLam : 0 < lambda) (hμSq : 0 ≤ μSq) :
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IsMinOn (potential μSq lambda) Set.univ φ ↔ ‖φ‖ ^ 2 = μSq /(2 * lambda) := by
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apply Iff.intro
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· intro h
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obtain ⟨φm, hφ⟩ := potential_bound_reached_of_μSq_nonneg hLam hμSq
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have hm := isMinOn_univ_iff.mp h φm
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rw [hφ] at hm
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have h1 := potential_bounded_below μSq lambda hLam φ
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rw [← potential_eq_bound_iff μSq lambda hLam φ]
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exact (Real.partialOrder.le_antisymm _ _ h1 hm).symm
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· intro h
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rw [← potential_eq_bound_iff μSq lambda hLam φ] at h
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exact potential_eq_bound_IsMinOn μSq lambda hLam φ h
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lemma IsMinOn_potential_iff_of_μSq_nonpos {μSq lambda : ℝ} (hLam : 0 < lambda) (hμSq : μSq ≤ 0) :
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IsMinOn (potential μSq lambda) Set.univ φ ↔ φ = 0 := by
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apply Iff.intro
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· intro h
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have h0 := isMinOn_univ_iff.mp h 0
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rw [(potential_eq_bound_iff_of_μSq_nonpos hLam hμSq 0).mpr (by rfl)] at h0
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have h1 := potential_bounded_below_of_μSq_nonpos hLam hμSq φ
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rw [← (potential_eq_bound_iff_of_μSq_nonpos hLam hμSq φ)]
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exact (Real.partialOrder.le_antisymm _ _ h1 h0).symm
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· intro h
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rw [← potential_eq_bound_iff_of_μSq_nonpos hLam hμSq φ] at h
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exact potential_eq_bound_IsMinOn_of_μSq_nonpos hLam hμSq φ h
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/-- Given a Higgs vector, a rotation matrix which puts the fst component of the
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vector to zero, and the snd componenet to a real -/
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def rotateMatrix (φ : higgsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
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![![φ 1 /‖φ‖ , - φ 0 /‖φ‖], ![conj (φ 0) / ‖φ‖ , conj (φ 1) / ‖φ‖] ]
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lemma rotateMatrix_star (φ : higgsVec) :
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star φ.rotateMatrix =
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![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
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simp [star]
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rw [rotateMatrix, conjTranspose]
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ext i j
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fin_cases i <;> fin_cases j <;> simp [conj_ofReal]
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lemma rotateMatrix_det {φ : higgsVec} (hφ : φ ≠ 0) : (rotateMatrix φ).det = 1 := by
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simp [rotateMatrix, det_fin_two]
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have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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field_simp
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rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
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rfl
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lemma rotateMatrix_unitary {φ : higgsVec} (hφ : φ ≠ 0) :
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(rotateMatrix φ) ∈ unitaryGroup (Fin 2) ℂ := by
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rw [mem_unitaryGroup_iff', rotateMatrix_star, rotateMatrix]
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erw [mul_fin_two, one_fin_two]
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have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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congr
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field_simp
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ext i j
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fin_cases i <;> fin_cases j <;> field_simp
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· rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
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rfl
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· ring_nf
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· ring_nf
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· rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
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rfl
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lemma rotateMatrix_specialUnitary {φ : higgsVec} (hφ : φ ≠ 0) :
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(rotateMatrix φ) ∈ specialUnitaryGroup (Fin 2) ℂ :=
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mem_specialUnitaryGroup_iff.mpr ⟨rotateMatrix_unitary hφ, rotateMatrix_det hφ⟩
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/-- Given a Higgs vector, an element of the gauge group which puts the fst component of the
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vector to zero, and the snd componenet to a real -/
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def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : guageGroup :=
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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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rep (rotateGuageGroup hφ) φ = ![0, ofReal ‖φ‖] := by
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rw [rep_apply]
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simp [rotateGuageGroup, rotateMatrix]
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ext i
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fin_cases i
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simp [mulVec, vecHead, vecTail]
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ring_nf
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simp only [Fin.mk_one, Fin.isValue, cons_val_one, head_cons]
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simp [mulVec, vecHead, vecTail]
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have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
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field_simp
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rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
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simp only [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
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Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
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rfl
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theorem rotate_fst_zero_snd_real (φ : higgsVec) :
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∃ (g : guageGroup), rep g φ = ![0, 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 [h]
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ext i
|
||
fin_cases i <;> rfl
|
||
· use rotateGuageGroup h
|
||
exact rotateGuageGroup_apply h
|
||
|
||
end higgsVec
|
||
end higgsVec
|
||
|
||
end
|
||
end StandardModel
|