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docs: Curated note for Higgs potential

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jstoobysmith 2025-03-04 06:21:04 +00:00
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12
PhysLean/Particles/StandardModel/HiggsBoson/Basic.lean
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@ -36,7 +36,8 @@ open SpaceTime
In other words, the target space of the Higgs field.
-/
/-- The complex vector space in which the Higgs field takes values. -/
/-- The vector space `HiggsVec` is defined to be the complex Euclidean space of dimension 2.
For a given spacetime point a Higgs field gives a value in `HiggsVec`. -/
abbrev HiggsVec := EuclideanSpace ℂ (Fin 2)
namespace HiggsVec
@ -80,14 +81,19 @@ We also define the Higgs bundle.
-/
TODO "Make `HiggsBundle` an associated bundle."
/-- The trivial vector bundle 𝓡² × ℂ². -/
/-- The `HiggsBundle` is defined as the trivial vector bundle with base `SpaceTime` and
fiber `HiggsVec`. Thus as a manifold it corresponds to `ℝ⁴ × ℂ²`. -/
abbrev HiggsBundle := Bundle.Trivial SpaceTime HiggsVec
/-- The instance of a smooth vector bundle with total space `HiggsBundle` and fiber `HiggsVec`. -/
instance : ContMDiffVectorBundle ⊤ HiggsVec HiggsBundle SpaceTime.asSmoothManifold :=
Bundle.Trivial.contMDiffVectorBundle HiggsVec
/-- A Higgs field is a smooth section of the Higgs bundle. -/
/-- The type `HiggsField` is defined such that elements are smooth sections of the trivial
vector bundle `HiggsBundle`. Such elements are Higgs fields. Since `HiggsField` is
trivial as a vector bundle, a Higgs field is equivalent to a smooth map
from `SpaceTime` to `HiggsVec`. -/
abbrev HiggsField : Type := ContMDiffSection SpaceTime.asSmoothManifold HiggsVec ⊤ HiggsBundle
/-- Given a vector in `HiggsVec` the constant Higgs field with value equal to that

9
PhysLean/Particles/StandardModel/HiggsBoson/PointwiseInnerProd.lean
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@ -109,12 +109,15 @@ lemma smooth_innerProd (φ1 φ2 : HiggsField) : ContMDiff 𝓘(ℝ, SpaceTime)
-/
/-- Given a `HiggsField`, the map `SpaceTime → ℝ` obtained by taking the square norm of the
pointwise Higgs vector. -/
/-- Given an element `φ` of `HiggsField`, `normSq φ` is defined as the
the function `SpaceTime → ℝ` obtained by taking the square norm of the
pointwise Higgs vector. In other words, `normSq φ x = ‖φ x‖ ^ 2`.
The notation `‖φ‖_H^2` is used for the `normSq φ` -/
@[simp]
def normSq (φ : HiggsField) : SpaceTime → ℝ := fun x => ‖φ x‖ ^ 2
/-- Notation for the norm squared of a Higgs field. -/
@[inherit_doc normSq]
scoped[StandardModel.HiggsField] notation "‖" φ1 "‖_H^2" => normSq φ1
/-!

