feat: Add properties of Weyl fermions
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jstoobysmith
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5 changed files with 743 additions and 55 deletions
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@ -88,6 +88,8 @@ import HepLean.SpaceTime.LorentzVector.NormOne
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import HepLean.SpaceTime.MinkowskiMetric
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import HepLean.SpaceTime.SL2C.Basic
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import HepLean.SpaceTime.WeylFermion.Basic
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import HepLean.SpaceTime.WeylFermion.Modules
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import HepLean.SpaceTime.WeylFermion.OverCat
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import HepLean.StandardModel.Basic
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import HepLean.StandardModel.HiggsBoson.Basic
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import HepLean.StandardModel.HiggsBoson.GaugeAction
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@ -26,6 +26,23 @@ open SpaceTime
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noncomputable section
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/-!
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## Some basic properties about SL(2, ℂ)
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Possibly to be moved to mathlib at some point.
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-/
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lemma inverse_coe (M : SL(2, ℂ)) : M.1⁻¹ = (M⁻¹).1 := by
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apply Matrix.inv_inj
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simp
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have h1 : IsUnit M.1.det := by
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simp
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rw [Matrix.inv_adjugate M.1 h1]
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· simp
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· simp
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/-!
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## Representation of SL(2, ℂ) on spacetime
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@ -4,44 +4,116 @@ 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 HepLean.Meta.Informal
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import HepLean.SpaceTime.SL2C.Basic
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import Mathlib.RepresentationTheory.Rep
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import HepLean.Tensors.Basic
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import HepLean.SpaceTime.WeylFermion.Modules
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import Mathlib.Logic.Equiv.TransferInstance
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/-!
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# Weyl fermions
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A good reference for the material in this file is:
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https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf
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-/
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/-!
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## The definition of Weyl fermion vector spaces.
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We define the vector spaces corresponding to different types of Weyl fermions.
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Note: We should prevent casting between these vector spaces.
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-/
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namespace Fermion
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noncomputable section
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informal_definition leftHandedWeyl where
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math :≈ "The vector space ℂ^2 carrying the fundamental representation of SL(2,C)."
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physics :≈ "A Weyl fermion with indices ψ_a."
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ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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informal_definition rightHandedWeyl where
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math :≈ "The vector space ℂ^2 carrying the conjguate representation of SL(2,C)."
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physics :≈ "A Weyl fermion with indices ψ_{dot a}."
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ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
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/-- The vector space ℂ^2 carrying the fundamental representation of SL(2,C).
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In index notation corresponds to a Weyl fermion with indices ψ_a.-/
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def leftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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toFun := fun M => {
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toFun := fun (ψ : LeftHandedModule) =>
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LeftHandedModule.toFin2ℂEquiv.symm (M.1 *ᵥ ψ.toFin2ℂ),
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map_add' := by
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intro ψ ψ'
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simp [mulVec_add]
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map_smul' := by
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intro r ψ
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simp [mulVec_smul]}
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map_one' := by
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ext i
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simp
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map_mul' := fun M N => by
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simp only [SpecialLinearGroup.coe_mul]
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ext1 x
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simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply, LinearEquiv.apply_symm_apply,
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mulVec_mulVec]}
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informal_definition altLeftHandedWeyl where
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math :≈ "The vector space ℂ^2 carrying the representation of SL(2,C) given by
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M → (M⁻¹)ᵀ."
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physics :≈ "A Weyl fermion with indices ψ^a."
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ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
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/-- The vector space ℂ^2 carrying the representation of SL(2,C) given by
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M → (M⁻¹)ᵀ. In index notation corresponds to a Weyl fermion with indices ψ^a. -/
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def altLeftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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toFun := fun M => {
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toFun := fun (ψ : AltLeftHandedModule) =>
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AltLeftHandedModule.toFin2ℂEquiv.symm ((M.1⁻¹)ᵀ *ᵥ ψ.toFin2ℂ),
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map_add' := by
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intro ψ ψ'
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simp [mulVec_add]
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map_smul' := by
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intro r ψ
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simp [mulVec_smul]}
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map_one' := by
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ext i
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simp
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map_mul' := fun M N => by
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ext1 x
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simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply,
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LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
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refine (congrFun (congrArg _ ?_) _)
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rw [Matrix.mul_inv_rev]
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exact transpose_mul _ _}
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informal_definition altRightHandedWeyl where
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math :≈ "The vector space ℂ^2 carrying the representation of SL(2,C) given by
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M → (M⁻¹)^†."
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physics :≈ "A Weyl fermion with indices ψ^{dot a}."
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ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
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/-- The vector space ℂ^2 carrying the conjugate representation of SL(2,C).
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In index notation corresponds to a Weyl fermion with indices ψ_{dot a}.-/
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def rightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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toFun := fun M => {
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toFun := fun (ψ : RightHandedModule) =>
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RightHandedModule.toFin2ℂEquiv.symm (M.1.map star *ᵥ ψ.toFin2ℂ),
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map_add' := by
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intro ψ ψ'
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simp [mulVec_add]
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map_smul' := by
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intro r ψ
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simp [mulVec_smul]}
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map_one' := by
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ext i
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simp
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map_mul' := fun M N => by
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ext1 x
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simp only [SpecialLinearGroup.coe_mul, RCLike.star_def, Matrix.map_mul, LinearMap.coe_mk,
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AddHom.coe_mk, LinearMap.mul_apply, LinearEquiv.apply_symm_apply, mulVec_mulVec]}
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/-- The vector space ℂ^2 carrying the representation of SL(2,C) given by
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M → (M⁻¹)^†.
