In the context of seesaw mechanism or Dirac and Majorana mass terms, one often see the following identity
¯ψcLψcR=¯ψRψL.
Here, I am using 4 components notation in the chiral basis. The convention for the charge conjugation is ψc=−iγ2ψ∗, and ψcL=(ψL)c.
The following is my effort of proving it.
¯ψcLψcR=¯iγ2ψ∗Liγ2ψ∗R=(iγ2ψ∗L)+γ0iγ2ψ∗R=ψTLiγ2γ0iγ2ψ∗R
=−ψTLγ2γ0γ2ψ∗R=ψTLγ2γ2γ0ψ∗R=−ψTLγ0ψ∗R=−ψL,iγ0ijψ∗R,j.
Now if
ψL,i and
ψR,i are anticommuting, then one have
−ψL,iγ0ijψ∗R,j=ψ∗R,jγ0jiψL,i=¯ψRψL.
Question:
Is the anticommuting assumption still true if ψR and ψL are two different species of fermion? (For example, ψL=χL)
Do we assume any two fermions are anticommuting even if they are two different fields in QFT?
This post imported from StackExchange Physics at 2014-05-04 11:37 (UCT), posted by SE-user Louis Yang