56
PhysLean/Particles/StandardModel/HiggsBoson/Potential.lean
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@ -31,7 +31,9 @@ open SpaceTime
-/
/-- The parameters of the Higgs potential. -/
/-- The structure `Potential` is defined with two fields, `μ2` corresponding
to the mass-squared of the Higgs boson, and `l` corresponding to the coefficent
of the quartic term in the Higgs potential. Note that `l` is usually denoted `λ`. -/
structure Potential where
/-- The mass-squared of the Higgs boson. -/
μ2 : ℝ
@ -42,7 +44,10 @@ namespace Potential
variable (P : Potential)
/-- The function corresponding to the Higgs potential. -/
/-- Given a element `P` of `Potential`, `P.toFun` is Higgs potential.
It is defined for a Higgs field `φ` and a spacetime point `x` as
`-μ² ‖φ‖_H^2 x + l * ‖φ‖_H^2 x * ‖φ‖_H^2 x`. -/
def toFun (φ : HiggsField) (x : SpaceTime) : ℝ :=
- P.μ2 * ‖φ‖_H^2 x + P.𝓵 * ‖φ‖_H^2 x * ‖φ‖_H^2 x
@ -156,6 +161,10 @@ lemma quadDiscrim_eq_zero_iff_normSq (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : S
field_simp
ring
/-- For an element `P` of `Potential`, if `l < 0` then the following upper bound for the potential
exists
`P.toFun φ x ≤ - μ2 ^ 2 / (4 * 𝓵)`. -/
lemma neg_𝓵_quadDiscrim_zero_bound (h : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
P.toFun φ x ≤ - P.μ2 ^ 2 / (4 * P.𝓵) := by
have h1 := P.quadDiscrim_nonneg (ne_of_lt h) φ x
@ -167,6 +176,10 @@ lemma neg_𝓵_quadDiscrim_zero_bound (h : P.𝓵 < 0) (φ : HiggsField) (x : Sp
ring_nf at h2 ⊢
exact h2
/-- For an element `P` of `Potential`, if `0 < l` then the following lower bound for the potential
exists
`- μ2 ^ 2 / (4 * 𝓵) ≤ P.toFun φ x`. -/
lemma pos_𝓵_quadDiscrim_zero_bound (h : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
- P.μ2 ^ 2 / (4 * P.𝓵) ≤ P.toFun φ x := by
have h1 := P.neg.neg_𝓵_quadDiscrim_zero_bound (by simpa [neg] using h) φ x
@ -196,6 +209,14 @@ lemma pos_𝓵_toFun_pos (h : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
(P.μ2 < 0 ∧ 0 ≤ P.toFun φ x) ∨ 0 ≤ P.μ2 := by
simpa using P.neg.neg_𝓵_toFun_neg (by simpa using h) φ x
/-- For an element `P` of `Potential` with `l < 0` and a real `c : ℝ`, there exists
a Higgs field `φ` and a spacetime point `x` such that `P.toFun φ x = c` iff one of the
following two conditions hold:
- `0 < μ2` and `c ≤ 0`. That is, if `l` is negative and `μ2` positive, then the potential
takes every non-positive value.
- or `μ2 ≤ 0` and `c ≤ - μ2 ^ 2 / (4 * 𝓵)`. That is, if `l` is negative and `μ2` non-positive,
then the potential takes every value less then or equal to its bound.
-/
lemma neg_𝓵_sol_exists_iff (h𝓵 : P.𝓵 < 0) (c : ℝ) : (∃ φ x, P.toFun φ x = c) ↔ (0 < P.μ2 ∧ c ≤ 0) ∨
(P.μ2 ≤ 0 ∧ c ≤ - P.μ2 ^ 2 / (4 * P.𝓵)) := by
refine Iff.intro (fun ⟨φ, x, hV⟩ => ?_) (fun h => ?_)
@ -250,6 +271,14 @@ lemma neg_𝓵_sol_exists_iff (h𝓵 : P.𝓵 < 0) (c : ℝ) : (∃ φ x, P.toFu
refine (quadratic_eq_zero_iff (ne_of_gt h𝓵).symm hdd _).mpr ?_
simp only [neg_neg, or_true, a]
/-- For an element `P` of `Potential` with `0 < l` and a real `c : ℝ`, there exists
a Higgs field `φ` and a spacetime point `x` such that `P.toFun φ x = c` iff one of the
