feat: Create fermion namespace
This commit is contained in:
jstoobysmith
parent
d713575b76
commit
cb389e2395
1 changed files with 62 additions and 58 deletions
|
|
@ -19,23 +19,25 @@ We define the vector spaces corresponding to different types of Weyl fermions.
|
|||
Note: We should prevent casting between these vector spaces.
|
||||
-/
|
||||
|
||||
informal_definition leftHandedWeylFermion where
|
||||
namespace Fermion
|
||||
|
||||
informal_definition leftHandedWeyl where
|
||||
math :≈ "The vector space ℂ^2 carrying the fundamental representation of SL(2,C)."
|
||||
physics :≈ "A Weyl fermion with indices ψ_a."
|
||||
ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
|
||||
|
||||
informal_definition rightHandedWeylFermion where
|
||||
informal_definition rightHandedWeyl where
|
||||
math :≈ "The vector space ℂ^2 carrying the conjguate representation of SL(2,C)."
|
||||
physics :≈ "A Weyl fermion with indices ψ_{dot a}."
|
||||
ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
|
||||
|
||||
informal_definition altLeftHandedWeylFermion where
|
||||
informal_definition altLeftHandedWeyl where
|
||||
math :≈ "The vector space ℂ^2 carrying the representation of SL(2,C) given by
|
||||
M → (M⁻¹)ᵀ."
|
||||
physics :≈ "A Weyl fermion with indices ψ^a."
|
||||
ref :≈ "https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf"
|
||||
|
||||
informal_definition altRightHandedWeylFermion where
|
||||
informal_definition altRightHandedWeyl where
|
||||
math :≈ "The vector space ℂ^2 carrying the representation of SL(2,C) given by
|
||||
M → (M⁻¹)^†."
|
||||
physics :≈ "A Weyl fermion with indices ψ^{dot a}."
|
||||
|
|
@ -47,25 +49,25 @@ informal_definition altRightHandedWeylFermion where
|
|||
|
||||
-/
|
||||
|
||||
informal_definition leftHandedWeylFermionAltEquiv where
|
||||
math :≈ "The linear equiv between leftHandedWeylFermion and altLeftHandedWeylFermion given
|
||||
by multiplying an element of rightHandedWeylFermion by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`."
|
||||
deps :≈ [`leftHandedWeylFermion, `altLeftHandedWeylFermion]
|
||||
informal_definition leftHandedWeylAltEquiv where
|
||||
math :≈ "The linear equiv between leftHandedWeyl and altLeftHandedWeyl given
|
||||
by multiplying an element of rightHandedWeyl by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`."
|
||||
deps :≈ [``leftHandedWeyl, ``altLeftHandedWeyl]
|
||||
|
||||
informal_lemma leftHandedWeylFermionAltEquiv_equivariant where
|
||||
math :≈ "The linear equiv leftHandedWeylFermionAltEquiv is equivariant with respect to the
|
||||
action of SL(2,C) on leftHandedWeylFermion and altLeftHandedWeylFermion."
|
||||
deps :≈ [`leftHandedWeylFermionAltEquiv]
|
||||
informal_lemma leftHandedWeylAltEquiv_equivariant where
|
||||
math :≈ "The linear equiv leftHandedWeylAltEquiv is equivariant with respect to the
|
||||
action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
|
||||
deps :≈ [``leftHandedWeylAltEquiv]
|
||||
|
||||
informal_definition rightHandedWeylFermionAltEquiv where
|
||||
math :≈ "The linear equiv between rightHandedWeylFermion and altRightHandedWeylFermion given
|
||||
by multiplying an element of rightHandedWeylFermion by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`"
|
||||
deps :≈ [`rightHandedWeylFermion, `altRightHandedWeylFermion]
|
||||
informal_definition rightHandedWeylAltEquiv where
|
||||
math :≈ "The linear equiv between rightHandedWeyl and altRightHandedWeyl given
|
||||
by multiplying an element of rightHandedWeyl by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`"
|
||||
deps :≈ [``rightHandedWeyl, ``altRightHandedWeyl]
|
||||
|
||||
informal_lemma rightHandedWeylFermionAltEquiv_equivariant where
|
||||
math :≈ "The linear equiv rightHandedWeylFermionAltEquiv is equivariant with respect to the
|
||||
action of SL(2,C) on rightHandedWeylFermion and altRightHandedWeylFermion."
|
||||
deps :≈ [`rightHandedWeylFermionAltEquiv]
|
||||
informal_lemma rightHandedWeylAltEquiv_equivariant where
|
||||
math :≈ "The linear equiv rightHandedWeylAltEquiv is equivariant with respect to the
|
||||
action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
|
||||
deps :≈ [``rightHandedWeylAltEquiv]
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -73,64 +75,66 @@ informal_lemma rightHandedWeylFermionAltEquiv_equivariant where
|
|||
|
||||
-/
|
||||
|
||||
informal_definition leftAltWeylFermionContraction where
|
||||
math :≈ "The linear map from leftHandedWeylFermion ⊗ altLeftHandedWeylFermion to ℂ given by
|
||||
summing over components of leftHandedWeylFermion and altLeftHandedWeylFermion in the
|
||||
informal_definition leftAltWeylContraction where
|
||||
math :≈ "The linear map from leftHandedWeyl ⊗ altLeftHandedWeyl to ℂ given by
|
||||
summing over components of leftHandedWeyl and altLeftHandedWeyl 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 ψ_a φ^a."