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In index notation this corresponds to a Weyl fermion with index `ψ^{dot a}`.-/
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def altRightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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toFun := fun M => {
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toFun := fun (ψ : AltRightHandedModule) =>
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AltRightHandedModule.toFin2ℂEquiv.symm ((M.1⁻¹).conjTranspose *ᵥ ψ.toFin2ℂ),
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map_add' := by
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intro ψ ψ'
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simp [mulVec_add]
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map_smul' := by
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intro r ψ
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simp [mulVec_smul]}
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map_one' := by
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ext i
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simp
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map_mul' := fun M N => by
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ext1 x
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simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply,
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LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
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refine (congrFun (congrArg _ ?_) _)
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rw [Matrix.mul_inv_rev]
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exact conjTranspose_mul _ _}
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/-!
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@ -49,20 +121,104 @@ informal_definition altRightHandedWeyl where
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-/
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informal_definition leftHandedWeylAltEquiv where
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math :≈ "The linear equiv between leftHandedWeyl and altLeftHandedWeyl given
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by multiplying an element of rightHandedWeyl by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`."
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deps :≈ [``leftHandedWeyl, ``altLeftHandedWeyl]
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/-- The morphism between the representation `leftHanded` and the representation
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`altLeftHanded` defined by multiplying an element of
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`leftHanded` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`. -/
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def leftHandedToAlt : leftHanded ⟶ altLeftHanded where
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hom := {
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toFun := fun ψ => AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ),
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map_add' := by
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intro ψ ψ'
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simp only [mulVec_add, LinearEquiv.map_add]
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map_smul' := by
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intro a ψ
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simp only [mulVec_smul, LinearEquiv.map_smul]
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rfl}
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comm := by
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intro M
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refine LinearMap.ext (fun ψ => ?_)
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change AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ M.1 *ᵥ ψ.val) =
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AltLeftHandedModule.toFin2ℂEquiv.symm ((M.1⁻¹)ᵀ *ᵥ !![0, 1; -1, 0] *ᵥ ψ.val)
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apply congrArg
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rw [mulVec_mulVec, mulVec_mulVec, SpaceTime.SL2C.inverse_coe, eta_fin_two M.1]
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refine congrFun (congrArg _ ?_) _
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rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
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Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ]
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simp
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informal_lemma leftHandedWeylAltEquiv_equivariant where
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math :≈ "The linear equiv leftHandedWeylAltEquiv is equivariant with respect to the
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action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
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deps :≈ [``leftHandedWeylAltEquiv]
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lemma leftHandedToAlt_hom_apply (ψ : leftHanded) :
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leftHandedToAlt.hom ψ =
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AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ) := rfl
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/-- The morphism from `altLeftHanded` to
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`leftHanded` defined by multiplying an element of
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altLeftHandedWeyl by the matrix `εₐ₁ₐ₂ = !![0, -1; 1, 0]`. -/
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def leftHandedAltTo : altLeftHanded ⟶ leftHanded where
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hom := {
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toFun := fun ψ =>
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LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ),
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map_add' := by
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intro ψ ψ'
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simp only [map_add]
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rw [mulVec_add, LinearEquiv.map_add]
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map_smul' := by
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intro a ψ
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simp only [LinearEquiv.map_smul]
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rw [mulVec_smul, LinearEquiv.map_smul]
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rfl}
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comm := by
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intro M
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refine LinearMap.ext (fun ψ => ?_)
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change LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ (M.1⁻¹)ᵀ *ᵥ ψ.val) =
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LeftHandedModule.toFin2ℂEquiv.symm (M.1 *ᵥ !![0, -1; 1, 0] *ᵥ ψ.val)
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rw [EquivLike.apply_eq_iff_eq, mulVec_mulVec, mulVec_mulVec, SpaceTime.SL2C.inverse_coe,
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eta_fin_two M.1]
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refine congrFun (congrArg _ ?_) _
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rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
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Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ]
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simp
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lemma leftHandedAltTo_hom_apply (ψ : altLeftHanded) :
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leftHandedAltTo.hom ψ =
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LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ) := rfl
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/-- The equivalence between the representation `leftHanded` and the representation
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`altLeftHanded` defined by multiplying an element of
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`leftHanded` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`. -/
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def leftHandedAltEquiv : leftHanded ≅ altLeftHanded where
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hom := leftHandedToAlt
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inv := leftHandedAltTo
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hom_inv_id := by
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ext ψ
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simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
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ModuleCat.id_apply]
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rw [leftHandedAltTo_hom_apply, leftHandedToAlt_hom_apply]
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rw [AltLeftHandedModule.toFin2ℂ, LinearEquiv.apply_symm_apply, mulVec_mulVec]
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rw [show (!![0, -1; (1 : ℂ), 0] * !![0, 1; -1, 0]) = 1 by simpa using Eq.symm one_fin_two]
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rw [one_mulVec]
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rfl
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inv_hom_id := by
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ext ψ
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simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
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ModuleCat.id_apply]
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rw [leftHandedAltTo_hom_apply, leftHandedToAlt_hom_apply, LeftHandedModule.toFin2ℂ,
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LinearEquiv.apply_symm_apply, mulVec_mulVec]
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rw [show (!![0, (1 : ℂ); -1, 0] * !![0, -1; 1, 0]) = 1 by simpa using Eq.symm one_fin_two]
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rw [one_mulVec]
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rfl
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lemma leftHandedAltEquiv_hom_hom_apply (ψ : leftHanded) :
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leftHandedAltEquiv.hom.hom ψ =
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AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ) := rfl
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lemma leftHandedAltEquiv_inv_hom_apply (ψ : altLeftHanded) :
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leftHandedAltEquiv.inv.hom ψ =
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LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ) := rfl
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informal_definition rightHandedWeylAltEquiv where
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math :≈ "The linear equiv between rightHandedWeyl and altRightHandedWeyl given
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by multiplying an element of rightHandedWeyl by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`"
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deps :≈ [``rightHandedWeyl, ``altRightHandedWeyl]
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deps :≈ [``rightHanded, ``altRightHanded]
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informal_lemma rightHandedWeylAltEquiv_equivariant where
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math :≈ "The linear equiv rightHandedWeylAltEquiv is equivariant with respect to the
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@ -74,37 +230,139 @@ informal_lemma rightHandedWeylAltEquiv_equivariant where
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## Contraction of Weyl fermions.
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-/
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open CategoryTheory.MonoidalCategory
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informal_definition leftAltWeylContraction where
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math :≈ "The linear map from leftHandedWeyl ⊗ altLeftHandedWeyl to ℂ given by
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/-- The bi-linear map corresponding to contraction of a left-handed Weyl fermion with a
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alt-left-handed Weyl fermion. -/
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def leftAltBi : leftHanded →ₗ[ℂ] altLeftHanded →ₗ[ℂ] ℂ where
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toFun ψ := {
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toFun := fun φ => ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ,
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map_add' := by
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intro φ φ'
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simp only [map_add]
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rw [dotProduct_add]
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map_smul' := by
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intro r φ
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simp only [LinearEquiv.map_smul]
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rw [dotProduct_smul]
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rfl}
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map_add' ψ ψ':= by
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refine LinearMap.ext (fun φ => ?_)
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simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
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rw [add_dotProduct]
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map_smul' r ψ := by
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refine LinearMap.ext (fun φ => ?_)
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simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
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rw [smul_dotProduct]
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rfl
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/-- The bi-linear map corresponding to contraction of a alt-left-handed Weyl fermion with a
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left-handed Weyl fermion. -/
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def altLeftBi : altLeftHanded →ₗ[ℂ] leftHanded →ₗ[ℂ] ℂ where
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toFun ψ := {
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toFun := fun φ => ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ,
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map_add' := by
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intro φ φ'
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simp only [map_add]
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rw [dotProduct_add]
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map_smul' := by
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intro r φ
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simp only [LinearEquiv.map_smul]
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rw [dotProduct_smul]
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rfl}
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map_add' ψ ψ':= by
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refine LinearMap.ext (fun φ => ?_)
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simp only [map_add, add_dotProduct, vec2_dotProduct, Fin.isValue, LinearMap.coe_mk,
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AddHom.coe_mk, LinearMap.add_apply]
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map_smul' ψ ψ' := by
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refine LinearMap.ext (fun φ => ?_)
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simp only [_root_.map_smul, smul_dotProduct, vec2_dotProduct, Fin.isValue, smul_eq_mul,
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LinearMap.coe_mk, AddHom.coe_mk, RingHom.id_apply, LinearMap.smul_apply]
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/-- The linear map from leftHandedWeyl ⊗ altLeftHandedWeyl to ℂ given by
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summing over components of leftHandedWeyl and altLeftHandedWeyl in the
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standard basis (i.e. the dot product)."
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physics :≈ "The contraction of a left-handed Weyl fermion with a right-handed Weyl fermion.
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In index notation this is ψ_a φ^a."
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deps :≈ [``leftHandedWeyl, ``altLeftHandedWeyl]
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standard basis (i.e. the dot product).
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Physically, the contraction of a left-handed Weyl fermion with a alt-left-handed Weyl fermion.
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In index notation this is ψ_a φ^a. -/
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def leftAltContraction : leftHanded ⊗ altLeftHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
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hom := TensorProduct.lift leftAltBi
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comm M := by
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apply TensorProduct.ext'
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intro ψ φ
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change (M.1 *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹ᵀ *ᵥ φ.toFin2ℂ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
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rw [dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
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simp
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informal_lemma leftAltWeylContraction_invariant where
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math :≈ "The contraction leftAltWeylContraction is invariant with respect to
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the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
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deps :≈ [``leftAltWeylContraction]
|
||||
@[simp]
|
||||
lemma leftAltContraction_hom_tmul (ψ : leftHanded) (φ : altLeftHanded) :
|
||||
leftAltContraction.hom (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ := by
|
||||
rw [leftAltContraction]
|
||||
erw [TensorProduct.lift.tmul]
|
||||
rfl
|
||||
|
||||
informal_definition altLeftWeylContraction where
|
||||
math :≈ "The linear map from altLeftHandedWeyl ⊗ leftHandedWeyl to ℂ given by
|
||||
/-- The linear map from altLeftHandedWeyl ⊗ leftHandedWeyl to ℂ given by
|
||||
summing over components of altLeftHandedWeyl and leftHandedWeyl in the
|
||||
standard basis (i.e. the dot product)."
|
||||
physics :≈ "The contraction of a left-handed Weyl fermion with a right-handed Weyl fermion.
|
||||
In index notation this is φ^a ψ_a ."
|
||||
deps :≈ [``leftHandedWeyl, ``altLeftHandedWeyl]
|
||||
standard basis (i.e. the dot product).
|
||||
Physically, the contraction of a alt-left-handed Weyl fermion with a left-handed Weyl fermion.