following two conditions hold:
- `μ2 < 0` and `0 ≤ c`. That is, if `l` is positive and `μ2` negative, then the potential
takes every non-negative value.
- or `0 ≤ μ2` and `- μ2 ^ 2 / (4 * 𝓵) ≤ c`. That is, if `l` is positive and `μ2` non-negative,
then the potential takes every value greater then or equal to its bound.
-/
lemma pos_𝓵_sol_exists_iff (h𝓵 : 0 < P.𝓵) (c : ℝ) : (∃ φ x, P.toFun φ x = c) ↔ (P.μ2 < 0 ∧ 0 ≤ c) ∨
(0 ≤ P.μ2 ∧ - P.μ2 ^ 2 / (4 * P.𝓵) ≤ c) := by
have h1 := P.neg.neg_𝓵_sol_exists_iff (by simpa using h𝓵) (- c)
@ -264,11 +293,16 @@ lemma pos_𝓵_sol_exists_iff (h𝓵 : 0 < P.𝓵) (c : ℝ) : (∃ φ x, P.toFu
-/
/-- The proposition on the coefficients for a potential to be bounded. -/
/-- Given a element `P` of `Potential`, the proposition `IsBounded P` is true if and only if
there exists a real `c` such that for all Higgs fields `φ` and spacetime points `x`,
the Higgs potential corresponding to `φ` at `x` is greater then or equal to`c`. I.e.
`∀ Φ x, c ≤ P.toFun Φ x`. -/
def IsBounded : Prop :=
∃ c, ∀ Φ x, c ≤ P.toFun Φ x
/-- If the potential is bounded, then `P.𝓵` is non-negative. -/
/-- Given a element `P` of `Potential` which is bounded,
the quartic coefficent `𝓵` of `P` is non-negative. -/
lemma isBounded_𝓵_nonneg (h : P.IsBounded) : 0 ≤ P.𝓵 := by
by_contra hl
rw [not_le] at hl
@ -313,7 +347,7 @@ lemma isBounded_𝓵_nonneg (h : P.IsBounded) : 0 ≤ P.𝓵 := by
rw [hφ] at hc2
linarith
/-- If `P.𝓵` is positive, then the potential is bounded. -/
/-- Given a element `P` of `Potential` with `0 < 𝓵`, then the potential is bounded. -/
lemma isBounded_of_𝓵_pos (h : 0 < P.𝓵) : P.IsBounded := by
simp only [IsBounded]
have h2 := P.pos_𝓵_quadDiscrim_zero_bound h
@ -401,6 +435,12 @@ lemma isMinOn_iff_field_of_μSq_nonneg_𝓵_pos (h𝓵 : 0 < P.𝓵) (hμ2 : 0
rw [P.isMinOn_iff_of_μSq_nonneg_𝓵_pos h𝓵 hμ2 φ x, ← P.quadDiscrim_eq_zero_iff_normSq
(Ne.symm (ne_of_lt h𝓵)), P.quadDiscrim_eq_zero_iff (Ne.symm (ne_of_lt h𝓵))]
/-- Given an element `P` of `Potential` with `0 < l`, then the Higgs field `φ` and
spacetime point `x` minimize the potential if and only if one of the following conditions
holds
- `0 ≤ μ2` and `‖φ‖_H^2 x = μ2 / (2 * 𝓵)`.
- or `μ2 < 0` and `φ x = 0`.
-/
theorem isMinOn_iff_field_of_𝓵_pos (h𝓵 : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
IsMinOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
(0 ≤ P.μ2 ∧ ‖φ‖_H^2 x = P.μ2 /(2 * P.𝓵)) ∨ (P.μ2 < 0 ∧ φ x = 0) := by
@ -415,6 +455,12 @@ lemma isMaxOn_iff_isMinOn_neg (φ : HiggsField) (x : SpaceTime) :
rw [isMaxOn_univ_iff, isMinOn_univ_iff]
simp_all only [Prod.forall, neg_le_neg_iff]
/-- Given an element `P` of `Potential` with `l < 0`, then the Higgs field `φ` and
spacetime point `x` maximizes the potential if and only if one of the following conditions
holds
- `μ2 ≤ 0` and `‖φ‖_H^2 x = μ2 / (2 * 𝓵)`.
- or `0 < μ2` and `φ x = 0`.
-/
lemma isMaxOn_iff_field_of_𝓵_neg (h𝓵 : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
IsMaxOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
(P.μ2 ≤ 0 ∧ ‖φ‖_H^2 x = P.μ2 /(2 * P.𝓵)) ∨ (0 < P.μ2 ∧ φ x = 0) := by

1
PhysLean/Relativity/Lorentz/MinkowskiMatrix.lean
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@ -18,6 +18,7 @@ open InnerProductSpace
# The definition of the Minkowski Matrix
-/
/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/
def minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ :=
LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin d) ℝ

2
README.md
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@ -12,7 +12,7 @@
A project to digitalize physics.
*(Formally called HepLean)*
*(Formerly called HepLean)*
## Aims of this project
🎯 __Digitalize__ results (meaning calculations, definitions, and theorems) from physics

2
docs/CuratedNotes/index.html
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@ -20,7 +20,7 @@ layout: default
<!-- End Wick's theorem -->
<!-- Higgs potential -->
<div class="note-card">
<h2 style="text-align: center;">Higgs potential 🚧</h2>
<h2 style="text-align: center;">Higgs potential</h2>
<p class="description">The formalization of properties of the Higgs potential.</p>
<a href="HiggsPotential" class="note-link">Read More →</a>
</div>

25
scripts/MetaPrograms/notes.lean
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@ -291,7 +291,7 @@ def perturbationTheory : Note where
]
def harmonicOscillator : Note where
title := "The Quantum Harmonic Oscillator in Lean 4"
title := "The Quantum Harmonic Oscillator"
curators := ["Joseph Tooby-Smith"]
parts := [
.h1 "Introduction",
@ -344,16 +344,33 @@ def harmonicOscillator : Note where
]
def higgsPotential : Note where
title := "The Higgs potential in Lean 4 🚧"
title := "The Higgs potential 🚧"
curators := ["Joseph Tooby-Smith"]
parts := [
.warning "This note is under construction (03-March-2025).",
.h1 "Introduction",
.p "The Higgs potential is a key part of the Standard Model of particle physics.
It is a scalar potential which is used to give mass to the elementary particles.
The Higgs potential is a polynomial of degree four in the Higgs field.",
.h1 "The Higgs field",
.name ``StandardModel.HiggsVec .incomplete,
.name ``StandardModel.HiggsVec .complete,
.name ``StandardModel.HiggsBundle .complete,
.name ``StandardModel.HiggsField .complete,
.h1 "The Higgs potential",
.name ``StandardModel.HiggsField.Potential .complete,
.name ``StandardModel.HiggsField.normSq .complete,
.name ``StandardModel.HiggsField.Potential.toFun .complete,
.h2 "Properties of the Higgs potential",
.name ``StandardModel.HiggsField.Potential.neg_𝓵_quadDiscrim_zero_bound .complete,
.name ``StandardModel.HiggsField.Potential.pos_𝓵_quadDiscrim_zero_bound .complete,
.name ``StandardModel.HiggsField.Potential.neg_𝓵_sol_exists_iff .complete,
.name ``StandardModel.HiggsField.Potential.pos_𝓵_sol_exists_iff .complete,
.h2 "Boundedness of the Higgs potential",
.name ``StandardModel.HiggsField.Potential.IsBounded .complete,
.name ``StandardModel.HiggsField.Potential.isBounded_𝓵_nonneg .complete,
.name ``StandardModel.HiggsField.Potential.isBounded_of_𝓵_pos .complete,
.h2 "Maximum and minimum of the Higgs potential",
.name ``StandardModel.HiggsField.Potential.isMaxOn_iff_field_of_𝓵_neg .complete,
.name ``StandardModel.HiggsField.Potential.isMinOn_iff_field_of_𝓵_pos .complete,
]
def notesToMake : List (Note × String) := [