Majorana and Weyl fermions
Last updated
Dec 29, 2024
# Majorana mass terms
We found the Dirac mass term
Ldirac mass=m(ψL†ψR+ψR†ψL),
to be Lorentz invariant, while terms like ψR†ψ are not.
It turns out there is another Lorentz invariant mass term
Majorana mass term
The Majorana mass term for Spin 21 particles is
LMaj=mψRTσ2ψR.
Lorentz invariance of Majorana mass term
The transformation of the Spinors ψR and ψR† was given as
δψR=21(iΘj+βj)σjψR,δψR†=21(−iΘj+βj)ψR†σj
We can now calculate the transformation property of the majorana mass term
δ(ψRTσ2ψR)=δ(ψRTσ2)ψR+ψRTσ2δψR
using
σjTσ2=−σ2σj
to be
δ(ψRTσ2ψR)=−21(iΘj+βj)ψRTσ2σjψR+21(iΘj+βj)ψRTσ2σjψR=0
Explicitly plugging in σ2 one can write
ψRTσ2ψR=−i(ψ1ψ2−ψ2ψ1).
However this term is only non-trivial if
ψ1ψ2=ψ2ψ1,
meaning the fermion components must not commute!
One finds that the fermion components must anti-commute, meaning they are
Grassmann numbers.
We can also write the mass term in terms of a four component spinor
ψ=(ψLiσ2ψL∗)
as
2mψˉψ=i2m(ψL†σ2ψL∗−ψLTσ2ψL)
These Majorana spinors are invariant under
Charge conjugation
Charge conjugation
Definition: Charge conjugation
> Charge conjugation is a discrete symmetry transformation, which replaces particles with anti particles, effectively reversign the sign...
12/29/2024
:
−iγ2ψ∗=−i(0σ2\-σ20)(ψL\iσ2ψL∗)=(ψL\iσ2ψL∗)=ψ
Because of this, Majorana fermions cant be charged under any U(1) gauge or any global symmetry, because ψ→eiαψ but ψc→e−iαψc which is a contradiction to ψ=ψc for α=0.
# Notation for Weyl spinors
One writes
ψ=ψL,ψ~=ψR,
And the indices are included as ψα and ψβ˙ so that a Dirac Spinor can be written as
ψ=(ψαψ~β˙)
Using this notation the Majorana mass term can be written as
Lmaj=ψRTσ2ψR=iψαϵαβψβ,
with the ϵ tensor ϵαβ=−iσ2αβ.
The ϵ tensor can raise and lower indices for Weyl spinors, like gμν can do for four vectors.
ψχ=ψαχα=ψαϵαβχβ=−χβϵαβψα=χβϵβαψα=χψ
\tilde{\psi}\tilde{\chi}=\tilde{\psi}^{\dot{\alpha}}\tilde{\chi}_\dot{\alpha}=-\tilde{\chi}_\dot{\alpha}\tilde{\psi}^{\dot{\alpha}}=\tilde{\chi}^\dot{\alpha}\tilde{\psi}_\dot{\alpha}=\tilde{\chi}\tilde{\psi}