2024-05-09 15:09:14 -04:00
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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 Mathlib.Data.Complex.Exponential
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import Mathlib.Geometry.Manifold.Instances.Real
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import Mathlib.LinearAlgebra.Matrix.ToLin
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/-!
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# Representations appearing in the Standard Model
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This file defines the basic representations which appear in the Standard Model.
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-/
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universe v u
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namespace StandardModel
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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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2024-05-09 15:16:38 -04:00
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/-- The 2d representation of U(1) with charge 3 as a map from U(1) to `unitaryGroup (Fin 2) ℂ`. -/
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2024-05-09 15:09:14 -04:00
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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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2024-05-09 15:16:38 -04:00
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/-- The 2d representation of U(1) with charge 3 as a homomorphism
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from U(1) to `unitaryGroup (Fin 2) ℂ`. -/
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2024-05-09 15:09:14 -04:00
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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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2024-05-09 15:16:38 -04:00
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/-- The fundamental representation of SU(2) as a homomorphism to `unitaryGroup (Fin 2) ℂ`. -/
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2024-05-09 15:09:14 -04:00
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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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end StandardModel
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