|
||||
In index notation this is φ^a ψ_a. -/
|
||||
def altLeftContraction : altLeftHanded ⊗ leftHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
|
||||
hom := TensorProduct.lift altLeftBi
|
||||
comm M := by
|
||||
apply TensorProduct.ext'
|
||||
intro φ ψ
|
||||
change (M.1⁻¹ᵀ *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1 *ᵥ ψ.toFin2ℂ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
|
||||
rw [dotProduct_mulVec, mulVec_transpose, vecMul_vecMul]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma altLeftContraction_hom_tmul (φ : altLeftHanded) (ψ : leftHanded) :
|
||||
altLeftContraction.hom (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ := by
|
||||
rw [altLeftContraction]
|
||||
erw [TensorProduct.lift.tmul]
|
||||
rfl
|
||||
|
||||
lemma leftAltContraction_apply_symm (ψ : leftHanded) (φ : altLeftHanded) :
|
||||
leftAltContraction.hom (ψ ⊗ₜ φ) = altLeftContraction.hom (φ ⊗ₜ ψ) := by
|
||||
rw [altLeftContraction_hom_tmul, leftAltContraction_hom_tmul]
|
||||
exact dotProduct_comm ψ.toFin2ℂ φ.toFin2ℂ
|
||||
|
||||
/-- A manifestation of the statement that `ψ ψ' = - ψ' ψ` where `ψ` and `ψ'`
|
||||
are `leftHandedWeyl`. -/
|
||||
lemma leftAltContraction_apply_leftHandedAltEquiv (ψ ψ' : leftHanded) :
|
||||
leftAltContraction.hom (ψ ⊗ₜ leftHandedAltEquiv.hom.hom ψ') =
|
||||
- leftAltContraction.hom (ψ' ⊗ₜ leftHandedAltEquiv.hom.hom ψ) := by
|
||||
rw [leftAltContraction_hom_tmul, leftAltContraction_hom_tmul,
|
||||
leftHandedAltEquiv_hom_hom_apply, leftHandedAltEquiv_hom_hom_apply]
|
||||
simp only [CategoryTheory.Monoidal.transportStruct_tensorUnit,
|
||||
CategoryTheory.Equivalence.symm_functor, Action.functorCategoryEquivalence_inverse,
|
||||
Action.FunctorCategoryEquivalence.inverse_obj_V, CategoryTheory.Monoidal.tensorUnit_obj,
|
||||
cons_mulVec, cons_dotProduct, zero_mul, one_mul, dotProduct_empty, add_zero, zero_add, neg_mul,
|
||||
empty_mulVec, LinearEquiv.apply_symm_apply, dotProduct_cons, mul_neg, neg_add_rev, neg_neg]
|
||||
ring
|
||||
|
||||
/-- A manifestation of the statement that `φ φ' = - φ' φ` where `φ` and `φ'` are
|
||||
`altLeftHandedWeyl`. -/
|
||||
lemma leftAltContraction_apply_leftHandedAltEquiv_inv (φ φ' : altLeftHanded) :
|
||||
leftAltContraction.hom (leftHandedAltEquiv.inv.hom φ ⊗ₜ φ') =
|
||||
- leftAltContraction.hom (leftHandedAltEquiv.inv.hom φ' ⊗ₜ φ) := by
|
||||
rw [leftAltContraction_hom_tmul, leftAltContraction_hom_tmul,
|
||||
leftHandedAltEquiv_inv_hom_apply, leftHandedAltEquiv_inv_hom_apply]
|
||||
simp only [CategoryTheory.Monoidal.transportStruct_tensorUnit,
|
||||
CategoryTheory.Equivalence.symm_functor, Action.functorCategoryEquivalence_inverse,
|
||||
Action.FunctorCategoryEquivalence.inverse_obj_V, CategoryTheory.Monoidal.tensorUnit_obj,
|
||||
cons_mulVec, cons_dotProduct, zero_mul, neg_mul, one_mul, dotProduct_empty, add_zero, zero_add,
|
||||
empty_mulVec, LinearEquiv.apply_symm_apply, neg_add_rev, neg_neg]
|
||||
ring
|
||||
|
||||
informal_lemma leftAltWeylContraction_symm_altLeftWeylContraction where
|
||||
math :≈ "The linear map altLeftWeylContraction is leftAltWeylContraction composed
|
||||
with the braiding of the tensor product."
|
||||
deps :≈ [``leftAltWeylContraction, ``altLeftWeylContraction]
|
||||
deps :≈ [``leftAltContraction, ``altLeftContraction]
|
||||
|
||||
informal_lemma altLeftWeylContraction_invariant where
|
||||
math :≈ "The contraction altLeftWeylContraction is invariant with respect to
|
||||
the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
|
||||
deps :≈ [``altLeftWeylContraction]
|
||||
deps :≈ [``altLeftContraction]
|
||||
|
||||
informal_definition rightAltWeylContraction where
|
||||
math :≈ "The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
|
||||
|
|
@ -112,7 +370,7 @@ informal_definition rightAltWeylContraction where
|
|||
standard basis (i.e. the dot product)."
|
||||
physics :≈ "The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
|
||||
In index notation this is ψ_{dot a} φ^{dot a}."
|
||||
deps :≈ [``rightHandedWeyl, ``altRightHandedWeyl]
|
||||
deps :≈ [``rightHanded, ``altRightHanded]
|
||||
|
||||
informal_lemma rightAltWeylContraction_invariant where
|
||||
math :≈ "The contraction rightAltWeylContraction is invariant with respect to
|
||||
|
|
@ -125,7 +383,7 @@ informal_definition altRightWeylContraction where
|
|||
standard basis (i.e. the dot product)."
|
||||
physics :≈ "The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
|
||||
In index notation this is φ^{dot a} ψ_{dot a}."
|
||||
deps :≈ [``rightHandedWeyl, ``altRightHandedWeyl]
|
||||
deps :≈ [``rightHanded, ``altRightHanded]
|
||||
|
||||
informal_lemma rightAltWeylContraction_symm_altRightWeylContraction where
|
||||
math :≈ "The linear map altRightWeylContraction is rightAltWeylContraction composed
|
||||
|
|
@ -137,4 +395,6 @@ informal_lemma altRightWeylContraction_invariant where
|
|||
the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
|
||||
deps :≈ [``altRightWeylContraction]
|
||||
|
||||
end
|
||||
|
||||
end Fermion
|
||||
|
|
|
|||
212
HepLean/SpaceTime/WeylFermion/Modules.lean
Normal file
212
HepLean/SpaceTime/WeylFermion/Modules.lean
Normal file
|
|
@ -0,0 +1,212 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Meta.Informal
|
||||
import HepLean.SpaceTime.SL2C.Basic
|
||||
import Mathlib.RepresentationTheory.Rep
|
||||
import HepLean.Tensors.Basic
|
||||
import Mathlib.Logic.Equiv.TransferInstance
|
||||
/-!
|
||||
|
||||
## Modules associated with Fermions
|
||||
|
||||
Weyl fermions live in the vector space `ℂ^2`, defined here as `Fin 2 → ℂ`.
|
||||
However if we simply define the Module of Weyl fermions as `Fin 2 → ℂ` we get casting problems,
|
||||
where e.g. left-handed fermions can be cast to right-handed fermions etc.
|
||||
To overcome this, for each type of Weyl fermion we define a structure that wraps `Fin 2 → ℂ`,
|
||||
and these structures we define the instance of a module. This prevents casting between different
|
||||
types of fermions.
|
||||
|
||||
-/
|
||||
|
||||
namespace Fermion
|
||||
noncomputable section
|
||||
|
||||
section LeftHanded
|
||||
|
||||
/-- The module in which left handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
|
||||
structure LeftHandedModule where
|
||||
/-- The underlying value in `Fin 2 → ℂ`. -/
|
||||
val : Fin 2 → ℂ
|
||||
|
||||
namespace LeftHandedModule
|
||||
|
||||
/-- The equivalence between `LeftHandedModule` and `Fin 2 → ℂ`. -/
|
||||
def toFin2ℂFun : LeftHandedModule ≃ (Fin 2 → ℂ) where
|
||||
toFun v := v.val
|
||||
invFun f := ⟨f⟩
|
||||
left_inv _ := rfl
|
||||
right_inv _ := rfl
|
||||
|
||||
/-- The instance of `AddCommMonoid` on `LeftHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommMonoid LeftHandedModule := Equiv.addCommMonoid toFin2ℂFun
|
||||
|
||||
/-- The instance of `AddCommGroup` on `LeftHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommGroup LeftHandedModule := Equiv.addCommGroup toFin2ℂFun
|
||||
|
||||
/-- The instance of `Module` on `LeftHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : Module ℂ LeftHandedModule := Equiv.module ℂ toFin2ℂFun
|
||||
|
||||
/-- The linear equivalence between `LeftHandedModule` and `(Fin 2 → ℂ)`. -/
|
||||
@[simps!]
|
||||
def toFin2ℂEquiv : LeftHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
|
||||
toFun := toFin2ℂFun
|
||||
map_add' := fun _ _ => rfl
|
||||
map_smul' := fun _ _ => rfl
|
||||
invFun := toFin2ℂFun.symm
|
||||
left_inv := fun _ => rfl
|
||||
right_inv := fun _ => rfl
|
||||
|
||||
/-- The underlying element of `Fin 2 → ℂ` of a element in `LeftHandedModule` defined
|
||||
through the linear equivalence `toFin2ℂEquiv`. -/
|
||||
abbrev toFin2ℂ (ψ : LeftHandedModule) := toFin2ℂEquiv ψ
|
||||
|
||||
end LeftHandedModule
|
||||
|
||||
end LeftHanded
|
||||
|
||||
section AltLeftHanded
|
||||
|
||||
/-- The module in which alt-left handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
|
||||
structure AltLeftHandedModule where
|
||||
/-- The underlying value in `Fin 2 → ℂ`. -/
|
||||
val : Fin 2 → ℂ
|
||||
|
||||
namespace AltLeftHandedModule
|
||||
|
||||
/-- The equivalence between `AltLeftHandedModule` and `Fin 2 → ℂ`. -/
|
||||
def toFin2ℂFun : AltLeftHandedModule ≃ (Fin 2 → ℂ) where
|
||||
toFun v := v.val
|
||||
invFun f := ⟨f⟩
|
||||
left_inv _ := rfl
|
||||
right_inv _ := rfl
|
||||
|
||||
/-- The instance of `AddCommMonoid` on `AltLeftHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommMonoid AltLeftHandedModule := Equiv.addCommMonoid toFin2ℂFun
|
||||
|
||||
/-- The instance of `AddCommGroup` on `AltLeftHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommGroup AltLeftHandedModule := Equiv.addCommGroup toFin2ℂFun
|
||||
|
||||
/-- The instance of `Module` on `AltLeftHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : Module ℂ AltLeftHandedModule := Equiv.module ℂ toFin2ℂFun
|
||||
|
||||
/-- The linear equivalence between `AltLeftHandedModule` and `(Fin 2 → ℂ)`. -/
|
||||
@[simps!]
|
||||
def toFin2ℂEquiv : AltLeftHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
|
||||
toFun := toFin2ℂFun
|
||||
map_add' := fun _ _ => rfl
|
||||
map_smul' := fun _ _ => rfl
|
||||
invFun := toFin2ℂFun.symm
|
||||
left_inv := fun _ => rfl
|
||||
right_inv := fun _ => rfl
|
||||
|
||||
/-- The underlying element of `Fin 2 → ℂ` of a element in `AltLeftHandedModule` defined
|
||||
through the linear equivalence `toFin2ℂEquiv`. -/
|
||||
abbrev toFin2ℂ (ψ : AltLeftHandedModule) := toFin2ℂEquiv ψ
|
||||
|
||||
end AltLeftHandedModule
|
||||
|
||||
end AltLeftHanded
|
||||
|
||||
section RightHanded
|
||||
|
||||
/-- The module in which right handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
|
||||
structure RightHandedModule where
|
||||
/-- The underlying value in `Fin 2 → ℂ`. -/
|
||||
val : Fin 2 → ℂ
|
||||
|
||||
namespace RightHandedModule
|
||||
|
||||
/-- The equivalence between `RightHandedModule` and `Fin 2 → ℂ`. -/
|
||||
def toFin2ℂFun : RightHandedModule ≃ (Fin 2 → ℂ) where
|
||||
toFun v := v.val
|
||||
invFun f := ⟨f⟩
|
||||
left_inv _ := rfl
|
||||
right_inv _ := rfl
|
||||
|
||||
/-- The instance of `AddCommMonoid` on `RightHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommMonoid RightHandedModule := Equiv.addCommMonoid toFin2ℂFun
|
||||
|
||||
/-- The instance of `AddCommGroup` on `RightHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommGroup RightHandedModule := Equiv.addCommGroup toFin2ℂFun
|
||||
|
||||
/-- The instance of `Module` on `RightHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : Module ℂ RightHandedModule := Equiv.module ℂ toFin2ℂFun
|
||||
|
||||
/-- The linear equivalence between `RightHandedModule` and `(Fin 2 → ℂ)`. -/
|
||||
@[simps!]
|
||||
def toFin2ℂEquiv : RightHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
|
||||
toFun := toFin2ℂFun
|
||||
map_add' := fun _ _ => rfl
|
||||
map_smul' := fun _ _ => rfl
|
||||
invFun := toFin2ℂFun.symm
|
||||
left_inv := fun _ => rfl
|
||||
right_inv := fun _ => rfl
|
||||
|
||||
/-- The underlying element of `Fin 2 → ℂ` of a element in `RightHandedModule` defined
|
||||
through the linear equivalence `toFin2ℂEquiv`. -/
|
||||
abbrev toFin2ℂ (ψ : RightHandedModule) := toFin2ℂEquiv ψ
|
||||
|
||||
end RightHandedModule
|
||||
|
||||
end RightHanded
|
||||
|
||||
section AltRightHanded
|
||||
|
||||
/-- The module in which alt-right handed fermions live. This is equivalent to `Fin 2 → ℂ`. -/
|
||||
structure AltRightHandedModule where
|
||||
/-- The underlying value in `Fin 2 → ℂ`. -/
|
||||
val : Fin 2 → ℂ
|
||||
|
||||
namespace AltRightHandedModule
|
||||
|
||||
/-- The equivalence between `AltRightHandedModule` and `Fin 2 → ℂ`. -/
|
||||
def toFin2ℂFun : AltRightHandedModule ≃ (Fin 2 → ℂ) where
|
||||
toFun v := v.val
|
||||
invFun f := ⟨f⟩
|
||||
left_inv _ := rfl
|
||||
right_inv _ := rfl
|
||||
|
||||
/-- The instance of `AddCommMonoid` on `AltRightHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommMonoid AltRightHandedModule := Equiv.addCommMonoid toFin2ℂFun
|
||||
|
||||
/-- The instance of `AddCommGroup` on `AltRightHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : AddCommGroup AltRightHandedModule := Equiv.addCommGroup toFin2ℂFun
|
||||
|
||||
/-- The instance of `Module` on `AltRightHandedModule` defined via its equivalence
|
||||
with `Fin 2 → ℂ`. -/
|
||||
instance : Module ℂ AltRightHandedModule := Equiv.module ℂ toFin2ℂFun
|
||||
|
||||
/-- The linear equivalence between `AltRightHandedModule` and `(Fin 2 → ℂ)`. -/
|
||||
@[simps!]
|
||||
def toFin2ℂEquiv : AltRightHandedModule ≃ₗ[ℂ] (Fin 2 → ℂ) where
|
||||
toFun := toFin2ℂFun
|
||||
map_add' := fun _ _ => rfl
|
||||
map_smul' := fun _ _ => rfl
|
||||
invFun := toFin2ℂFun.symm
|
||||
left_inv := fun _ => rfl
|
||||
right_inv := fun _ => rfl
|
||||
|
||||
/-- The underlying element of `Fin 2 → ℂ` of a element in `AltRightHandedModule` defined
|
||||
through the linear equivalence `toFin2ℂEquiv`. -/
|
||||
abbrev toFin2ℂ (ψ : AltRightHandedModule) := toFin2ℂEquiv ψ
|
||||
|
||||
end AltRightHandedModule
|
||||
|
||||
end AltRightHanded
|
||||
|
||||
end
|
||||
end Fermion
|
||||
197
HepLean/SpaceTime/WeylFermion/OverCat.lean
Normal file
197
HepLean/SpaceTime/WeylFermion/OverCat.lean
Normal file
|
|
@ -0,0 +1,197 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import Mathlib.CategoryTheory.Category.Basic
|
||||
import Mathlib.CategoryTheory.Types
|
||||
import Mathlib.CategoryTheory.Monoidal.Category
|
||||
import Mathlib.CategoryTheory.Comma.Over
|
||||
import Mathlib.CategoryTheory.Core
|
||||
import HepLean.SpaceTime.WeylFermion.Basic
|
||||
/-!
|
||||
|
||||
## Category over color
|
||||
|
||||
-/
|
||||
|
||||
namespace IndexNotation
|
||||
open CategoryTheory
|
||||
|
||||
def OverColor (C : Type) := CategoryTheory.Core (CategoryTheory.Over C)
|
||||
|
||||
instance (C : Type) : Groupoid (OverColor C) := coreCategory
|
||||
|
||||
namespace OverColor
|
||||
|
||||
namespace Hom
|
||||
|
||||
variable {C : Type} {f g h : OverColor C}
|
||||
|
||||
def toEquiv (m : f ⟶ g) : f.left ≃ g.left where
|
||||
toFun := m.hom.left
|
||||
invFun := m.inv.left
|
||||
left_inv := by
|
||||
simpa only [Over.comp_left] using congrFun (congrArg (fun x => x.left) m.hom_inv_id)
|
||||
right_inv := by
|
||||
simpa only [Over.comp_left] using congrFun (congrArg (fun x => x.left) m.inv_hom_id)
|
||||
|
||||
@[simp]
|
||||
lemma toEquiv_comp (m : f ⟶ g) (n : g ⟶ h) : toEquiv (m ≫ n) = (toEquiv m).trans (toEquiv n) := by
|
||||
ext x
|
||||
simp [toEquiv]
|
||||
rfl
|
||||
|
||||
lemma toEquiv_symm_apply (m : f ⟶ g) (i : g.left) :
|
||||
f.hom ((toEquiv m).symm i) = g.hom i := by
|
||||
sorry
|
||||
end Hom
|
||||
|
||||
end OverColor
|
||||
|
||||
end IndexNotation
|
||||
|
||||
namespace Fermion
|
||||
|
||||
noncomputable section
|
||||
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
open TensorProduct
|
||||
open IndexNotation
|
||||
|
||||
inductive Color
|
||||
| upL : Color
|
||||
| downL : Color
|
||||
| upR : Color
|
||||
| downR : Color
|
||||
|
||||
def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
|
||||
match c with
|
||||
| Color.upL => altLeftHanded
|
||||
| Color.downL => leftHanded
|
||||
| Color.upR => altRightHanded
|
||||
| Color.downR => rightHanded
|
||||
|
||||
def colorToRepCongr {c1 c2 : Color} (h : c1 = c2) : colorToRep c1 ≃ₗ[ℂ] colorToRep c2 where
|
||||
toFun := Equiv.cast (congrArg (CoeSort.coe ∘ colorToRep) h)
|
||||
invFun := (Equiv.cast (congrArg (CoeSort.coe ∘ colorToRep) h)).symm
|
||||
map_add' x y := by
|
||||
subst h
|
||||
rfl
|
||||
map_smul' x y := by
|
||||
subst h
|
||||
rfl
|
||||
left_inv x := Equiv.symm_apply_apply (Equiv.cast (congrArg (CoeSort.coe ∘ colorToRep) h)) x
|
||||
right_inv x := Equiv.apply_symm_apply (Equiv.cast (congrArg (CoeSort.coe ∘ colorToRep) h)) x
|
||||
|
||||
lemma colorToRepCongr_comm_ρ {c1 c2 : Color} (h : c1 = c2) (M : SL(2, ℂ)) (x : (colorToRep c1) ) :
|
||||
(colorToRepCongr h) ((colorToRep c1).ρ M x) = (colorToRep c2).ρ M ((colorToRepCongr h) x) := by
|
||||
subst h
|
||||
rfl
|
||||
|
||||
def obj (f : OverColor Color) : Rep ℂ SL(2, ℂ) := Rep.of {
|
||||
toFun := fun M => PiTensorProduct.map (fun x => (colorToRep (f.hom x)).ρ M),
|
||||
map_one' := by
|
||||
simp
|
||||
map_mul' := fun M N => by
|
||||
simp
|
||||
ext x : 2
|
||||
simp only [LinearMap.compMultilinearMap_apply, PiTensorProduct.map_tprod, LinearMap.mul_apply]}
|
||||
|
||||
lemma obj_ρ (f : OverColor Color) (M : SL(2, ℂ)) : (obj f).ρ M =
|
||||
PiTensorProduct.map (fun x => (colorToRep (f.hom x)).ρ M) := rfl
|
||||
|
||||
lemma obj_ρ_tprod (f : OverColor Color) (M : SL(2, ℂ)) (x : (i : f.left) → CoeSort.coe (colorToRep (f.hom i))) :
|
||||
(obj f).ρ M ((PiTensorProduct.tprod ℂ) x) =
|
||||
PiTensorProduct.tprod ℂ (fun i => (colorToRep (f.hom i)).ρ M (x i)) := by
|
||||
rw [obj_ρ]
|
||||
change (PiTensorProduct.map fun x => (colorToRep (f.hom x)).ρ M) ((PiTensorProduct.tprod ℂ) x) =
|
||||
(PiTensorProduct.tprod ℂ) fun i => ((colorToRep (f.hom i)).ρ M) (x i)
|
||||
rw [PiTensorProduct.map_tprod]
|
||||
