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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