Chiral representation
CHIRALITY PROJECTORS AND CHIRAL REPRESENTATION
Let us, for a minute, go back to the notion of chirality. Previously, the chirality projectors
have been introduced, which project onto the left- and right-handed components of the
spinors, respectively. A careful analysis of the Dirac spinor in terms of the two two-component
spinors reveals that they are eigenstates w.r.t. the two projection operators, i.e.
Thus, the identification of the Dirac spinor as a bispinor, consisting of the two SU(2)
spinor representations,
from above is valid only in another representation,
named the chiral representation. Of course, the anticommutation relation etc. of the γ matrices
remains to be valid, their specific form however changes. In particular,
Evidently, in this representation, the L- and R-projectors project onto the two basic spinors.
Also, the parity and charge conjugation transformations are still realised as before, i.e.
It is easy to see that in this representation, under parity transformations the χ and the η
components are transformed into each other, whereas under charge conjugations they are mapped onto
themselves. Therefore, in this representation, the upper and the lower two components each carry
particle states of the same parity but with different charge. This is in contrast to the standard
Dirac representation used so far, where the opposite statement was true. It is straightforward to
check, however, that both representations are connected by a linear transformation, just a change of
base in the space with Dirac (spinor) indices.
PHYSICAL SIGNIFICANCE: MASSTERM
To understand a little bit better the physical significance of this new base, let us check
the massterm of the Dirac Lagrangian. It reads
With help of the relation
the barred spinor can be decoposed as
In other words,
Using the properties of the projectors, namely
yields for the massterm
i.e. the left- and right-handed components mix and therefore have the same mass.
More general, introducing complex masses, the Dirac mass term reads
In contrast, for Majorana spinors, the mass terms decouple the &xi, and the η component,
which is not such a big surprise since for Majorana particles these two Dirac components describe
two completely different particles.