|
||||
deps :≈ [``leftHandedWeylFermion, ``altLeftHandedWeylFermion]
|
||||
deps :≈ [``leftHandedWeyl, ``altLeftHandedWeyl]
|
||||
|
||||
informal_lemma leftAltWeylFermionContraction_invariant where
|
||||
math :≈ "The contraction leftAltWeylFermionContraction is invariant with respect to
|
||||
the action of SL(2,C) on leftHandedWeylFermion and altLeftHandedWeylFermion."
|
||||
deps :≈ [``leftAltWeylFermionContraction]
|
||||
informal_lemma leftAltWeylContraction_invariant where
|
||||
math :≈ "The contraction leftAltWeylContraction is invariant with respect to
|
||||
the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
|
||||
deps :≈ [``leftAltWeylContraction]
|
||||
|
||||
informal_definition altLeftWeylFermionContraction where
|
||||
math :≈ "The linear map from altLeftHandedWeylFermion ⊗ leftHandedWeylFermion to ℂ given by
|
||||
summing over components of altLeftHandedWeylFermion and leftHandedWeylFermion in the
|
||||
informal_definition altLeftWeylContraction where
|
||||
math :≈ "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 :≈ [``leftHandedWeylFermion, ``altLeftHandedWeylFermion]
|
||||
deps :≈ [``leftHandedWeyl, ``altLeftHandedWeyl]
|
||||
|
||||
informal_lemma leftAltWeylFermionContraction_symm_altLeftWeylFermionContraction where
|
||||
math :≈ "The linear map altLeftWeylFermionContraction is leftAltWeylFermionContraction composed
|
||||
informal_lemma leftAltWeylContraction_symm_altLeftWeylContraction where
|
||||
math :≈ "The linear map altLeftWeylContraction is leftAltWeylContraction composed
|
||||
with the braiding of the tensor product."
|
||||
deps :≈ [``leftAltWeylFermionContraction, ``altLeftWeylFermionContraction]
|
||||
deps :≈ [``leftAltWeylContraction, ``altLeftWeylContraction]
|
||||
|
||||
informal_lemma altLeftWeylFermionContraction_invariant where
|
||||
math :≈ "The contraction altLeftWeylFermionContraction is invariant with respect to
|
||||
the action of SL(2,C) on leftHandedWeylFermion and altLeftHandedWeylFermion."
|
||||
deps :≈ [``altLeftWeylFermionContraction]
|
||||
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]
|
||||
|
||||
informal_definition rightAltWeylFermionContraction where
|
||||
math :≈ "The linear map from rightHandedWeylFermion ⊗ altRightHandedWeylFermion to ℂ given by
|
||||
summing over components of rightHandedWeylFermion and altRightHandedWeylFermion in the
|
||||
informal_definition rightAltWeylContraction where
|
||||
math :≈ "The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
|
||||
summing over components of rightHandedWeyl and altRightHandedWeyl in the
|
||||
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 :≈ [``rightHandedWeylFermion, ``altRightHandedWeylFermion]
|
||||
deps :≈ [``rightHandedWeyl, ``altRightHandedWeyl]
|
||||
|
||||
informal_lemma rightAltWeylFermionContraction_invariant where
|
||||
math :≈ "The contraction rightAltWeylFermionContraction is invariant with respect to
|
||||
the action of SL(2,C) on rightHandedWeylFermion and altRightHandedWeylFermion."
|
||||
deps :≈ [``rightAltWeylFermionContraction]
|
||||
informal_lemma rightAltWeylContraction_invariant where
|
||||
math :≈ "The contraction rightAltWeylContraction is invariant with respect to
|
||||
the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
|
||||
deps :≈ [``rightAltWeylContraction]
|
||||
|
||||
informal_definition altRightWeylFermionContraction where
|
||||
math :≈ "The linear map from altRightHandedWeylFermion ⊗ rightHandedWeylFermion to ℂ given by
|
||||
summing over components of altRightHandedWeylFermion and rightHandedWeylFermion in the
|
||||
informal_definition altRightWeylContraction where
|
||||
math :≈ "The linear map from altRightHandedWeyl ⊗ rightHandedWeyl to ℂ given by
|
||||
summing over components of altRightHandedWeyl and rightHandedWeyl in the
|
||||
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 :≈ [``rightHandedWeylFermion, ``altRightHandedWeylFermion]
|
||||
deps :≈ [``rightHandedWeyl, ``altRightHandedWeyl]
|
||||
|
||||
informal_lemma rightAltWeylFermionContraction_symm_altRightWeylFermionContraction where
|
||||
math :≈ "The linear map altRightWeylFermionContraction is rightAltWeylFermionContraction composed
|
||||
informal_lemma rightAltWeylContraction_symm_altRightWeylContraction where
|
||||
math :≈ "The linear map altRightWeylContraction is rightAltWeylContraction composed
|
||||
with the braiding of the tensor product."
|
||||
deps :≈ [``rightAltWeylFermionContraction, ``altRightWeylFermionContraction]
|
||||
deps :≈ [``rightAltWeylContraction, ``altRightWeylContraction]
|
||||
|
||||
informal_lemma altRightWeylFermionContraction_invariant where
|
||||
math :≈ "The contraction altRightWeylFermionContraction is invariant with respect to
|
||||
the action of SL(2,C) on rightHandedWeylFermion and altRightHandedWeylFermion."
|
||||
deps :≈ [``altRightWeylFermionContraction]
|
||||
informal_lemma altRightWeylContraction_invariant where
|
||||
math :≈ "The contraction altRightWeylContraction is invariant with respect to
|
||||
the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
|
||||
deps :≈ [``altRightWeylContraction]
|
||||
|
||||
end Fermion
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue