116 lines
5 KiB
Text
116 lines
5 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 as described in the file 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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import HepLean.SpaceTime.Basic
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import HepLean.Meta.Informal.Basic
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/-!
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# The Standard Model
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This file defines the basic properties of the standard model in particle physics.
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-/
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/-! TODO: Redefine the gauge group as a quotient of SU(3) x SU(2) x U(1) by a subgroup of ℤ₆. -/
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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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/-- The global gauge group of the Standard Model with no discrete quotients.
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The `I` in the Name is an indication of the statement that this has no discrete quotients. -/
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abbrev GaugeGroupI : Type :=
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specialUnitaryGroup (Fin 3) ℂ × specialUnitaryGroup (Fin 2) ℂ × unitary ℂ
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informal_definition gaugeGroupℤ₆SubGroup where
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physics :≈ "The subgroup of the un-quotiented gauge group which acts trivially on
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all particles in the standard model. "
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math :≈ "The ℤ₆-subgroup of ``GaugeGroupI with elements (α^2 * I₃, α^(-3) * I₂, α), where `α`
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is a sixth complex root of unity."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI]
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informal_definition GaugeGroupℤ₆ where
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physics :≈ "The smallest possible gauge group of the Standard Model."
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math :≈ "The quotient of ``GaugeGroupI by the ℤ₆-subgroup `gaugeGroupℤ₆SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₆SubGroup]
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informal_definition gaugeGroupℤ₂SubGroup where
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physics :≈ "The ℤ₂subgroup of the un-quotiented gauge group which acts trivially on
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all particles in the standard model. "
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math :≈ "The ℤ₂-subgroup of ``GaugeGroupI derived from the ℤ₂ subgroup of `gaugeGroupℤ₆SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₆SubGroup]
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informal_definition GaugeGroupℤ₂ where
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physics :≈ "The guage group of the Standard Model with a ℤ₂ quotient."
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math :≈ "The quotient of ``GaugeGroupI by the ℤ₂-subgroup `gaugeGroupℤ₂SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₂SubGroup]
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informal_definition gaugeGroupℤ₃SubGroup where
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physics :≈ "The ℤ₃-subgroup of the un-quotiented gauge group which acts trivially on
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all particles in the standard model. "
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math :≈ "The ℤ₃-subgroup of ``GaugeGroupI derived from the ℤ₃ subgroup of `gaugeGroupℤ₆SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₆SubGroup]
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informal_definition GaugeGroupℤ₃ where
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physics :≈ "The guage group of the Standard Model with a ℤ₃-quotient."
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math :≈ "The quotient of ``GaugeGroupI by the ℤ₃-subgroup `gaugeGroupℤ₃SubGroup`."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``StandardModel.gaugeGroupℤ₃SubGroup]
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/-- Specifies the allowed quotients of `SU(3) x SU(2) x U(1)` which give a valid
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gauge group of the Standard Model. -/
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inductive GaugeGroupQuot : Type
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/-- The element of `GaugeGroupQuot` corresponding to the quotient of the full SM gauge group
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by the sub-group `ℤ₆`. -/
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| ℤ₆ : GaugeGroupQuot
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/-- The element of `GaugeGroupQuot` corresponding to the quotient of the full SM gauge group
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by the sub-group `ℤ₂`. -/
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| ℤ₂ : GaugeGroupQuot
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/-- The element of `GaugeGroupQuot` corresponding to the quotient of the full SM gauge group
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by the sub-group `ℤ₃`. -/
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| ℤ₃ : GaugeGroupQuot
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/-- The element of `GaugeGroupQuot` corresponding to the full SM gauge group. -/
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| I : GaugeGroupQuot
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informal_definition GaugeGroup where
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physics :≈ "The (global) gauge group of the Standard Model given a choice of quotient."
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math :≈ "The map from `GaugeGroupQuot` to `Type` which gives the gauge group of the Standard Model
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for a given choice of quotient."
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ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
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deps :≈ [``GaugeGroupI, ``gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup, ``gaugeGroupℤ₃SubGroup,
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``GaugeGroupQuot]
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/-!
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## Smoothness structure on the gauge group.
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-/
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informal_lemma gaugeGroupI_lie where
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math :≈ "The gauge group `GaugeGroupI` is a Lie group.."
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deps :≈ [``GaugeGroupI]
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informal_lemma gaugeGroup_lie where
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math :≈ "For every q in ``GaugeGroupQuot the group ``GaugeGroup q is a Lie group."
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deps :≈ [``GaugeGroup]
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informal_definition gaugeBundleI where
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math :≈ "The trivial principal bundle over SpaceTime with structure group ``GaugeGroupI."
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deps :≈ [``GaugeGroupI, ``SpaceTime]
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informal_definition gaugeTransformI where
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math :≈ "A global section of ``gaugeBundleI."
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deps :≈ [``gaugeBundleI]
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end StandardModel
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