85 lines
3.6 KiB
Text
85 lines
3.6 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 HepLean.Meta.Informal
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/-!
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# Weyl fermions
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-/
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/-!
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## The definition of Weyl fermion vector spaces.
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-/
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informal_definition leftHandedWeylFermion 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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informal_definition rightHandedWeylFermion 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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informal_definition altLeftHandedWeylFermion 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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informal_definition altRightHandedWeylFermion 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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/-!
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## Equivalences between Weyl fermion vector spaces.
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-/
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informal_definition leftHandedWeylFermionAltEquiv where
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math :≈ "The linear equiv between leftHandedWeylFermion and altLeftHandedWeylFermion given
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by multiplying an element of rightHandedWeylFermion by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`."
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deps :≈ [`leftHandedWeylFermion, `altLeftHandedWeylFermion]
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informal_lemma leftHandedWeylFermionAltEquiv_equivariant where
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math :≈ "The linear equiv leftHandedWeylFermionAltEquiv is equivariant with respect to the
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action of SL(2,C) on leftHandedWeylFermion and altLeftHandedWeylFermion."
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deps :≈ [`leftHandedWeylFermionAltEquiv]
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informal_definition rightHandedWeylFermionAltEquiv where
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math :≈ "The linear equiv between rightHandedWeylFermion and altRightHandedWeylFermion given
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by multiplying an element of rightHandedWeylFermion by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`"
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deps :≈ [`rightHandedWeylFermion, `altRightHandedWeylFermion]
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informal_lemma rightHandedWeylFermionAltEquiv_equivariant where
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math :≈ "The linear equiv rightHandedWeylFermionAltEquiv is equivariant with respect to the
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action of SL(2,C) on rightHandedWeylFermion and altRightHandedWeylFermion."
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deps :≈ [`rightHandedWeylFermionAltEquiv]
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/-!
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## Contraction of Weyl fermions.
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-/
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informal_definition leftAltWeylFermionContraction where
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math :≈ "The linear map from leftHandedWeylFermion ⊗ altLeftHandedWeylFermion to ℂ given by
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summing over components of leftHandedWeylFermion and altLeftHandedWeylFermion 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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ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
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deps :≈ [``leftHandedWeylFermion, ``altLeftHandedWeylFermion]
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informal_lemma leftAltWeylFermionContraction_invariant where
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math :≈ "The contraction leftAltWeylFermionContraction is invariant with respect to
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the action of SL(2,C) on leftHandedWeylFermion and altLeftHandedWeylFermion."
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deps :≈ [`leftAltWeylFermionContraction]
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