feat: Add some basic properties for the higgs field

HepLean/StandardModel/HiggsField.lean

@@ -0,0 +1,106 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+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
+/-!
+# The Higgs field
+
+This file defines the basic properties for the higgs field in the standard model.
+
+-/
+universe v u
+namespace StandardModel
+
+open Manifold
+open Matrix
+open Complex
+open ComplexConjugate
+
+/-- The space-time (TODO: Change to Minkowski. Move.) -/
+abbrev spaceTime := EuclideanSpace ℝ (Fin 4)
+
+abbrev guageGroup : Type := specialUnitaryGroup (Fin 2) ℂ × unitary ℂ
+
+/-- The trivial vector bundle 𝓡² × ℂ². (TODO: Make associated bundle.) -/
+abbrev higgsBundle := Bundle.Trivial spaceTime (Fin 2 → ℂ)
+
+
+instance : SmoothVectorBundle (Fin 2 → ℂ) higgsBundle (𝓡 4)  :=
+  Bundle.Trivial.smoothVectorBundle (Fin 2 → ℂ) 𝓘(ℝ, spaceTime)
+
+/-- A higgs field is a smooth section of the higgs bundle. -/
+abbrev higgsFields : Type := SmoothSection (𝓡 4) (Fin 2 → ℂ) higgsBundle
+
+/-- -/
+@[simps!]
+noncomputable def higgsRepMap (g : guageGroup) : (Fin 2 → ℂ) →ₗ[ℂ] (Fin 2 → ℂ) where
+  toFun S :=  (g.2 ^ 3) • (g.1.1 *ᵥ S)
+  map_add' S T := by
+    simp [Matrix.mulVec_add, smul_add]
+  map_smul' a S := by
+    simp [Matrix.mulVec_smul]
+    exact smul_comm  _ _ _
+
+
+/-- The representation of the SM guage group acting on `ℂ²`. -/
+noncomputable def higgsRep : Representation ℂ guageGroup (Fin 2 → ℂ) where
+  toFun := higgsRepMap
+  map_mul' U V := by
+    apply LinearMap.ext
+    intro S
+    simp only [higgsRepMap, Prod.snd_mul, Submonoid.coe_inf, Prod.fst_mul, Submonoid.coe_mul,
+      LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply, LinearMap.map_smul_of_tower,
+      mulVec_mulVec]
+    simp  [mul_pow, smul_smul, mul_comm]
+  map_one' := by
+    apply LinearMap.ext
+    intro S
+    simp only [higgsRepMap, LinearMap.mul_apply, AddHom.coe_mk, LinearMap.coe_mk]
+    change 1 ^ 3 • (1 *ᵥ _) = _
+    rw [one_pow, Matrix.one_mulVec]
+    simp only [one_smul, LinearMap.one_apply]
+
+
+
+namespace higgsFields
+
+
+/-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/
+def isConst (Φ : higgsFields) : Prop := ∀ x y, Φ x = Φ y
+
+/-- Given a vector `ℂ²` the constant higgs field with value equal to that
+section. -/
+noncomputable def const (φ : Fin 2 → ℂ) : higgsFields where
+  toFun := fun _ => φ
+  contMDiff_toFun := by
+    intro x
+    rw [Bundle.contMDiffAt_section]
+    exact smoothAt_const
+
+
+lemma const_isConst (φ : Fin 2 → ℂ) : (const φ).isConst := by
+  intro x _
+  simp [const]
+
+lemma isConst_iff_exists_const (Φ : higgsFields) : Φ.isConst ↔ ∃ φ, Φ = const φ := 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
+
+end higgsFields
+
+end StandardModel