|
||||
def morToLinearEquiv {f g : OverColor Color} (m : f ⟶ g) : (obj f).V ≃ₗ[ℂ] (obj g).V :=
|
||||
(PiTensorProduct.reindex ℂ (fun x => colorToRep (f.hom x)) (OverColor.Hom.toEquiv m)).trans
|
||||
(PiTensorProduct.congr (fun i => colorToRepCongr (OverColor.Hom.toEquiv_symm_apply m i)))
|
||||
|
||||
lemma morToLinearEquiv_tprod {f g : OverColor Color} (m : f ⟶ g)
|
||||
(x : (i : f.left) → CoeSort.coe (colorToRep (f.hom i))) :
|
||||
morToLinearEquiv m (PiTensorProduct.tprod ℂ x) =
|
||||
PiTensorProduct.tprod ℂ (fun i => (colorToRepCongr (OverColor.Hom.toEquiv_symm_apply m i ))
|
||||
(x ((OverColor.Hom.toEquiv m).symm i))) := by
|
||||
rw [morToLinearEquiv]
|
||||
simp
|
||||
change (PiTensorProduct.congr fun i => colorToRepCongr _)
|
||||
((PiTensorProduct.reindex ℂ (fun x => CoeSort.coe (colorToRep (f.hom x)))
|
||||
(OverColor.Hom.toEquiv m)) ((PiTensorProduct.tprod ℂ) x)) = _
|
||||
rw [PiTensorProduct.reindex_tprod, PiTensorProduct.congr_tprod]
|
||||
rfl
|
||||
|
||||
def mor {f g : OverColor Color} (m : f ⟶ g) : obj f ⟶ obj g where
|
||||
hom := (morToLinearEquiv m).toLinearMap
|
||||
comm M := by
|
||||
ext x : 2
|
||||
refine PiTensorProduct.induction_on' x ?_ (by
|
||||
intro x y hx hy
|
||||
simp only [CategoryTheory.Functor.id_obj, map_add, hx, ModuleCat.coe_comp,
|
||||
Function.comp_apply, hy])
|
||||
intro r x
|
||||
simp only [CategoryTheory.Functor.id_obj, PiTensorProduct.tprodCoeff_eq_smul_tprod,
|
||||
_root_.map_smul, ModuleCat.coe_comp, Function.comp_apply]
|
||||
apply congrArg
|
||||
change (morToLinearEquiv m) (((obj f).ρ M) ((PiTensorProduct.tprod ℂ) x)) =
|
||||
((obj g).ρ M) ((morToLinearEquiv m) ((PiTensorProduct.tprod ℂ) x))
|
||||
rw [morToLinearEquiv_tprod, obj_ρ_tprod]
|
||||
erw [morToLinearEquiv_tprod, obj_ρ_tprod]
|
||||
apply congrArg
|
||||
funext i
|
||||
rw [colorToRepCongr_comm_ρ]
|
||||
|
||||
open CategoryTheory
|
||||
|
||||
def colorFun : OverColor Color ⥤ Rep ℂ SL(2, ℂ) where
|
||||
obj := obj
|
||||
map := mor
|
||||
map_id f := by
|
||||
simp only
|
||||
ext x
|
||||
refine PiTensorProduct.induction_on' x ?_ (by
|
||||
intro x y hx hy
|
||||
simp only [CategoryTheory.Functor.id_obj, map_add, hx, ModuleCat.coe_comp,
|
||||
Function.comp_apply, hy])
|
||||
intro r x
|
||||
simp only [CategoryTheory.Functor.id_obj, PiTensorProduct.tprodCoeff_eq_smul_tprod,
|
||||
_root_.map_smul, Action.id_hom, ModuleCat.id_apply]
|
||||
apply congrArg
|
||||
rw [mor]
|
||||
erw [morToLinearEquiv_tprod]
|
||||
apply congrArg
|
||||
funext i
|
||||
rfl
|
||||
map_comp {X Y Z} f g := by
|
||||
simp only
|
||||
ext x
|
||||
refine PiTensorProduct.induction_on' x ?_ (by
|
||||
intro x y hx hy
|
||||
simp only [CategoryTheory.Functor.id_obj, map_add, hx, ModuleCat.coe_comp,
|
||||
Function.comp_apply, hy])
|
||||
intro r x
|
||||
simp
|
||||
apply congrArg
|
||||
rw [mor, mor, mor]
|
||||
simp
|
||||
change (morToLinearEquiv (CategoryTheory.CategoryStruct.comp f g)) ((PiTensorProduct.tprod ℂ) x) =
|
||||
(morToLinearEquiv g) ((morToLinearEquiv f) ((PiTensorProduct.tprod ℂ) x))
|
||||
rw [morToLinearEquiv_tprod, morToLinearEquiv_tprod]
|
||||
erw [morToLinearEquiv_tprod]
|
||||
apply congrArg
|
||||
funext i
|
||||
simp only [colorToRepCongr, Function.comp_apply, Equiv.cast_symm, LinearEquiv.coe_mk,
|
||||
Equiv.cast_apply, cast_cast, cast_inj]
|
||||
rfl
|
||||
|
||||
|
||||
|
||||
end
|
||||
end Fermion
|
||||
Loading…
Add table
Add a link
Reference in a new issue